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• CONTAMOS CON ATRACTIVOS PLANES DE FINANCIAMIENTO ENGANCHE DESDE EL 20%, PLAZOS HASTA 48 MESES!
• NO TE PREOCUPES SI NO CUENTAS CON COMPROBANTE DE INGRESOS TAMBIEN TENEMOS UN PLAN IDEAL PARA TI!
¡MENCIONA QUE LO VISTE POR INTERNET Y OBTEN UNA ATRACTIVA BONIFICACION!!
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{
"redpajama_set_name": "RedPajamaC4"
}
| 346
|
<html><head><title>CameraFollow</title></head>
<body bgcolor="#EFF1F0" link="#3A3966" vlink="#000000" alink="#000000">
<font face="Verdana, sans-serif" size="2"><p align="center"><b><font size="4">CameraFollow()</font></b></p>
<p><b>Syntax</b></p><blockquote>
<font color="#3A3966"><b>CameraFollow</b></font>(#Camera, ObjectID, Angle, Height, Distance, RotationPercent, PositionPercent [, Mode])</blockquote>
</blockquote>
<b>Description</b><br><blockquote>
Follow the specified object in a smooth manner, using interpolation.
</blockquote><p><b>Parameters</b></p><blockquote>
<style type="text/css">
table.parameters { border-spacing: 0px; border-style: none; border-collapse: collapse; }
table.parameters td { border-width: 1px; padding: 6px; border-style: solid; border-color: gray; vertical-align: top; font-family:Arial; font-size:10pt; }
</style>
<table width="90%" class="parameters">
<tr><td width="10%"><i>#Camera</i></td>
<td width="90%">
The camera to use.
</td></tr>
<tr><td><i>ObjectID</i></td>
<td>
The object to follow. It can be one of the following type:
<pre><font face="Courier New, Courier, mono"size="2"> - Entity : use <a href="../entity/entityid.html">EntityID()</a> to get a valid ID.
- Light : use <a href="../light/lightid.html">LightID()</a> to get a valid ID.
- Node : use <a href="../node/nodeid.html">NodeID()</a> to get a valid ID.
- ParticleEmitter: use <a href="../particle/particleemitterid.html">ParticleEmitterID()</a> to get a valid ID.
- BillboardGroup : use <a href="../billboard/billboardgroupid.html">BillboardGroupID()</a> to get a valid ID.
- Text3D : use <a href="../text3d/text3did.html">Text3DID()</a> to get a valid ID.
</font></pre>
</td></tr>
<tr><td><i>Angle</i></td>
<td>
The angle of the camera relative to the followed object.
</td></tr>
<tr><td><i>Height</i></td>
<td>
The absolute height of the camera.
</td></tr>
<tr><td><i>Distance</i></td>
<td>
The distance of the camera relative to the followed object.
</td></tr>
<tr><td><i>RotationPercent</i></td>
<td>
Value to apply when the camera rotate to get it again in the correct angle. Valid values are from 0 to 1.
</td></tr>
<tr><td><i>PositionPercent</i></td>
<td>
Value to apply when the camera moves to get it again in the correct position. Valid values are from 0 to 1.
When value is 0, the camera doesn't move. When value is 1, the camera is set to the final position, without interpolation.
</td></tr>
<tr><td><i>Mode (optional)</i></td>
<td>
It can be one of the following value:
<pre><font face="Courier New, Courier, mono"size="2"> <font color="#924B72">#True</font> : the camera will automatically look at the followed entity (default).
<font color="#924B72">#False</font>: the camera will not automatically look at the followed entity.
</font></pre>
</td></tr>
</table>
</blockquote><p><b>Return value</b></p><blockquote>
None.
</blockquote><p><b>See Also</b></p><blockquote>
<a href="createcamera.html">CreateCamera()</a>
</Blockquote><p><b>Supported OS </b><Blockquote>All</Blockquote></p><center><- <a href=camerafixedyawaxis.html>CameraFixedYawAxis()</a> - <a href="index.html">Camera Index</a> - <a href="cameraid.html">CameraID()</a> -><br><br>
</body></html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,076
|
Q: How to Build an Interaction GUI that Interrupts Calculation? I would like to construct a GUI using just the Mathematica language that allows me to start and stop a calculation interactively with the press of buttons. I have set up the following to show what I wish to do, although you will notice that I can start the calculation, but I can't stop it.
Pressing Start will begin a count-up of num and dynamic update of the field to show num. I have a limit of 20 set so that the loop does not go on forever. I wish to be able to hit Stop along the way and halt the loop. Obviously, I need help figuring out a good way to do this through Dynamic, Monitor or some other construct.
DIAG2 := CreateDialog[
Module[{buttonpar, startbutton, stopbutton, startcalc, stopcalc,
labels, fields, num, numlim, calc},
numlim = 20;
calc = False;
labels = {"num"};
fields = {InputField[Dynamic[num], Number, Enabled -> False]};
startcalc[] := (
num = 0;
calc = True;
Print["START"];
While[calc && (num <= numlim), Pause[1]; num = num + 1;
num = num++]
);
stopcalc[] := (
calc = False;
Print["STOP"];
);
buttonpar = {ImageSize -> All,
BaseStyle -> {"Evaluate", 12, Bold}};
startbutton =
Button["START", startcalc[], Evaluate@buttonpar,
Method -> "Queued", Enabled -> True];
stopbutton =
Button["STOP", stopcalc[], Evaluate@buttonpar,
Method -> "Queued", Enabled -> True];
Panel[TableForm[{startbutton, stopbutton, labels, fields}]]
],
Modal -> False, WindowTitle -> "START/STOP TEST"
];
I run it with this:
DIAG2;
Thank you for your attention and help with this.
Ben
==================== UPDATE! ====================
Thanks to Gustavo's answer, I have modified the code to the following, which does what I had originally wanted. Now I have also include a Reset button as well as using a dynamic variable en for the Enabled parameter. This sets the Reset and Start buttons as executable at the beginning, but then they are greyed out after Start. Stop does the opposite.
I would still appreciate any further advice or comments on executing this GUI, perhaps with improvements, listing any potential problems, etc. Thank you!
DIAG2 := CreateDialog[
Module[
{buttonpar, resetbutton, startbutton, stopbutton, resetcalc,
startcalc, stopcalc, labels, fields, num, numlim, en},
numlim = 100;
num = 0;
en = True;
labels = {"num"};
fields = {InputField[Dynamic[num], Number, Enabled -> False]};
resetcalc[] := (
num = 0;
Print["RESET"];
);
startcalc[] := (
Print["START"];
en = False;
While[num <= numlim, Pause[1]; num = num + 1; num = num++];
);
stopcalc[] := (
FrontEndTokenExecute["EvaluatorAbort"];
en = True;
Print["STOP"];
);
buttonpar = {ImageSize -> All,
BaseStyle -> {"Evaluate", 12, Bold}};
resetbutton[en_] :=
Button["RESET", resetcalc[], Evaluate@buttonpar,
Method -> "Queued", Enabled -> en];
startbutton[en_] :=
Button["START", startcalc[], Evaluate@buttonpar,
Method -> "Queued", Enabled -> en];
stopbutton[en_] :=
Button["STOP", stopcalc[], Evaluate@buttonpar,
Method -> "Preemptive", Enabled -> ! en];
Panel[
TableForm[{resetbutton[Dynamic[en]], startbutton[Dynamic[en]],
stopbutton[Dynamic[en]], labels, fields}]]
],
Modal -> False, WindowTitle -> "START/STOP TEST"
];
Best Regards,
Ben
A: You could add to your GUI a button like this:
Button[
"Abort Operation",
FrontEndTokenExecute["EvaluatorAbort"]
]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,934
|
\section{Introduction}
In the past few decades, our understanding of the role of symmetries in quantum phases has been deepened very much. Even if the ground states have the same symmetries and thus cannot be distinguished from the Landau-Ginzburg-type phase transitions with spontaneous symmetry breaking, quantum many-body systems can have numerous distinct phases. For example, the Haldane phase \cite{HaldanePhysLett, Haldane, AKLT1, AKLT2} emerging in spin-1 chains cannot be characterized by any local order parameter associated with symmetry breaking, but it possesses a non-local string order\cite{denNijs, KennedyTasaki1, KennedyTasaki2} and is still a distinct quantum phase from featureless product states. Now the Haldane phase is recognized as a typical example of symmetry-protected topological (SPT) phases.\cite{Chen, SPTbook} SPT phases are characterized by non-degenerate gapped ground states without symmetry breaking which cannot be adiabatically connected to site-product states under some symmetry constraint. Since the SPT phases can be connected to trivial product states if symmetry-breaking perturbation is allowed, the presence of symmetries is indispensable for SPT phases. In fact, the Haldane phase is distinguished from product states if either time-reversal, spatial inversion, or spin dihedral symmetry is present in the system.\cite{GuWen, Pollmann1, Pollmann2} The existence of string order is also understood from a modern perspective in connection with the symmetry protection of the Haldane phase by the spin dihedral symmetry.
The concept of SPT phases is applicable to ground states of quantum many-body systems. Hence, topological phase transitions between SPT phases (and a trivial phase) are necessarily quantum phase transitions triggered by tuning of parameters of the Hamiltonian. From this perspective, ultracold atoms with great tunability of system parameters \cite{Bloch_review} are a promising candidate for direct observation of such quantum phase transitions. For example, by engineering artificial gauge fields, transitions between topologically trivial and nontrivial band structures of non-interacting systems have been observed using fermionic\cite{Jotzu} and bosonic\cite{Aidelsburger} atoms. Since interactions can be easily introduced to atoms, an intriguing prospect in this field is realization of SPT phases with strong correlations. It potentially provides a versatile platform to study exotic phase transitions arising from the topological nature of quantum systems.
In this paper, we propose an experimentally feasible scheme to realize SPT phase transitions induced by strong interactions using ultracold fermionic atoms loaded in optical lattice. Our model is based on one of the prototypical models of strongly correlated fermions: the Kondo lattice model.\cite{Coleman_book} Using an Abelian bosonization approach, we show that a one-dimensional (1D) version of the Kondo lattice model has several distinct quantum phases including a SPT phase and identify what symmetries protect them. We demonstrate that ultracold alkaline-earth-like atoms (AEA) in optical lattices can realize the SPT phase and access the associated quantum phase transitions.
In our setup, the phase transitions are triggered by Kondo effect which is induced by laser irradiation to the atoms using a recently proposed scheme.\cite{NakagawaKawakami} In this scheme, the laser field couples with the spin degrees of freedom of atoms and thereby realizes a tunable anisotropic spin exchange interaction. This feature enables us to engineer the quantum phase transitions with high controllability in sharp contrast to solid state realizations, where the strength of exchange interactions is intrinsic to the materials and is usually fixed. Furthermore, we show that the anisotropic exchange interaction realizes the Kondo effect with an ``unusual" spin state different from ordinary Kondo singlet. The unusual Kondo state is certainly distinct from the ordinary Kondo state by comparing their symmetry eigenvalues in terms of the spin $\pi$ rotation around $x$ or $y$ axis. Owing to this feature, we point out that the topological phase transition of this system is protected not only by its topological nature, but also by the symmetry eigenvalues of the spin states.
Besides providing the experimental setup, the other main aim of this paper is to provide a description of a crossover of SPT phases from interacting fermions to spin chains, using the bosonization approach.
In the Kondo lattice systems, interplay of mobile charges and their exchange coupling to localized spins leads to a variety of quantum phases with or without magnetic order.\cite{TsunetsuguSigristUeda, Tsvelik, FujimotoKawakami1, FujimotoKawakami2, Zachar1, LeHur, Zachar2, Garcia, PivovarovQi, FyeScalapino, Tsunetsugu, YuWhite, Shibata3, Shibata1, Shibata2, Shibata4, Peters, Silva-Valencia, Tsvelik2015, Tsvelik2016} The SPT phase that we focus on emerges in the 1D Kondo lattice with ferromagnetic exchange coupling (the double exchange model) and has been shown to approach the Haldane phase in the strong coupling limit.\cite{Tsunetsugu, Shibata3}
However, the main difference between the Kondo lattice and the Haldane spin chains is the existence of the charge degrees of freedom. In this case, the SPT phase is no longer treated as a bosonic spin system, but must be treated as fermions. Correspondingly, when the charge fluctuations cannot be neglected, the time-reversal and spin dihedral symmetries no longer protect the Haldane phase, and only the inversion symmetry remains as the protecting symmetry. This phenomenon was previously studied using Hubbard ladders\cite{AnfusoRosch, MoudgalyaPollmann}, but here we provide alternative derivation in the present Kondo lattice setup by the bosonization method. This method transparently captures how the SPT phase composed of interacting fermions changes into the bosonic SPT phase (the Haldane phase) as the charge degrees of freedom are frozen by taking the strong coupling limit. As a result, the topological phase of the 1D Kondo lattice fits into the $\mathbb{Z}_4$ classification of interacting fermionic SPT phases protected by the inversion symmetry in addition to the charge conservation.\cite{YouXu, Shiozaki1, Shiozaki2} The fermionic aspects of the SPT phase in the present setup can be contrasted to previous studies on realization of correlated SPT phases in cold alkaline-earth atoms,\cite{Kobayashi, Kobayashi2, Nonne1, Nonne2, Nonne3, Nonne4, Nonne5, Bois, Tanimoto, Capponi} where only the strong-coupling limit and thus spin-chain models were considered.
The organization of this paper is as follows. In Sec.\ \ref{Model}, we describe our setup used in this paper and derive a 1D Kondo lattice model as an effective low-energy theory of this system. Before analyzing the Kondo lattice model, we first examine the corresponding impurity problems in Sec.\ \ref{Imp} to obtain some intuition for the problem.
In Sec.\ \ref{Bosonization}, we proceed to an analysis of the 1D Kondo lattice model using Abelian bosonization and derive a set of renormalization group (RG) equations. Based on the RG equations, we determine the phase diagram of the system in Sec.\ \ref{Phasediagram}. In Sec.\ \ref{SymProt}, we elucidate what symmetries protect the quantum phases and describe the crossover of the SPT phase by the bosonization method.
Finally, we conclude this paper in Sec.\ \ref{Conclusion} with discussions for experimental detections.
\section{Model\label{Model}}
We start by introducing our setup and model used in this paper. Our setup utilizes a recently proposed scheme to realize the Kondo lattice using specific properties of AEA such as $^{171}$Yb, $^{173}$Yb, and $^{87}$Sr in optical lattices. \cite{NakagawaKawakami} AEA have an electronic ground state and a long-lived excited state denoted by $^1S_0$ and $^3P_0$, respectively. We consider ultracold AEA in 1D optical lattice and assign fermionic annihilation operators of the $^1S_0$ state at lattice site $j$ to $c_{j\sigma}$, and those of the $^3P_0$ state to $f_{j\sigma}$. Here the spin indices $\sigma= -I, \cdots, I$ come from the nuclear spin degrees of freedom of atoms. Since the polarizability to light is different for each state, we can load these atoms in an optical lattice with state-dependent lattice depth. This leads to state-dependent Wannier orbitals and gives transfer integrals $t_c , t_f$ to each state. Thus the model Hamiltonian can be written in the most general form as \cite{Gorshkov, NakagawaKawakami}
\begin{align}
H=&\sum_{j, \sigma}(-t_c c_{j\sigma}^\dag c_{j+1,\sigma}-t_f f_{j\sigma}^\dag f_{j+1,\sigma}+\mathrm{h.c.})+\sum_{j,\sigma}\varepsilon_f^{(0)}n_{fj\sigma}\notag\\
&+U\sum_{j,\sigma<\sigma'} n_{cj\sigma}n_{cj\sigma'}+U_{ff}\sum_{j,\sigma<\sigma'}n_{fj\sigma}n_{fj\sigma'}\notag\\
&+U_{cf}\sum_{j,\sigma,\sigma'}n_{cj\sigma}n_{fj\sigma'}+V_{\mathrm{ex}}\sum_{j,\sigma,\sigma'}c_{j\sigma}^\dag f_{j\sigma'}^\dag c_{j\sigma'}f_{j\sigma}\notag\\
&+\sum_{j,\sigma,\sigma'}(\bm{V}\cdot\bm{\sigma}_{\sigma\sigma'}e^{i\bm{K}\cdot\bm{R}_j-i\omega t}f_{j\sigma}^\dag c_{j\sigma'}+\mathrm{h.c.}),
\label{Hamil}
\end{align}
where $n_{cj\sigma}=c_{j\sigma}^\dag c_{j\sigma}$ and $n_{fj\sigma}=f_{j\sigma}^\dag f_{j\sigma}$ count the number of particles at site $j$. $\varepsilon_f^{(0)}$ denotes the excitation energy of the $^3P_0$ state from the $^1S_0$ state. The specific values of interaction parameters $U, U_{ff}, U_{cf}, V_{\mathrm{ex}}$ depend on $s$-wave scattering lengths in corresponding collision channels and the details of optical lattice setups, namely, the Wannier-function overlaps and the trap potential for confining the atoms in one direction.\cite{Gorshkov, RZhang} Since the $s$-wave scattering lengths are independent of the nuclear spin states, the interactions possess SU($N=2I+1$) symmetry\cite{Gorshkov, Cazallila} as confirmed by experiments.\cite{Taie1, Taie2, Cappellini, Scazza, Zhang} Hereafter we assume that $U>0, U_{ff}>0$. In principle, there exist additional terms originating from a magnetic field,\cite{RZhang} but for simplicity we take the zero-field limit and avoid the complication.
The last term in Eq.\ \eqref{Hamil} is an important ingredient for our model. This term represents optical transitions between the $^1S_0$ state and the $^3P_0$ state allowed by dipole coupling with the help of hyperfine interactions.\cite{Porsev} From the Wigner-Eckart theorem, we find that the matrix elements are the inner product of a three-component vector $\bm{V}$ (which is proportional to the electric field component of the optical field) and Pauli matrices of the nuclear spin.\cite{Beri, NakagawaKawakami} $\bm{K}$ and $\omega$ are the wave number and the frequency of the optical field, respectively. We here consider a $\pi$-polarized laser field with $\bm{V}=(0,0,V)$, which does not break the time-reversal symmetry.
The explicit time dependence in the hybridization term of Eq.\ \eqref{Hamil} is eliminated by a gauge transformation $f_{j\sigma}\to e^{-i\omega t}f_{j\sigma}$. After this transformation, the energy level of the $^3P_0$ state is effectively shifted, and we replace $\varepsilon_f^{(0)}$ with $\varepsilon_f\equiv \varepsilon_f^{(0)}-\omega$. Besides the trivial time dependence due to the gauge transformation, the system is assumed to be an equilibrium state with temperature $T$ and chemical potential $\mu$. In this paper, we mainly consider the case of $T=0$ and focus on quantum phase transitions that the system exhibits.
We assume that the lattice potential is sufficiently deep for the $^3P_0$ state to suppress inelastic collisions which cause loss of atoms, and thus $t_f\ll U_{ff}$. On the other hand, the lattice potential for the $^1S_0$ state is shallow to allow the hopping between sites. To simplify the original model \eqref{Hamil}, we consider a limiting case in which the Kondo limit is achieved: $\varepsilon_f\ll \mu \ll \varepsilon_f+U_{ff}$ and $|\bm{V}|$ is sufficiently small. In this case, since the occupation number of the $^3P_0$ state in low-energy states is one at each site, we can restrict ourselves to the Hilbert subspace with $\sum_\sigma n_{fj\sigma}=1$ and derive an effective low-energy Hamiltonian using the Schrieffer-Wolff transformation.\cite{SW} The resulting low-energy theory leads to the Kondo lattice (or Kondo-Heisenberg) model
\begin{align}
H_{\mathrm{eff}}=&-t_c\sum_{j,\sigma}(c_{j\sigma}^\dag c_{j+1,\sigma}+\mathrm{h.c.})+U\sum_{j,\sigma<\sigma'} n_{cj\sigma}n_{cj\sigma'}\notag\\
&+\sum_{j,\sigma,\sigma'}(V_{\mathrm{ex}}-\sigma\sigma'J)c_{j\sigma}^\dag f_{j\sigma'}^\dag c_{j\sigma'}f_{j\sigma}\notag\\
&+J_H\sum_{j,\sigma,\sigma'}f_{j\sigma}^\dag f_{j\sigma'} f_{j+1,\sigma'}^\dag f_{j+1,\sigma}
\label{Heff}
\end{align}
where $J=2V^2(\frac{1}{|\varepsilon_f-\mu|}+\frac{1}{\varepsilon_f-\mu+U_{ff}})>0$ and $J_H=4t_f^2/U_{ff}>0$. We note that when $\sum_\sigma n_{fj\sigma}=1$, the interaction $U_{cf}$ can be incorporated into the chemical potential and therefore we omit this term from $H_{\mathrm{eff}}$.
The effective Hamiltonian \eqref{Heff} contains an effective Kondo interaction $V_{\mathrm{ex}},J$ between the two orbitals and the Heisenberg interaction $J_H$ between $^3P_0$ states. While the spin-exchanging collision $V_{\mathrm{ex}}$ is fully symmetric, the optically induced Kondo coupling $J$ breaks the spin SU($N$) symmetry due to the polarization-spin coupling in the last term in Eq.\ \eqref{Hamil}. For general $N$, this Kondo coupling is somewhat complicated, but the case of $N=2$ is simple. For $N=2$, we can rewrite the Kondo coupling as
\begin{align}
&\sum_{j,\sigma,\sigma'}(V_{\mathrm{ex}}-\sigma\sigma'J)c_{j\sigma}^\dag f_{j\sigma'}^\dag c_{j\sigma'}f_{j\sigma}\notag\\
=&-J_\perp\sum_j(S_{cj}^xS_{fj}^x+S_{cj}^yS_{fj}^y)-J_z\sum_jS_{cj}^zS_{fj}^z\notag\\
&+\mathrm{potential\; term},
\label{JK}
\end{align}
where
\begin{subequations}
\begin{align}
J_\perp&\equiv V_{\mathrm{ex}}+J/4,\label{Jperp}\\
J_z&\equiv V_{\mathrm{ex}}-J/4.\label{Jz}
\end{align}
\end{subequations}
The ``potential term" can be absorbed into the chemical potential. The spin operators are defined by $\bm{S}_{cj}=\frac{1}{2}\sum_{\sigma,\sigma'}c_{j\sigma}^\dag\bm{\sigma}_{\sigma\sigma'}c_{j\sigma'}$ and $\bm{S}_{fj}=\frac{1}{2}\sum_{\sigma,\sigma'}f_{j\sigma}^\dag\bm{\sigma}_{\sigma\sigma'}f_{j\sigma'}$, where $\bm{\sigma}$ is the three-component Pauli matrices. This interaction is just an anisotropic XXZ-type exchange coupling between the $^1S_0$ and the $^3P_0$ states.
Hereafter, we analyze the low-energy effective model \eqref{Heff} for $N=2$. Experimentally, this case can be realized using two specific spin states $\sigma$ and $-\sigma$ selected from $2I+1$ nuclear spins of AEA.
\section{Kondo impurity\label{Imp}}
Before studying the full Kondo lattice Hamiltonian \eqref{Heff}, it is helpful to gain some insights from what happens when a single atom in the $^3P_0$ state is immersed into the Fermi sea of $^1S_0$ atoms as an impurity. Here we summarize known basic results\cite{Coleman_book, Anderson, LeeToner, FurusakiNagaosa} and extend them to obtain a phase diagram in Fig.\ \ref{fig_imp} (b) which is important for later analysis. Let us consider the following Kondo impurity problems:
\begin{align}
H_{3D}=-t_c&\sum_{\langle i,j\rangle,\sigma}(c_{i\sigma}^\dag c_{j\sigma}+\mathrm{h.c.})\notag\\
&-J_\perp(S_{c0}^xS_{\mathrm{imp}}^x+S_{c0}^yS_{\mathrm{imp}}^y)-J_zS_{c0}^zS_{\mathrm{imp}}^z,\\
H_{1D}=-t_c&\sum_{j,\sigma}(c_{j,\sigma}^\dag c_{j+1,\sigma}+\mathrm{h.c.})+U\sum_j n_{cj\uparrow}n_{cj\downarrow}\notag\\
&-J_\perp(S_{c0}^xS_{\mathrm{imp}}^x+S_{c0}^yS_{\mathrm{imp}}^y)-J_zS_{c0}^zS_{\mathrm{imp}}^z.\label{H1D}
\end{align}
In both cases, a single impurity spin is located at $j=0$. The impurity interacts with itinerant fermions living in 3D (or 1D) lattices via anisotropic Kondo couplings. In the 1D case, we have introduced the interaction between itinerant fermions and consider a metallic Tomonaga-Luttinger-liquid region away from half filling. If we set a high-energy cutoff (the bandwidth) as $D$, the RG equations for the 3D case are\cite{Anderson}
\begin{subequations}
\begin{align}
\frac{dJ_\perp}{d\ell}&=-\rho_0 J_\perp J_z,\\
\frac{dJ_z}{d\ell}&=-\rho_0J_\perp^2,
\end{align}
\end{subequations}
where $d\ell=-d\ln D$. Here $\rho_0$ is the density of states at the Fermi energy. The flow diagram is depicted in Fig.\ \ref{fig_imp} (a). The system has two fixed points characterized by growth of Kondo coupling with different signs of $J_\perp$. The fixed point with $J_\perp\to-\infty, J_z\to-\infty$ corresponds to the ordinary Kondo effect with isotropic antiferromagnetic interactions. However, an important aspect arises from the other fixed point in Fig. \ref{fig_imp} (a) for the present setup in cold atoms. As found from Eqs.\ \eqref{Jperp} and \eqref{Jz}, when the laser-induced Kondo coupling is sufficiently strong, we reach the fixed point with $J_\perp\to\infty, J_z\to-\infty$. The nature of this fixed point can be extracted from a transformation
\begin{equation}
(S_{\mathrm{imp}}^x, S_{\mathrm{imp}}^y, S_{\mathrm{imp}}^z)\rightarrow(-S_{\mathrm{imp}}^x, -S_{\mathrm{imp}}^y, S_{\mathrm{imp}}^z),
\label{anomKondotfm}
\end{equation}
which is equivalent to flipping the sign of $J_\perp$. Note that this transformation keeps the commutation relation intact. Since the singlet state $\ket{\downarrow}_c\ket{\uparrow}_f-\ket{\uparrow}_c\ket{\downarrow}_f$ is transformed into $\ket{\downarrow}_c\ket{\uparrow}_f+\ket{\uparrow}_c\ket{\downarrow}_f$ by this procedure, we find that the fixed point describes the Kondo effect with Kondo ``singlet" $\ket{\downarrow}_c\ket{\uparrow}_f+\ket{\uparrow}_c\ket{\downarrow}_f$.
\begin{figure}[t]
\includegraphics[width=8.5cm]{Fig1.pdf}
\caption{(a) RG flow for the Kondo impurity in 3D. (b) Phase diagram of the Kondo impurity in 1D. We set $J_{\perp F}=J_{\perp B}=J_\perp, J_{zF}=J_{zB}=J_z$, $v_F=1$, and $g_2=0.5$. The broken line indicates the isotropic line on which $J_\perp=J_z$ is satisfied.}
\label{fig_imp}
\end{figure}
The 1D case was studied by Refs.\ \onlinecite{LeeToner, FurusakiNagaosa}, and the situation is somewhat different from 3D. In 1D, the forward scattering off the impurity and the backward one are distinguished. Hence we must double the coupling constants for the Kondo coupling: $J_{\perp F}, J_{\perp B}, J_{zF}, J_{zB}$ where the subscript $F$ ($B$) denotes the forward (backward) process. Then the RG equations are given by
\begin{subequations}
\begin{align}
\frac{dJ_{\perp F}}{d\ell}&=-\frac{1}{2\pi v_F}(J_{\perp F}J_{zF}+J_{\perp B}J_{zB}),\label{impRG1D_1}\\
\frac{dJ_{\perp B}}{d\ell}&=-\frac{1}{2\pi v_F}(J_{\perp F}^2+J_{\perp B}^2),\\
\frac{dJ_{zF}}{d\ell}&=\frac{1}{2\pi v_F}(g_2 J_{\perp B}-J_{\perp F}J_{zB}-J_{\perp B}J_{zF}),\\
\frac{dJ_{zB}}{d\ell}&=\frac{1}{2\pi v_F}(g_2 J_{zB}-2J_{\perp F}J_{\perp B}),\label{impRG1D_4}
\end{align}
\end{subequations}
where $g_2$ denotes the matrix element of the forward scattering process between itinerant fermions due to the Hubbard repulsion in Eq.\ \eqref{H1D}. $v_F$ is the Fermi velocity.
By integrating Eqs.\ \eqref{impRG1D_1} - \eqref{impRG1D_4} numerically, we obtain a phase diagram in Fig.\ \ref{fig_imp} (b), although the flow diagram was shown only for the isotropic ($J_\perp=J_z$) case in Ref.\ \onlinecite{FurusakiNagaosa}. The phase (K) shows the ordinary Kondo effect and the phase (K') shows the ``unusual" Kondo effect as in the 3D case. A peculiar point in 1D is the existence of a new phase (F) where the exchange coupling grows to strong coupling starting from bare \textit{ferromagnetic} interactions. This fixed point appears only when $g_2>0$ is included,\cite{FurusakiNagaosa} and therefore we need to consider the Hubbard repulsion in Eq.\ \eqref{H1D}. At the fixed point, the coupling constants grow as $J_{\perp F}\to-\infty, J_{\perp B}\to\infty, J_{zF}\to-\infty, J_{zB}\to\infty$. Note that the signs are negative for the forward processes and positive for the backward ones. From this observation, it turns out that the fixed point describes growth of nearest-neighbor antiferromagnetic Kondo coupling which leads to a Kondo singlet state with the adjacent sites of the impurity, while the onsite Kondo coupling is kept finite.\cite{FurusakiNagaosa} The phase (F') is not important for the later discussions, but the nature of this phase is also understood by the transformation \eqref{anomKondotfm}. In the subsequent sections, we show that the phase diagram of 1D Kondo lattice has similarity to the 1D impurity case.
\section{Renormalization group analysis of 1D anisotropic Kondo lattice\label{Bosonization}}
Let us now proceed to the analysis of the 1D Kondo lattice model. Hereafter we consider the Hamiltonian \eqref{Heff} for $N=2$ with the half-filling condition for $^1S_0$ states. To analyze the low-energy behavior of the system, we apply Abelian bosonization\cite{Giamarchi} to the Hamiltonian using the following identity:
\begin{align}
c_{j\sigma}=\frac{1}{\sqrt{2\pi}}(&\eta_{R\sigma}e^{ik_Fx}e^{i(\theta_{1\sigma}(x)-\phi_{1\sigma}(x))}\notag\\
&+\eta_{L\sigma}e^{-ik_Fx}e^{i(\theta_{1\sigma}(x)+\phi_{1\sigma}(x))})
\end{align}
where $x=ja$ is the continuum space variable and the boson fields $\phi, \theta$ satisfy a commutation relation $[\phi_{1\sigma}(x),\nabla\theta_{1\sigma'}(y)]=i\pi\delta_{\sigma\sigma'}\delta(x-y)$. In the above expression, the boson field $\phi$ is compactified as $\phi\sim\phi+2\pi$.
The Fermi momentum $k_F$ is fixed at $k_F=\pi/2a$ due to the half-filling condition. $\eta_{R/L\sigma}$ is a Klein factor expressed in terms of Majorana fermions satisfying $\{\eta_{\alpha},\eta_{\beta}\}=2\delta_{\alpha\beta}$, which ensures the anticommutation relation between the right mover and the left mover. Similarly, we introduce the boson fields $\phi_{2\sigma}, \theta_{2\sigma}$ for the $f_{j\sigma}$ fermions. Following standard calculations detailed in Appendix \ref{App_bos}, we obtain
\begin{widetext}
\begin{align}
H_{\mathrm{eff}}=&H_0+H_{\mathrm{int}},\label{H_bos}\\
H_0=&\frac{1}{2\pi}\int dx(u_{1c}K_{1c}(\nabla\theta_{1c})^2+\frac{u_{1c}}{K_{1c}}(\nabla\phi_{1c})^2
+\sum_{\nu=\pm}\frac{1}{2\pi}\int dx(u_{\nu}K_{\nu}(\nabla\theta_{\nu})^2+\frac{u_{\nu}}{K_{\nu}}(\nabla\phi_{\nu})^2),\\
H_{\mathrm{int}}=&g_U\int dx \cos(2\sqrt{2}\phi_{1c}
-g_{K\perp F+
\int dx\cos 2\phi_+\cos 2\theta_
-g_{K\perp F-
\int dx\cos 2\phi_-\cos 2\theta_-\notag\\
&-g_{K\perp B
\int dx\sin\sqrt{2}\phi_{1c}\cos 2\theta_-
-g_{KzB+
\int dx\sin\sqrt{2}\phi_{1c}\cos 2\phi_
-g_{KzB-
\int dx\sin\sqrt{2}\phi_{1c}\cos 2\phi_-,
\label{Hint}
\end{align}
\end{widetext}
where
\begin{subequations}
\begin{gather}
u_{1c}=2t_c a\sqrt{1+\frac{U}{2\pi t_c}},\\
u_\pm=2t_c a\sqrt{\bigl(1-\frac{U}{2\pi t_c}\bigr)\bigl(1\mp\frac{\alpha J_z}{2\pi u}\bigr)},\\
K_{1c}=1/\sqrt{1+\frac{U}{2\pi t_c}},\\
K_\pm=\frac{1}{\sqrt{1\mp\frac{\alpha J_z}{2\pi u}}},
\end{gather}
\end{subequations}
and the coupling constants are
\begin{subequations}
\begin{gather}
g_{U}=\frac{U}{2\pi^2\alpha},\\
g_{K\perp F+}=g_{K\perp F-}=\frac{1}{2m}g_{K\perp B}=\frac{J_\perp}{2\pi^2\alpha},\\
g_{KzB+}=g_{KzB-}=\frac{mJ_z}{2\pi^2\alpha}.
\end{gather}
\end{subequations}
Here $\alpha$ denotes the short-range cutoff and $m=\langle \sin\sqrt{2}\phi_{2c}\rangle$ is the expectation value of the gapped charge mode of localized $f$ fermions. The new boson fields for the charge mode (of $^1S_0$ state) $\phi_{1c},\theta_{1c}$ and the total/relative spin modes $\phi_\pm, \theta_\pm$ are defined as
\begin{subequations}
\begin{align}
\phi_{1c}\equiv&\frac{1}{\sqrt{2}}(\phi_{1\uparrow}+\phi_{1\downarrow}),\\
\theta_{1c}\equiv&\frac{1}{\sqrt{2}}(\theta_{1\uparrow}+\theta_{1\downarrow}),\\
\phi_\pm\equiv&\frac{1}{2}(\phi_{1\uparrow}-\phi_{1\downarrow}\pm(\phi_{2\uparrow}-\phi_{2\downarrow})),\\
\theta_\pm\equiv&\frac{1}{2}(\theta_{1\uparrow}-\theta_{1\downarrow}\pm(\theta_{2\uparrow}-\theta_{2\downarrow})).
\end{align}
\end{subequations}
For later convenience, we name each term in Eq.\ \eqref{Hint} as $H_U, H_{K\perp F+},H_{K\perp F-}, H_{K\perp B}, H_{KzB+},$ and $H_{KzB-}$, where the subscripts correspond to those of the coupling constants (see Appendix \ref{App_bos}).
The low-energy behavior of the model \eqref{H_bos} is deduced from perturbative RG analysis in terms of $H_{\mathrm{int}}$. Since the unperturbed theory $H_0$ is free bosons and thus is a conformal field theory (CFT), the RG equations can be derived from the CFT data of the free boson theory, i.e. scaling dimensions and operator-product-expansion coefficients.\cite{Fradkin} After some calculations, we arrive at a set of RG equations when the cutoff is changed from $\alpha$ to $e^{d\ell}\alpha$, as
\begin{subequations}
\begin{align}
\frac{dK_{1c}}{d\ell}=&-K_{1c}^2(2\tilde{g}_{U}+2\tilde{g}_{K\perp B}^2+\tilde{g}_{KzB+}^2+\tilde{g}_{KzB-}^2),\label{RGK1c}\\
\frac{dK_{+}}{d\ell}=&-K_+^2(2\tilde{g}_{K\perp F+}^2+2\tilde{g}_{KzB+}^2),\\
\frac{dK_{-}}{d\ell}=&-K_-^2(2\tilde{g}_{K\perp F-}^2+2\tilde{g}_{KzB-}^2)\notag\\
&+2\tilde{g}_{K\perp F+}^2+2\tilde{g}_{K\perp F-}^2+4\tilde{g}_{K\perp B}^2,
\end{align}
and
\begin{align}
\frac{d\tilde{g}_{U}}{d\ell}=&(2-2K_{1c})\tilde{g}_{U}+\tilde{g}_{K\perp B}^2+\tilde{g}_{KzB+}^2+\tilde{g}_{KzB-}^2,\label{RGU}\\
\frac{d\tilde{g}_{K\perp F+}}{d\ell}=&(2-K_+-\frac{1}{K_-})\tilde{g}_{K\perp F+}-\tilde{g}_{K\perp B}\tilde{g}_{KzB+},\label{RGKperpFp}\\
\frac{d\tilde{g}_{K\perp F-}}{d\ell}=&(2-K_--\frac{1}{K_-})\tilde{g}_{K\perp F-}-\tilde{g}_{K\perp B}\tilde{g}_{KzB-},\\
\frac{d\tilde{g}_{K\perp B}}{d\ell}=&(2-\frac{1}{2}K_{1c}-\frac{1}{K_-})\tilde{g}_{K\perp B}\notag\\
&-\tilde{g}_{K\perp F+}\tilde{g}_{KzB+}-\tilde{g}_{K\perp F-}\tilde{g}_{KzB-}+\frac{1}{2}\tilde{g}_{U}\tilde{g}_{K\perp B},\\
\frac{d\tilde{g}_{KzB+}}{d\ell}=&(2-\frac{1}{2}K_{1c}-K_+)\tilde{g}_{KzB+}\notag\\
&-\tilde{g}_{K\perp F+}\tilde{g}_{K\perp B}+\frac{1}{2}\tilde{g}_{U}\tilde{g}_{KzB+}\label{RGKz2p},\\
\frac{d\tilde{g}_{KzB-}}{d\ell}=&(2-\frac{1}{2}K_{1c}-K_-)\tilde{g}_{KzB-}\notag\\
&-\tilde{g}_{K\perp F-}\tilde{g}_{K\perp B}+\frac{1}{2}\tilde{g}_{U}\tilde{g}_{KzB-},\label{RGKz2m}
\end{align}
\end{subequations}
up to the second order perturbation theory. Here the dimensionless coupling constants are defined by $\tilde{g}_{\alpha}\equiv \frac{1}{\pi}g_{\alpha}a^{2-\Delta_\alpha}$, where $\Delta_\alpha$ is the scaling dimension of the perturbation.
\section{Phase diagram\label{Phasediagram}}
The zero-temperature phase diagram of the system is determined by fixed points derived from the RG equations \eqref{RGK1c} - \eqref{RGKz2m}. Numerical solutions of the RG equations indicate the phase diagram summarized in Fig.\ \ref{fig_phase}. In calculating Fig.\ \ref{fig_phase}, we have set $\tilde{g}_U=0.1$ and the initial values of the coupling constants as $\tilde{g}_{K\perp F\pm}=\frac{1}{2}\tilde{g}_{K\perp B}=\tilde{g}_{K\perp}$ and $\tilde{g}_{KzB\pm}=\tilde{g}_{Kz}$. The phase diagram is fully symmetric with respect to the sign of $\tilde{g}_{K\perp}$. As seen from scaling dimensions, the low-energy behavior is mainly governed by relevant terms $H_{K\perp B}, H_{KzB+}$, and $H_{KzB-}$. Each phase is characterized by the most divergent interactions as follows:
\begin{center}
\begin{table}[h]
\begin{tabular}{lll}
(K) & $\tilde{g}_{K\perp B}\to-\infty,$ & $\tilde{g}_{KzB+}\to-\infty$ \\
(K') & $\tilde{g}_{K\perp B}\to+\infty,$ & $\tilde{g}_{KzB+}\to-\infty$ \\
(Top) & $\tilde{g}_{K\perp B}\to+\infty,$ & $\tilde{g}_{KzB+}\to+\infty$ \\
(Top') & $\tilde{g}_{K\perp B}\to-\infty,$ & $\tilde{g}_{KzB+}\to+\infty$ \\
(N1) & $\tilde{g}_{KzB+}\to-\infty,$ & $\tilde{g}_{KzB-}\to-\infty$\\
(N2) & $\tilde{g}_{KzB+}\to+\infty,$ & $\tilde{g}_{KzB-}\to+\infty$
\end{tabular}
\end{table}
\end{center}
The phase boundary between (K') and (Top) [or (K) and (Top')] is signaled by the change of the sign of $\tilde{g}_{KzB+}$. On the other hand, the transitions to the phase (N1) or (N2) are determined by competition between $H_{K\perp B}$ and $H_{KzB-}$, which cannot be minimized simultaneously. Since the renormalization is stopped around $\tilde{g}(\ell)\sim 1$, we determine those phase boundaries by examining which of $\tilde{g}_{K\perp B}$ and $\tilde{g}_{KzB-}$ first grows to unity. We note that the role of less relevant $H_U, H_{K\perp F\pm}$ terms is the shift of phase boundaries. If we truncate the RG equations up to the tree level, the phase boundary between the phase (K') and the phase (Top) is located at $\tilde{g}_{Kz}=0$. Thus the generation of effective couplings due to less relevant interactions significantly shifts the phase boundaries. We note that the precise positions of phase boundaries depend on the Luttinger parameter.
Qualitatively, our weak-coupling calculation by the perturbative RG approach reproduces the phase diagram of 1D anisotropic Kondo lattice obtained by strong coupling expansion and exact diagonalization of a small cluster.\cite{Shibata3} Although Ref.\ \onlinecite{Shibata3} explained each phase based on spin-chain pictures in the strong coupling limit, we here point out that the phase diagram has some resemblance with the impurity phase diagram in Fig.\ \ref{fig_imp} (b) except for the appearance of the phases (N1) and (N2) which denote N\'{e}el orders.
This resemblance can be understood to some extent by comparing the RG equations \eqref{impRG1D_1}-\eqref{impRG1D_4} and \eqref{RGKperpFp}-\eqref{RGKz2m}. Hence, our weak-coupling approach provides a complementary understanding of the phase diagram in Ref.\ \onlinecite{Shibata3}. In the following subsections, we explain the details of each phase, keeping in mind the connection to the impurity physics.
\begin{figure}[h]
\includegraphics[width=5.5cm]{Fig2.pdf}
\caption{Phase diagram of the 1D anisotropic Kondo lattice model. The broken line indicates the isotropic line on which $J_\perp=J_z$ is satisfied.}
\label{fig_phase}
\end{figure}
\begin{figure}
\includegraphics[width=7.5cm]{Fig3.pdf}
\caption{Schematic pictures of the phases of the 1D Kondo lattice. The red (blue) balls illustrate atoms in the $^1S_0$ ($^3P_0$) state loaded in a shallow (deep) optical lattice potential. In the figure of the phase (Top), the singlet formation is represented by the central site for clarity of illustration.}
\label{fig_state}
\end{figure}
\subsection{Kondo insulator}
The phases (K), (K'), (Top), and (Top') are described by pinning of $\phi_{1c}, \phi_+$ and $\theta_-$ to their potential minimum, leading to disordered ground states with an energy gap. The phase (K) corresponds to the growth of on-site antiferrromagnetic Kondo coupling, which means the formation of the Kondo insulator.\cite{TsunetsuguSigristUeda} The strong coupling picture of this phase is illustrated in Fig.\ \ref{fig_state}, where the Kondo singlet at each site opens the energy gaps in charge and spin sectors.
We note that the Kondo coupling effectively generates the Hubbard repulsion between conductive fermions due to Eq.\ \eqref{RGU}. Hence, even if the bare Hubbard interaction is switched off, the Kondo insulator cannot be distinguished from the Mott insulating state at least in the low-energy region. At the strong coupling limit, the Kondo insulating state approaches to the rung-singlet state if we regard the system as a spin-1/2 ladder.
\subsection{Laser-induced Kondo insulator}
With sufficiently strong laser coupling, the phase (K') is realized owing to Eqs.\ \eqref{Jperp} and \eqref{Jz}. This phase is also a Kondo insulator, but is composed of the ``unusual" Kondo effect described in Sec.\ \ref{Imp} by a strong coupling fixed point with anisotropic Kondo coupling. As in Sec.\ \ref{Imp}, a physical picture of this Kondo insulator is obtained by a unitary transformation
\begin{equation}
f_{j\sigma}\to \mathrm{sgn}(\sigma)f_{j\sigma},
\label{lattice_AK_tfm}
\end{equation}
which flips the sign of $S_{fj}^x, S_{fj}^y$ and maps the Kondo singlet $\ket{\downarrow}_c\ket{\uparrow}_f-\ket{\uparrow}_c\ket{\downarrow}_f$ to $\ket{\downarrow}_c\ket{\uparrow}_f+\ket{\uparrow}_c\ket{\downarrow}_f$. Thus, in the strong coupling limit, the phase (K') is described by an insulating state where the $^1S_0$ state and the $^3P_0$ state form the ``Kondo singlet" $\ket{\downarrow}_c\ket{\uparrow}_f+\ket{\uparrow}_c\ket{\downarrow}_f$ at each site (Fig. \ref{fig_state}). The unusual Kondo singlet has total spin 1 with $S_c^z+S_f^z=0$, and therefore the expectation value of total spin is nonzero in the $x, y$ plane: $\langle(S_{cj}^x+S_{fj}^x)^2+(S_{cj}^y+S_{fj}^y)^2\rangle\neq0$. In the language of spin systems, this phase is very similar to the so-called large-D phase\cite{Shibata3} where a strong single-ion anisotropy favors the $S^z=0$ state in spin-1 systems.\cite{ChenHidaSanctuary}
\subsection{Topological phase}
The phase (Top) in Fig.\ \ref{fig_phase} is a nontrivial topological phase protected by the spatial inversion symmetry, whose topological aspects are described in the next section. This phase includes the case of isotropic ferromagnetic Kondo coupling indicated by the broken line in Fig.\ \ref{fig_phase}. This phase is smoothly connected to the Haldane phase in spin ladders\cite{StrongMillis1, StrongMillis2, Shelton, Lecheminant} in the strong coupling limit $U\to\infty$ (or $J_\perp, J_z\to\infty$).\cite{Tsunetsugu} An intuitive picture of this fixed point can be obtained by considering a nearest-neighbor Kondo coupling
\begin{equation}
\tilde{H}_K\equiv -\tilde{J}\sum_j(\bm{S}_{c,j-1}+\bm{S}_{c,j+1})\cdot\bm{S}_{f,j}
\end{equation}
in addition to the original on-site Kondo coupling. The bosonized Hamiltonian is changed as
\begin{subequations}
\begin{align}
H_{K\perp F}&\to \frac{J_\perp+\tilde{J}}{J_\perp}H_{K\perp F},\\
H_{K\perp B}&\to \frac{J_\perp-\tilde{J}}{J_\perp}H_{K\perp B},\\
H_{Kz F}&\to \frac{J_z+\tilde{J}}{J_z}H_{Kz F},\\
H_{Kz B}&\to \frac{J_z-\tilde{J}}{J_z}H_{Kz B}.
\end{align}
\end{subequations}
Thus, the fixed point is equivalent to growth of the \textit{nearest-neighbor antiferromagnetic} Kondo coupling, similarly to the phase (F) appearing in the 1D Kondo impurity problem in Sec.\ \ref{Imp}, while the on-site Kondo coupling is ferromagnetic and kept finite. An intuitive picture is illustrated in Fig.\ \ref{fig_state}. The formation of the non-local Kondo singlets is reminiscent of 1D topological Kondo insulators\cite{Alexandrov, Lobos1, Lobos2, Hagymasi} realized by a $p$-wave Kondo coupling. In fact, the low-energy effective theory is the same as that of the 1D topological Kondo insulators.\cite{Lobos1}
We note that the nature of the phase (Top') is related to the topological phase (Top) via the transformation \eqref{lattice_AK_tfm}, although this phase cannot be realized because the coupling constants cannot be manipulated into the corresponding parameter region, since $J$ is always positive in Eqs.\ \eqref{Jperp} and \eqref{Jz}.
\subsection{N\'{e}el order}
The phases (N1) and (N2) which appear near the ``Ising line" $J_\perp=0$ have an antiferromagnetic N\'{e}el order with spontaneously broken spin flip symmetry. The ordered spin patterns are illustrated in Fig.\ \ref{fig_state}. To understand the appearance of the N\'{e}el order, it is useful to consider the case of $J_\perp=0$. In this case, the remaining perturbation terms are $H_{U}, H_{KzB+},$ and $H_{KzB-}$, which are all relevant for $U>0$ and thus lock the fields $\phi_{1c}, \phi_{+}, \phi_{-}$ at their potential minimum. The locking of $\phi_\pm$ leads to the nonzero expectation value of $N_{c,f}^z(x)$ [Eq.\ \eqref{Ncz}], implying the emergence of the N\'{e}el ordering. Since the pinning of $\phi_{1c}, \phi_{+}, \phi_{-}$ opens the energy gap and the gap cannot be collapsed by infinitesimal perturbation, the N\'{e}el order should persist to some threshold value of $J_\perp$. However, the threshold value should not exceed $|J_z|$, since at the isotropic line $|J_{\perp}|=|J_z|$ we obtain the Kondo insulating phases or the topological phases by non-Abelian bosonization.\cite{FujimotoKawakami1, FujimotoKawakami2}
The existence of the N\'{e}el order can also be naturally understood from the corresponding impurity problem. When the Kondo coupling is completely Ising-like with vanishing $J_\perp$, we do not have the Kondo effect and the impurity ground state is doubly degenerate where the spins of conduction electrons and the impurity align ferromagnetically in $J_z>0$ and antiferromagnetically in $J_z<0$. Thus the residual impurity entropy $\ln 2$ should be washed out by spin ordering in the case of Kondo lattice systems.
\section{Symmetry protection\label{SymProt}}
All the quantum phases of the 1D anisotropic Kondo lattice described in Sec.\ \ref{Phasediagram} have energy gaps both in charge and spin excitations. While the N\'{e}el orders can be characterized by spontaneous breaking of the spin flip symmetry, rest four phases have the same symmetries and cannot be characterized by spontaneous symmetry breaking. In this section, we describe the roles of various symmetries in the system and provide conditions to distinguish these four phases as different quantum phases.
\subsection{Protection by spatial inversion symmetry: a crossover from a fermionic SPT phase to a bosonic SPT phase}
First, we describe what symmetry protects the topological phase (Top). The topological phase approaches the Haldane phase in spin chains in the strong coupling limit $U\to\infty$. Hence the topological phase of the 1D Kondo lattice is expected to be stable under either time-reversal, spatial inversion, or spin dihedral symmetry, if $U$ is sufficiently large and the charge degrees of freedom are frozen in the low-energy part of the Hilbert space. However, if $J_\perp, J_z$ and $U$ are small compared to the kinetic energy $t_c$, we can no more regard the system as bosonic (spin) systems and must treat it as interacting fermions. It was previously shown\cite{AnfusoRosch, MoudgalyaPollmann} that the Haldane phase with mobile charge degrees freedom is unstable and can be adiabatically connected to a trivial band insulator by only breaking inversion symmetry, even if the time-reversal and spin rotation symmetries are preserved. This fact stems from that the charge fluctuations in the low-energy Hilbert space mix the integer-spin representation of the original spin chain and that of half-odd-integer spin, invalidating the proof of the symmetry protection of the Haldane phase. Hence the only protecting symmetry of the topological phase is the inversion symmetry. Under the inversion symmetry, the degeneracy of the entanglement spectrum, which is a fingerprint of the SPT phase, still persists.\cite{MoudgalyaPollmann} A similar degenerate structure of the entanglement spectrum is also observed in 1D topological Kondo insulator\cite{Hagymasi} and 1D periodic Anderson model with Hund coupling,\cite{Hagymasi2} indicating the existence of the SPT phase. A related study on a three-leg Hubbard ladder has been also performed.\cite{Nourse}
Here we show that the above difference between the fermionic and the bosonic SPT phases is captured by the bosonization method in the present Kondo lattice system. To apply the symmetry protection argument to the present Kondo lattice system, we summarize the symmetry transformation of bosonized fields for each symmetry of the system in Table \ref{SymTfm}. Let us first consider the strong coupling limit $U\to\infty$. In this case, the charge mode $\phi_{1c}$ is completely frozen to the potential minimum of the Umklapp term $H_U$. The remaining degrees of freedom are the total and relative spin modes $\phi_\pm, \theta_\pm$, and they are equivalent to the effective theory of the corresponding spin ladder system.\cite{Shelton, Lecheminant} Hence the proof of symmetry protection can be performed in parallel with the case of the spin ladder\cite{Fuji} (see Appendix \ref{App_SPT} for the description of SPT phases by bosonization).
The gapped phases are characterized by the expectation values of the boson fields $\phi_+$ and $\theta_-$.
To connect the topological phase with the trivial phases, a shift of the expectation value of $\phi_+$ must take place, which breaks the time-reversal, spatial inversion, and spin dihedral symmetries. Hence the topological phase is protected by those three symmetries. However, the situation is changed if we consider a weakly interacting regime. If the Hubbard interaction $U$ is sufficiently small, the Umklapp term is less relevant than the Kondo couplings $H_{K\perp B}, H_{KzB+}$, and $H_{KzB-}$. Thus the low-energy behavior is mainly governed by the Kondo couplings. In this case, we can adiabatically connect the topological phase (Top) and the ordinary Kondo insulator (K) without closing the energy gap, by shifting the expectation value of the charge mode. This is done by adding the following perturbation:
\begin{align}
&g_{K\perp B}'\int dx\cos\sqrt{2}\phi_{1c}\cos 2\theta_-\notag\\
+&g_{KzB+}'\int dx\cos\sqrt{2}\phi_{1c}\cos 2\phi_+
\label{ptb_SPT_bos}
\end{align}
which is generated by an artificial Kondo coupling
\begin{equation}
H'_K=J'\sum_{j,\sigma,\sigma'}c_{j\sigma}^\dag\bm{\sigma}_{\sigma\sigma'}c_{j+1\sigma'}\cdot \bm{S}_{fj}+\mathrm{h.c.}.
\label{ptb_SPT}
\end{equation}
The shift of the expectation value of $\phi_{1c}$ by $\pi$ is equivalent to the sign reversal of the Kondo couplings $H_{K\perp B}, H_{KzB+}$, and $H_{KzB-}$, and thus this procedure connects the topological phase with the trivial Kondo insulator.
As seen from Eq.\ \eqref{ptb_SPT} or Table \ref{SymTfm}, this perturbation only breaks the inversion symmetry, and preserves the other symmetries. In the present system, the charge U(1) symmetry prohibits vertex operators which involve the field $\theta_{1c}$.
Thus, the only possible way to connect the topological phase and the trivial phase using the charge degrees of freedom is the shift of the expectation value of $\phi_{1c}$ accompanied by the breaking of inversion symmetry. From these observations, we conclude that the topological phase is protected only by the inversion symmetry (under the assumption of the charge conservation).
From the above argument, we can interpret the crossover from the fermionic SPT phase (protected by the inversion symmetry only) to the bosonic SPT phase (the Haldane phase, protected by the time-reversal, inversion, and spin dihedral symmetries) via the bosonization language. In the weakly interacting regime, the low-energy behavior of the charge mode is mainly determined by the Kondo coupling rather than the Umklapp scattering due to the Hubbard repulsion. In this case, we can connect the topological phase and the trivial phase by shifting the pinning position of the charge mode with breaking the inversion symmetry, while the time-reversal and the spin rotation symmetries are kept intact. However, this shift cannot be reconciled with minimization of the Umklapp term $H_{U}$. Hence if we gradually increase the Hubbard repulsion $U$, the above procedure fails to work at some point. After that, the topological phase and the trivial phase are separated by a quantum phase transition if the time-reversal or the spin dihedral symmetry is present. We note that the perturbation \eqref{ptb_SPT_bos} vanishes if the charge mode is frozen at the potential minimum of the Umklapp term, $2\sqrt{2}\phi_{1c}=\pi$.
Finally, let us clarify where the topological phase of the Kondo lattice stands in the classification of SPT phases of interacting fermions. In non-interacting systems, topological insulators protected by the inversion symmetry in 1D are classified\cite{ShiozakiSato, LuLee, Hughes} by integer $\mathbb{Z}$, which means that there are infinitely many different topological phases. However, when we allow interactions as perturbation to systems, a part of nontrivial topological phases can be connected to the trivial phase and free-fermion classification of topological phases is reduced to its subgroup.\cite{FidkowskiKitaev1, FidkowskiKitaev2, Turner, Morimoto, Kapustin} In the case of inversion-symmetric topological insulators, the classification is performed by several methods\cite{YouXu, Shiozaki1, Shiozaki2} and is argued to reduce from $\mathbb{Z}$ to $\mathbb{Z}_4$ in the interacting case. Since the Haldane phase is classified by $\mathbb{Z}_2$, two copies of them can be deformed into the trivial phase. Using the fact that the topological phase of the 1D Kondo lattice approaches the Haldane phase in the strong coupling limit, we can also deform the two copies of the model \eqref{H_bos} into a trivial phase. Thus we conclude that the topological phase in 1D Kondo lattice is specified by an integer $2\in\mathbb{Z}_4=\{0,1,2,3\}$.
\begin{center}
\begin{table*}
\caption{Symmetry transformation in bosonization. The transformation on boson fields $\phi_{2s}, \theta_{2s}$ is the same as that on $\phi_{1s}, \theta_{1s}$ in the table.\label{SymTfm}}
\begin{ruledtabular}
\begin{tabular}{lcccc}
Symmetry operation & \multicolumn{2}{c}{Transformation law} & \multicolumn{2}{c}{Transformation on boson fields} \\ \hline
Translation & $c_\sigma(x)\to c_\sigma(x+a),$ & $\bm{S}_f(x)\to \bm{S}_f(x+a)$ & $\phi_{1c}(x)\to\phi_{1c}(x+a)-\sqrt{2}k_F a,$ & $\theta_{1c}(x)\to\theta_{1c}(x+a)$ \\
& & & $\phi_{1s}(x)\to\phi_{1s}(x+a),$ & $\theta_{1s}(x)\to\theta_{1s}(x+a)$\\
Charge U(1) & $c_\sigma\to e^{i\varphi}c_\sigma$ & & $\phi_{1c}\to\phi_{1c},$ & $\theta_{1c}\to\theta_{1c}+\varphi$\\
& & & $\phi_{1s}\to\phi_{1s},$ & $\theta_{1s}\to\theta_{1s}$\\
Time reversal & $c_\sigma\to \sum_{\sigma'}(i\sigma_y)_{\sigma\sigma'}c_{\sigma'},$ & $\bm{S}_f\to -\bm{S}_f$ & $\phi_{1c}\to\phi_{1c},$ & $\theta_{1c}\to -\theta_{1c}+\frac{\pi}{\sqrt{2}}$ \\
& & & $\phi_{1s}\to -\phi_{1s},$ & $\theta_{1s}\to\theta_{1s}-\frac{\pi}{\sqrt{2}}$ \\
Spatial inversion & $c_\sigma(x)\to c_{\sigma}(a-x),$ & $\bm{S}_f(x)\to\bm{S}_f(a-x)$ & $\phi_{1c}(x)\to -\phi_{1c}(a-x)+\sqrt{2}k_F a,$ & $\theta_{1c}(x)\to\theta_{1c}(a-x)$ \\
& & & $\phi_{1s}(x)\to -\phi_{1s}(a-x),$ & $\theta_{1s}(x)\to\theta_{1s}(a-x)$ \\
$\pi$ rotation around $x$ axis & \multicolumn{2}{c}{$S_{c,f}^x\to S_{c,f}^x,S_{c,f}^{y,z}\to -S_{c,f}^{y,z}$} & $\phi_{1c}\to\phi_{1c},$ & $\theta_{1c}\to\theta_{1c}$ \\
& & & $\phi_{1s}\to -\phi_{1s},$ & $\theta_{1s}\to -\theta_{1s}$ \\
$\pi$ rotation around $y$ axis & \multicolumn{2}{c}{$S_{c,f}^y\to S_{c,f}^y,S_{c,f}^{x,z}\to -S_{c,f}^{x,z}$} & $\phi_{1c}\to\phi_{1c},$ & $\theta_{1c}\to\theta_{1c}$ \\
& & & $\phi_{1s}\to -\phi_{1s},$ & $\theta_{1s}\to -\theta_{1s}+\frac{\pi}{\sqrt{2}}$ \\
Spin U(1) & \multicolumn{2}{c}{$S_{c,f}^x\to S_{c,f}^x\cos\varphi+S_{c,f}^y\sin\varphi,$} & $\phi_{1c}\to\phi_{1c},$ & $\theta_{1c}\to\theta_{1c}$ \\
& \multicolumn{2}{c}{$S_{c,f}^y\to -S_{c,f}^x\sin\varphi+S_{c,f}^y\cos\varphi$} & $\phi_{1s}\to \phi_{1s},$ & $\theta_{1s}\to \theta_{1s}+\varphi$ \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{center}
\subsection{Protection by spin $\pi$ rotation symmetries around the $x$ or $y$ axis}
Besides the topological protection described in the previous subsection, the gapped phases in this system are also protected by spin $\pi$ rotation symmetries around the $x$ or $y$ axis. This gives a distinction between the laser-induced Kondo insulator (K') with the ``unusual" spin state and the two phases (Top) and (K) composed of the ordinary Kondo singlet state. Although this fact is not related to the discussion of SPT phases, we describe the mechanism of this symmetry protection for completeness. This fact arises from the symmetry eigenvalues of the spin $\pi$ rotation symmetries. In the description of SPT phases using matrix product states, the symmetry eigenvalues correspond to phase factors which are not related to topological phases and provide distinction between ``trivial" phases (see Appendix \ref{App_SPT}).
To calculate the phase factors, we use a strong coupling limit $|J_\perp|, |J_z|\to\infty$, since the phase factors cannot change unless the energy gap collapses. The strong coupling limit of the topological phase is continuously connected to the Haldane phase of the spin-1 Heisenberg model, and therefore we obtain $\vartheta_{x}=\vartheta_y=0$ (see the notation in Appendix \ref{App_SPT}). In the strong coupling limit of the (ordinary) Kondo insulator and the laser-induced Kondo insulator, the ground states are site-product states of on-site Kondo singlets. The Kondo singlet is $\ket{\downarrow}_c\ket{\uparrow}_f-\ket{\uparrow}_c\ket{\downarrow}_f$ for the former phase, and $\ket{\downarrow}_c\ket{\uparrow}_f+\ket{\uparrow}_c\ket{\downarrow}_f$ for the latter phase, respectively. Since the spin $\pi$ rotational operation $R_x$ around the $x$ axis satisfies
\begin{align}
R_{x}(\ket{\downarrow}_c\ket{\uparrow}_f-\ket{\uparrow}_c\ket{\downarrow}_f)&=+(\ket{\downarrow}_c\ket{\uparrow}_f-\ket{\uparrow}_c\ket{\downarrow}_f),\\
R_{x}(\ket{\downarrow}_c\ket{\uparrow}_f+\ket{\uparrow}_c\ket{\downarrow}_f)&=-(\ket{\downarrow}_c\ket{\uparrow}_f+\ket{\uparrow}_c\ket{\downarrow}_f),
\end{align}
and the same holds for $R_y$, we obtain $\vartheta_{x}=\vartheta_y=0$ for the ordinary Kondo insulator and $\vartheta_{x}=\vartheta_y=\pi$ for the laser-induced Kondo insulator. By comparing $\vartheta_{x}, \vartheta_y$ of each phase, we conclude that the laser-induced Kondo insulating phase (K') is distinct from the ordinary Kondo insulator (K) and the topological phase (Top), protected by the spin $\pi$ rotation symmetry around the $x$ or $y$ axis. To connect the distinct phases, we must close the energy gap or break the symmetry. In fact, at the phase boundary between the topological phase and the laser-induced Kondo insulator, the spin gap of $\phi_+$ is collapsed. At the boundary between the ordinary and the laser-induced Kondo insulators, the N\'{e}el order intervenes, signaling the symmetry breaking. Thus, the phase diagram obtained in Sec.\ \ref{Phasediagram} is consistent with the symmetry protection.
If the spin $\pi$ rotation symmetries are broken, we can adiabatically connect the laser-induced Kondo insulator and the ordinary Kondo insulator. To check this, let us consider a unitary transformation\cite{MaruyamaHatsugai} $U(\gamma)^\dag H U(\gamma)$ with
\begin{equation}
U(\gamma)=\exp\Bigl[i\gamma\sum_j S_{fj}^z\Bigr],
\end{equation}
which changes the Kondo coupling into
\begin{align}
&U(\gamma)^\dag H_K U(\gamma)\notag\\
=&-J_\perp\cos\gamma\sum_j (S_{cj}^x S_{fj}^x+S_{cj}^y S_{fj}^y)-J_z\sum_j S_{cj}^z S_{fj}^z\notag\\
&-J_\perp\sin\gamma\sum_j(S_{cj}^x S_{fj}^y-S_{cj}^y S_{fj}^x).
\label{tfm_SPt}
\end{align}
The rest of the Hamiltonian is unchanged. As seen easily, the spin $\pi$ rotation symmetry around the $x$ or $y$ axis is broken in the transformed Hamiltonian except for $\gamma=0,\pi$. Since $U(\gamma)$ is unitary, the energy spectra of $H$ and $U(\gamma)^\dag H U(\gamma)$ are identical. Thus we can connect the ordinary Kondo insulator at $\gamma=0$ and the laser-induced Kondo insulator at $\gamma=\pi$ without closing the energy gap by changing $\gamma$ continuously.
We can also show the symmetry protection using the bosonization language. Let us focus on a parameter region near the phase boundary between the topological phase and the laser-induced Kondo insulator. In that region, the relevant perturbation for the gap generation in terms of the scaling dimensions is $H_{K\perp B}$ and $H_{KzB+}$, and the low-energy behavior is governed by these terms, making the fields $\phi_{1c}, \phi_+$, and $\theta_-$ locked at their potential minimum. Here we note that the $H_{K\perp B}$ term does not change its sign between the two phases, but the $H_{KzB+}$ term does. Hence the difference between the two phases is the pinning position of the total spin mode $\phi_+$. To adiabatically connect the two phases preserving the energy gap, we must shift the expectation value of $\phi_+$ by allowing a perturbation term like
\begin{equation}
g_{KzB+}'\int dx \sin\sqrt{2}\phi_{1c}\sin2\phi_+.
\label{ptb_SPt}
\end{equation}
We note that in the present system an additional spin U(1) symmetry forbids perturbations containing the dual field $\theta_+$. However, the shift of the expectation value of $\phi_+$ necessarily breaks the spin $\pi$ rotation symmetry as inferred from Table \ref{SymTfm}.
Hence the quantum phase transition between the topological phase and the laser-induced Kondo insulator is protected by the spin $\pi$ rotation symmetry, being consistent with the analysis of symmetry eigenvalues.
To connect the ordinary Kondo insulator and the laser-induced Kondo insulator, we must shift the expectation value of $\theta_-$. This procedure also breaks the spin $\pi$ rotation symmetries. The required perturbation can be obtained by bosonization of the last term in Eq.\ \eqref{tfm_SPt}.
\section{Discussions and Conclusion\label{Conclusion}}
We have shown that cold-atom realization of the Kondo lattice model offers a platform to investigate a 1D SPT phase and an associated quantum phase transition with high controllability. By utilizing the spin-exchanging collisions with the help of the laser-induced mixing of internal states, ultracold AEA in optical lattice can realize the Kondo lattice with tunable anisotropic Kondo couplings, which is hard to be realized in solid state experiments. Since the sign of the bare exchange coupling $V_{\mathrm{ex}}$ can be controlled using the confinement-induced resonance specific to 1D optical lattices,\cite{RZhang} a large portion of the phase diagram in Fig.\ \ref{fig_phase} can be accessed in this system. If we start from ferromagnetic $V_{\mathrm{ex}}>0$, the SPT phase transition from the topological phase to the laser-induced Kondo insulating state is possible. This phase transition is protected by the inversion symmetry and the spin $\pi$ rotation symmetries around the $x$ or $y$ axis, and the only former symmetry stands for the topological properties. On the other hand, if we switch on the laser coupling starting from antiferromagnetic $V_{\mathrm{ex}}<0$, the ordinary Kondo insulator is first changed into the N\'{e}el order, and finally turns into the laser-induced Kondo insulator. This reentrant Kondo transitions associated with the N\'{e}el order are stable (at least $T=0$) if the spin $\pi$ rotation symmetries are preserved.
We have also demonstrated the topological phase of the 1D Kondo lattice is protected only by the inversion symmetry when the charge fluctuations cannot be ignored, while the Haldane phase in the strong coupling limit is also protected by the time-reversal and spin dihedral symmetries. The change of the nature of the topological phase from fermionic to bosonic SPT phases leads to an intriguing consequence in the fate of edge states of the topological phase. In the strong coupling regime, the Haldane phase has spin-1/2 zero-energy states at the edge of the system. The edge states are magnetically active, and have been detected by applying magnetic fields.\cite{Hagiwara,Glarum} On the other hand, in the weak-coupling regime, the SPT phase is protected only by the inversion symmetry. This means that the zero-energy edge state is absent in general, since the edges generically break the inversion symmetry. Thus it is implied that the edge states gradually decrease their excitation energies with increasing the Hubbard interaction $U$, and finally they turn into the zero-energy state at some threshold value of $U$. Such ``interaction-induced" edge states are one possible hallmark of the crossover from fermionic SPT phases to bosonic ones.
Observation of such a crossover using the present cold-atom setup is intriguing but may be a challenging issue. To detect a clear signature of the edge states, it is appropriate to create an interface between the topologically nontrivial phase and the trivial phase,\cite{Leder, Goldman} since the true edge of the atomic cloud is usually a metallic state due to a harmonic confinement potential. In our setup, the interface can be easily created, since the topological-trivial phase transition is caused by the laser irradiation, which can be performed in a spatially varying manner. The interface-localized edge modes are, in principle, detected by combining a magnetic field and spin-resolved quantum gas microscopy, by which antiferromagnetic correlations were recently observed in the Fermi-Hubbard model.\cite{Parsons, Boll, Cheuk, Mazurenko}
\begin{acknowledgments}
We are grateful to Ken Shiozaki for useful discussions, and acknowledge Tsuneya Yoshida, Takahiro Morimoto, and Akira Furusaki for helpful comments at the early stage of this work. This work was supported by JSPS KAKENHI (Grants No.\ JP16K05501 and No.\ JP14J01328) and a Grand-in-Aid for Scientific Research on Innovative Areas (Grant No.\ JP15H05855). M.\ N.\ was supported by a JSPS Research Fellowship for Young Scientists and RIKEN Special Postdoctoral Researcher Program.
\end{acknowledgments}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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layout: post
title: My Goals as a Developer
---
I worked for a Chicago-based startup for over three (3) years. During that time, I worked closely with many different developers. Being able to witness first-hand the solutions they were able to craft, inspired me into starting a career as one of them.
In synthesis, as a beginner, my current goals are:
* To find a group of people or company, where I can apply what I have learned during bootcamp
* To polish my skills by starting new projects
* To expand my knowledge by reading more material and taking it to practice
* To learn and master the basics of Javascript, in addition to its many popular frameworks such as AngularJS and Ember.js
Ultimately, my main goal is to be able to garner enough knowledge to be able to help others create beautiful and efficient web applications.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 410
|
Q: Vuex: Having a problem using a getter inside a view I have a module named 'tasks' in store which has this getter -
getTaskById: state => id => state.idToTask[id]
seems quite straightforward and basic, idToTask is just an object with ids as keys.
Now in the view (the component that displays the task) - it gets taskId as prop from router, I use mapGetters like this -
methods: {
...mapGetters('tasks', ['getTaskById'])
}
and I have a computed:
task() {
return this.getTaskById(this.taskId)
}
I really don't know what went wrong here but I get error of task being undefined (in the template) for some reason...
And I'll just say ahead that I've tried
this.$store.getters['tasks/getTaskById'](taskId)
but it does not seem to work as well...
Any help?
A: Disregard my comment, mapGetters should go in computed, not methods.
computed: {
...mapGetters('tasks', ['getTaskById'])
}
https://vuex.vuejs.org/guide/getters.html
https://github.com/vuejs/vuex/issues/1136
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,084
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\section{Introduction}
Recent research has suggested deep neural networks are dramatically over-parametrized. In natural language processing alone, most state-of-the-art neural networks have computational and memory complexities that scale with the size of the vocabulary. Practitioners have developed numerous methods to reduce the complexity of these models---either before, during, or after training---while retaining existing performance. Some of these methods include quantization \cite{gong2014compressing, hubara2017quantized}, and different flavors of pruning \cite{zhu2017prune,liu2018rethinking,frankle2018the,gale2019state}.
In particular, the Lottery Ticket Hypothesis \cite{frankle2018the} proposes that small, sparse subnetworks are embedded within large, over-parametrized neural networks. When trained in isolation, these subnetworks can achieve commensurate performance using the same initialization as the original model. The lottery ticket training procedure is formalized as an iterative three-stage approach: (1) train an over-parametrized model with initial parameters $\theta_0$; (2) prune the trained model by applying a mask $m \in \{0,1\}^{|\theta|}$ identified by a sparsification algorithm; (3) reinitialize the sparse subnetwork by resetting its non-zero weights to the initial values ($m \odot \theta_0$) and retrain it. These three stages are repeated for multiple rounds. If the final subnetwork achieves similar (or better) test performance in comparison to the original network, a winning \textit{lottery ticket} has been identified.
Evidence of the existence of winning tickets has been empirically shown on a range of tasks, including computer vision, reinforcement learning, and natural language processing \cite{frankle2018the, yu2019play}. However, the merits of lottery ticket training has recently been called into question. In particular, (1) whether keeping the same initialization (e.g., $\theta_0$) is crucial for acquiring tickets \cite{liu2018rethinking}; and (2) if tickets can generalize across multiple datasets \cite{morcos2019one}.
Our paper investigates the efficacy of lottery tickets when the data distribution changes. We define multiple data domains such that their input distributions are varied. Then, we consider whether subnetworks obtained in a source domain $\mathcal{D}_s$ can be used to specify and train subnetworks in a target domain $\mathcal{D}_t$ where $s \ne t$. Inspired by \citet{liu2018rethinking}, we also experiment with different initialization methods at transfer-time, probing at the importance of initial (source domain) values in disparate target domains. We find that subnetworks obtained through lottery ticket training do not completely overfit to particular input distributions, showing some generalization potential when distributional shifts occur. In addition, we discover a \textit{phase transition} point, at which subnetworks reset to their initial values show better and more stable generalization performance when transferred to an arbitrary target domain.
In summary, our contributions are (1) continuing the line of work on the Lottery Ticket Hypothesis \cite{frankle2018the}, showing that tickets exist in noisy textual domains; (2) performing comprehensive experiments pointing towards the transferability of lottery tickets under distributional shifts in natural language processing; and (3) publicly releasing our code and datasets to promote further discussion on these topics\footnote{https://github.com/facebookresearch/pytext}.
\section{Related Work}
There is a large body of work on transfer learning for neural networks \cite{deng2013sparse, yosinski2014how, liu2017sparse, zoph2018learning, kornblith2019better}. Most of these works focus on improving the transferred representation across tasks and datasets. The representation from a source dataset is fine-tuned or learned collaborately on a target dataset. In contrast, we focus on understanding whether the \textit{architecture} can be transferred and retrained, and whether transferring the initialization is required. Our work is also related to Neural Architecture Search (NAS) \cite{zoph2018learning, liu2018darts, elsken2018neural}. The goal of NAS is to identify well-performing neural networks automatically. Network pruning can be viewed as a form of NAS, where the search space is the sparse topologies within the original over-parameterized network \cite{liu2018rethinking, gale2019state, frankle2018the}.
Iterative magnitude pruning \cite{frankle2018the, frankle2019stable} is a recently proposed method for finding small, sparse subnetworks from large, over-parameterized neural networks that can be trained in isolation to reach a similar (or better) test accuracy. To obtain these re-trainable sparse subnetworks, \citet{frankle2018the} uses an iterative pipeline that involves training a model, removing ``redundant'' network connections identified by a sparsification algorithm, re-training the subnetwork with the remaining connections. In particular, the experiments in \citet{frankle2018the} show it is critical to re-initialize the subnetworks using the \textit{same} initial values after each round of the iterative pipeline.
However, the importance of re-using the original initialization is questioned in \citet{liu2018rethinking}, where the authors show that competitive performance of the sparse subnetworks can be achieved with random initialization as well. \citet{morcos2019one} investigate the transferability of lottery tickets across multiple optimizers and datasets for supervised image classification, showing that tickets can indeed generalize \cite{morcos2019one}. Beyond the differences between our domain, task, and datasets, our work carries an important distinction. In \citet{morcos2019one}, the authors refer to the \textit{transfer of initialization} as both the \textit{transfer of the sparse topologies} and the \textit{transfer of the initial values} of the subnetworks. Therefore, it is unclear whether the \textit{sparse topology} alone can be transferred across datasets or the topology combined with the initial values must be exploited jointly to achieve transferability. In our work, we decouple this question by investigating the influence of different initialization strategies on the sparse architecture during the process of finding the winning tickets and after the transfer to other domains.
\section{Task and Datasets}
\paragraph{Distributional Shifts}
Let $(x^s_i, y^s_i) \in \mathcal{X} \times \mathcal{Y}$ denote a pair of training samples from domain $\mathcal{D}_s$. Let $f(x; \theta)$ be a function (e.g., deep neural network) that maps an input from $\mathcal{X}$ to the label space $\mathcal{Y}$, parameterized by $\theta$. In this work, the sparsity of $\theta$ is induced by the lottery ticket training process \cite{frankle2018the}. To model distributional shifts, we characterize each domain $\mathcal{D}_i$ as a dataset from the Amazon Reviews corpus \cite{mcauley2013hidden}. The differences in unigram frequencies, semantic content, and random noise mimic the type of distributional shifts that occur in machine learning.
\paragraph{Subword Vocabulary} We ensure each domain $\mathcal{D}$ shares an identical support on $\mathcal{X}$ by encoding the inputs using a vocabulary common across all datasets. Word-level vocabularies may introduce problems during domain transfer as certain words potentially only appear within a particular domain. On the other end of the spectrum, character-level vocabularies ameliorate this issue but may not contain enough expressive power to model the data. We elect to use a subword vocabulary, balancing the out-of-vocabulary and effectiveness problems introduced by the word- and character-level vocabularies, respectively. Technical details for creating the shared subword vocabulary are presented in \S\ref{method:vocab}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.4]{jsd.png}
\caption{Jenson-Shannon Divergence scores on subword unigram distributions for each domain pair $(\mathcal{D}_i, \mathcal{D}_{i'})$. Domains include Books (B), Electronics (E), Movies (M), CDs (C), and Home (H). Values are scaled by $1e^5$ for presentation.}
\label{fig:jsd}
\end{figure}
\paragraph{Divergence Scores} Given an identical support for all data distributions, we now quantify the distributional shifts between our domains using Jenson-Shannon Divergence (JSD). JSD is a symmetric measure of similarity between two (continuous) probability distributions $p$ and $q$ with a proxy, averaged distribution $m = \frac{1}{2}(p+q)$:
\begin{equation}
\mathrm{JSD}(p||q) = \frac{1}{2}\mathrm{KL}(p||m) + \frac{1}{2}\mathrm{KL}(q||m)
\label{eq:jsd}
\end{equation}
where $\mathrm{KL}(p||q)$ in Eq. \ref{eq:jsd} denotes the Kullback-Leibler divergence, defined as:
\begin{equation}
\mathrm{KL}(p||q) = \int_{-\infty}^{\infty} p(x)\log \frac{p(x)}{q(x)}dx
\end{equation}
Figure \ref{fig:jsd} displays the divergence scores between our datasets. On average, there is high disagreement with respect to the prevalence and usage of subwords in each domain, with Electronics$\rightarrow$Home the most similar and CDs$\rightarrow$Home the most dissimilar.
\paragraph{Sentiment Analysis} Finally, we introduce our base task for experimentation. Our models are evaluated on a binary sentiment analysis task constructed from five categories in the Amazon Reviews corpus: books (B), electronics (E), movies (M), CDs (C), and home (H). The dataset originally provides fine-grained sentiment labels ($1$ through $5$) so we group $1$, $2$ as negative and $4$, $5$ as positive. Following \citet{peng2018cross}, reviews with neutral ratings ($3$) are discarded. We sample 20K train, 10K validation, and 10K test samples from each category, ensuring there is an equal distribution of positive and negative reviews.
\section{Methods}
In this section, we discuss our technical methods. First, we describe the subword vocabulary creation process (\S\ref{method:vocab}). Second, we cover the underlying model used in the sentiment analysis task (\S\ref{method:model}). Third, we detail the lottery ticket training and transferring methods (\S\ref{method:tickets}).
\subsection{Vocabulary} \label{method:vocab}
We use the SentencePiece\footnote{https://github.com/google/sentencepiece} library to create a joint subword vocabulary for our datasets \cite{kudo2018sentencepiece}. The subword model is trained on the concatenation of all five training datasets (100K sentences) using the byte-pair encoding algorithm \cite{sennrich2016neural}. We set the vocabulary size to 8K. The final character coverage is 0.9995, ensuring minimal out-of-vocabulary problems during domain transfer.
\subsection{Model} \label{method:model}
We use convolutional networks (CNN) as the underlying model given their strong performance on numerous text classification tasks \cite{kim2014convolutional, mou2016transferable, gehring2017convolutional}. Let $V$ and $n$ represent the vocabulary of the corpus and maximum sequence length, respectively. Sentences are encoded as an integer sequence $t_1, \cdots, t_n$ where $t_i \in V$. The embedding layer replaces each token $t_i$ with a vector $\mathbf{t}_i \in \mathbb{R}^d$ that serves as the corresponding $d$-dimensional embedding. The vectors $\mathbf{t}_1, \cdots, \mathbf{t}_n$ are concatenated row-wise to form a token embedding matrix $\mathbf{T} \in \mathbb{R}^{n \times d}$.
Our model ingests the embedding matrix $\mathbf{T}$, then performs a series of convolutions to extract salient features from the input. We define a convolutional filter $\mathbf{W} \in \mathbb{R}^{h \times d}$ where $h$ represents the \textit{height} of the filter. The filter is not strided, padded, or dilated, Let $\mathbf{T}[i:j] \in \mathbb{R}^{h\times d}$ represent a sub-matrix of $\mathbf{T}$ extracted from rows $i$ through $j$, inclusive. The feature map $\mathbf{c} \in \mathbb{R}^{n-h+1}$ is induced by applying the filter to each possible window of $h$ words, i.e.,
\begin{equation}
c_i = f\Big( \big\langle \mathbf{T}[i:i+h],\mathbf{W}\big\rangle_{\fro} + b\Big)
\end{equation}
for $1\leq i \leq n-h+1$, where $b \in \mathbb{R}$ is a bias term, $f$ is a non-linear function, and the Frobenius inner product is denoted by $\langle \mathbf{A},\mathbf{B}\rangle_{\fro} = \sum_{i=1}^h \sum_{j=1}^d \mathbf{A}_{ij} \mathbf{B}_{ij}$. 1-max pooling \cite{collobert2011natural} is applied on $\mathbf{c}$, defined as $\hat{c} = \textnormal{max}\{\mathbf{c}\}$. This is performed to propagate the maximum signal throughout the network and reduce the dimensionality of the input.
The process described above creates \textit{one} feature from \textit{one} convolution with window $h$ followed by a pooling operation. To extract multiple features, the model uses several convolutions with varying $h$ to obtain features from different sized $n$-grams in the sequence. The convolutional (and pooled) outputs are concatenated along the channel dimension, then fed into a one-layer MLP to obtain a distribution over the $c$ classes.
\subsection{Lottery Tickets} \label{method:tickets}
\subsubsection{Initialization}
\label{sec:init}
The embedding matrix is initialized from a unit Gaussian, $\mathbf{T} \sim \mathcal{N}(0,1)$. The convolutional and MLP layers use He initialization \cite{he2015delving}, whose bound is defined as
\begin{equation}
b = \sqrt{\frac{6}{(1+a^2) \times \mathrm{fan\_in}}}
\end{equation}
where $a$ and $\mathrm{fan\_in}$ are parameters calculated for each weight. The resulting weights have values uniformly sampled from $\mathcal{U}(-b,b)$.
\subsubsection{Training}
\label{sec:train}
We use iterative pruning with alternating cycles of training and pruning to obtain the tickets \cite{han2015learning, frankle2018the}.
For clarity, we define a \textit{round} as training a network for a fixed number of epochs. We begin with a seed round $r_0$ where the model does not undergo any pruning, then begin to procure tickets in a series of lottery ticket training rounds.
In each successive round $r_{i>0}$, a fraction $p$ of the weights that survived round $r_{i-1}$ are pruned (according to a sparsification algorithm, discussed below) to obtain a smaller, sparser subnetwork; this is denoted by $f(x;m_i\odot \theta_i)$ where $m_i$ and $\theta_i$ represent the sparse mask and weights at round $r_i$. The weights $\theta_i$ of this subnetwork are set according to an \textit{initialization strategy} and the subnetwork is re-trained to convergence. We refer to the \textit{sparsity} as the fraction of weights in the network that are exactly zero. In each round, we prune $p\%$ of the weights in the model. Therefore, the resulting ticket has sparsity $1-(1 - p\%)^{r_{total}}$, where $r_{total}$ is the total number of lottery ticket training rounds.
Next, we discuss the sparsification algorithm used to prune weights in each round $r_i$. Let $\mathbf{p}_i$ denote the vectorized collection of trainable parameters in layer $i \ge 0$, with the embedding layer as layer $0$. After re-training the (sub-)networks in each round, we apply the $\ell_0$ projection on the parameters in each layer, i.e.
\begin{equation}
\argmin_{\mathbf{p}} ||\mathbf{p}-\mathbf{p}_i||^{2}_{2}
\label{eq:10}
\end{equation}
subject to $\card(\mathbf{p}) \le k_i$, where $\card(\mathbf{p})$ denotes the number of non-zeros in $\mathbf{p}$. The optimization problem in Eq. \ref{eq:10} can be solved analytically by sorting the elements of $\mathbf{p}_i$ with respect to their absolute values and picking the top $k_i$ elements with the largest magnitude \cite{jain2017non, zhu2017prune}. We use the sparsity hyperparameter $p$ introduced above to decide $k_i$ for each layer. Let $\mathrm{len}(\mathbf{p}_i)$ denote the total number of trainable parameters in layer $i$. We set $k_i = p\% \times \mathrm{len}(\mathbf{p}_i)$ for each layer. In accordance with our training procedure, once a weight is pruned, it is no longer a trainable parameter; hence, $\mathrm{len}(\mathbf{p}_i)$ is strictly decreasing after each round.
\begin{figure}[t]
\centering
\includegraphics[scale=0.15]{transfer/transfer.png}
\caption{Visualization of the subnetwork transfer process. Purple denotes elements from the source domain, while blue denotes elements from the target domain. Tickets are composed of two elements: (1) the sparsified mask ($m_i$) and (2) the initial parameter values ($\theta_i$). During transfer, we create subnetworks in the source domain with the mask borrowed from the source domain, but with potentially different parameters. We use $\theta_i'$ to denote that these parameters are set according to some \textit{initialization strategy}, which we discuss further in our experiments (\S\ref{sec:exp}).}
\label{fig:clarify}
\end{figure}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=0.325\textwidth]{transfer/books.png}
\includegraphics[width=0.325\textwidth]{transfer/movies.png}
\includegraphics[width=0.325\textwidth]{transfer/electronics.png}
\includegraphics[width=0.325\textwidth]{transfer/cds.png}
\includegraphics[width=0.325\textwidth]{transfer/home.png}
\end{center}
\label{fig:books}
\caption{Results obtaining lottery tickets on the Books, Movies, Electronics, CDs, and Home categories of the Amazon Reviews dataset \cite{mcauley2013hidden}. Experiments are repeated five times, where the solid lines represent the mean and shaded regions represent the standard deviation. Note that the $x$-axis ticks are \textit{not} uniformly spaced.}
\label{fig:obtain}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.325\textwidth]{transfer/books_cds.png}
\includegraphics[width=0.325\textwidth]{transfer/electronics_home.png}
\includegraphics[width=0.325\textwidth]{transfer/cds_books.png}
\includegraphics[width=0.325\textwidth]{transfer/books_electronics.png}
\includegraphics[width=0.325\textwidth]{transfer/electronics_movies.png}
\includegraphics[width=0.325\textwidth]{transfer/cds_home.png}
\includegraphics[width=0.325\textwidth]{transfer/books_home.png}
\includegraphics[width=0.325\textwidth]{transfer/electronics_cds.png}
\includegraphics[width=0.325\textwidth]{transfer/cds_movies.png}
\end{center}
\label{fig:books}
\caption{Results transferring lottery tickets on nine transfer tasks constructed from the five categories of the Amazon Reviews dataset \cite{mcauley2013hidden}. Experiments are repeated five times, where the solid lines represent the mean and shaded regions represent the standard deviation. Note that the $x$-axis ticks are \textit{not} uniformly spaced.}
\label{fig:transfer}
\end{figure*}
\subsubsection{Transferring}
The lottery ticket training procedure outlined in \S\ref{sec:train} yields a batch of subnetworks $f(x^s;m_1\odot \theta), \cdots, f(x^s;m_n\odot \theta)$ where $x^s$ represents the inputs from a \textit{source} domain $\mathcal{D}_s$ and $m_i$ represents the sparse mask used to prune weights at round $r_i$. During transfer, we construct a new batch of subnetworks $f(x^t;m_1\odot \theta'), \cdots, f(x^t;m_n\odot \theta')$ to be evaluated on inputs from a (non-identical) \textit{target} domain $\mathcal{D}_t$ with masks derived from the \textit{source} domain. The change in parameter notation ($\theta \rightarrow \theta'$) implies that the subnetworks evaluated in a disparate domain can potentially use a different \textit{transfer} initialization strategy. We clarify this process in Figure \ref{fig:clarify}. In contrast, \citet{morcos2019one} transfers the entire ticket (sparse masks and initial values) to the target domain. Finally, using the new batch of subnetworks, we evaluate each subnetwork $f(x^t;m_i \odot \theta')$ in the target domain for $r_{total}$ rounds. Unlike the canonical ticket training rounds, we do not (additionally) sparsify the subnetworks during transfer. All in all, our transfer task is designed to answer the following question: can the \textit{sparse masks} found in a source domain using lottery ticket training (\S\ref{method:tickets}) be transferred to a target domain with \textit{different initialization strategies} to match the performance of a ticket obtained in same target domain?
\section{Experiments} \label{sec:exp}
\subsection{Settings}
\label{sec:expsetting}
Our CNN uses three filters ($h \in [3,4,5]$), each with $127$ channels, and ReLU activation \cite{nair2010rectified}. We fix the maximum sequence length to $500$ subwords. The embeddings are $417$-dimensional and trained alongside the model. We opt not to use pre-trained embeddings to ensure the generalizability of our results. Additionally, we regularize the embeddings with dropout \cite{srivastava2014dropout}, $p=0.285$. The MLP contains one hidden layer with a dimension of $117$. Hyperparameters were discovered using Bayesian hyperparameter optimization \cite{snoek2012practical} on the Books validation set. The models are trained with a batch size of $32$ for a maximum of $15$ epochs. Early stopping is used to save iterative model versions that perform well on a development set. We use the Adam optimizer \cite{kingma2014adam} with a learning rate of $1e^{-3}$ and $\ell_2$ regularization with a weight of $1e^{-5}$.
\subsection{Obtaining Tickets} \label{exp:obtain}
First, we use the lottery ticket training procedure outlined in \S\ref{sec:train} to obtain tickets for our five datasets with $p=35\%$ and $r_{total}=20$. We compare the test performance of the subnetworks using the following baselines:
\begin{itemize}
\item \textsc{Full-Model:} This baseline evaluates the performance of the original network \textit{without} any pruning. In other words, we train a model for a seed round $r_0$, then record its performance.
\item \textsc{Ticket-Reset:} The values of the subnetwork are reset to their \textit{original values} before training. This initialization strategy was used in the earliest formation of the Lottery Ticket Hypothesis \cite{frankle2018the}.
\item \textsc{Ticket-Random:} The values of the subnetwork are reset to \textit{random values} drawn from the initialization distribution(s) of the original network. We sample weights from the distributions outlined in \S\ref{sec:init} to initialize the subnetworks.
\end{itemize}
The results are shown in Figure \ref{fig:obtain}. For all datasets, \textsc{Ticket-Reset} shows the best performance, notably outperforming \textsc{Full-Model} in early stages of sparsification (0-90\%) for the Books, Electronics, and Home datasets. This demonstrates that deep neural networks---especially those for sentiment analysis---are highly over-parameterized, and the sparsity induced by lottery ticket training can help to increase performance. This observation is consistent with \citet{louizos2018learning}, which also showed sparse networks fashion a regularization effect that results in better generalization performance. In addition, we observe that \textsc{Ticket-Reset} and \textsc{Ticket-Random} have similar test performance until about 96\% sparsity. This casts some doubt around whether the initial values truly matter for sparse models as the randomly sampled values seem to fit sparse masks well.
However, a \textit{phase transition} occurs in the high sparsity regime, where the differences between \textsc{Ticket-Reset} and \textsc{Ticket-Random} are significantly enlarged. The performance of \textsc{Ticket-Random} becomes highly unstable and drops off much faster than \textsc{Ticket-Reset} after 96\% sparsity. In contrast, \textsc{Ticket-Reset} remains relatively stable---even with sparsity levels over 99.9\%---pointing towards the enigmatic importance of original values in extreme levels of sparsity.
\subsection{Transferring Tickets}
Next, we use the lottery ticket transferring procedure outlined in \S\ref{method:tickets} to transfer (obtained) subnetworks from a \textit{source} domain to a non-identical \textit{target} domain. Identical to the previous experiment, we use $r_{total}=20$. We compare the test performance of the \textit{transferred} subnetworks using the following baselines:
\begin{itemize}
\item \textsc{Ticket-Target:} This baseline is comprised of the subnetworks obtained in the target domain using lottery ticket training. We borrow the values for this baseline (without modification) from the \textsc{Ticket-Reset} subnetworks shown in Figure \ref{fig:obtain}, albeit from the domain of interest.
\item \textsc{Masks-Reset:} Under this initialization strategy, the masks obtained in the source domain is used on the target domain and the subnetwork is trained from the \textit{same} initial values as in the source domain.
\item \textsc{Masks-Random:} Under this initialization strategy, \textit{only} the masks are used from the subnetwork obtained in the source domain. The parameters are randomly initialized from the distributions outlined in \S\ref{sec:init} before training on the target domain.
\end{itemize}
The results are shown in Figure \ref{fig:transfer}. Both \textsc{Masks-Reset} and \textsc{Masks-Random} show signs of generalization in the early stages of sparsification. Most notably, subnetworks obtained in the CDs domain are extremely robust; both the \textsc{Masks-Reset} and \textsc{Masks-Random} results show stronger performance than \textsc{Ticket-Target}, even in sparsity levels over 99\%. This is relatively surprising as the \textsc{Full-Model} in \S\ref{exp:obtain} achieved the worst performance in the CDs domain. Further inspection of representations learned in this domain will be required to understand its strong ticket performance, which may or may not be a coincidence.
We see a 3-5\% dropoff in performance (up to 90\% sparsity) from tickets identified from the Books and Electronics tasks after transferring. These results together imply that tickets are not completely immune to distributional shifts, although the degradation in test accuracy is not substantial until reaching high sparsity. Nevertheless, we notice the accuracies of \textsc{Masks-Reset} and \textsc{Masks-Random} stay relatively stable from 0-90\% sparsity; they only begin to steadily decline after this point.
Finally, we compare the performance of \textsc{Masks-Reset} and \textsc{Masks-Random}. In the Books tasks, \textsc{Masks-Random} performs better overall in comparison to \textsc{Masks-Reset}. Its performance is slightly worse in the Electronics and CDs tasks, although it is relatively comparable to \textsc{Masks-Reset} up to 96\%. Similar to the results in \S\ref{exp:obtain}, we notice a \textit{phase transition} point where the initial values (e.g., \textsc{Masks-Reset}) play a much bigger role in maintaining stability and performance in the deeper stages of sparsification.
\section{Discussion}
In this section, we briefly recap our findings, highlighting key points observed through our ticket procuring and transfer experiments. For each section, we also touch on areas for future work.
\paragraph{Evidence of transferability of winning tickets in natural language processing.}
Our experiments show that ``winning tickets'' can indeed be identified in a sentiment task formulated from noisy, user-generated datasets. Moreover, the ``winning tickets'', up to extreme level of sparsity (e.g., $\> 90\%$), can be transferred across domains without much loss in accuracy. The fact that tickets can be obtained in noisy environments shows its prominence across multiple data sources. However, our work only considers a binary sentiment analysis task. Future work can explore other tasks such as multi-class text classification, language modeling, and machine translation.
\paragraph{Randomly initialized tickets are strong baselines.} Consistent with the observations in \citet{liu2018rethinking}, initializing tickets to their \textit{original values} before training is not necessarily required for strong performance. In our experiments, we show that in high sparsity conditions (up to 90\%), there is no noticeable difference between the performance of the \textit{originally} and \textit{randomly} initialized subnetworks. Although the sparse masks build on top of each other from round $r_i$ to $r_{i+1}$, randomly initialized subnetworks are still able to settle in a local minima with comparable performance to that of the originally initialized subnetworks. However, our work fixes the optimizer and learning rate across experiments. It may be possible that randomly initialized subnetworks using varying optimization reach better minima.
\paragraph{A \textit{phase transition} point largely influences ticket performance.} As alluded to above, there is almost no difference in performance when considering originally and randomly initialized subnetworks. However, our experiments point towards a crucial turning point---the \textit{phase transition}---in which the initialization begins to matter. In particular, especially in extreme levels of sparsity (e.g., 99.99\%) originally initialized networks exhibit less variance than randomly initialized tickets in test accuracy. However, the specific sparsity at which the phase transition happens is dataset-dependent. Understanding why this occurs and its relation with other models, datasets, and optimization algorithms can further unveil and explain the phenomena behind lottery tickets.
\section{Applications in Federated Learning}
Federated learning is a scenario where a centralized model is trained over decentralized data, distributed across millions (if not billions) of clients (e.g., electronic devices) \cite{jakub2016federated,bonawitz2019towards}. Crucially, the clients are not allowed to exchange \textit{data} with the central server or each other. Instead, each client can fine-tune a model for a couple of iterations on their own data, then send their (encrypted) parameters or gradients to a server for aggregation. This ``collaborative learning" setup effectively maintains a level of user privacy by ensuring the data always stays on-device. However, this poses several challenges for optimization; as the centralized server does not have access to the data distribution of each client, any neural architecture selection has to be done on either (a) a \textit{different} data source the server has access to or (b) on each individual client. Since (b) is generally quite expensive, the server usually maintains some seed data, as alluded to in (a).
With the transferability of lottery tickets, the server can procure lottery tickets on server-accessible data, then retrain the tickets on client data under the federated learning framework. While there may be a large performance drop when transferring \textit{extremely} sparse networks, our results show that clients can still re-train \textit{moderately} sparse networks with commensurate performance. We believe that this ``sparsify and transfer" procedure has two immediate benefits: (1) past work---including the original incarnation of the lottery ticket hypothesis---has shown that sparse networks can be, under certain conditions, easier to optimize \cite{frankle2018the, morcos2019one, gale2019state}; and (2) sparser sub-networks have significantly less capacity than their large, over-parameterized counterparts, which can alleviate client-server communication costs (e.g., model uploading and downloading) \cite{jakub2016federated,sattler2019robust}.
\section{Conclusion}
The Lottery Ticket Hypothesis \cite{frankle2018the} posits that large, over-parameterized networks contain small, sparse subnetworks that can be re-trained in isolation with commensurate test performance. In this paper, we examine whether these tickets are robust against distributional shifts. In particular, we set up domain transfer tasks with the Amazon Reviews dataset \cite{mcauley2013hidden} to obtain tickets in a \textit{source} domain and transfer them in a disparate \textit{target} domain. Moreover, we experiment with the \textit{transfer} initialization of the networks, determining if resetting to initial values (obtained in the source domain) are required for strong performance in the target domain. Our experiments show that tickets (under several initialization strategies) can be transferred across different text domains without much loss up to a very high level of sparsity.
In addition, there is a lot of debate on whether initial value resetting is critical to achieve commensurate test performance. While \citet{frankle2018the, frankle2019stable} present evidence supporting the importance of resetting, \citet{gale2019state, liu2018rethinking} show that sparse re-trainable subnetworks can be found independent of resetting. Our experiments show that this is \textit{not} a yes or no question. Specifically, we show there is a \textit{phase transition} related to sparsity. Resetting is not critical before extreme levels of sparsity (i.e., below 99\%), but the effect of resetting is magnified in high sparsity regimes. Finally, we demonstrate the practical applications of our results in federated learning.
\section*{Acknowledgments}
Thanks to Veselin Stoyanov and our anonymous reviewers for their helpful comments.
\bibliographystyle{acl_natbib_nourl}
\section{Experiments}
\subsection{Hyperparameter Settings}
Our CNN uses three filters ($h \in [3,4,5]$), each with $127$ channels, and ReLU activation \cite{nair2010rectified}. We fix the maximum sequence length to $500$ words. The embeddings are $417$-dimensional and trained alongside the model. We opt not to use pre-trained embeddings to ensure the generalizability of our results. Additionally, we regularize the embeddings with dropout \cite{srivastava2014dropout}, $p=0.285$. The MLP contains one hidden layer with a dimension of $117$. Hyperparameters were discovered using Bayesian hyperparameter optimization \cite{snoek2012practical} on the Books validation set.
The model is trained with a batch size of $32$ for a maximum of $15$ epochs. Early stopping is used to save iterative model versions that perform well on a development set. We use the Adam optimizer \cite{kingma2014adam} with a learning rate of $1e^{-3}$ and $\ell_2$ regularization with a weight of $1e^{-5}$.
\bibliographystyle{acl_natbib_nourl}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Saskatchewan is full of history, from its noteworthy people to ancient landmarks to the famous stories of old that have shaped the province. As the umbrella organization for all of the public museums in the province, the Museums Association of Saskatchewan (MAS) helps to preserve this history.
MAS is a non-profit, collective organization of over 250 member museums and a total membership of over 400, including individuals and associates. Their members span the nine sport, culture and recreation districts throughout the province and include flagship museums such as the Saskatchewan Western Development Museum, Royal Saskatchewan Museum and MacKenzie Art Gallery, all the way to small rural volunteer-run community museums.
To ensure people can access the memories of the provinces, MAS works at strengthening Saskatchewan museums through a variety of education, advisory, resource and networking programs.
One of the association's main purposes is to provide professional development and training so its members can provide stewardship and access for the heritage community in the province. MAS hosts many training events, both in person and online. In 1988, MAS produced the first set of standards for museums in the province and is currently finalizing the fifth edition.
To be more accessible to visitors in this digital age, MAS is also encouraging its members to explore new platforms.
Fitch said that without Saskatchewan Lotteries funding through SaskCulture Inc., MAS likely wouldn't exist.
For more information on the Museums Association of Saskatchewan visit www.saskmuseums.org.
SaskCulture - thanks to the Saskatchewan Lotteries Trust Fund for Sport, Culture and Recreation - provides funding to a wide range of cultural groups and activities in the province, including museums. Together, with organizations, such as the Museums Association of Saskatchewan, we are building a culturally vibrant Saskatchewan. Culture Builds Community! Learn more at www.saskculture.sk.ca.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 5,990
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#ifndef TARGET_H_INCLUDED
#define TARGET_H_INCLUDED
class TargetWindow;
class TargetClient : public wxWindow
{
TargetClient(wxWindow *parent);
~TargetClient(void);
static const unsigned int m_maxHistorySize = 400;
unsigned int m_minLength;
unsigned int m_maxLength;
struct
{
double ra;
double dec;
} m_history[m_maxHistorySize];
unsigned int m_nItems; // # of items in the history
unsigned int m_length; // # of items to display
double m_zoom;
double m_refCircleRadius;
void AppendData(const GuideStepInfo& step);
void OnPaint(wxPaintEvent& evt);
friend class TargetWindow;
DECLARE_EVENT_TABLE()
};
class TargetWindow : public wxWindow
{
OptionsButton *m_lengthButton;
wxCheckBox *m_enableRefCircle;
wxSpinCtrlDouble *m_refCircleRadius;
TargetClient *m_pClient;
bool m_visible;
public:
TargetWindow(wxWindow *parent);
~TargetWindow(void);
void AppendData(const GuideStepInfo& step);
void SetState(bool is_active);
void UpdateControls(void);
void OnButtonClear(wxCommandEvent& evt);
private:
void OnButtonLength(wxCommandEvent& evt);
void OnMenuLength(wxCommandEvent& evt);
void OnButtonZoomIn(wxCommandEvent& evt);
void OnButtonZoomOut(wxCommandEvent& evt);
void OnCheckBoxRefCircle(wxCommandEvent& event);
void OnRefCircleRadius(wxSpinDoubleEvent& event);
DECLARE_EVENT_TABLE()
};
#endif // TARGET_H_INCLUDED
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,500
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Q: Trying to load .dll file in Python. Getting OSError: [WinError 126]. What am I doing wrong? import os
import ctypes
os.path.dirname(os.path.abspath("Python_ESS_2-SWITCH.py"))
h = ctypes.WinDLL("ess_64.dll")
Both my OS and Python working directories are in the folder with the dll file I'm interested in.
The error that comes back is:
347 ##
348 if handle is None:
--> 349 self._handle = _dlopen(self._name, mode)
350 else:
351 self._handle = handle
OSError: [WinError 126] The specified module could not be found
When I go into the ctypes module that the error traces back to I can see that the module that cannot be loaded is LoadLibrary:
if _os.name in ("nt", "ce"):
print("in the if")
from _ctypes import LoadLibrary as _dlopen
There's another thread that suggested some solutions but they have not yielded results for me. If anyone has any ideas I'd be very greatful.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,194
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Le Van Hool A360 est un autobus à plancher bas fabriqué et commercialisé par le constructeur belge Van Hool de 1995 à 2003. Il sera remplacé sur les chaînes de montage par le Van Hool NewA360.
Historique
Après avoir produit les Van Hool A500 et A600, dérivés du prototype Van Hool A280, le constructeur belge mit au point le premier prototype des Van Hool A300 en 1991. Ce modèle allait jeter les bases de tous les autres autobus urbains créés par Van Hool jusque dans les années 2000.
L'A300 se distingue de ces prédécesseurs par une nouvelle carrosserie, plus rectangulaire, et un plancher plus bas obtenu en disposant le compartiment moteur au-dessus du niveau du plancher. Il est par conséquent plus aisé d'embarquer que sur un autobus de la génération précédente.
Dès le début des années 1990, ce premier modèle de la nouvelle génération commença à se vendre en Belgique et à l'exportation, ainsi qu'une version midibus (le A308) et une version articulée (AG300 ou AGG30).
En 1994, Van Hool créa sur la base de ce châssis une version au moteur installé horizontalement à l'arrière : le Van Hool A360. Dans la foulée, les prototypes des A320 et A330 seront produits en 1997. Ces trois versions, à moteur arrière, supprimaient l'encombrement du milieu de la salle par le compartiment moteur.
L'A360, qui est surtout destiné aux dessertes rurales et périurbaines, a le plancher surélevé au-dessus du moteur mais reprend, à l'avant, le plancher bas du Van Hool A300 ; à l'intérieur, en arrière de la seconde porte, se trouve un escalier pour atteindre la partie arrière. Contrairement aux A300, A320 et A330, il n'y a pas eu d'A360 à trois portes.
Les A320 et A330, développés ultérieurement, avaient un plancher plat. Le Van Hool A320 possédait, comme l'A360, un moteur installé horizontalement, d'où un léger surhaussement du plancher à l'arrière ; le Van Hool A330 avait, lui, un moteur vertical, installé à gauche, ce qui permettait un véritable plancher plat sur toute sa longueur.
L'A360 sera produit en série de 1995 à 2003.
À partir de 2002, le Van Hool NewA360 remplaça les A360 sur les chaînes de montage.
Caractéristiques
Exploitants
Notes et références
Voir aussi
Van Hool
Van Hool NewA360
Van Hool A600
Autobus Van Hool
Poids lourd des années 1990
Poids lourd des années 2000
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{\label{}}
\section{\label{sec:intro}Introduction}
The supersymmetry~\cite{susy}(SUSY) is a potential symmetry between the fermions and the bosons of the standard model (SM) of particles proposing that each fermion has a bosonic superpartner and each boson has a fermionic superpartner.
The main attraction of the SUSY models is that they solve the hierarchy and the fine-tuning problems of SM.
Given that no sparticles have been observed it must be that SUSY is broken such that the sparticles are heavier than the particles.
One of the most studied SUSY models is the minimally supersymmetric standard model (MSSM) of particles.
This model is an extension of SM postulating the conservation of the R-parity.
This conservation law results in the pair prodcution of the sparticles and their decay into an odd number of sparticles.
This means further that the lightest supersymmetric particle (LSP) is stable and interacts very weakly with the matter making it a dark matter candidate.
Another postulate of MSSM is the soft SUSY breaking.
There are several theoretical approaches to this, but this report will mention only a gravity mediated SUSY breaking (mSUGRA) case.
The strategy used in every search for SUSY is geared towards selecting events with very energetic observables: leptons, photons, jets, missing transverse energy ($E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}) due to neutrinos and LSP.
The event selection and the methods used in these searches~\cite{Call} are optimized on the simulation of a set of benchmark points defined by certain values of the SUSY parameters.
These points are chosen such that a good coverage of the phase space is obtained beyond the current experimental and theoretical limits.
\section{\label{sec:det}The CMS detector and event clean-up}
The SUSY searches described in this report are using simulated 10~TeV proton-proton collisions followed by the simulation of the detector response to the resulting particles.
The detector used in these studies is the general purpose CMS~\cite{cms} detector installed at LHC.
The simulation of this detector is made using the software package GEANT4~\cite{geant} that takes into account a detailed description of the geometry of the detectors, the materials and the magnetic field.
In order to perform any kind of analysis with the data obtained from this detector, all the observables (i.e. jets, electrons, muons, photons, $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}) have to be well understood.
Given its discriminating power in any search for SUSY, $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} needs special attention.
Besides potential undetected new particles, the sources for $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} are many: cracks in the detector, cosmic rays, beam halo, electronic noise, and even SM processes (e.g. semileptonic decay of hadrons).
At CMS it has been studied that the effect of the cosmic rays and of the beam halo can be minimized by selecting events with significant average electromagnetic fraction ($EEMF>0.1$) and average charged fraction ($ECHF >0.175$).
\begin{equation}
EEMF=\frac{\sum_{jets} P_{T}^{jets}EMF^{jets}}{\sum_{jets}P_{T}^{jets}}
\end{equation}
\begin{equation}
ECHF=\frac{\sum_{jets}(\sum_{tracks}P_{T}^{tracks})/P_{T}^{jets}}{N_{jets}}
\end{equation}
\section{\label{sec:cmssusy}Search for SUSY program at CMS}
The strategy employed by the CMS collaboration to search for SUSY follows three main directives: model independent classification of the searches based on the topology of the final state, data-driven estimation of the SM backgrounds, and optimization of the event selection based on a set of benchmark signal points.
Depending on the physics objects content of the final state, there are eight main SUSY searches at CMS: exclusive n-jet, inclusive 3-jets, photon+$E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}, single lepton, same-sign dilepton, opposite-sign dilepton, trilepton, and dilepton+photon analysis.
In this report, only the hadronic searches (i.e. exclusive n-jet and inclusive 3-jets) are discussed.
For each of these analyses the estimation of the various SM backgrounds is performed using techniques that rely as little as possible on the detector simulation.
Some of these techniques are described briefly in Section~\ref{sec:smbg}.
In order to enhance the sensitivity of these searches the selection criteria used in these searches are lightly optimized using a set of benchmark SUSY signal points not excluded by theoretical and other hadron colliders experimental constraints.
The distribution of some of these points is illustrated in Fig.~\ref{fig:cmssusy} for the mSUGRA scenario.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{pic/susy_m0m12.eps}
\caption{The mSUGRA benchmark points used by the CMS collaboration to optimize the selection criteria applied in the SUSY searches.}\label{fig:cmssusy}
\end{center}
\end{figure}
In the following sections the details of the inclusive hadronic search and of the exclusive multi-jet search are discussed.
\subsection{\label{sec:allhad}Inclusive hadronic search}
In this analysis we look for an excess over the SM background in a final state that contains at least three jets and significant $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}.
The trigger used to collect the data contains requirements on the scalar sum of transverse energies ($E_{T}$) of the jets (HT) and on the magnitude of the vectorial sum of the transverse momentum ($p_{T}$) of the jets (MHT).
\begin{equation}
HT = \sum_{jets} E_{T}^{jets}
\end{equation}
\begin{equation}
\vec{MHT} = \sum_{jets} \vec{p}_{T}^{jets}
\end{equation}
At Level 1 we ask for HT $>$ 200 GeV while at the High Level Trigger (HLT) the HT $>$ 350 GeV and MHT $>$ 100 GeV.
In addition to the event clean-up described in Section~\ref{sec:det}, the offline analysis requires at least three jets with $E_{T}$ $>$ 30 GeV and pseudo-rapidity ($\eta$) $<$ 3 in absolute value.
The electron or photon contamination of these jets is reduced by requiring the electromagnetic fraction of the jet energy to be $<$ 0.9.
The QCD background can be significantly reduced by selecting a certain region in the $\delta\phi_{1}$-$\delta\phi_{2}$ plane defined such that $R_{1}> 0.5$ and $R_{2}> 0.5$, where
\begin{equation}
R_{1(2)}=\sqrt{\delta\phi_{2(1)}^{2}+(\pi-\delta\phi_{1(2)})^{2}}
\end{equation}
and $\delta\phi_{1(2)}$ is the difference in the polar angles of the leading(subleading) jet and $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}.
The contribution from top quark pair production as well as from W boson production is reduced by requiring no isolated tracks in the event.
To reduce the overall SM background, the leading jet is required to have $E_{T}$ $>$ 180 GeV and $|\eta|<$ 1.7, while the second leading jet should have $E_{T}$ $>$ 110 GeV.
For further background reduction we require $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} $>$ 200 GeV and $M_{eff}>$ 500 GeV, where $M_{eff}$ is the sum of $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} and the $E_{T}$ of the second, third and fourth jets.
This is a prototypical event selection in this analysis that has been optimized using MC samples generated at 14 TeV collisions and described in~\cite{Call}.
For less energetic collisions, the values of the kinematic cuts should be lower than shown above and are currently under study at CMS.
\subsubsection{\label{sec:smbg}SM background}
An important part of the searches for SUSY is the evaluation of the SM background.
The SM processes have a much larger cross-section than the potential SUSY signal.
For the LM1 benchmark point ($m_{0}=60$ GeV, $m_{1/2}=250$ GeV, tan$\beta=10$, $A_{0}=0$, $\mu>0$) the leading order (LO) cross-section at 14~TeV is about 42~pb.
The corresponding LO cross-sections for the most important SM backgrounds are: $\approx$5.6E10~pb for QCD events, $\approx$15E3~pb for events with a $Z$-boson produced in association with jets, and $\approx$800~pb for $t\bar{t}$~ events.
In order to minimize the systematic uncertainties due to potential differences between data and simulation, the strategy behind the evaluation of these backgrounds relies on data driven methods.
Also with these methods there is no need to calibrate the simulation to the data.
The most important SM background is due to QCD processes.
Using the cuts on the angle between the leading jets and $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}(see Section~\ref{sec:allhad}), about 80$\%$ of the QCD events can be rejected, while keeping about 90$\%$ of the SUSY events.
The remaining QCD content can be evaluated directly from data via two proposed methods: smearing method and ABCD method.
The smearing method, developed at ATLAS~\cite{Asm}, relies on the parameterization of a response function from multijet events with high $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} values and $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} aligned with one of the jets.
The response function is defined for each jet as
\begin{equation}
R=\frac{1-p_{T}^{jet}cos(jet,E_{\rm{T}}\hspace{-1.1em}/\hspace{0.7em})}{|p_{T}^{jet}+E_{\rm{T}}\hspace{-1.1em}/\hspace{0.7em}|^{2}}
\end{equation}
and it used to smear the jets from events with low $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}.
The smearing of the jets will result in artificially created $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} used to estimate the real $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} distribution.
The normalization is obtained from the multijet data events with low $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}.
The ABCD method, pursued at CMS, uses two uncorrelated observables ($E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} variable and the minimum azimuthal angle between its direction and that of the three leading jets).
The plane formed by these two variables is split in four regions: A, B, C and D, such that the SUSY signal is contained in region C.
Given that the two observables are uncorrelated, the number of QCD events in region C can be derived from the number of QCD events in the other regions: $C=DxB/A$.
For this method to work it is important to have little SUSY content in region A, B and D.
Another important SM background is produced by $Z \to \nu\nu$~+jets events.
This background is irreducible and there are few proposed ways to measure it.
One method relies on the measurement of $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} from $Z \to \mu\mu$~+jets events where the muons are ignored.
The normalization is set by taking into account the theoretical ratio of cross-sections between the $Z \to \nu\nu$~and $Z \to \mu\mu$~processes.
Another method, developed at CMS~\cite{znuphot}, relies on $\gamma$+jets events with the photon being ignored in the event.
This method benefits from a larger data sample, while the disadvantage is due to the uncertainty on the normalization.
The plot in Figure~\ref{fig:3} shows the comparison between the $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} distribution from $Z \to \nu\nu$~+jets MC sample and the corresponding distributions using $\gamma$+jets events.
The $Z \to \nu\nu$~+jets events background can also be estimated from leptonic $W$ decays where the lepton is ignored.
This method~\cite{znuphot} also benefits from a larger statistics compared to $Z \to \mu\mu$~+jets, but less than $\gamma$+jets events.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{pic/cms-znunu.eps}
\caption{The comparison between the $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} distribution from $Z \to \nu\nu$~+jets MC sample and the corresponding distributions using $\gamma$+jets events.}\label{fig:3}
\end{center}
\end{figure}
\subsubsection{Sensitivity reach}
In Fig.~\ref{fig:cmssens} it is shown the 5-sigma discovery limit in the mSUGRA m$_{0}$-m$_{1/2}$ plane for the final state with at least three jets and $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}.
These curves are shown assuming 100~pb$^{-1}$ of good data collected by CMS for 14 TeV collisions (light color) and 10 TeV collisions (dark).
The limits obtained from the Tevatron experiments (solid, dark) and from LEP (dashed lines) are also shown.
The curve for the 10 TeV collisions has been obtained using the same event selection used for the 14 TeV case.
We conclude that for this final state there is a very good chance of discovering SUSY with several tens of pb$^{-1}$ of understood data.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{pic/cms_allhad_sens.eps}
\caption{Discovery potential in the inclusive 3 jets + $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} final state at CMS.}\label{fig:cmssens}
\end{center}
\end{figure}
\subsection{Exclusive two jets channel}
It expected that in the early days the $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em} observable will be poorly understood.
As studied in~\cite{djet}, a new discriminating variable ($\alpha_{T}$) is used instead of $E_{\rm{T}}\hspace{-1.1em}/$\hspace{0.7em}: the ratio between the transverse energy of the second leading jet and the transverse invariant mass of the two leading jets.
The search is performed in events with the two leading jets having $p_{T}$ $>$ 50 GeV/c, and no hard third jet ($p_{T}$ $<$ 50 GeV/c) or hard leptons ($p_{T}$ $<$ 10 GeV/c).
The dominant QCD dijet background can be dramatically reduced by requiring $\alpha_{T}>0.55$ as it can be seen in Fig.~\ref{fig:alphaT}.
An additional cut on the HT $>$ 500 GeV keeps the SM backgrounds under control.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{pic/cms_alphaT.eps}
\caption{Distribution of the $\alpha_{T}$ variable calculated in events with twojets.}\label{fig:alphaT}
\end{center}
\end{figure}
The remaining backgrounds as listed in Fig.~\ref{fig:alphaT} are estimated with a data-driven method using an ABCD-like technique where the variables of interest are $\alpha_{T}$ and $|\eta|$ of the leading jet.
As it can be seen in Fig.~/ref{fig:alphaTeta}, the signal (LM1) is expected to have the leading jet mostly in the central region ($|\eta|<$ 2.5), while the SM background extends more forward in pseudo-rapidity.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{pic/cms_alphaT_eta1.eps}
\caption{Distribution of the pseudo-rapidity of the leading jet for SUSY signal (LM1) and various SM backgrounds.}\label{fig:alphaTeta}
\end{center}
\end{figure}
The signal region (C) is defined by $\alpha_{T}>0.55$ and $|\eta_{1}|<$2.5, while the control regions are defined as follows: region A has $\alpha_{T}<0.55$ and $|\eta_{1}|>$2.5, region B has $\alpha_{T}>0.55$ and $|\eta_{1}|>$2.5, and region D has $\alpha_{T}<0.55$ and $|\eta_{1}|<$2.5
The amount of background in the signal region is then equal to $Dx B/A$.
Fig.~\ref{fig:alphaTbg} shows that for background (triangles) the fraction $B/A$ is constant as a function of the pseudo-rapidity of the leading jet, while adding signal (squares) introduces a dependence.
It has to be noted here that relaxing the HT cut results in a larger signal contamination in regions A and B which will create larger systematic uncertainties on the background estimate.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{pic/cms_alphaT_bckg.eps}
\caption{The ratio of the number of events with $\alpha_{T}>0.55$ to the number of events with $\alpha_{T}<0.55$ as a function of the pseudo-rapidity of the leading jet. The triangles correspond to the SM background events while the squares represent the mix of signal (LM1) and the various SM backgrounds.}\label{fig:alphaTbg}
\end{center}
\end{figure}
With this background estimation method, the signal to background ratio is expected to be of order 6 and SUSY signal to be $\approx$44 events for 100~pb$^{-1}$ of data.
\subsubsection{Generalization to n-jets}
The techniques used in the search performed on the sample of events with exactly two jets can be applied on the sample with more jets by generalizing the definition of $\alpha_{T}$.
This can be done by combining the jets into two groups such that the difference ($\Delta H_{T}^{min}$) between the HT calculated in each group is minimized.
The generalized $\alpha_{T}$ is defined as $(H_{T}-\Delta H_{T}^{min})/2M_{T}$, where HT is computed using all the jets used in the analysis and $M_{T}$ is their transverse invariant mass.
In Fig.~\ref{fig:alphaTn} it is shown that the dominant QCD background can be effectively reduced by cutting on this generalized variable.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{pic/alphaT_njet.eps}
\caption{Distribution of the $\alpha_{T}$ variable calculated in events with n jets.}\label{fig:alphaTn}
\end{center}
\end{figure}
As detailed in~\cite{njet}, this analysis has been performed in sample of events with 3, 4, 5, and 6 jets in the final state.
The trigger used to select these events requires at L1 that the leading jet has $E_{T}$ $>$ 70 GeV and at HLT $E_{T}$ $>$ 110 GeV.
The jets used in this analysis are required to have $E_{T}$ $>$ 50 GeV and $|\eta|<$3, while the leading two jets must have $E_{T}$ $>$ 100 GeV and the leading jet $|\eta|<$2.
In order to eliminate leptonic events the muons and electrons in these events should have $p_{T}$ $<$ 10 GeV/c.
The remaining backgrounds are further reduced by requiring the events to have HT $>$ 350 GeV.
The contribution from the QCD background is diminished by requiring that $\alpha_{T}>$ 0.55.
To reduce the QCD contribution to large $\alpha_{T}>$ values, it is required that the ratio of MHT calculated using jets with $E_{T}$ $>$ 50 GeV to the MHT using jets with $E_{T}$ $>$ 30 GeV is less than 1.25.
As in the case of the exclusive two jets analysis, the signal to background ratio is expected to be of order 6 and SUSY signal to be $\approx$52 events for 100~pb$^{-1}$ of data.
\subsection{Conclusion}
The CMS collaboration has a search for SUSY program covering all possible final states and it is using a variety of tools and methods to this end.
In particular, the searches for SUSY in the fully hadronic channels are well advanced and the most sensitive.
This sensitivity is in part due to the data-driven approaches to estimate the SM backgrounds thus minimizing the systematic uncertainties.
The studies described in this report indicate that SUSY can be discovered in the fully hadronic final state with less than 100~pb$^{-1}$ of understood data.
\bigskip
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Logan Evans is a commercial pilot from Sitka. He stands next to his plane after flying up Lynn Canal, the longest fjord in North America, known for icy winds and bumpy flights. Evans' company, Harris Air, started flying up the Canal earlier this year, challenging Alaska Sea Planes for their monopoly in the area.
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"redpajama_set_name": "RedPajamaC4"
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What is essential is invisible to the eye
Author Allison J. GongPublished on 2018-10-22 2020-08-09 Leave a comment on What is essential is invisible to the eye
When I teach sponge biology to students of invertebrate zoology, I spend a lot of time describing them as phenomenal filter feeders, and suspect that most other professors do the same. There really are no animals that come close to possessing sponges' ability to remove very small particles from the water. Sponges have this ability despite the fact that their bodies are extraordinarily simple. I can draw pictures on the board to diagram the variety of sponge body types, but I've always wanted to show students how these bodies actually work.
Thing is, from the outside sponges just aren't that interesting. Some grow into large, conspicuous tube or vase shapes, but most occur as crusts of varying thickness and color. For example:
Sponge in display tank at Seymour Marine Discovery Center
Sponge in aquarium at Long Marine Lab
Not much to write home about, is it?
But as with most things invertebrate, sponges are more complex than they appear to be at first glance. And of course their complexity can be best appreciated when you observe sponges under the microscope. That's what I've been doing over the past few weeks: making wet mounts of living sponge and looking at them under the compound microscope. I'm still figuring out the best way to take photos through the scope, and trying to find the magic combination of lighting, magnification, and depth of field to obtain the clearest images.
Let's take a step back and review some basic sponge fundamentals. Sponges are animals in the phylum Porifera. Their bodies are characterized by a lack of true tissues; in other words, a sponge's body consists of various types of cells that do not form permanent connections. The different types of cells have different functions, but most of the cells retain the characteristic of totipotency, the ability to differentiate into another cell type as needed.
The sponge cells that do the filtering are called choanocytes. They form the lining of the sponge's body cavity. Choanocytes consist of a cell body and a collar region of microvilli that form a ring. From the center of the ring protrudes a single flagellum, whose undulations travel from base to tip. The choanocytes are arranged so that the flagella face into the body cavity, and their collective beating draws water through the body. The flagella also capture food particles, which are phagocytosed by the cell.
Ascon body type of a sponge. Arrows indicate direction of water flow.
© Sinauer Associates, Inc.
In its simplest tubular form, a sponge can be visualized as a miniature vase, with a single body cavity called a spongocoel ('sponge cavity') which is lined with choanocytes. Water enters the sponge through many microscopic pores on the outer skin of the body, is filtered by the choanocytes, and exits through a single opening called the osculum. This system works, but the efficiency of filtering is limited by the surface area of the choanocyte layer lining the spongocoel, and very few sponges have this body type.
Now if you imagine making invaginations into the choanocyte layer and continue the choanocytes into the channels you create, you could increase the filtering surface area of a sponge without having to increase its overall body size. Continue this maneuver to its logical end and you'd end up with something that resembles a cluster of grapes. The skin of the grapes would represent the layer of choanocytes, all oriented so that their flagella face the hollow interior of the grape, which would correspond to what we call a choanocyte chamber. This type of body plan has a vastly expanded surface area to volume ratio compared to the tubular form, and these sponges achieve the largest sizes. Incidentally, natural selection has used this exact same strategy to maximize the respiratory exchange surface area of your lungs: gas exchange occurs in the alveoli, which are tiny thin-walled sacs where oxygen diffuses into and carbon dioxide diffuses out of capillaries. The total respiratory surface area of your lungs is about 70 m2—i.e., roughly equivalent to one side of a standard tennis court, without the doubles lanes—all tucked neatly into the volume of your thoracic cavity.
The canals leading into and out of each choanocyte chamber are smaller than the chamber itself, and this arrangement takes advantage of some fundamental fluid dynamics: a given volume of water flows faster through a tube with a narrow diameter and slower through a tube with a wider diameter. Water travels relatively fast through the narrow canals on either end of a choanocyte chamber and slows down significantly within the chamber proper. This gives the choanocytes time to capture all of the food particles in the water stream, and speeds the water to the outside of the body once it has been filtered.
Now we can get back to the animals themselves. Their external appearance may not look like much, but sponges are very interesting when viewed under a microscope. I've been taking samples and squashing them under coverslips for a close look.
Here's a view under darkfield lighting:
Piece of a living sponge, viewed with darkfield lighting
The clear-ish objects that look like the back roads of a map are spicules. They provide a bit of skeletal support for the sponge's body and help to deter predators--who would want to bite a mouthful of glass splinters?
When I switched to higher magnification and phase-contrast lighting I could see hollow spherical structures that vaguely resembled blackberries. I felt a thrill of excitement to realize that these were probably choanocyte chambers, and I was looking at the choanocytes themselves!
Interior of sponge body
Here's another view at the same magnification, which shows more clearly the cells of the chamber:
Choanocyte chambers
The chambers themselves closely resemble the blastula stage of early animal embryology. Like a blastula, a choanocyte chamber is a hollow ball of cells; unlike a blastula, which has a ciliated outer surface, a choanocyte chamber consists of flagellated cells with the flagella oriented towards the inner hollow space. At a bit less than 40 µm in diameter, the chambers are about half the size of my sea urchin blastulae.
Remember how I said that the structure of the choanocyte chambers is similar to that of our alveoli? You may not be able to visualize the alveoli in your lungs, but this photo shows how the chambers resemble a cluster of grapes.
Because it's impossible to see the three-dimensional structure of the chambers from the single plane of focus you get with a photograph, I shot some video while focusing up and down through the sample on the slide.
They really do look like grapes, don't they?
Published on 2018-10-22 2020-08-09 Author Allison J. GongCategories Marine biology, Marine invertebrates, PhotographyTags marine biology, marine invertebrates, microscopy, sponges
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
Since Turing's initial insights into reaction-diffusion driven morphogenesis \cite{turing1952chemical}, a substantial research effort has elucidated various mathematical and biophysical aspects of such symmetry-breaking instabilities leading from homogeneity to patterned states \cite{de1991turing, cross1993pattern, maini2012turing, kondo2010reaction, green2015positional,woolley2014visions}. An important and well-studied aspect of these instabilities is the underlying geometry, which can influence both the stability of a homogeneous state, as well as the subsequent mode selection of emergent patterns \cite{Murray2003}. However, one less well-studied aspect of geometry is the coupling between layered spatial domains, which can arise in a variety of settings and is the primary object of interest in this paper.
Reaction diffusion processes arise in a diversity of layered settings, from bulk-surface membrane-cytosol interactions \cite{halatek2018review, ratz2014symmetry, kretschmer2016pattern, spill2016effects, cusseddu2018coupled} to epithelial-mesenchymal couplings in developing skin \cite{cruywagen1992tissue, shaw1990analysis}. Synthetic experiments involving pattern formation in monolayers also exhibit clearly stratified regions of cells and culture medium \cite{sekine2018synthetic}. Additionally, many experiments involving bacterial pattern-formation are performed using colonies on the surface of a substrate, such as agar \cite{budrene1991complex, budrene1995dynamics}. Such systems either use natural chemotaxis mechanisms to initiate spatial pattern formation of the bacterial density itself \cite{tyson1999minimal}, or instead use synthetic bacteria re-engineered to express additional quorum-sensing pathways that spatially coordinate patterns in gene expression \cite{Basu2005bandpass, Tabor2009edge, grant2016orthogonal}. Other examples are synthetically reconstituted protein interaction systems with bulk-membrane coupling such as the Min system \cite{loose2008exp, kretschmer2016pattern, frey2018review}, where molecular interactions \cite{denk2018syth,glock2019synth}, or {\it in vitro} system geometries \cite{wu2016geometry, brauns2020bulk, halatek2018box}, are modified to stimulate changes in the observed protein patterns.
Examples of particular contemporary interest include the use of bacterial colonies as exemplars of synthetic multicellular communication and self-organisation \cite{balagadde2008synthetic, dalchau2012towards, Payne2013rings, grant2016orthogonal, karig2018stochastic}, for example using modified \emph{E. coli} with engineered quorum-sensing signalling on the surface of an agar plate \cite{grant2016orthogonal, Payne2013rings, Cao2016scale,Cao2017pressure}. Some of these systems take advantage of the geometry of colony growth and nutrient diffusion to influence pattern formation \cite{Payne2013rings,Cao2016scale,Cao2017pressure} while in other systems the bacteria are confined \cite{grant2016orthogonal, boehm2018}, but the signalling molecules can diffuse into the inert agar layer below the chemically active colonies. The impact of this leaching on the prospects of a Turing instability in experimentally-relevant geometries has not been fully characterised and is a key motivation for our study.
Our first objective is to develop a two-domain model of reaction-diffusion processes coupled in a stratified bi-layer and to determine conditions for the Turing instability, on the assumption that the upper region is sufficiently substantive in the transverse direction to merit continuum modelling. Such a model is also applicable to a variety of other settings beyond multi-layered bacterial pattern formation, such as developing skin. Our second objective is to focus on the Turing instability for multi-layered bacterial systems, where signalling molecules only diffuse in the lower (agar) layer and especially where the upper layer is asymptotically thin relative to the scale of the pattern and the depth of the lower layer. The main biological motivation is to determine to what extent the diffusive bulk helps, or hinders, the ability of an engineered system to exhibit Turing-type patterning.
In terms of model development, domain-coupled reaction-diffusion systems broadly fall into three major types: instantaneously coupled, bulk-surface models and bulk-bulk models. The first type are models where the components occupy the same physical space (or the reactions occur in thin regions where a homogenisation approximation is sensible) \cite{yang2002spatial, epstein2007coupled, yang2003oscillatory, fujita2013pattern}. Such models are essentially just larger reaction-diffusion systems with linear coupling between subsystems, and amenable to block-matrix analysis in the study of Turing instabilities \cite{catlla2012instabilities}, but do not capture the spatial separation of the domains. When applied to layered media, these models effectively assume vertical transport between distinct layers (such as that of Figure~\ref{Schematic}) is instantaneous. However, considering physical scales representative of synthetic pattern formation experiments using {\it E. coli} \cite{grant2016orthogonal,boehm2018}, and summarised below in Table~\ref{tab0}, one has an agar block with a depth of a few millimetres, say three, and a diffusion rate on the scale of $4\times 10^{-10}$ m$^2$ s$^{-1}$. Thus the timescale for vertical transport is in the region of 375 minutes, which is short compared to the timescales on which experimental measurements of the equilibrated system are recorded (1500-3000 minutes, \cite{grant2016orthogonal,boehm2018}) but far from instantaneous.
Hence, such models are inappropriate for the motivating examples here.
A second class of model considers bulk-surface coupling, where one component is confined to the boundary of the main bulk domain, and reactants flow between the two regions, such as in the case of proteins diffusing in the cytoplasm and binding on the cell membrane \cite{ratz2014symmetry, madzvamuse2015stability, spill2016effects, cusseddu2018coupled, paquin2018pattern, halatek2018review, frey2018review}. There is substantial recent interest in such models, from very theoretical results on existence and fast-reaction limiting behaviour \cite{ratz2015turing, anguige2017global, hausberg2018well}, to spike dynamics \cite{gomez2018linear} and a myriad of applications to understanding cell polarity \cite{thalmeier2016geometry, kretschmer2016pattern, halatek2018review, gesele2020ellipse}. One particularly well studied example is the pole-to-pole Min protein oscillation in {\it E. coli}, which has the biological function of guiding the cell division machinery to midcell \cite{kretschmer2016pattern}. Such intracellular protein patterning systems have been studied experimentally and theoretically in a wide range of system geometries, such as spherical \cite{klunder2013sphere, levine2005membrane}, elliptical \cite{halatek2012ellipse, wu2016geometry, gesele2020ellipse}, and planar membrane geometries \cite{halatek2018box}. A striking feature of these examples is that the geometry itself has a major impact on pattern formation and pattern selection, which has been confirmed experimentally \cite{wu2016geometry, brauns2020bulk}.
More generally, the Turing instability has also been studied in the context of membrane-cytosol models \cite{ratz2014symmetry, madzvamuse2015stability}.
Overall, linear stability analysis (as used by Turing) is highly applicable to such membrane-cytosol systems because the nonlinear interactions are typically restricted to the lower-dimensional membrane surface. The dynamics in the extended bulk are typically linear such that a general solution (or a good approximation) can be obtained analytically and used to satisfy the linearised reactive boundary condition. This is justified because the membrane can be considered as a surface with no transverse extent, and so transverse gradients only play a role in the cytosolic layer close to the membrane surface.
However, in multi-layered cellular systems the transverse lengthscales are at least that of many cells and hence transverse gradients cannot be neglected {\it a priori}, and thus should be accommodated in the modelling.
Models accounting for this represent the final class, with two separated spatial domains with an interface and suitable coupling boundary conditions. From the perspective of pattern formation, this kind of model has only been subject to recent numerical exploration \cite{vilaca2019numerical}, though it is used in the derivation of the second class -- bulk-surface models -- given appropriate distinguished limits and scaling assumptions (for example, \cite{chapman2016reactive, fussell_hybdrid_2018}).
Hence, we will develop models of the latter type, with an exploration of the conditions for the Turing instability, and their detailed study in the context of a stratified model with an inert underlying agar layer. Turing instabilities of reaction-diffusion systems have been studied on a variety of complex spatial domains such as compact manifolds \cite{varea1999turing, chaplain2001spatio}, networks \cite{asllani2014theory, ide2016turing,nakao2010turing}, and many of the aforementioned complex system geometries \cite{halatek2012ellipse, klunder2013sphere, halatek2018box}. The primary difficulty in such cases, compared to the textbook example of a continuous line, is determining the corresponding set of eigenfunctions and eigenvalues of the spatial transport operators, which for some system geometries do not need to coincide between domains (e.g. in the surface-bulk elliptical case \cite{halatek2012ellipse}). In such cases approximate solutions for the system's eigenfunctions need to be derived that are orthogonal in the patterning layer.
Examples that deviate even further from the classical case are growing domains \cite{crampin1999reaction, plaza2004effect, KrauseAnisotropy2018, SanchezGarduno2018} and spatially heterogeneous reaction-diffusion processes \cite{benson1998unravelling, page2003pattern, page2005complex, haim2015non, kolokolnikov2018pattern}, for which the canonical approach does not work. In such cases, novel approaches to pattern-forming instabilities have recently been developed for growth \cite{madzvamuse2010stability, van_gorder_growth_2019} and heterogeneity \cite{krause_WKB} under certain simplifications, but such analyses are quite different to the classical case. In a similar direction, as part of our objective in exploring the Turing instability for layered reaction-diffusion systems, we will aim to demonstrate a much richer diversity of structure in the resulting dispersion relations (and hence instability conditions), compared to classical counterparts.
As an outline, in Section~\ref{modelling} we present a two-domain layered model, where each domain consists of closed two-dimensional rectangular regions, coupled through a single shared boundary (See Figure~\ref{Schematic}) and briefly discuss how it can be reduced to a variety of other models. We focus on a special case of a two-domain model where we assume linear coupling and no reactions in the second (bulk) region, but note that our analysis can be applied with relatively simple modifications to more general cases. In Section~\ref{GenLinAnalysis} we develop an approach to linear stability analysis of homogeneous states. In Section~\ref{Asymptotics} we derive a variety of asymptotic results regarding our dispersion relation, especially considering limits that are of particular relevance for synthetic pattern formation in {\it E. coli} colonies. We further explore these results and other parameter regimes numerically in Section~\ref{Numerics}. Finally we discuss our results in Section~\ref{Discussion}.
\section{Two-Region Model}\label{modelling}
We consider a layered two-domain model where each domain is governed by a different reaction-diffusion system. We consider several interacting species in these two domains, which we write as $\Omega = \Omega_S \bigcup \Omega_B$ where we refer to $\Omega_B = [0,L]\times [0,H]$ as the bulk region, and $\Omega_S = [0,L]\times [H,H+H_\varepsilon]$ as the surface region (see Figure~\ref{Schematic}). We write $\bm{\hat{u}_B} \in \mathbb{R}^n$ for the concentrations of reactants in the bulk, and $\bm{\hat{u}_S}\in \mathbb{R}^n$ for the concentrations of reactants in the surface region. For simplicity, we consider a simple one-dimensional lateral geometry (orthogonal to the direction of the coupling condition), but note that the geometric details in the lateral direction(s) can be easily extended to much more complicated geometries, as long as eigenfunctions of the Laplacian in these directions are separable from the transverse coordinate $y$. We only consider reactions on the surface layer and assume the bulk only permits diffusion.
\begin{figure}[ht] \centering
\subfloat[Cells grown atop agar]{\includegraphics[width=0.35\linewidth]{Fig1aCells_Agar.pdf}}
\subfloat[Geometry of the system]{\input{geometric_setup.tikz}}
\caption{Example experimental system under consideration. (a) Here we consider cells growing in culture on top of a solid reservoir of nutrients, such as agar. (b) The surface (cellular) region denoted $\Omega_S$ has height $H_\varepsilon$ and contains both reaction and diffusion terms, whereas the bulk (nutrient) region $\Omega_B$ is of height $H$ and is assumed to have no reactions, but permits diffusion. Both have lateral extent $L$, with no-flux conditions on all boundaries except for the interface between the two regions, where a coupling condition is applied.\label{Schematic}}
\end{figure}
We have the following equations for the species concentrations in the bulk and surface regions:
\begin{equation}\label{eqSdim}
\frac{\partial \bm{\hat{u}_S}}{\partial \hat{t}} = \bm{\hat{D}_S}\nabla^2 \bm{\hat{u}_S}+\bm{\hat{f}_S}(\bm{\hat{u}_S}), \quad \hat{x} \in [0,L], \quad \hat{y} \in [H,H+H_\varepsilon],
\end{equation}
\begin{equation}\label{eqBdim}
\frac{\partial \bm{\hat{u}_B}}{\partial \hat{t}} = \bm{\hat{D}_B}\nabla^2 \bm{\hat{u}_B} , \quad \hat{x} \in [0,L], \quad \hat{y} \in [0,H],
\end{equation}
where $\bm{\hat{D}_S}, ~\bm{\hat{D}_B}$ are positive definite diagonal matrices. We further specify Neumann (no-flux) boundary conditions on the outer boundaries as,
\begin{equation}\label{BC1dim}
\frac{\partial \bm{\hat{u}_S}}{\partial \hat{x}} = \frac{\partial \bm{\hat{u}_B}}{\partial \hat{x}} = \bm{0}, \text{ for } \hat{x}=0, L,
\end{equation}
\begin{equation}\label{BC2dim}
\frac{\partial \bm{\hat{u}_S}}{\partial \hat{y}} = \bm{0}, \text{ for } \hat{y}=H+H_\varepsilon, \quad \frac{\partial \bm{\hat{u}_B}}{\partial \hat{y}} = \bm{0}, \text{ for } \hat{y}=0,
\end{equation}
and lastly coupling conditions on the interior boundary which conserve fluxes and take the form,
\begin{equation}\label{couplingdim}
\bm{\hat{D}_S}\frac{\partial \bm{\hat{u}_S}}{\partial \hat{y}} = \hat{\eta}\bm{\hat{g}}(\bm{\hat{u}_S}, \bm{\hat{u}_B}), \quad \bm{\hat{D}_B}\frac{\partial \bm{\hat{u}_B}}{\partial \hat{y}} = \hat{\eta}\bm{\hat{g}}(\bm{\hat{u}_S}, \bm{\hat{u}_B}), \text{ for } \hat{y}=H,
\end{equation}
where $\bm{\hat{g}}$ is a given function determining the transport between the surface and the bulk region, and $\hat{\eta}$ is a rate of transport across the boundary. Essentially, all of the forthcoming analysis can be carried out with a general $\bm{\hat{g}}$, as linearisation will also linearise this function. For brevity and concreteness, we will henceforth assume a linear transport law, so that we have \begin{eqnarray}\label{geqn}
\bm{\hat{g}} = \bm{\hat{u}_S}-\bm{\hat{u}_B}.
\end{eqnarray}
We non-dimensionalise the above model via concentration, time and length scales corresponding to the reaction kinetics and a unit lengthscale $\hat{L}$, respectively. Specifically, we define $\bm{\hat{u}_S} = \bm{U}\bm{{u}_S}$, $\bm{\hat{u}_B} = \bm{U}\bm{{u}_B}$, where $\bm{U}$ is a diagonal matrix of concentration scales. Equally, we set $\hat{t} = \tau {t}$, where $\tau$ is the timescale of the fastest reaction in the surface and bulk, and $\bm{\hat{x}} = \hat{L} \bm{x}$. The dimensional scalings are then chosen such that
\begin{eqnarray}\label{sceqn}
\bm{\hat{f_S}}(\bm{\hat{u}_S}) =(1/\tau) \bm{U}\bm{{f}_S}(\bm{{u}_S}), ~~~ \bm{\hat{g}}(\bm{\hat{u}_S},\bm{\hat{u}_B}) = \bm{U}{\bm{g}}(\bm{{u}_S},\bm{{u}_B}).
\end{eqnarray}
We define new dimensionless groupings $h = H/\hat{L}$, $\varepsilon = H_\varepsilon/\hat{L}$, $\bm{D_S} = \tau\bm{\hat{D}_S}/(\hat{L}^2)$, $\bm{{D}_B} = \tau\bm{\hat{D}_B}/(\hat{L}^2)$, $\tilde{L} = L/\hat{L}$ and $\eta = \tau\hat{\eta}/\hat{L}$. The nondimensional system is written as
\begin{equation}\label{eqS}
\frac{\partial \bm{u_S}}{\partial t} = \bm{D_S}\nabla^2 \bm{u_S}+\bm{f_S}(\bm{u_S}), \quad x \in [0,\tilde{L}], \quad y \in [h,h + \varepsilon],
\end{equation}
\begin{equation}\label{eqB}
\frac{\partial \bm{u_B}}{\partial t} = \bm{D_B}\nabla^2 \bm{u_B}, \quad x \in [0,\tilde{L}], \quad y \in [0,h],
\end{equation}
\begin{equation}\label{BC1}
\frac{\partial \bm{u_S}}{\partial x} = \frac{\partial \bm{u_B}}{\partial x} = \bm{0}, \text{ for } x=0\text{ and } \tilde{L},
\end{equation}
\begin{equation}\label{BC2}
\frac{\partial \bm{u_S}}{\partial y} = \bm{0}, \text{ for } y=h+\varepsilon, \quad \frac{\partial \bm{u_B}}{\partial y} = \bm{0}, \text{ for } y=0,
\end{equation}
\begin{equation}\label{coupling}
\bm{D_S}\frac{\partial \bm{u_S}}{\partial y} = \eta\bm{g}(\bm{u_S}, \bm{u_B}), \quad \bm{D_B}\frac{\partial \bm{u_B}}{\partial y} = \eta\bm{g}(\bm{u_S}, \bm{u_B}), \text{ for } y=h.
\end{equation}
There are several distinguished limits of the nondimensional system \eqref{eqS}-\eqref{coupling} that reduce the model to different cases already present in the literature. In the limit $\varepsilon \to 0$, one can consider either scaling $\eta \sim O(\varepsilon)$ or scaling $\bm{f_S} \sim O(\varepsilon^{-1})$ in order to reduce the system to a bulk-surface model, which is well-studied in the literature (though primarily in radial geometries) \cite{levine2005membrane, ratz2014symmetry, ratz2015turing, madzvamuse2015stability,cusseddu2018coupled, gomez2018linear, paquin2018pattern}. The second scaling, indicating that the surface timescale is rapid, can be related to assumptions regarding rapid surface reactions used to justify reactive boundary conditions from the microscopic viewpoint \cite{chapman2016reactive}. Finally another limit is the case of infinite permeability, $\eta \to \infty$, wherein the concentrations and fluxes are continuous across the interface. In this case, the system can be seen as a single domain model with a step function heterogeneity, which has been studied extensively as an example of spatially heterogeneous reaction-diffusion systems \cite{benson1998unravelling, page2003pattern, stephetero}. Nonetheless, pattern formation in the system above, as well as several other distinguished limits, has not been analysed yet in the literature.
In Table~\ref{tab0}, we give the dimensional parameter scales to be considered in our framework, taken from the key motivating example of synthetic patterning in {\it E. coli} bacterial colonies on an agar substrate \cite{grant2016orthogonal,boehm2018}. While such experiments can be conducted with a variety of settings, an overall restriction on the variation of these parameters is motivated by the range of the physical scales in these studies. Here, bacteria are plated in squares of about 1 mm (Methods, \cite{boehm2018}) with patterning cells considered in an 8$\times$8 grid in one study (Supplementary Information, \cite{grant2016orthogonal}) and more generally the patterning fields are observed across about 22 such squares (Fig 5B, \cite{boehm2018} and Fig 3E \cite{grant2016orthogonal}).
Thus we consider a range of $\hat{L} \sim 8-22$ mm.
For the diffusion matrices, the infinity (max) norm $\| \cdot \|_\infty$ is presented, i.e. the maximum value of the matrix's components.
From Grant et al., (Supplementary Material, Tables S8, S9, \cite{grant2016orthogonal}) diffusion coefficients have been estimated in the range $\| \bm{\hat{D}_S}\|_\infty \sim 10^{-10}$ m$^2$s$^{-1} - 10^{-9}$ m$^2$s$^{-1}$ by model fitting to exemplar results.
As this is also the scale of diffusion (or slightly more than the scale) for the signalling molecule EGF in water \cite{diffsize}, the same scale is used for $\| \bm{\hat{D}_B}\|_\infty$.
Similarly, in the parameter fitting by Grant et. al., a reaction timescale on the scale of the faster reactions is such that $1/\tau \sim 8.4\times10^{-5}$ s$^{-1} - 10^{-3}$ s$^{-1}$, with the range arising from the use of different model kinetics in parameter fitting. We further assume 10-50 layers of bacteria, with an {\it E. coli} bacterium size scale of about $10^{-6}$ m$-2\times 10^{-6}$ m \cite{bactsize}, and hence a surface depth on the scale of $H_\varepsilon\sim 10^{-5}$ m$- 10^{-4}$ m. Finally, the depth of the bulk is highly variable and easily changed upwards from the millimetre scale and so $H$ is taken with the range of $1-10$ mm; estimates for the interfacial permeability, $\hat{\eta}$, are currently unavailable. These dimensional parameter estimates generate the non-dimensional scales of Table~\ref{tab1}, which will guide the asymptotic and numerical investigations presented below.
\begin{table}[ht] \centering
\begin{tabular}{ccc}
\toprule
Parameter & Range & Justification \\
\midrule
$\hat{L}$ & $8\times 10^{-3}$ m$- 2.2\times10^{-2}$ m & See text \\
$\| \bm{\hat{D}_S}\|_\infty, \|\bm{\hat{D}_B} \|_\infty $ & $10^{-10}$ m$^2$ s$^{-1}- 10^{-9}$ m$^2$s$^{-1}$ & Table~S8, \cite{grant2016orthogonal} \\
$1/\tau $ & $ 8.4\times 10^{-5}$ s$^{-1}-1.0\times10^{-3}$ s$^{-1}$ & Tables S8,~S9, \cite{grant2016orthogonal} \\
$H_\varepsilon$ & $10^{-5}$ m$- 10^{-4}$ m & See text \\
$H$ & $10^{-3}$m $- 10^{-2}$ m & See text \\
$\hat{\eta}$ & Unknown & $-$ \\
\bottomrule
\end{tabular}
\caption{Numerical scales of various dimensional parameters and parameter groupings in SI units, based on patterning in synthetic pattern formation with {\it E. coli} bacterial colonies, using physical scales motivated by the studies of Grant et al.~\cite{grant2016orthogonal} and Boehm et al.~\cite{boehm2018}.}
\label{tab0}
\end{table}
\begin{table}[ht] \centering
\begin{tabular}{cc}
\toprule
Parameter & Typical Value/Range \\
\midrule
$ \varepsilon = H_\varepsilon/\hat{L}$ & $4.5\times 10^{-4}-1.3\times 10^{-2} $ \\
$h = H/\hat{L}$ & $0.045-1.3$ \\
$ \tilde{L}=L/\hat{L}$ & 1 \\
$\varepsilon_* = \varepsilon \| \bm{D_S}^{-1}\bm{J} \|^{1/2}_\infty$ & $ 1.1\times10^{-3}- 0.87$ \\
$h_* = h \| \bm{D_S}^{-1}\bm{J} \|^{1/2}_\infty$ & $ 0.11-87$ \\
$\varepsilon_*^2/3$ & $4.0 \times10^{-7}-0.3$ \\
$\|\bm{D_S}\|_\infty = \tau\| \bm{\hat{D}_S}\|_\infty/\hat{L}^2 $ & $ 2\times 10^{-4}- 0.2$ \\
$\| \bm{D_B}\|_\infty \sim \|\bm{D_S}\|_\infty$ & $ 2\times 10^{-4}- 0.2$ \\
$\eta = \tau\hat{\eta}/\hat{L}$ & Unknown \\
\bottomrule
\end{tabular}
\caption{Numerical scales of various non-dimensional parameters and parameter groupings, motivated by the physical scales of synthetic pattern formation with {\it E. coli} bacterial colonies, in the studies of Grant et al.~\cite{grant2016orthogonal} and Boehm et al.~\cite{boehm2018}. For matrices, the infinity (max) norm $\| \cdot \|_\infty$ is used, which is the modulus of the matrix component with largest magnitude. For the non-dimensional matrix Jacobian, this norm is taken to be of order unity as the timescale is non-dimensionalised relative to $\tau$, a representative timescale associated with a fast reaction in the system. The non-dimensional lengthscale, $\tilde{L}$, is retained symbolically throughout the presentation to facilitate determining the impact of this scale, though it is unity for these scalings. The parameter scales $\varepsilon_*$ and $h_*$ as well as the range of $\varepsilon_*^2/3$ are presented as they will be important in the asymptotic analyses below. \label{tab1}}
\end{table}
\bk
\section{Linear Stability Analysis}
\label{GenLinAnalysis}
For a linear stability analysis of homogeneous equilibria of \eqref{eqS}-\eqref{coupling}, we require the steady states to this system, which arise from specifying
\begin{equation*}
\bm{f_S}(\bm{u_S^*}) = \bm{g}(\bm{u_S^*},\bm{u_B^*}) = \bm{u_S^*}-\bm{u_B^*}=\bm{0},\end{equation*}
so that the surface reactions determine the spatially-homogeneous steady state concentration in both regions, and our simple constitutive choice of $\bm{g}$ implies that these concentrations must be equal. We will focus exclusively on the case of an absence of reactions in the bulk, as motivated by the underlying inert agar layer in synthetic pattern formation with {\it E. coli} bacterial experiments \cite{grant2016orthogonal,boehm2018} and which requires only a root of the surface kinetics for there to be a steady state.
We proceed by considering perturbations to this steady state of the form $$\bm{u_S} = \bm{u_S^*}+\sigma \bm{w_S}(x,y,t) , ~~~~~~~~\bm{u_B} = \bm{u_B^*}+\sigma \bm{w_B}(x,y,t),$$ where $|\sigma| \ll 1$, and in general the bulk and surface perturbations are $n$-dimensional functions, where $n$ is the number of species. We substitute these perturbations into equations \eqref{eqS}-\eqref{coupling} to find, from equations \eqref{eqS}-\eqref{eqB}, that the perturbations will satisfy
\begin{equation}\label{eqwS}
\frac{\partial \bm{w_S}}{\partial t} = \bm{D_S}\nabla^2 \bm{w_S}+\bm{J_{S}}\bm{w_S}, \quad x \in [0,\tilde{L}], \quad y \in [h,h+\varepsilon],
\end{equation}
\begin{equation}\label{eqwB}
\frac{\partial \bm{w_B}}{\partial t} = \bm{D_B}\nabla^2 \bm{w_B}, \quad x \in [0,\tilde{L}], \quad y \in [0,h],
\end{equation}
where the Jacobian, $\bm{J_{S}} = \partial \bm{f_S}/\partial \bm{u_S} \in \mathbb{R}^{n\times n}$, is evaluated at the steady state concentrations. We also have the coupling condition from equation~\eqref{coupling} given by,
\begin{equation}\label{couplingw}
\bm{D_S}\frac{\partial \bm{w_S}}{\partial y} = \eta(\bm{w_S}-\bm{w_B}), \quad -\bm{D_B}\frac{\partial \bm{w_B}}{\partial y} = \eta(\bm{w_B}-\bm{w_S}), \text{ for } y=h.
\end{equation}
\subsection{Spatially homogeneous perturbations}\label{shp}
We now consider the appropriate generalisation of stability in the absence of transport, as is typically assumed in a Turing-type analysis. However, unless the reaction kinetics are the same in both domains, which is not true in our setting, then spatially homogeneous perturbations are not consistent with equations \eqref{eqwS}-\eqref{couplingw}. Such perturbations will not remain homogeneous under time evolution due to the coupling condition \eqref{couplingw}, except in the mathematically fine-tuned case where the homogeneous surface perturbation is along an eigenvector of $\bm{J_S}$ with eigenvalue $0$.
Previous studies of more complex systems (beyond those considered in textbook Turing models) also highlight that, when generalising the conditions that arise from the stability of the homogeneous steady state with respect to spatially homogeneous perturbations, one must also consider a perturbation with respect to the zero mode(s) of the transport operator \cite{klika2018domain}. However, given the assumption of completeness, i.e.~that separable solutions in $x$, $y$ and $t$ span the space of possible solutions, as generally used in linear stability theory, the existence of zero modes of the transport operator also requires mathematical fine tuning. In particular, with $\nabla^2 \bm{u_S}^0=0$ for the zero mode of the transport operator acting in the surface layer, $\bm{u_S}^0$, one has
$$
\bm{u_S}^0 ={\bm A}\cos(k_qx)\cosh(k_q(y-(h+\varepsilon))), ~~~ k_q =q\pi/\tilde{L},
$$ for a general $\bm{A}$ and $q$ a natural number on enforcing the zero flux boundary conditions. There is a directly analogous expression for the zero mode of the transport operator within the bulk region. However, after rearrangement, the interfacial condition at $y=h$ requires
\begin{equation*}
D_SD_B\sinh(k_q\varepsilon)\sinh(k_qh)=\eta \left(D_B \cosh(k_q\varepsilon)\sinh(k_qh)-D_S\sinh(k_q\varepsilon)\cosh(k_qh)
\right).
\end{equation*}
One possible solution occurs for $k_q=0$, which generates a spatially homogeneous mode that has already been considered above. Satisfying this equation for other $k_q$ requires mathematical fine tuning as $k_q$ is already constrained to a set of zero measure and all other parameters are either geometrical or biophysical in origin.
Hence to summarise, in contrast to the textbook Turing case, unless the surface and bulk kinetics are the same, constraints on the parameters do not arise from the constraint of stability to homogeneous perturbations; instead this stability always holds, at least in the absence of a mathematical fine tuning of parameters and the possibility of such fine-tuning is neglected below.
\subsection{Spatially inhomogeneous perturbations}
To proceed, we \bk
assume a separable solution in $x$, $y$ and $t$ for the linearised system \eqref{eqwS}-\eqref{couplingw}. With the usual assumption of a uniform temporal growth rate $\lambda$, the perturbation ansatz for a single mode of a separable solution is
\begin{equation}\label{expansions}
\bm{w_S} = e^{\lambda t} \bm{s}(y)\cos \left( k_qx\right), \, \bm{w_B} = e^{\lambda t} \bm{b}(y)\cos \left( k_qx\right),
\end{equation}
where $k_q = q \pi /\tilde{L} $ for $q$ a natural number (including $0$). Assuming completeness of such a set of modes, then linear superposition entails that an arbitrary function may be expanded via a weighted linear sum of individual modes. Hence, the question of stability of a linear perturbation reduces to the same question for single modes, as in the standard textbook analysis \citep{Murray2003}, without the need for the modes to be orthogonal. Noting that homogeneous modes are not feasible, as shown above, we proceed to consider whether any heterogeneous modes exhibit instability ($\Re(\lambda)>0$). Furthermore we note that even in the absence of completeness, $\Re(\lambda)>0$ still provides a sufficient condition for instability, though it is not strictly necessary. We will see later that our conditions are not refuted by comparisons with numerics, and so we anticipate that the set of modes we construct is at least generic if not a complete basis. We note that for each $q$, there may be many distinct $\lambda$, and corresponding to each distinct pair of $(q,\lambda)$ we will have possibly different eigenfunctions $\bm{s},$ and $\bm{b}$. We will suppress this dependence in the following, but it is important to keep in mind that the following analysis applies for a given pair $(q,\lambda)$. As we are looking for modes which grow in time, leading to instability, we will impose $\Re(\lambda)>0$ in the following. Substituting these expansions into \eqref{eqwS}-\eqref{eqwB} we find that a given mode satisfies,
\begin{equation}\label{eqwS1}
\lambda \bm{s} = \bm{D_S}(-k_q^2+\partial_y^2) \bm{s}+\bm{J_{S}}\bm{s}, \quad y \in [h,h+\varepsilon],
\end{equation}
\begin{equation}\label{eqwB1}
\lambda \bm{b} = \bm{D_B}(-k_q^2+\partial_y^2) \bm{b}, \quad y \in [0,h].
\end{equation}
After multiplying these equations by the inverse of the diffusion matrices and rearranging, we find
\begin{equation}\label{eqwS2}
\bm{s}'' = (k_q^2\bm{I_n}+\bm{D_S}^{-1}(\lambda \bm{I_n}-\bm{J_{S}}))\bm{s}, \quad y \in [h,h+\varepsilon],
\end{equation}
\begin{equation}\label{eqwB2}
\bm{b}'' = (k_q^2\bm{I_n}+\lambda \bm{D_B}^{-1})\bm{b}, \quad y \in [0,h],
\end{equation}
where $'$ denotes the ordinary derivative with respect to $y$. These spatial functions are required to satisfy the external boundary conditions $\bm{b}'(0)=\bm{s}'(h+\varepsilon)=\bm{0}$ and the coupling conditions which read,
\begin{equation}\label{couplingw2}
\bm{D_S}\bm{s'} = \eta(\bm{s_\textbf{q}}-\bm{b_\textbf{q}}), \quad \bm{D_B}\bm{b_\textbf{q}'} = \eta(\bm{s_\textbf{q}}-\bm{b_\textbf{q}}), \text{ for } y=h.
\end{equation}
To find suitable $(\lambda,q)$ that solve the coupled problem \eqref{eqwS2}-\eqref{couplingw2}, we will make use of the matrix-valued function defined by $\cosh(\bm{M}) = (\exp(\bm{M})+\exp(\bm{-M}))/2$, for some matrix $\bm{M}$, as well as $\sinh(\bm{M}) = (\exp(\bm{M})-\exp(\bm{-M}))/2$. We recall the differentiation identity $\cosh(y\bm{M})' = \bm{M}\sinh(y\bm{M})$, which follows from this definition. We now seek to take the square-root of the matrices on the right-hand side of equations \eqref{eqwS2} and \eqref{eqwB2} and thus define
\begin{equation}\bm{M_{S}}^2 = (k_q^2\bm{I_n}+\bm{D_S}^{-1}(\lambda\bm{I_n}-\bm{J_{S}})), \quad \text{and} \quad
\bm{M_{B}}^2= (k_q^2\bm{I_n}+\lambda\bm{D_B}^{-1}).
\end{equation}
We next consider solutions to equations \eqref{eqwS2} and \eqref{eqwB2} via hyperbolic matrix functions. As we will observe (e.g.~equation~\eqref{detcond} and the resulting dispersion relation), these matrices will always be in terms of functions that can be expressed in terms of even powers of $\bm{M_{B}}$ and $\bm{M_{S}}$, and thus functions of $\bm{M_{B}}^2$ and $\bm{M_{S}}^2$. This dependence on the squares of these matrices follows as if $f:\mathbb{C}\to\mathbb{C}$ is analytic, then a matrix-valued function can be defined via a power series in the matrix argument \cite{higham2008functions}.
The hyperbolic functions we will use are meromorphic with poles away from $0$, and hence the ambiguity in defining the square root matrices, $\bm{M_{S}}$ and $\bm{M_{B}}$ does not play a role. Without loss of generality, we will consider the principal square roots of the matrices for definiteness, so that eigenvalues of $\bm{M_{B}}$ and $\bm{M_{S}}$ are the square roots with positive (or possibly zero) real parts of the eigenvalues of $\bm{M_{B}}^2$ and $\bm{M_{S}}^2$.
Proceeding, we then have the following solutions to equations \eqref{eqwS2} and \eqref{eqwB2} given by the hyperbolic matrix functions:
\begin{equation}\label{eigenfunctions}
\bm{s} = \cosh((y-h-\varepsilon)\bm{M_{S}})\bm{\alpha}, \quad \bm{{b}} = \cosh(y\bm{M_{B}})\bm{\beta},
\end{equation}
for some nonzero constant vectors $\bm{\alpha}, \bm{\beta}$. We note these functions satisfy the no-flux conditions at the top and bottom boundaries by construction. We now use the coupling conditions \eqref{couplingw} to determine a condition for nontrivial $\bm{\alpha}$ and $\bm{\beta}$. These read,
\begin{equation}\label{couplingwS}
-\bm{D_S}\bm{M_{S}}\sinh(\varepsilon\bm{M_{S}})\bm{\alpha} = \eta(\cosh(\varepsilon\bm{M_{S}})\bm{\alpha}-\cosh(h\bm{M_{B}})\bm{\beta}),
\end{equation}
\begin{equation}\label{couplingwB}
\bm{D_B}\bm{M_{B}}\sinh(h\bm{M_{B}})\bm{\beta} = \eta(\cosh(\varepsilon\bm{M_{S}})\bm{\alpha}-\cosh(h\bm{M_{B}})\bm{\beta}).
\end{equation}
We then have, writing equations \eqref{couplingwS} and \eqref{couplingwB} as a $2n\times 2n$ block matrix, the following condition for nontrivial solutions to this system:
\begin{equation}\label{detcond}
\det{\begin{pmatrix} \eta\cosh(\varepsilon\bm{\bm{M_{S}}})+\bm{D_S}\bm{M_{S}}\sinh(\varepsilon\bm{M_{S}}) & -\eta \cosh(h\bm{M_{B}}) \\
\eta \cosh(\varepsilon \bm{M_{S}}) & -\eta\cosh(h\bm{M_{B}})-\bm{D_B}\bm{M_{B}}\sinh(h\bm{M_{B}})
\end{pmatrix}} = 0.
\end{equation}
As this condition involves transcendental functions of $\lambda$, we note that in general for a fixed spatial mode $q$, there will be infinitely many values of $\lambda$ for which equation~\eqref{detcond} is satisfied. Equivalently, $q$ only differentiates between eigenmodes in the $x$ direction, but cannot do so in $y$, and so these eigenmodes must be captured via multiplicity in $\lambda$.
While the condition given by equation~\eqref{detcond} is in principle computable, it is difficult to use to gain insight into Turing-like instabilities. Even simplifying the determinant condition is nontrivial, as the four blocks will not in general commute, so we now exploit the assumption of no reactions in the bulk to simplify this condition. We have that $\bm{M_{B}}^2$ is diagonal, and from our assumption that $\Re(\lambda)> 0$, we have that its eigenvalues have positive real part. Therefore, the elements of $\cosh(\bm{M_{B}})$ are given by the hyperbolic cosine of the diagonal elements of $\bm{M_{B}}$, and since these are all positive definite, $\cosh(h\bm{M_{B}})$ is invertible.
Now we define the matrices $\bm{A} = \eta\cosh(\varepsilon\bm{M_{S}})+\bm{D_S}\bm{M_{S}}\sinh(\varepsilon\bm{M_{S}}) $, $\bm{B} = -\eta \cosh(h\bm{M_{B}})$, $\bm{C} = \eta \cosh(\varepsilon \bm{M_{S}})$, $\bm{D} = -\eta\cosh(h\bm{M_{B}})-\bm{D_B}\bm{M_{B}}\sinh(h\bm{M_{B}})$. By the above argument, we have that $\bm{B}$ is invertible. We then have that \eqref{detcond} can be written (by exchanging rows and using the Schur complement) as,
\begin{equation}\label{detcondsimp}
\det{\begin{pmatrix} \bm{A} & \bm{B} \\
\bm{C} & \bm{D}
\end{pmatrix}} = (-1)^n\det(\bm{B})\det(\bm{C}-\bm{D}\bm{B}^{-1}\bm{A}) = 0.
\end{equation}
Noting that $\bm{D}\bm{B}^{-1} = \bm{I_n}+\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})/\eta$, we have that
\begin{align}
\bm{C}-\bm{D}\bm{B}^{-1}\bm{A} =& \eta \cosh(\varepsilon \bm{M_{S}})\nonumber - \left( \bm{I_n}+\frac{1}{\eta}\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})\right)(\eta\cosh(\varepsilon\bm{M_{S}})+\bm{D_S}\bm{M_{S}}\sinh(\varepsilon\bm{M_{S}}))\nonumber\\
=& -\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})\left(\cosh(\varepsilon \bm{M_{S}})+\frac{1}{\eta}\bm{D_S}\bm{M_{S}}\sinh(\varepsilon \bm{M_{S}})\right)\nonumber -\bm{D_S}\bm{M_{S}}\sinh(\varepsilon \bm{M_{S}}),\nonumber
\end{align}
so equation~\eqref{detcondsimp} is equivalent to,
\begin{equation}\label{detconfFull}
\det\left(\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})\left(\cosh(\varepsilon \bm{M_{S}})+\frac{1}{\eta}\bm{D_S}\bm{M_{S}}\sinh(\varepsilon \bm{M_{S}})\right)+\bm{D_S}\bm{M_{S}}\sinh(\varepsilon \bm{M_{S}})\right)=0.
\end{equation}
We note that the Turing instability conditions for the surface system in isolation -- neglecting spatial structure in $y$ -- are precisely that the growth rates $\lambda$ computed from $\det(\bm{M_{S}})=0$ have negative real part for $k_0=0$, and positive real part for some $k_q > 0$, and so this matrix encodes directly the classical case in this way. Furthermore, for a fixed $q$, and with fixed model parameters, we expect that condition \eqref{detconfFull} admits infinitely many distinct values of $\lambda$. The intuition for this is that in the uncoupled case ($\eta=0$), the surface domain is a rectangle and, hence, the surface eigenfunctions $\bm{s}(y)$ are also cosines of different spatial eigenvalues, which can vary independently from $k_q$. However, we know of no method to compute analytical expressions for such spatial eigenvalues in the coupled case, and so instead use condition \eqref{detconfFull} to compute $\lambda$ directly, remaining aware of the inherent multiplicity. To further understand the dispersion relation given by \eqref{detconfFull}, and how it relates to classical conditions for Turing instabilities, we now pursue several asymptotic reductions.
\section{Instability Conditions in Thin Surface Regimes}\label{Asymptotics}
In this section we compute instability conditions from equation~\eqref{detconfFull} for a variety of distinguished limits modelling a thin surface region, as motivated by synthetic patterning in bacterial populations. First, we mention even simpler reductions of the system, as a consistency check of our dispersion relation. We show that patterning is equivalent in the limit of decoupling the interaction of the surface and bulk regions, that is for sufficiently small $\eta \ll 1$. This is pursued in Appendix~\ref{App},
where the classical Turing conditions are recovered as the surface system becomes isolated, as required. In addition, in Appendix~\ref{App}, we also demonstrate that no patterning can occur for classical Turing kinetics once all diffusion coefficients are equal in each of the regions, a direct analogue of the well known result that the classical Turing instability requires differential transport.
Noting that the full system is too rich to investigate in generality and that the non-dimensional surface depth parameters, $\varepsilon$ and $\varepsilon_*$ are small in Table~\ref{tab1} for the motivating example of synthetic pattern formation in {\it E. coli} colonies, we proceed below to studying pattern formation instabilities with thin surface asymptotics. In the experimental setting of Grant et al.~\cite{grant2016orthogonal}, the bacterial layer is always relatively thin, owing to transport constraints in the bacteria, though the agar layer can take different bulk heights. For this reason, after first introducing a thin surface limit of the dispersion relation \eqref{detcond} in Section~\ref{surfaceasy}, we consider subsequent limits of large or small bulk thickness, $h$, in Section~\ref{bulkasy}. We anticipate that the permeability of the interface, $\eta$, is large in these experiments but do not have quantitative estimates, and so also consider our asymptotics across varying values of this parameter. In Section~\ref{tsar}, we derive asymptotic results under a regular asymptotic assumption on $k$ and $\lambda$ (i.e.~that they remain comparable with non-asymptotic terms in the dispersion relation), and collect these results in Table~\ref{tablelims}. Finally in Section~\ref{ftsar}, we give an example of distinguished limits where this asymptotic assumption breaks down.
Throughout the following, we implicitly assume that the surface Jacobian, ${\bm J_S}$, has elements that are of the same order and thus of the order of $\|{\bm J_S}\|_\infty$, so that $\|{\bm J_S}{\bm A}\|_\infty$ is of the same scale as $\|{\bm J_S}\|_\infty \|{\bm A}\|_\infty$ for any matrix ${\bm A}$ considered.
\subsection{Thin Surface Limits $\left(\varepsilon_*^2/3 \ll 1\right)$}\label{surfaceasy}
Here, we consider an asymptotically thin-surface, requiring $\varepsilon\|\bm{M_{S}}\|_\infty=H_{\varepsilon}\|\bm{M_{S}}\|_\infty/\tilde{L}\ll 1$. First note that in the thin layer limit below, the surface Jacobian ${\bm J_S}$ only appears via
\begin{eqnarray}\label{dsms} \bm{D_S}\bm{M_{S}}^2 = k_q^2\bm{D_S}+ \lambda\bm{I_n}-\bm{J_{S}}.
\end{eqnarray}
In addition, given patterning (i.e.~$\Re(\lambda)>0$), the matrix $ \bm{J_S}$ cannot be dominated by the terms $ \lambda \bm{I_n}$
or $k_q^2\bm{D_S}$ within $ \bm{D_S}\bm{M_{S}}^2, $ since then the reaction kinetics are subleading in the requirements for patterning, which thus contain only terms associated with pure diffusion at leading order. However, pure diffusion cannot induce patterning, as demonstrated in Appendix \ref{AppB}. Thus we conclude that, given patterning
\begin{eqnarray}\label{cns} \mbox{max}(|\lambda|, k_q^2\|\bm{D_S} \| _\infty) \sim O(\| \bm{J_{S}}\| _\infty ),
\end{eqnarray}
and also that
$||\bm{M_S}||_\infty$ has an upper bound ( and in particular the $k_q^2$ term is in fact bounded).
Noting the boundedness of $\bm{M_S}$ we have that $\cosh(\varepsilon\bm{M_S})$ is invertible, as for $\varepsilon$ sufficiently small this matrix has a determinant which is asymptotically $1+\varepsilon^2 \textrm{trace}(\bm{M_S}^2)/2>0$. In addition, for sufficiently small $\varepsilon$, we have the the Taylor expansion
\begin{eqnarray}\label{tanhexp}\tanh(\varepsilon {\bm M_S}) = \varepsilon {\bm M_S}\left(1+O\left( \varepsilon^2 \|{\bm M_S}\|_\infty^2/3\right)\right),
\end{eqnarray}
where $O( \varepsilon^2 \|{\bm M_S}\|_\infty^2/3 )$ means the same scale as $\varepsilon^2 \|{\bm M_S}\|_\infty^2/3 $, {\it or smaller}, as the surface thickness tends to zero. Thus by right muyltiplying \eqref{detcond} by $\cosh(\varepsilon\bm{M_S})^{-1}$ and Taylor expanding we obtain (to leading order) the relation
\begin{equation} \label{smalleps}
\det\left(\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})+\varepsilon\left(\frac{1}{\eta}\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})+\bm{I_n}\right)\bm{D_S}\bm{M_{S}}^2\right)=0,
\end{equation}
providing $\varepsilon^2\|\bm{M_{S}}\|^2_\infty/3\ll 1$ (ensuring the invertibility of $\cosh(\varepsilon \bm{M_S})$ and the validity of the Taylor expansion above).
Furthermore, noting that
$\varepsilon_* = \varepsilon \| \bm{D_{S}}^{-1}\bm{J}_S\|_\infty^{1/2}$ together with the relations (\ref{cns}),
which give the maximum scale of $k_q^2$ and show that
$|\lambda| \| \bm{D_{S}}^{-1}\| \sim O(\| \bm{D_{S}}^{-1}\|_\infty \|\bm{J}_S\|_\infty )\sim O(\| \bm{D_{S}}^{-1}\bm{J}_S\|_\infty ),$
we have
\begin{eqnarray}
\varepsilon^2\|\bm{M_{S}}\|^2_\infty \sim \mbox{max}\left(\varepsilon^2 k_q^2, \varepsilon^2\| \bm{D_{S}}^{-1}\bm{J_S} \| _\infty \right)
\sim
\mbox{max}\left(\varepsilon^2 \frac{\| {\bm J_{S}} \| _\infty }{\| \bm{D_{S}} \| _\infty }, \varepsilon^2 \| {\bm D_{S}}^{-1} \| _\infty \| {\bm J_{S}} \| _\infty \right)\sim\varepsilon^2 \| \bm{D_{S}}^{-1}\bm{J}_S\|_\infty =\varepsilon^2_* , \label{reliq1}
\end{eqnarray}
\bk using $ \| {\bm D_{S}}^{-1} \| _\infty\geq 1/ \| {\bm D_{S}} \| _\infty.$
The latter inequality is immediate in the two species case on writing ${\bm{D_S}} = \mbox{diag}(a,a\xi)$ with $\xi \leq 1,$ as then $\| {\bm D_{S}}^{-1} \| _\infty = 1/(a\xi) \geq 1/a =1/\| {\bm D_{S}} \| _\infty,$ with a trivial generalisation to higher number of species. Hence, for conditions associated with patterning, the relative error in the leading order thin surface approximation arising from equation (\ref{tanhexp}) is $\varepsilon_*^2/3 $ and thus we require $\varepsilon_*^2/3 \ll 1.$
Despite the very large range of potential parameters in Table~\ref{tab1}, the scales for synthetic patterning in bacterial colonies are consistent with this bound.
\subsection{Consideration of bulk depth $h$}\label{bulkasy}
Noting $\bm{D_S} \approx \bm{D_B}$ at least for the parameter estimates of Tables \ref{tab0}, \ref{tab1}, and also relations (\ref{cns}), (\ref{reliq1}) we also have
\begin{eqnarray}\label{cns1} \|\bm{M_{B}}^2\| _\infty = \| k_q^2 \bm{I_n}+\lambda\bm{D_B}^{-1}\| _\infty \sim O( \|\bm{D_S}^{-1} \bm{J_{S}}\|_\infty), ~~~~~~~~~~ \|\bm{D_{B}}\bm{M_{B}}^2\| _\infty = \| k_q^2 \bm{D_B}+\lambda\bm{I_n}\| _\infty \sim O( \| \bm{J_{S}}\|_\infty) .
\end{eqnarray}
Hence an appropriate scale for the largest component of $h\bm{M_B}$ is
$h_* = h \| \bm{D_S}^{-1}\bm{J_S} \|^{1/2}_\infty$, which ranges from small to large in Table~\ref{tab1} and thus we proceed to consider simplifications of the expression $ \bm{M_B}\tanh(h \bm{M_{B}})$ within the instability condition (\ref{detconfFull}) for small and large values of $h_*$.
For the small $h_*$ limit a Taylor series expansion immediately gives
$\bm{M_{B}}\tanh(h \bm{M_{B}})\sim h \bm{M_{B}}^2$, with relative corrections on the scale of $h_*^2/3$ and we also have
\begin{eqnarray}
\label{hs1}
\|\bm{M_{B}}\tanh(h \bm{M_{B}})\|_\infty\sim h \|\bm{M_{B}}^2\|_\infty \sim h_* \|\bm{M_{B}} \|_\infty ~~~~ \mbox{for}~~ h_*\ll1 .
\end{eqnarray}
For large $h_*$ simplifications, first note that $\bm{M_B}^2$ is diagonal, with diagonal components that have positive real parts since $\Re(\lambda)>0$ as we require instability.
Furthermore, similar to the synthetic patterning explored in experimental studies \cite{grant2016orthogonal,boehm2018}, we are interested in lateral patterning (in the $x$-direction of Fig. \ref{Schematic}), thus, we take $k_q^2>0$ and enforce $k_q^2\geq \pi/\tilde{L}$ by wavemode selection, which bounds the real part of $\bm{M_B}^2$ away from zero.
For $z\in\mathbb{C}$ with $\Re(z)\neq 0$, we have the limit
$$ z\tanh(z) \rightarrow \mbox{Sign}(\Re(z))z \mbox{ ~~~as~~~ } |z|\rightarrow \infty ,$$
as may be deduced by writing $z$ in terms of its real and imaginary parts, with subsequent use of the properties of trigonometric and hyperbolic functions.
In addition we have, without loss of generality, defined $\bm{M_B}$ by the diagonal matrix with {\it positive} semi-definite real part for the square root of the diagonals of $\bm{M_B}^2$,
and in fact no such square root has zero real part since the diagonals of $\bm{M_B}^2$ have positive real part. Consequently, at leading order we have in the large $h_*$ limit that
$h\bm{M_B}\tanh(h \bm{M_{B}}) \rightarrow h\bm{M_B}$ and thus $\bm{M_B}\tanh(h \bm{M_{B}}) \rightarrow \bm{M_B}$, with
\begin{eqnarray}\label{hs2}
\|\bm{M_B}\tanh(h \bm{M_{B}})\|_\infty \sim \|\bm{M_B}\|_\infty ~~~~ \mbox{for}~~ h_*\gg 1.
\end{eqnarray}
Finally, with this definition of $\bm{M_B}$, which is diagonal with terms whose real parts are bound away from zero, we also have that the diagonal elements, and hence the matrix norm, do not blow up on taking the hyperbolic tangent (all of its singularities lie on the imaginary axis) and thus $ \| \tanh(h \bm{M_{B}})\|_\infty \sim O(1) $ for $h_* \sim O(1)$. This may be summarised together with equations \eqref{hs1} and \eqref{hs2} via
\begin{eqnarray}\label{hs3}
\|\bm{M_B}\tanh(h \bm{M_{B}})\|_\infty \sim \mbox{min}(h_*\|\bm{M_B}\|_\infty , \|\bm{M_B}\|_\infty) = \mbox{min}(h_*,1) \|\bm{M_B}\|_\infty.
\end{eqnarray}
We are now in a position to consider the small $\varepsilon_*$, thin surface, limit of the instability condition given by equation (\ref{detconfFull}), considering the full range of values of $h_*$, which is a measure of the non-dimensional depth of the bulk relative to the patterning lengthscale.
We also consider the case $h_* \sim O(\varepsilon_*)$ for relative completeness, even though Tables \ref{tab0}, \ref{tab1} highlight that $h_*\gg \varepsilon_*$ is anticipated for experiments with synthetic pattern formation within bacterial populations.
\subsection{Thin surface asymptotic regimes with $k_q^2 \|\bm{D_S}\| _\infty,~|\lambda| \sim $~ord$(\|\bm{J_S}\| _\infty)$}\label{tsar}
An example of patterning when $k_q^2 \|\bm{D_S}\| _\infty,~|\lambda| \ll \mbox{ord}(\|\bm{J_S}\| _\infty) $ is given in the next subsection, but here we consider thin surface asymptotics with $\varepsilon_*^2/3 \ll 1$ on fixing $k_q^2 \|\bm{D_S}\| _\infty,~|\lambda| \sim $~ord$(\|\bm{J_S}\| _\infty),$
where ord$(\|\bm{J_S}\| _\infty)$ is defined to mean both $O(\|\bm{J_S}\| _\infty)$ and not $o(\|\bm{J_S}\| _\infty).$ Hence we are considering pattern formation that occurs on the timescales of the kinetics with a lengthscale associated with the timescale of the kinetics and the (largest) diffusion scale and, as previously noted, this simplifies the instability condition \eqref{detconfFull} at leading order to
\begin{equation}\label{smalleps1}
\det\left(\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})+\varepsilon\left(\frac{1}{\eta}\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})+\bm{I_n}\right)\bm{D_S}\bm{M_{S}}^2\right)=0,
\end{equation}
where the scale of the non-dimensional permeability $\eta$ is unknown, and the possible values of $h_*=h \|\bm{D_S}^{-1}\bm{J_S}\|^{{ 1/2}}_\infty\sim h \|\bm{M_{B}}\|_\infty \ $
are wide-ranging. Hence, there are several nontrivial distinguished limits, which we proceed to document. Where possible, we will also relate these limits to the isolated surface case, where $\lambda$ is determined by the dispersion relation $\det(\bm{M_S^2})=\det(\lambda\bm{I_n}+k_q^2\bm{D_S}-\bm{J_{S}})=0$, in order to understand the impact of the bulk on the classical single-domain situation.
\vspace{6pt}\noindent{\bf Case I $h_* \ll \varepsilon_* \ll 1$:} Noting that $h_* \ll \varepsilon_*$ is equivalent to $h \ll \varepsilon$ by definition, in this limit equation \eqref{smalleps1} reduces to,
\begin{equation}\label{smallepshleps}
\det\left(\left(\frac{1}{\eta}\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})+\bm{I_n}\right)\bm{D_S}\bm{M_{S}}^2\right)=\det\left(\frac{1}{\eta}\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})+\bm{I_n}\right)\det\left(\bm{D_S} \right)\det\left( \bm{M_{S}}^2\right)=0.
\end{equation}
However the determinant with the hyperbolic tangent term cannot generate a root with $\Re(\lambda)>0$ and thus patterning. In particular,
in Appendix \ref{AppA2}, following equation (\ref{ztz}), it is shown that when $\Re (z^2)>0$ one also has $\Re (z\tanh(z))>0.$ With $\Re(\lambda)>0$ for patterning, let $z^2=h^2(k_q^2+\lambda /d_B)$, where $d_B$ is a bulk diffusion coefficient. Thus $z^2$ is an eigenvalue of $h^2\bm{M_{B}}^2$, and all eigenvalues of this matrix are of this form. Furthermore, we have $\Re(z^2)>0$ where $z$ is an eigenvalue of $h\bm{M_{B}}$ and satisfies $\Re (z\tanh(z))>0.$ However for the hyperbolic tangent term in equation (\ref{smallepshleps}) to generate a root, at least one eigenvalue of $h\bm{M_{B}}$, that is one such $z$, must satisfy $\Re (z\tanh(z))<0, $ a contradiction, thus showing there are no roots from the determinant involving the hyperbolic tangent. Hence, noting $ \bm{D_S} $ is positive definite,
the only roots are those of the isolated Turing modes, independent of $\eta$, and determined purely from $\det\left(\bm{M_{S}}^2\right)=0$.
\vspace{6pt}\noindent{\bf Case II $\varepsilon_* \ll 1$, $h_*/ \varepsilon_*=h/\varepsilon = \hat{h} \sim \mathrm{ord}(1)$:} This limit corresponds to the entire domain being thin with respect to the lengthscale in the $x$ direction ($\tilde{L}$). In this case we have,
\begin{equation}\label{smallepsh}
\det\left(\hat{h}\bm{D_B}\bm{M_{B}}^2+\left(\frac{\varepsilon\hat{h}}{\eta}\bm{D_B}\bm{M_{B}}^2+\bm{I_n}\right)\bm{D_S}\bm{M_{S}}^2\right)=0.
\end{equation}
Equation \eqref{smallepsh} is a slight modification of the isolated surface Turing conditions in 1-D given by $\det(\bm{M_{S}}^2)=0$, and can similarly be written as an $n$th order polynomial in $\lambda$. Further, if
$\eta \ll \varepsilon \| \bm{D_B} \bm{M_{B}}^2\|_\infty \sim \mbox{ord}(\varepsilon \| \bm{J_S} \|_\infty)$, then the conditions for instability are precisely those for an isolated surface. Similarly, if $\eta = \mbox{ord}(\varepsilon\| \bm{D_B} \bm{M_{B}}^2\|_\infty)\sim \mbox{ord}(\varepsilon\| \bm{J_S} \|_\infty)$, then we are left with a `quadratic' dispersion relation, which does not simplify from the form given in \eqref{smallepsh} (`quadratic' meaning this dispersion relation will give a polynomial of order $2n$ for $\lambda$, compared to the standard $n$th order polynomial). In general such a relation could lead to quite different values of $\lambda$ from the isolated case, though we will not analyse it further here.
If $\eta \gg \varepsilon\| \bm{D_B} \bm{M_{B}}^2\|_\infty\sim \mbox{ord}(\varepsilon\| \bm{J_S} \|_\infty) $, we then have the instability condition,
\begin{equation}\label{smallepshleta}
\det\left(\lambda(1+\hat{h})\bm{I_n}+k_q^2(\hat{h}\bm{D_B}+\bm{D_S})-\bm{J_{S}}\right) = 0,
\end{equation}
which can be seen as a homogenisation, or averaging, of the bulk and surface layers. Such an averaged dispersion relation has the potential to increase the ability of the system to pattern compared to the isolated case by, e.g., introducing, or increasing, the differential diffusion between species.
In some other (experimentally relevant) cases this averaged system will decrease the ability of the system to pattern compared to the isolated case. For instance, the necessary differential diffusion for Turing patterning may be due to, for example, substrate binding {\gr \cite{korvasova}} that is only present in the surface system. In an inert bulk region, there are fewer physical scenarios where differential diffusion is likely as most biological proteins are roughly the same size. In such a case, we have that $\bm{D_B}=c_B\bm{I}$, so that \eqref{smallepshleta} can be rearranged to given,
\begin{equation}\label{smallepshletaDB=I}
\det\left(\left(\lambda(1+\hat{h})+k_q^2\hat{h}c_B\right)\bm{I_n}+k_q^2\bm{D_S}-\bm{J_{S}}\right) = 0,
\end{equation}
which we can see as a shrinking and shifting to the left a root $\lambda$ coming from the isolated case. Effectively then, such a scenario leads to a smaller instability region in parameter space, subject to the wavemode selection constraint that $k_q=q\pi/\tilde{L}$, for a natural number $q$.
Another plausibly relevant case of equation~\eqref{smallepshleta} is if $\bm{D_S}=\bm{D_B}$, i.e.~the surface and bulk diffusivities are the same. Here, the dispersion relation
is that of the classic case except $\lambda$ and $k_q^2$ are both scaled by $(1+\hat{h})$. Hence the allowed values of $(\lambda,k_q^2)$ are those of the classic case divided by $(1+\hat{h})$, which shrinks the range of the allowed patterning wavenumbers relative to the classic case and thus leads to a smaller Turing space compared to the isolated surface system, though again subject to the wavemode selection constraint.
\vspace{6pt}\noindent
{\bf Case III $\varepsilon_* \ll h_* $ : } This case proceeds similarly regardless of whether $h_* \ll 1, ~h_*\sim\mbox{ord}(1)$ or $h_*\gg 1$.
Noting
$\bm{D_S} \approx \bm{D_B}$,
$ \|\bm{D_S}\bm{M_{S}}^2\|_\infty \sim \| \bm{J_S} \|_\infty , $ from equations (\ref{dsms}) and (\ref{cns}),
$\|\bm{M_{B}} \| _\infty \sim O( \|\bm{D_S}^{-1} \bm{J_{S}}\|^{1/2}_\infty) $
by square rooting the first of relations (\ref{cns1}),
and equation (\ref{hs3}), that is $\|\bm{M_B}\tanh(h \bm{M_{B}})\|_\infty \sim \mbox{min}(h_*,1) \|\bm{M_B}\|_\infty \sim \mbox{min}(h_*,1) \|\bm{D_S}^{-1} \bm{J_{S}}\|^{1/2}_\infty $,
we have
$$\frac{\varepsilon\| \bm{D_S} \bm{M_{S}}^2\|_\infty}{ \| \bm{D_B} \bm{M_{B}}\tanh(h \bm{M_{B}})\|_\infty } \sim
\frac{\varepsilon \| \bm{J_S} \|_\infty}{\|\bm{D_S}\|_\infty\|\bm{D_S}^{-1} \bm{J_{S}}\|^{1/2}_\infty \mbox{min}(h_*,1)} \sim
\frac{\varepsilon \| \bm{D_S}^{-1} \bm{J_S} \|_\infty}{\| \bm{D_S}^{-1} \|_\infty\|\bm{D_S}\|_\infty\|\bm{D_S}^{-1} \bm{J_{S}}\|^{1/2}_\infty \mbox{min}(h_*,1)}.
$$
Noting $ \| {\bm D_{S}}^{-1} \| _\infty \| {\bm D_{S}} \| _\infty \geq 1,$ as deduced just below equation (\ref{reliq1}), and $ \varepsilon_* = \varepsilon\| \bm{D_S}^{-1} \bm{J_S} \|_\infty^{1/2} $ we thus have
$$
\frac{\varepsilon\| \bm{D_S} \bm{M_{S}}^2\|_\infty}{ \| \bm{D_B} \bm{M_{B}}\tanh(h \bm{M_{B}})\|_\infty }
\lesssim \frac{\varepsilon_*}{ \mbox{min}(h_*,1)}
\sim \mbox{max}\left(\varepsilon_*,\frac{\varepsilon_*}{h_*}\right) \ll 1.
$$
Hence the final term from relation \eqref{smalleps}, that is $\varepsilon\bm{D_S}\bm{M_{S}}^2$, may always be dropped relative to the first term, that is $ \bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})$.
This reveals that the instability condition simplifies to
\begin{equation}\label{smallepsord1h}
\det\left(\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}})\left(\frac{\varepsilon }{\eta}\bm{D_S}\bm{M_{S}}^2+\bm{I_n}\right)\right) = \det(\bm{D_B}\bm{M_{B}}\tanh(h\bm{M_{B}}))\det\left(\frac{\varepsilon }{\eta}\bm{D_S}\bm{M_{S}}^2+\bm{I_n}\right) = 0.
\end{equation}
The hyperbolic tangent does not contribute to instability, by analogous reasoning to Case I. Thus, there is no instability unless $\eta$ is concomitantly small alongside $\varepsilon\| \bm{D_S} \bm{M_{S}}^2\|_\infty\sim \varepsilon\| \bm{J_S} \|_\infty $. Writing out $\bm{M_{S}}^2$, we see that the impact of the bulk on the surface system is simply to shift the eigenvalues to the left in the complex plane by the quantity $\eta/\varepsilon $, and hence the Turing space for this system is strictly smaller than the Turing space for an isolated one-dimensional system with surface kinetics.
Collecting all of these various limits together in Table~\ref{tablelims}, we can see a pattern emerging. As $\eta$ or $h_*$ is increased, we observe a trend of moving from the isolated surface system to a reduced, or average system, and eventually, for $h_* / \varepsilon_* \sim h/\varepsilon \gg 1$ and $\eta \gg \varepsilon\| \bm{J_S} \|_\infty $, to no patterning being permitted. While it is not true in general that the Averaged case or the Quadratic case correspond to a reduced ability for a system to pattern, we anticipate that this is the case for most standard Turing systems, and hence there is a broadly monotonic decrease on the ability of a system to pattern as the bulk becomes larger or the boundary more permeable. This has concomitant implications for prospective multilayered Turing systems, such as the experimental studies involving bacterial patterning which motivate this study.
\begin{table}[ht]
\centering
\begin{tabular}{cccc}
\toprule
& Case I. & Case II. & Case III. \\
& ($h_* \ll \varepsilon_* \ll 1$) & ($\varepsilon_* \ll 1$, $\varepsilon_*/h_*=\varepsilon/h \sim \mbox{ord}(1)$) & ($\varepsilon_* \ll h_*$) \\
\midrule
$ \eta \ll \varepsilon \| \bm{J_S} \|_\infty $& Isolated & Isolated &Isolated \\
$\eta \sim \mbox{ord}\ ( \varepsilon \| \bm{J_S} \|_\infty)$ &Isolated &Quadratic condition, Eqn (\ref{smallepsh})& Reduced instability \\
$ \eta \gg \varepsilon \| \bm{J_S} \|_\infty$&Isolated &Averaged condition, Eqn (\ref{smallepshleta}) &No instabilities \\
\bottomrule
\end{tabular}
\caption{Thin-surface limits obtained in different asymptotic regimes given
$k_q^2 \|\bm{D_S}\| _\infty,~|\lambda| \sim $~ord($\| \bm{J_S} \|_\infty $). Note that moving left to right corresponds to an increasing size of $h_* $, and moving top to bottom corresponds to increasing scales of $\eta$. No instabilities: $\det(\bm{D_{B}}\bm{M_{B}}\tanh(h\bm{M_{B}}))=0$; Isolated (1-D) surface: $\det(\bm{M_S}^2)=0$; Quadratic $\lambda$: $\det(\hat{h}\bm{D_B}\bm{M_{B}}^2+(\hat{h}c\bm{D_B}\bm{M_{B}}^2+\bm{I_n})\bm{D_S}\bm{M_{S}}^2)=0$, $c = \varepsilon/\eta \sim \mbox{ord}(\| \bm{J_S} \|_\infty^{-1})$; Reduced instability: $\det(c\bm{D_S}\bm{M_{S}}^2+\bm{I_n})=0$, $c = \varepsilon/\eta \sim \mbox{ord}(\| \bm{J_S} \|_\infty^{-1})$; Averaged condition: equation~\eqref{smallepshleta}. }\label{tablelims}
\end{table}
\subsection{Further thin surface asymptotic regimes with $ |\lambda|, k_q^2 \|\bm{D_S}\| _\infty\sim \mbox{ord}\left(\varepsilon^{1/2} \|\bm{J_S}\| _\infty\right)$} \label{ftsar}
There exist nontrivial asymptotic limits which are not described by $|\lambda|, k^2_q \|\bm{D_S}\| _\infty \sim$~ord$(\|\bm{J_S}\| _\infty)$, which can lead to instabilities not captured in Table~\ref{tablelims}, as we now show.
In particular, with $|\lambda|, k_q^2\|\bm{D_S}\| _\infty \sim \mbox{ord}(\varepsilon^{1/2}\|\bm{J_S}\| _\infty)$ and $\bm{D_B}\approx \bm{D_S}$ we then have $\varepsilon \bm{D_S}\bm{M_S}^2 \sim -\varepsilon \bm{J_S} + \mbox{ord}(\varepsilon^{3/2}\|\bm{J_S}\| _\infty),$ $\|\bm{D_B}\bm{M_B}^2\|_\infty \sim \mbox{ord}( |\lambda|, k_q^2 \|\bm{D_B}\| _\infty)\sim \mbox{ord}(\varepsilon^{1/2}\|\bm{J_S}\| _\infty)$ and finally $\|h\bm{M_B}\|_\infty \sim \mbox{ord}(h\varepsilon^{1/4}\|\bm{D_S}^{-1} \bm{J_S}\| _\infty^{1/2})
\sim \mbox{ord}(h \varepsilon_*/\varepsilon^{3/4})$. Hence from equation~\eqref{smalleps}, after expanding $\tanh(z) \sim z\left(1+ \mbox{ord}\left(z^2/3\right)\right)$ for small $z$, we have at leading order
\begin{equation}\label{smallepslambdakq}
\det\left(h\bm{D_B}\bm{M_{B}}^2+\frac{\varepsilon h}{\eta}\bm{D_B}\bm{M_{B}}^2\bm{D_S}\bm{M_{S}}^2+ \varepsilon\bm{D_S}\bm{M_{S}}^2\right)=0,
\end{equation}
with relative corrections of $ h ^2\varepsilon_*^{2} /(3\varepsilon^{3/2}) $
which is required to be much less than unity.
Further, noting we have already assumed $h \sim \mbox{ord}(\varepsilon^{1/2})$, we thus additionally require $ \varepsilon_*^{2}/(3\varepsilon^{1/2}) \ll 1$ for equation (\ref{smallepslambdakq}) to hold. For the range of parameters detailed in Table~\ref{tab1}, we have $ \varepsilon_*^{2}/(3\varepsilon^{1/2}) \in[1.8\times 10^{-5},2.3]$, and thus we have equation (\ref{smallepslambdakq}) is typically valid for parameters associated with synthetic patterning in bacterial colonies, but not always.
We proceed by noting that the first and third terms of equation (\ref{smallepslambdakq}) are $\mbox{ord}(\varepsilon\|\bm{J_S}\| _\infty)$ and the second is $\mbox{ord}\left(\varepsilon^2\|\bm{J_S}\| _\infty^2/\eta\right)$.
Writing
$$
\lambda=\varepsilon^{1/2} \mu, ~~~~~~~ h=\varepsilon^{1/2}\hat{h}, ~~~~~~~ k_q^2 \bm{D_B}= \varepsilon^{1/2} K_q^2\bm{D_B} ~~~~~~~ \mbox{with} ~~~~|\mu|, K_q^2\|\bm{D_B}\|_\infty \sim~\mbox{ord}(\|\bm{J_S}\| _\infty), ~~~~ \hat{h} \sim~\mbox{ord}(1).$$
We then have $h\bm{D_B}\bm{M_B}^2 = \varepsilon \hat{h}( K_q^2\bm{D_B}+\mu\bm{I_n}),$ and can factor an $\varepsilon $ from equation~\eqref{smallepslambdakq} to obtain,
\begin{equation}\label{smallepslambdakq2}
\det\left(\hat{h}(K_q^2\bm{D_B}+\mu\bm{I_n})-\frac{\varepsilon \hat{h}}{\eta}\left (K_q^2\bm{D_B}+\mu\bm{I_n}\right )\bm{J_{S}}- \bm{J_S}\right)=0,
\end{equation}
which, in general, can admit {\it nontrivial instabilities due to the coupling of the surface and the bulk.} In particular, when $\eta \gg \varepsilon \|\bm{J_S}\| _\infty$, so that the second term is no longer retained in the leading order, we find that the growth rates $\mu$ are given as the eigenvalues of $\bm{J_S}/\hat{h}-K_q^2\bm{D_B}$.
This matrix resembles the classical isolated-surface case except with a scaling of the kinetics by $\hat{h}$ and the appearance of the bulk diffusion parameters, rather than those in the surface. Hence, we can use usual methods (e.g.~the Routh-Hurwitz criterion) to determine parameters that lead to instability in this case, noting that any values of $\lambda$ associated with instability will be of modulus $\mbox{ord}(\varepsilon^{1/2})$, and hence will be associated with slow growing modes. Additionally, we anticipate that such modes will also exhibit small amplitude patterns, as is typical due to center-manifold reduction near Turing-type bifurcations \cite{cross1993pattern}, and hence may not be visible against experimental noise.
While other distinguished limits may exist which do not fall into the classifications given in Table~\ref{tablelims}, for brevity we do not pursue a systematic classification of these here. In the next section we will show that almost all numerically computed dispersion relations given by condition \eqref{detconfFull} fall within the asymptotics given in Table~\ref{tablelims},
with the exception of the case given in equation~\eqref{smallepslambdakq2} which was found numerically first, and subsequently motivated the above scaling.
\section{Numerical Exploration of Example Systems}\label{Numerics}
As an example of these dynamics we consider the Schnakenberg kinetics for surface reactants $\bm{u_S} = (u_S, v_S)$ given by $$ \bm{f_S}(u_S, v_S) = \left(a-u_S+u_S^2v_S,b-u_S^2v_S\right)$$ with $a \geq 0$, $b > 0$. The spatially homogeneous steady state is given by $\bm{u_S^*} = \bm{u_B^*} = \left(a+b, b/(a+b)^2\right)$. Unless otherwise stated, we will assume equal diffusion coefficients between the surface and the bulk given by the diagonal matrices $\bm{D_S}=\bm{D_B}=\text{diag}(d_u, d_v)$. Without bulk reactions, and given linear interfacial conditions as summarised by equation (\ref{coupling}) with the relations (\ref{geqn}) and (\ref{sceqn}), we can immediately apply condition \eqref{detconfFull} to determine whether, or not, we expect a solution to pattern, and then compare these predictions with numerical simulations of the full nonlinear system.
Numerically computing $\lambda$ from condition \eqref{detconfFull} is substantially more involved than typical Turing-type analyses (e.g.~for polynomial dispersion relations \cite{Murray2003}) due to the transcendental nature of this determinant condition. In particular, we expect that for any given wavemode in the $x$ direction given by $k_q=q\pi/\tilde{L}$, for a natural number $q$, we have infinitely many distinct values of $\lambda$. These essentially correspond to the wavemodes in the $y$ direction which we have found only implicitly in our construction of the dispersion relation. So to determine if, for a given set of parameters, condition \eqref{detconfFull} admits a value of $\lambda$ with $\Re(\lambda)>0$ we resort to numerical heuristics. While fast general-purpose methods exist for rootfinding of polynomials over the complex numbers \cite{verschelde1999algorithm}, we are unaware of similar methods for more complicated functions. In lieu of this, we developed a set of numerical heuristics to accurately determine whether or not a value of $\lambda$ with $\Re(\lambda)>0$ exists, and tested this against full numerical simulations. We make use of the Matlab function \texttt{PatternSearch} as well as a deflation algorithm based on Muller's method to find many candidate roots with positive real part \cite{muller1956method, conte2017elementary}, and then discard any which are spurious. Throughout this section, we denote the largest such root by $\max(\Re(\lambda))$, noting that even in the classical case this maximum is needed as there are generically $n$ distinct values of $\lambda$.
\begin{figure} \centering
\begin{tabular}{cc}
\subfloat[Case I: $h=10^{-3}, \varepsilon=10^{-2}$] {\includegraphics[width=0.45\textwidth]{figures/Fig2ah0p001eps0p01.eps}} &
\subfloat[Case II: $h=10^{-2}, \varepsilon=10^{-2}$] {\includegraphics[width=0.45\textwidth]{figures/Fig2bh0p01eps0p01.eps}} \\
\subfloat[Case III: $h=3\times 10^{-2}, \varepsilon=10^{-3}$]{\includegraphics[width=0.45\textwidth]{figures/Fig2ch0p03eps0p001.eps}} &
\subfloat[Case III: $h=3\times 10^{-2}, \varepsilon=10^{-3}$, $\bm{D_B}=\bm{I_2}$]{\includegraphics[width=0.45\textwidth]{figures/Fig2dh0p03eps0p001DB1.eps}}
\end{tabular}
\includegraphics[width=0.9\textwidth]{figures/Legend}
\caption{ Dispersion relations in the $x$ coordinate computed via \eqref{detconfFull} for a continuous variable $k_q$, using the parameters $a=0.1$, $b=2$, with surface diffusion parameters $d_u=10^{-3}$, and $d_v=10^{-1}$. In (a) -- (c) we take $\bm{D_B}=\bm{D_S}$, though in (d) we set $\bm{D_B}=d_u\bm{I_2}$, corresponding to equal bulk diffusion between species. The solid lines correspond to $\max(\Re(\lambda))$ for different values of $\eta$ for the bulk-surface condition, whereas the dashed line corresponds to the single-domain classical case. For (c) we anticipate there is an instability for relatively low $k_q/\pi$ and large $\eta$ due to surface-bulk interaction instabilities, as exemplified in Section~\ref{ftsar} for $h \sim \mbox{ord}(\varepsilon^{1/2})$ and $ \varepsilon_*^{2}/(3\varepsilon^{1/2} ) \ll 1$.}
\label{dispfig2}
\end{figure}
We first consider numerical constructions of dispersion relations for the small-asymptotic limits described in the previous section. Here we consider $\Re(\lambda)$ as a continuous function of the spectral parameter in the $x$ direction, $k_q$, as is commonly done \cite{Murray2003}. For $\varepsilon, \varepsilon_* \ll 1$, we have that the isolated reaction-diffusion system can admit growth rates $\lambda$ comparable to a classical one dimensional reaction-diffusion system, which we will denote by $\lambda_C$ (which can be computed in the standard way \cite{Murray2003}). We can then consider the maximum value of $\Re(\lambda)$ (across all values of $\lambda$ found from condition \eqref{detconfFull}), and compare this to the isolated case. We have confirmed these dispersion relations against full numerical simulations by simulating on a domain of lateral size $\tilde{L}$ such that a particular mode $k_q = q\pi/\tilde{L}$ is admissible, and observing a patterned solution.
We plot these dispersion curves in Figure~\ref{dispfig2} for a variety of the geometric and coupling parameters. As anticipated, the coupling strength $\eta$ and geometric parameters $h, h_*$ and $\varepsilon, \varepsilon_*$ each influence the shape of these dispersion curves greatly. We now compare these curves to the predictions in Table~\ref{tablelims}. For Case I ($h \ll \varepsilon$), we see that $\max(\Re(\lambda))$ is almost unchanged to the standard case up to small corrections not captured by the asymptotics. In Case II ($h \sim \varepsilon$), we observe approximate equivalence of the dispersion curve to the isolated case for small $\eta$, and an apparent change in the dispersion relation for increasing $\eta$. The Case III behaviour ($h \gg \varepsilon$) is consistent with the asymptotics of Table~\ref{tablelims} whenever $\Re(\lambda)>0$ except for $\eta \sim \varepsilon \|{\bm J_S}\|_\infty$ and $\eta \gg \varepsilon\|{\bm J_S}\|_\infty$ at relatively small values of $k_q/\pi$. Given these constraints, this mismatch is anticipated to be due to the interaction between the surface kinetics and the bulk diffusion, as described in Section~\ref{ftsar} given the thin surface approximation
$\varepsilon_*^2/(3\epsilon^{1/2})\ll 1$ with $k_q^2||\bm{D_S}||_\infty, |\lambda|\sim$ord$(\varepsilon^{1/2}||\bm{J_S}||_\infty)$. As a consistency test of this suggested mechanism, in Figure~\ref{dispfig2}(d) we replace $\bm{D_B}$ by a scaled identity matrix so that differential diffusion in the bulk is no longer present, and we see that all of the dispersion curves, for smaller values of $k_q/\pi$ and $\eta$ sufficiently large, fall below the axis as expected. This is true for different scalar multiples of the identity, such as $\bm{D_B} = d_v\bm{I_2}$ where the dispersion curves were even more stable. We remark that considering other parameters demonstrates that this nontrivial bulk-surface interaction can lead to a non-monotonic behaviour of the dispersion relation with respect to $\eta$.
As in the classical case, we expect that for sufficiently large domains, any region where $\Re(\lambda)>0$ should admit a patterned state. We confirmed this using $\tilde{L}=100$ for each of the dispersion curves, finding that they admitted patterned solutions for long time simulations if and only if $\Re(\lambda)>0$ for some region in $k_q$-space. Similar to the classical case, the layered model is always observed to stable at $k_q=0$ though with a local maximum at this point, in contrast to the behaviour of the classical Turing instability dispersion relation.
To compare these dispersion relations against numerical simulations of the full nonlinear system, we compute a heterogeneity functional determining how far a solution is from a homogeneous state \cite{berding1987heterogeneity}. For simplicity, and because the surface layer is of
primary interest in synthetic pattern formation within
bacterial colonies, we only consider the heterogeneity of the activator in the surface. We define the heterogeneity functional as
\begin{equation}\label{hetero}
F_h (u_S) = c\int_0^1 \int_h^{h+\varepsilon} \left(\frac{\partial u_S}{\partial x}\right)^2+\left(\frac{\partial u_S}{\partial y}\right)^2 \mathrm{d}y\mathrm{d}x,
\end{equation}
where $c >0$ is simply a positive definite (dimensional) scaling parameter. Note that $F_h(u_S) \geq 0$ and for $u_S \in C^1$, $F(u_S) = 0$ if and only if $u_S$ is spatially homogeneous. While we do not anticipate this metric to be quantitatively comparable to $\max(\Re(\lambda))$, we note that near the boundary of a Turing instability, the amplitude of patterns and their growth rates in time both scale with the distance from the bifurcation point, typically as a square root of the growth rate \cite{cross1993pattern}. Hence this functional should at least qualitatively scale with the growth of $\max(\Re(\lambda))$ near the onset of instability. The value $c$ is taken so that $F_h(u_S)=\max(\Re(\lambda))$ when $\eta=0$ for scaling purposes. We note that these plots are intended to demonstrate qualitative, rather than quantitative, behaviour near the onset of instability. In particular, we anticipate quantitative disagreement between $F_h(u_S)$ and $\max(\Re(\lambda))$ when $\eta$ is large, though the functional will still indicate whether or not $\max(\Re(\lambda))$ predicts pattern formation, as well as the scaling of pattern heterogeneity as a function of $\max(\Re(\lambda))$ near the onset of instability.
To use this heterogeneity functional, the full system \eqref{eqS}-\eqref{coupling} was solved until a final time of $t=10^5$ to ensure a good representation of the steady state pattern. The initial data were taken to be $u_0 = {u^*}(1+\xi_u(x,y))$ and $v_0 = {v^*}(1+\xi_v(x,y))$ with $\xi_u$ and $\xi_v$ random fields such that at each value of $(x,y)$, they are independently and identically distributed normal random variables with zero mean and variance $10^{-4}$. The equations were simulated using the COMSOL Multiphysics\textsuperscript{\tiny\textregistered} software \cite{COMSOL} with at least $2\times 10^4$ second-order triangular finite elements. A non-uniform mesh was constructed such that the surface region $\Omega_S$ was resolved with at least $10$ distinct triangular elements in any vertical cross-section. Convergence was checked in spatial and temporal discretisations, and a relative tolerance of $10^{-5}$ was given to the adaptive timestepping algorithm.
\begin{figure} \centering
\begin{tabular}{cc}
\subfloat[$h=1, \varepsilon=10^{-3}$]{\includegraphics[width=0.4\textwidth]{figures/Fig3a.eps}} &
\subfloat[$h=1, \varepsilon=10^{-2}$]{\includegraphics[width=0.4\textwidth]{figures/Fig3b.eps}} \\
\subfloat[$h=10^{-2}, \varepsilon=10^{-2}$]{\includegraphics[width=0.4\textwidth]{figures/Fig3c.eps}} &
\subfloat[$h=10^{-1}, \varepsilon=10^{-2}$]{\includegraphics[width=0.4\textwidth]{figures/Fig3d.eps}} \\
\end{tabular}
\caption{Non-trivial dependency of Turing instabilities on geometric parameters. Plots of $\max(\Re(\lambda))$ given by equation~\eqref{detconfFull} in blue computed across 250 values of $\eta$ for different parameter combinations, and plots of $F_h(u_S)$ given by \eqref{hetero} in red asterisks for 100 values of $\eta$. The other parameters were taken as $a=0.1$, $b=2$, $d_u=10^{-3}$, $d_v=10^{-1}$, $\tilde{L}=1$. The constant $c$ in $F_h$ was fixed per set of parameters/panel to match the maxima of $\max(\Re(\lambda))$ and $F_h$ across $\eta$ to qualitatively compare these metrics. The parameter sets corresponding to $h=10^{-1}, \varepsilon=10^{-1}$ and $h=1, \varepsilon=10^{-1}$ gave qualitatively the same results as in panel (c) with $\max(\Re(\lambda))>0$ for all $\eta$. }
\label{stabfig}
\end{figure}
In Figure~\ref{stabfig} we give examples of this heterogeneity functional across the ranges of the geometric parameters $\varepsilon$, $h$, and $\eta$, alongside predictions from the instability condition \eqref{detconfFull}. As anticipated by the asymptotics, for very small $\varepsilon$ (Figure~\ref{stabfig}(a)), we see the system fails to support spatial patterns for $\eta \geq 3.9 \times 10^{-4}$. Additionally, we see a jump in the value of the heterogeneity between $\eta = 8\times 10^{-5}$ and $\eta = 10^{-4}$. We plot values of $u_B$ in Figure~\ref{h1eps1e-3} across this jump to demonstrate that this discontinuity in the value of the spatial heterogeneity $F_h(u_S)$ for these parameters is due to different nonlinear modes emerging as parameters are varied, and so it is sensible that it is not captured in the linear analysis. Other discontinuities in the plots of the heterogeneity functional in Figure~\ref{stabfig} are similarly due to different patterned states being selected, and we do not further explore pattern multistability or dependence on initial data here.
\begin{figure} \centering
\includegraphics[width=0.8\textwidth]{figures/fig4.png}
\caption{One-dimensional plots of $u_S$ corresponding to parameters in Figure~\ref{stabfig}(a) for two values of $\eta$ in the top two panels, and plots of the corresponding $u_B$ below (with $\tilde{L}=1$ in all cases). The surface concentration $u_S$ is effectively homogeneous in the $y$ direction, and so is essentially a one-dimensional pattern, shown above. Note that the bulk concentrations are almost homogeneous, whereas the surface concentrations are not (compare the scales of $u_S$ and $u_B$). }
\label{h1eps1e-3}
\end{figure}
In Figures \ref{stabfig}(b) and (d) we see a region of intermediate values of $\eta$ for which no patterning occurs, and more broadly across all of Figure~\ref{stabfig} we see that a minimal value of $\max(\Re(\lambda))$ occurs approximately for $\eta$ within the range $ (10^{-3}, 1)$. We show examples of the mode selection process from Figure~\ref{stabfig}(d) in Figure~\ref{h0p1eps1e-2}. For small $\eta=10^{-4}$, we see stable multiple-spike solutions that are essentially confined to the surface. As $\eta$ increases further to $10^{-1}$, a single-spike solution is observed, at a smaller amplitude as the dispersion relation has just crossed the instability threshold given in Figure~\ref{stabfig}(d). Further increases to large $\eta$ lead to stable spike solutions that remain essentially vertically homogeneous in the surface, but have small transverse variations in the bulk due to the change in reaction kinetics across the interface, as illustrated for $\eta=10^5$. Further increasing $\eta$ sharpens these spike solutions across the domain, but does not impact the number of modes. Besides the discontinuities in the heterogeneity due to nonlinear mode selection, there is often a good match between the linear analysis (e.g.~value of $\max(\Re(\lambda)$) and the heterogeneity, which can be expected near to the Turing bifurcation points in simpler settings due to the existence of normal forms of the pattern amplitude \cite{cross1993pattern}.
\begin{figure} \centering
\includegraphics[width=\textwidth]{figures/fig5.png}
\caption{Plots of $u_S$ and $u_B$ corresponding to parameters in Figure~\ref{stabfig}(d) for three values of $\eta$, and $\tilde{L}=1$. Here, $\varepsilon=10^{-2}$ and $h=10^{-1}$.}
\label{h0p1eps1e-2}
\end{figure}
In all of Figure~\ref{stabfig} we observe that $\max(\Re(\lambda))$ appears to asymptotically approach a fixed value for either $\eta \to 0$ (which corresponds to the static Turing conditions) or $\eta \to \infty$, with the latter always being smaller than the former, though this may just be a feature of the parameters explored here. However, in \ref{stabfig}(c) (and the other cases noted in the caption), we observe that an instability occurs for all values of $\eta$, which is confirmed by numerical simulations of the full system.
As a further example which helps visualise the impact of varying the geometric parameters and coupling constant $\eta$, we observe patterns primarily confined to the surface but with some interaction with the bulk in Figure~\ref{h1eps0p1}. Again some mode selection effects are present (two vs three spot solutions for small and largeer values of $\eta$ respectively), though due to generic aspects of multistability in two spatial dimensional systems \cite{dewel1995pattern}, we suspect these depend somewhat on initial data, rather than just parameter values. Finally in Figure~\ref{h0p1eps0p1} we give an example where no change in the number of unstable modes was apparent for variation in $\eta$, though the structure of the solution does change.
\begin{figure} \centering
\includegraphics[width=\textwidth]{figures/fig6.png}
\caption{Plots of $u_S$ and $u_B$ corresponding to parameters in Figure~\ref{stabfig} with $h=0.5$ and $\varepsilon=10^{-1}$ for three values of $\eta$, and $\tilde{L}=1$. }
\label{h1eps0p1}
\end{figure}
\begin{figure} \centering
\includegraphics[width=\textwidth]{figures/fig7.png}
\caption{Plots of $u_S$ and $u_B$ corresponding to parameters in Figure~\ref{stabfig} except that $h=10^{-1}$ and $\varepsilon=10^{-1}$ for three values of $\eta$, and $\tilde{L}=1$.}
\label{h0p1eps0p1}
\end{figure}
Within Turing-unstable regimes, the surface largely drives the structure of the modes and hence the patterns can be thought of as quasi-one-dimensional (Figures \ref{h1eps0p1}-\ref{h0p1eps0p1}). The permeability $\eta$ does control how much structure there is, both in the bulk in general and in the surface modes' variation in the $y$ direction, though in all cases the largest spatial variation is along the lateral coordinate $x$. For the largest permeability we explored ($\eta=10^5$), we see that the sizes of the surface and bulk can have a significant impact on the relative shape of the solutions in the bulk region (cf Figures \ref{h0p1eps1e-2}(c), \ref{h1eps0p1}(c), and \ref{h0p1eps0p1}(c)). In particular we see that the deepest part of the bulk ($y=0$) in Figure \ref{h0p1eps1e-2}(c) and \ref{h0p1eps0p1}(c) maintain a fairly distinct periodic pattern between high and low activator concentrations, whereas the larger bulk in Figure \ref{h1eps0p1}(c) is substantially more homogeneous at $y=0$. We also note that for intermediate values of $\eta$, Figures \ref{h1eps0p1}(b) and \ref{h0p1eps0p1}(b) have the largest visual gradients in the activator in the surface layer, consistent with the intermediate-$\eta$ values having significant impacts on the predicted values of $\max(\Re{\lambda})$ in Figure \ref{dispfig2}. This further demonstrates nontrivial impacts of the bulk geometry on the structure of emergent patterns, and such leeching into the bulk may be useful to help quantify its impact in synthetic systems.
\section{Discussion}\label{Discussion}
Motivated by recent interest in a range of biological contexts, we have developed and analysed a general class of reaction-diffusion models of pattern formation in stratified media, though with an absence of reactions in the bulk and a linear coupling between the layers. We have derived a criterion for pattern-forming instability in such media, given by equation~\eqref{detcond}. In Appendix \ref{App} we showed that the absence of differential transport within each layer entails no patterning for these systems, in direct analogy to the classical Turing instability. We have also demonstrated a range of interesting behaviours via asymptotic reductions in thin domains, and numerical simulations.
In particular, this setting of a linearly coupled system with no reactions in the bulk with a thin surface layer is also of significant biological interest, as several groups are using bacterial colonies on inert substrates as a medium for engineered pattern formation via synthetic biology \cite{grant2016orthogonal,boehm2018,karig2018stochastic}. However, as far as we are aware, there is little theoretical understanding of how the inert substrate impacts the surface reaction-diffusion systems in these kinds of geometries. Additionally, to accurately model the real complexity of these experimental systems we would need to account for intracellular (i.e.~non-diffusible) proteins which play a role in the reactions, as our reaction-diffusion framework only captures the dynamics of diffusible signalling molecules.
Nevertheless, even in the simplified setting of an inert bulk and a thin surface, the computed instability criteria are much richer than in the classical case. For instance, the nine distinguished limits for $|\lambda|, k_q^2\|\bm{D_S}\|_\infty \sim \mbox{ord}(\| \bm{J_S} \|_\infty)$ given in Table~\ref{tablelims} demonstrate a variety of behaviours not predicted by analysing the surface reaction-diffusion system alone, as is typical in applications. In addition, these distinguished limits, though emergent from a complex multi-parameter system, depend on only three non-dimensional parameter groupings, $\varepsilon_*,h_*$ and $\eta/( \varepsilon \| \bm{J_S} \|_\infty )$. The first two of these respectively are the surface and bulk depth relative to the lateral lengthscale, i.e. the basic geometry. The final grouping is $\eta/( \varepsilon \| \bm{J_S} \|_\infty )=\tau \hat{\eta}/(H_\varepsilon \| \bm{J_S} \|_\infty )$.
Noting that $\tau$ is chosen such that $\| \bm{J_S} \|_\infty\sim\mbox{ord}(1)$, one can deduce more generally that $\tau/\| \bm{J_S} \|_\infty$ is the dimensional timescale of surface reaction.
Hence the final parameter grouping is the ratio of the interface permeability to the surface velocity scale, $\| \bm{J_S} \|_\infty H_\varepsilon/\tau$, with the latter in turn given by the ratio of the surface depth and reaction timescale.
We further note that our instability condition \eqref{detconfFull}, recovers the usual features of Turing instabilities, such as requiring differential diffusion for their onset, and reducing to the polynomial dispersion relation when the bulk becomes uncoupled from the surface. The explicit coupling between bulk diffusion and surface reactions given by \eqref{smallepslambdakq2} when $|\lambda|, k_q^2\|\bm{D_S}\|_\infty \sim \mbox{ord}(\varepsilon^{1/2}\| \bm{J_S} \|_\infty)$ suggests additional distinguished limits from those in Table~\ref{tablelims}; the associated instabilities possess slower growth rates, but nonetheless highlight substantial and non-trivial impacts of the bulk on the system's ability to pattern. We anticipate that there are other examples of nontrivial surface-bulk coupling driven instabilities, as suggested in the discussion of the Averaged and Quadratic cases in Table~\ref{tablelims}, but leave investigation of these to further work.
Broadly, our asymptotic and numerical results on thin surfaces suggest that the presence of the inert bulk generally decreases the ability of the surface system to undergo a Turing instability compared to an isolated system. The exceptional cases, such as the homogenised limit \eqref{smallepshleta} and the explicit coupling in equation~\eqref{smallepslambdakq2}, can in principle lead to larger Turing spaces, though we have shown in some realistic cases such as equal bulk diffusions ($\bm{D_B}=\bm{I_n}$) that these do not enlarge the Turing space. Note that in systems where diffusion varies significantly between domains (e.g. non-diffusible proteins in the surface) the parameter space that admits pattern formation can increase with increasing bulk size (see, for instance \cite{halatek2018box, brauns2020bulk}). Exploring such interplays will be the focus of future work.
Our results suggest that experiments should aim to design large and robust parameter regimes using classical criteria for pattern-formation (e.g.~using design approaches such as in \cite{dalchau2012towards}), as diffusion into the bulk region will likely decrease the size of such Turing spaces. We have shown that even in cases where the broad influence of the bulk is to decrease the ability of the system to pattern, such a decrease will be non-monotonic in the geometric and transport parameters of the bulk region in general, as illustrated with the non-dimensional bulk depth, $h$ and permeability, $\eta$. Many of the parameters may not be controllable, though one can often choose an agar height $h$ above a certain minimal threshold. The results in Table \ref{tablelims} broadly suggest that the agar layer should be made as thin as possible to limit the impact on a system's ability to pattern. There may also be opportunities to decrease the permeability into the bulk, $\eta$, by using thicker filter paper or modifying the pore size or density, which would also reduce the negative impact of the bulk on pattern formation, though due to metabolic constraints (as the agar is primarily a nutrient) this too may be somewhat limited. We do note that there are important experimental controls in the genetic circuits encoded in the nonlinear reaction kinetics, which we have only caricatured in this study by considering the two-species case with only diffusible morphogens. Finally, we have shown instances of instability such that the bulk domain is a necessary component to drive an otherwise stable surface system to a patterned state (e.g.~equation~\eqref{smallepslambdakq2} and the following discussion), though we leave systematic analysis of such instabilities for future work. This route to instability does not contradict the preceding suggestions about reducing $\eta$ and $h$, as it is likely inadmissible for bacterial pattern formation on agar. Such an experimental setting entails that it is reasonable to assume $\bm{D_B} \propto \bm{I}$, and hence by \eqref{smallepslambdakq2} we see that bulk diffusion will not drive an instability in this case.
We remark that the mode selection phenomena we have illustrated (e.g.~in Figure \ref{h0p1eps1e-2}) can be understood in the context of finite-size effects, which are well-studied in the classical case \cite{Murray2003}. Namely given a dispersion relation for $\Re(\lambda_C(k_q))$, where $\lambda_C$ is the growth rate of a classical Turing mode, one can tune the geometry to select different spatial eigenvalues $k_q$ to give rise to non-monotonic effects as, for instance, the domain size is increased. However, here the effects are more subtle as we cannot explicitly compute the relationship between $\lambda$ and eigenvalues of the full spatial operator, and so can only implicitly observe these effects. Nevertheless, these mode selection effects appear to be more prevalent compared to classical cases as they require very small domains and other fine-tuning \cite{Murray2003}. Additionally, in our setting mode selection effects appear to be more prevalent across a wide range of geometric parameters, whereas the classical cases have been studied almost entirely in terms of a scalar length, and are generally restricted in parameter regimes where they occur. In particular, we conjecture that the non-monotonic dependence of $\max(\Re(\lambda))$ on $\eta$ seen in Figure \ref{stabfig} is due to these effects, as we see different modes being excited on either side of this region in Figure \ref{h0p1eps1e-2}.
There are numerous extensions of these results that are worth pursuing. In the example setting of bacterial colony formation on an agar substrate, one might need to augment the bulk evolution with a degradation reaction. We remark that such a simple addition leads to substantial complexity as, if the surface equilibrium is nonzero, then there does not exist a homogeneous equilibrium across the whole coupled system (a degradation reaction in the bulk by itself will always lead to a homogeneous zero equilibrium concentration). We anticipate that the mathematical structure in this case will be even more intricate.
A simpler addition, also of relevance to bacterial patterning on agar, would be the inclusion of non-diffusible reactants in the surface region. This approach would also pave the way to account for all gene regulatory dynamics in a quantitative model based on mass action kinetics. In such a case, we can apply techniques to incorporate the impact of such reactants on the surface reaction kinetics directly (in the linearised system) \cite{klika2012influence}.
Along similar lines, more complicated transport functions $\bm{g}$ across the membrane can be studied, again leading to new possibilities of differential transport, which can easily be added to the analysis implemented here. We have also assumed that the same number of species diffuse throughout both domains, but in principle one can generalize this by introducing different coupling functions $\bm{g}$ for the surface and bulk boundary conditions, presently given in equation \eqref{coupling}. Such an analysis is broadly similar though there are several key details to account for, so we leave this for further work.
There are many biological examples of physical layered media with reactions in multiple different spatial domains, such as in the epithelial-mesenchymal coupling during the development of the skin in mammals \cite{vilaca2019numerical}. For example, in the study of hair follicle morphogenesis, a substantial amount of biochemical research has implicated Turing-type instabilities in the formation of follicle primordia \cite{mou2006generation}. More recently, it has been suggested that a simple activator-inhibitor system is insufficient to capture the dynamical complexity in hair follicle patterning, and so suggestions have been made that such patterns arise due to many coupled processes, which will undoubtedly occur across the different domains of the epithelium and the developing mesenchyme \cite{glover2017hierarchical}. Similar remarks can be made about many kinds of skin and other organ patterning events across a range of species, suggesting that general methodologies for stratified reaction-diffusion systems would be useful to elucidate underlying physico-chemical mechanisms. A related layered system is the synthetic pattern formation studied in a monolayer of HEK293 cells grown beneath a culture medium \cite{sekine2018synthetic}, where presumably bulk diffusion plays a significant role in transporting signalling molecules.
While we have explored exemplar reaction-diffusion systems in such coupled domains, there are more general transport mechanisms that could be studied. Both chemotaxis and a range of mechanical taxis, as well as mechanical forces, could be included in such a model. We note that a numerical study \cite{vilaca2019numerical} has made some progress towards such a model. The linear stability analysis for such problems is involved, but the approach presented here generalises to these settings. Of course, in the absence of a homogeneous steady state, one must develop new methods for the analysis of pattern-forming instabilities. This has been done recently for heterogeneous steady states \cite{krause_WKB}, but extending such an analysis to these coupled geometries is nontrivial. Mathematically, the limit of $\eta \to \infty$ can be thought of as a step function heterogeneity, as explored in \cite{stephetero}, so that the systems studied here are also in some sense a generalisation of piecewise-constant reaction-diffusion problems, providing another perspective on heterogeneous reaction-diffusion systems.
Another related generalization would be to study discrete or hybrid discrete-continuum formulations of these kinds of layered media, such as the recent hybrid Turing-type model proposed in \cite{macfarlane2020hybrid}. Turing's original paper contained a study of discrete cells \cite{turing1952chemical}, which was later extended in \cite{othmer1971instability} and more recently in \cite{nakao2010turing} to reaction-diffusion systems on discrete networks. Such a formulation has been extended to consider multiplex networks, themselves a model of discrete layered media \cite{gomez2013diffusion}, within which Turing pattern formation has also been studied \cite{asllani2014turing, kouvaris2015pattern}. Such systems deserve exploration on their own, in addition to relating them to spatially continuous analogues of Turing systems in stratified media.
Finally we mention that one could generalise from our setting of two planar domains to many more coupled domains, or to more complicated geometric settings, including those relevant for more realistic models of development, such as in the blastula stage or later stages of epithelial-mesenchymal development on complicated morphologies. While our approach may be generalisable to very different geometric settings, the dispersion relation we have found in this simple case is already somewhat difficult to analyse, and full numerical simulations may be more expedient. Nevertheless, analytically tractable results for this family of problems are valuable in understanding the role of coupled domain structures in pattern formation, as such scenarios are ubiquitous in biological settings.
\begin{acknowledgements}
A.L.K. and E.A.G. are grateful for support from BBSRC grant BB/N006097/1; V.K. is grateful for support from the European Regional Development Fund-Project 'Center for Advanced Applied Science' (no. CZ.02.1.01/0.0/0.0/16\_019/0000778) and the Mathematical Institute at the University of Oxford. In compliance with BBSRC's open access initiative, the data in this paper is available from http://dx.doi.org/xx.xxxx/xxxxxxxxxxxxxxxxxx.
\end{acknowledgements}
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Daniel Nestor e Nenad Zimonjić erano i detentori del titolo ma hanno deciso di non partecipare insieme. Nestor ha giocato insieme a Leander Paes perdendo al secondo turno contro Pablo Cuevas e David Marrero, Zimonjić ha giocato insieme a Marcin Matkowski perdendo al secondo turno contro Jamie Murray e John Peers.
Cuevas e Marrero hanno vinto il titolo battendo in finale Marcel Granollers e Marc López 6-4, 7-5.
Teste di serie
Le prime 4 teste di serie hanno ricevuto un bye per il secondo turno.
Bob Bryan / Mike Bryan (secondo turno)
Ivan Dodig / Marcelo Melo (quarti di finale)
Jean-Julien Rojer / Horia Tecău (quarti di finale)
Marcin Matkowski / Nenad Zimonjić (secondo turno)
Marcel Granollers / Marc López (finale)
Daniel Nestor / Leander Paes (secondo turno)
Alexander Peya / Bruno Soares (secondo turno)
Simone Bolelli / Fabio Fognini (secondo turno)
Tabellone
Parte finale
Parte alta
Parte bassa
Collegamenti esterni
Internazionali d'Italia 2015
ATP World Tour 2015
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Property + Business Information
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PEXA
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The Search People supports Domestic Violence services through Dancing CEOs
The Search People CEO Rafe Berding joined ten of Queensland's corporate movers and shakers as they swapped the boardroom for the ballroom to raise a record $578,377.80 to help victims of domestic violence.
The cast of prominent CEOs put their pirouettes and pride on the line for the annual Dancing CEOs competition on 15 May 2021 to support the Women's Legal Service's provision of vital legal and social support services for vulnerable women and children.
CEO of The Search People, Rafe Berding said he was excited to take to the stage after months of preparation.
"I have taken the dancing lessons and gained confidence rehearsing in my office countless times, but I know it'll be a completely different feeling when I take to stage next month," Mr Berding said.
"I'll be the first to say I've got two left feet, but I know this event is about more than dancing for all of us competing.
"It's an honour to help raise funds for the Women's Legal Service to support the amazing work they do," he said.
Mr Berding and his competitors had extra time to master their moves after the planned 2020 Dancing CEO event was postponed due to the pandemic.
Fellow competitor, founder and owner of beauty clinic Brazilian Beauty, Francesca Webster, said her intimate understanding of the challenge's women face inspired her to don her dancing shoes.
"I've competed in Dancing CEOs, and I know how great the event is and the impact it has, but this year is particularly special to me," Ms Webster said.
"It was only in the lead-up to this year's competition that I shared my story of survival.
"Being out of my relationship and having the confidence to use my voice to help others makes me feel better than I have for years," she said.
"This year, I'm dancing for the one in six women who, like myself included, have experienced physical or sexual violence at the hands of their partner.
The information in this publication is intended for general and/ or product information purposes only. It does not serve as specific advice to any particular person or organisation and should not be relied upon as such. Any information contained is general in nature and does not take into account any person's or organisation's situation, circumstances or individual needs. Before acting on anything held within you should consider professional advice and the information's appropriateness to you, having regard to your objectives and needs.
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\section{Introduction}
\label{intro}
The ability to enumerate and classify all of the mechanically stable
(MS) packings of frictionless particles is important for understanding
glass transitions~\cite{stillinger0} in atomic, molecular, and
colloidal systems, and the structural and mechanical properties of
particulate materials such as granular media, foams, and emulsions.
For example, if all MS packings in a given system are known, one can
measure accurately the frequency with which each MS packing occurs,
and determine how the packing frequencies and materials properties
depend on the preparation history~\cite{ning,gao}. Further, MS packing
frequencies are important for identifying the appropriate statistical
mechanical ensemble for weakly perturbed granular
materials~\cite{song}. However, since the number of MS packings grows
exponentially with the number of particles~\cite{stillinger}, exact
enumeration of static packings is prohibitive for even modest system
sizes~\cite{shattuck}. Thus, one of the most important outstanding
questions in the area of disordered particulate materials is
determining how the packing-generation protocol influences the
distribution of MS packings and their structural and mechanical
properties.
Previous work has suggested that the positional order of MS packings
of frictionless spheres increases monotonically with packing fraction
and contact number in dense packings~\cite{torquato,kansal}. However,
the MS packings in these previous studies were created using
monodisperse systems, which are prone to
crystallization~\cite{torquato3}, and prepared using the
Lubachevsky-Stillinger compression algorithm~\cite{ls}, which is a
thermalized packing-generation protocol. In addition, these prior
studies did not distinguish the distribution of isostatic MS packings
(in which the number of degrees of freedom matches the number of
constraints~\cite{witten}) from the distribution of hyperstatic
packings (with more contacts than degrees of freedom). Later work
characterized bidisperse systems, which are less prone to
crystallization, but focused on microphase-separated states, not
amorphous, isostatic packings~\cite{donev}. However, recent studies
on systems composed of 3D monodisperse, frictionless, spherical
particles have pointed out that amorphous, isostatic packings can
exist over a finite range of packing fraction in the large-system
limit, with no correlation between positional order and packing
fraction~\cite{berthier2,chaudhuri}. Moreover,
simulations~\cite{vagberg} and experiments ~\cite{lechenault} on
two-dimensional systems also suggest a finite range of jamming onsets
rather than a single packing fraction in the large system limit.
Further, the body of work on jammed particulate systems has emphasized
the concept of point J, {\it i.e.} that there is a single packing
fraction at which jamming occurs in the large system
limit~\cite{longJ,kamien}. Since amorphous, isostatic packings can
exist over a finite range of packing fractions, the onset of jamming
should not be classified as a point in the jamming phase
diagram, but rather as a region of finite extent. It has
also been argued that the wide distribution of packing fractions at
which the onset of jamming occurs in small periodic
systems~\cite{longJ} is related to the finite range of packing
fractions over which amorphous, isostatic packings occur in the large
system limit~\cite{bible}. However, it has not been proved that these
two effects are directly connected.
A number of overarching questions related to the connection between
positional order, isostaticity, and material properties of static
packings remain open. For example, can isostatic or nearly isostatic
packings possess significant positional order and if so, what are the
fundamental differences in the normal modes and mechanical
properties between those that do and do not possess significant positional
order? This question is particularly important since recent studies
have emphasized that {\it amorphous}, isostatic packings possess an
excess of low-frequency normal modes~\cite{liu,mao} over that
for harmonic, ordered solids.
In addition, previous work has drawn a strong contrast between
amorphous packings and configurations with crystalline
order~\cite{torquato2}. However, how different are the structural and
mechanical properties of amorphous versus partially ordered
particulate systems? For example, it is possible that the amorphous
regions in the interstices between ordered domains in partially
crystalline materials dominate the structural and mechanical
properties, in which case their properties would be similar to
amorphous packings. At the very least, one would assume that there is
not a strong difference between the mechanical properties of isostatic
and only slightly hyperstatic packings that possess significant positional
order.
In this article, we describe extensive computer simulations of
collections of frictionless, bidisperse disks with short-range
repulsive interactions to address two important, open questions:
1. What is the range of packing fractions over which amorphous,
isostatic static packings occur with similar structural and mechanical
properties, and 2. How do the structural and mechanical properties of
static packings change with the deviation in the contact number at
jamming onset from the isostatic value, $z_J - z_{\rm
iso}$~\cite{foot2}? Using two distinct packing-generation protocols,
we construct scatter plots for more than $10^4$ static packings
characterized by the contact number, packing fraction, measures of
positional order, and mechanical properties. The first protocol
involves thermally quenching equilibrated liquid configurations to
zero temperature over a range of thermal quench rates $r$ followed by
compression and decompression in small steps to reach packing
fractions $\phi_J$ at jamming onset. For the second, we seed the
system with initial configurations that promote micro- and
macrophase-separated packings followed by compression and
decompression to $\phi_J$.
Our main results are fourfold: 1. Isostatic, amorphous packings exist
over a finite range of packing fraction from $\phi_{\rm min}$ to
$\phi_{\rm max}$ in the large system limit, with similar structural
and mechanical properties. 2. In agreement with previous
calculations, we obtain $\phi_{\rm min} \approx 0.84$ for $r > r^*$,
where $r^*$ is the rate above which $\phi_J$ is insensitive to
rate. In contrast, $\phi_{\rm max}$ depends sensitively on quench
rate, system size, and boundary conditions. 3) The amorphous,
isostatic packings coexist with an abundance of hyperstatic,
microphase- and macrophase-separated packings. 4) When considering the
full ensemble of static frictionless packings, the packings possess
structural and mechanical properties that span a continuous range from
amorphous to partially ordered to ordered in contrast to the results
and interpretations of recent studies~\cite{makse,radin}.
The remainder of the manuscript will be organized as follows. In
Sec.~\ref{protocol}, we describe the computational system we consider
and the two protocols we employ to generate static frictionless disk
packings. In Sec.~\ref{characterize}, we present our results, which
include characterizations of the structural (packing fraction, contact
number, and several order parameters to detect positional and
compositional order) and mechanical (shear modulus and eigenvalues of
the dynamical matrix~\cite{gao}) properties of more than $10^4$ static
packings and comparisons of these properties for isostatic and
hyperstatic configurations. Finally, in Sec.~\ref{conclusions}, we
provide our conclusions and promising future research directions.
\section{Packing-Generation Protocols}
\label{protocol}
We focus on well-characterized two-dimensional systems composed of $N$
bidisperse disks ($50$-$50$ by number), each of mass $m$, with
diameter ratio $d=\sigma_l/\sigma_s=1.4$~\cite{harrowell,longJ,donev},
within square, periodic simulation cells with side length $L$. We
consider frictionless particles that interact through the
finite-range, purely repulsive spring potential. The total potential
energy per particle is given by
\begin{equation}
\label{interaction}
V = \frac{\epsilon}{2N} \sum_{i>j} \left( 1 - \frac{r_{ij}}{\sigma_{ij}}
\right)^{2} \Theta \left( 1 - \frac{r_{ij}}{\sigma_{ij}} \right),
\end{equation}
where $r_{ij}$ is the center-to-center separation between disks $i$
and $j$, $\epsilon$ is the characteristic energy scale of the
interaction, $\Theta(x)$ is the Heaviside function, and $\sigma_{ij} =
(\sigma_{i}+\sigma_{j})/2$ is the average diameter. We simulated a
range of system sizes from $N=256$ to $8192$ particles to assess
finite size effects. Energy, length, and time scales are measured in
units of $\epsilon$, $\sigma_s$, and $\sigma_s \sqrt{m/\epsilon}$,
respectively.
The packing fraction $\phi_J$ at which jamming occurs and the
structural and mechanical properties of static packings can depend
strongly on the packing-generation protocol employed. Our goal is to
generate static frictionless MS packings that span the range of contact numbers
from the isostatic value $z_{\rm iso} = 4$ to the hexagonal crystal
value $z_{\rm xtal} = 6$ and the range of positional order from
amorphous to phase-separated and from partially crystalline to
crystalline states. To accomplish this, we investigate two distinct classes of
packing-generation protocols: 1) thermal quenching from liquid initial
conditions coupled with compression and decompression steps, which
typically generates amorphous configurations and 2) compression and
decompression steps from initial conditions that promote micro- or
macrophase separation~\cite{phase}.
{\it Protocol 1: Thermal quenching from liquid initial conditions}
In this algorithm, we prepare equilibrated, liquid configurations at
high temperature $T_0 = 10^{-3}$ and in molecular dynamics (MD)
simulations quench them to a very low final temperature $T_f=10^{-16}
\simeq 0$ at fixed packing fraction $0.8 \leq \phi_i < \phi_{\rm xtal} =
\pi/2\sqrt{3}$~\cite{foot4} over a time interval $t$ by rescaling the particle
velocities so that the kinetic temperature $T = N^{-1} \sum_i m
v_i^2/2$ obeys
\begin{equation}
T(t) = T_{0}e^{-rt},
\label{eq2}
\end{equation}
where $r$ is the thermal quench rate, which is varied over five orders
of magnitude $10^{-5} \le r \le 1$. We generated $50$ equilibrated,
independent liquid configurations at $T_0$ at each $\phi_i$ by writing
out configurations every $10 \tau$, where $\tau$ is a decay time
obtained from the self-intermediate scattering function at wavenumbers
corresponding to the first peak in the structure factor~\cite{isf}.
After reaching a local potential energy minimum at each initial
packing fraction $\phi_i$ and thermal quench rate $r$, we input the
configurations into an `athermal' algorithm (`packing finder') that
searches for the nearest static packing in configuration space with
infinitesimal particle overlaps. The algorithm has been described in
detail in previous work~\cite{gao}. Briefly, we successively increase
or decrease the diameters of the grains (while maintaining the
diameter ratio $d$), with each compression or decompression step
followed by conjugate gradient minimization of $V$. The system is
decompressed when the total potential energy per particle at a local
minimum is nonzero, {\it i.e.} there are finite particle overlaps. If
the potential energy of the system is zero and gaps exist between
particles, the system is compressed. The increment by which the
packing fraction is changed at each compression or decompression step
is gradually decreased. Numerical details of the algorithm are the
same as in Ref.~\cite{gao}. When this algorithm terminates, we obtain
a static packing defined by the particle positions $\{ {\vec r}_1,
{\vec r}_2,\ldots, {\vec r}_N\}$ and packing fraction $\phi_J$. Since
we use an energy tolerance (per particle) $V_{\rm tol}/\epsilon =
10^{-16}$ for the termination of the energy minimization and
compression/decompression scheme in the packing finder, the positions
and packing fraction at jamming are extremely accurate with errors at
one part in $10^8$.
{\it Protocol 2: Compression and decompression steps from initial
conditions that promote order} We will see below in
Sec.~\ref{characterize} that Protocol $1$ produces amorphous,
isostatic packings. Thus, we seek an algorithm that will generate
static packings with variable positional and compositional order. To
bias the system toward micro- and macrophase-separated configurations,
we seed the packing finder with particular sets of initial conditions.
We first divided the unit cell into $s\times s$ equal-sized
partitions, where $s$ is an even integer that ranged from $2$ to $26$,
and placed approximately $N/s^2$ large or small particles in
alternating partitions to create a checkerboard-like pattern. The
particles were placed randomly in each partition. The initial
configuration is then input into the packing finder to yield a static
packing. In the large $s$ limit, we expect amorphous static packings,
while at intermediate and small $s$, we expect micro- and
macrophase-separated packings. To generate static packings near
$\phi_{\rm xtal}$ we also divided the unit cell into two partitions
and placed the large (small) particles on a hexagonal lattice in a
region with area $A_L=d^2/(1+d^2)$ ($1-A_L$) and then applied the
packing finder.
\section{Structural and Mechanical Properties}
\label{characterize}
After generating static packings using the two packing-generation
protocols described above, we contrast them by calculating several
structural and mechanical properties. The structural
characterizations include the packing fraction, contact number, and
compositional and positional order parameters. For the packing
fraction at jamming onset, we calculate
\begin{equation}
\label{packing_fraction}
\phi_J = \frac{N\pi}{8} \left( \frac{ \sigma_s}{L}\right)^2 \left( 1 + d^2 \right)
\end{equation}
including all $N$ particles. For the contact number at jamming, we
sum up all overlapping pairs ($r_{ij} \le \sigma_{ij}$) of
particles, $z_J = N_c/N'$, where $N'=N-N_r$, $N_r$ is the number of
rattler particles with fewer than three contacts, and $N_c$ only
includes overlapping pairs among the $N'$ particles within the `true'
contact network. It is crucial to perform an error analysis on
the contact number $z_J$, which is described in Appendix~\ref{error}.
\begin{figure}
\scalebox{0.45}{\includegraphics{phi_logr_wfit.eps}}
\caption{Average packing fraction $\langle \phi_J \rangle$ obtained
from Protocol $1$ as a function of the negative logarithm of the
thermal quench rate $r$ for $N=1024$. Data points at each rate
represent an average over typically $300$ static, amorphous
packings. The dashed line shows the scaling $\langle \phi_J \rangle
\sim [\log_{10} (r - r^*)]^{\mu}$, where $\mu \sim 0.5$ and $r^*
\approx 0.03$ is the thermal quench rate above which $\langle \phi_J
\rangle \approx 0.841$ is independent of $r$. }
\label{rate}
\end{figure}
\paragraph*{Packing Fraction}
We show results for the average packing fraction $\langle \phi_J
\rangle$ versus thermal quench rate $r$ over five orders of magnitude
obtained from Protocol $1$ in Fig.~\ref{rate}. For large rates $r >
r^* \approx 0.03$, the average packing fraction $\langle \phi_J
\rangle \rightarrow 0.841$ is independent of rate, which agrees with
studies that employ athermal compression/decompression
packing-generation algorithms~\cite{longJ,ning}. For $r < r^*$,
$\langle \phi_J \rangle$ increases approximately as $[\log_{10}
(r-r^*)]^{0.5}$ with decreasing rate. We emphasize that all packings
used to present the data in Fig.~\ref{rate} are amorphous and
isostatic. Since $\langle \phi_J \rangle$ increases so slowly, it is
not possible to approach $\phi_{\rm xtal}$ using protocol
$1$. Using an extrapolation, we estimate that rates
below $10^{-45}$ are required to reach $\phi_{\rm xtal}$, and thus we
employed Protocol $2$, not $1$, to generate compositionally and
positionally ordered packings.
\begin{figure}
\scalebox{0.45}{\includegraphics{phi_z_scatter_inset.eps}}
\caption{Scatter plot of the contact number $z_J$ versus the packing
fraction at jamming onset $\phi_J$. The open circles indicate static
packings that were generated using Protocol $1$ for $N=1024$, while
all other symbols indicate static packings generated using Protocol
$2$. The open squares, diamonds, and triangles correspond to
$N=1024$, $2048$, and $4096$, respectively, for all partitions $s$ and
systems with two partitions and random particle placements. The filled
squares, diamonds, upward triangles, and downward triangles correspond
to $N=1024$, $2048$, $4096$, and $8192$, respectively, for the systems with
two partitions and initial crystal lattice positions. The black cross
indicates the values $z_J=6$ and $\phi_J=\pi/2\sqrt{3}$ for the
hexagonal crystal. The labels (a)-(d) correspond to the images in
Fig.~\ref{picture}. The inset shows the system-size dependence for
systems with two partitions and random initial positions at $N=256$
(leftward triangles), $1024$ (squares), and $4096$ (upward
triangles).}
\label{scatter}
\end{figure}
\begin{figure}
\scalebox{0.4}{\includegraphics{systemimages_thermal_seeded_vertical_small.eps}}
\caption{Images of representative static packings from the scatter
plot in Fig.~\ref{scatter} with (a) $\phi_J=0.837$, $z_J=3.99$, (b)
$\phi_J=0.853$, $z_J=4.00$, (c) $\phi_J=0.846$, $z_J=4.04$, (d)
$\phi_J=0.860$, $z_J=4.41$, and (e) $\phi_J=0.892$, $z_J \simeq 4.1$. (See
Appendix~\ref{error}.)}
\label{picture}
\end{figure}
\paragraph*{Contact Number}
In Fig.~\ref{scatter}, we display a scatter plot of the contact number
$z_J$ versus $\phi_J$ for all static packings (where the contact
number is insensitive to the definition of `contact') generated using
Protocols $1$ and $2$. (See Appendix~\ref{error} for a discussion of
the sensitivity of the contact number on the definition of contacting
particles.) Fig.~\ref{scatter} shows several compelling
features. First, nearly all of the static packings obtained from
Protocol $1$ (open circles) are isostatic with $z_J = 4$, but they
occur over a range of packing fractions $\phi_{\rm min} \le \phi_J \le
\phi_{\rm max}$, where $\phi_{\rm min} = 0.837$ and $\phi_{\rm max} =
0.853$. As shown in Appendix~\ref{error} $\phi_{\rm max}$ is likely
only a lower bound for the largest packing fraction at which isostatic
packings can occur in these systems. Second, we find a cluster of
data points for Protocol $2$, for which the average $z_J$ is strongly
correlated---varying roughly linearly---with $\phi_J$. The cluster
originates near $\phi_J \approx 0.84$, $z_J = z_{\rm iso} = 4$. In
the inset to Fig.~\ref{scatter}, we show that the width of the cluster
of data points from Protocol $2$ narrows with increasing system size,
but the approximate linear relationship between the average $z_J$ and
$\phi_J$ is maintained. Images of five representative packings from
the scatter plot in Fig.~\ref{scatter} are displayed in
Fig.~\ref{picture}.
\begin{figure}
\scalebox{0.4}{\includegraphics{z_SS_LL.eps}}
\caption{Scatter plot of the fraction of contacts between two large
$f_{ll}$ or two small particles $f_{ss}$ versus packing fraction
$\phi_J$ for all static packings from both protocols. The diamonds
(circles) and triangles (squares) display data from Protocol $1$ ($2$)
for $f_{ll}$ and $f_{ss}$, respectively.}
\label{ss}
\end{figure}
\paragraph*{Compositional Order}
We now describe measurements of the compositional and positional order
for static packings. For the compositional order, we quantify the
fraction of overlapping pairs ($r_{ij} \le \sigma_{ij}$) that involve
two small $f_{ss}$ or large $f_{ll}$ particles. A scatter plot of
$f_{ll}$ and $f_{ss}$ versus $\phi_J$ for static packings generated
from both protocols is shown in Fig.~\ref{ss}. The packings from
Protocol $1$ show no signs of phase separation with $f_{ss} + f_{ll}
\approx f_{sl} \approx 0.5$ for all packings. In contrast, Protocol
$2$ generates static packings with a range of compositional order as shown in
Fig.~\ref{picture} (c)-(e). For example, at the largest $\phi_J$,
the system displays macrophase separation with $f_{ss} + f_{ll} \approx 1$ and
$f_{sl} \approx 0$. We find similar results when we define contacting
pairs as those with $r_{ij} \le r_{\rm min} \sigma_{ij}$, where
$r_{\rm min}$ is set by the first minimum in $g(r)$.
\begin{figure}
\scalebox{0.8}{\includegraphics{Q6_thermal_microphase.eps}}
\caption{Scatter plot of the (a) global and (b) local bond
orientational order parameters, $\psi_6^g$ and $\psi_6^l$, versus
packing fraction for static packings from protocol $1$ (squares)
and $2$ (circles).}
\label{Q6}
\end{figure}
\paragraph*{Bond Orientational Order}
To quantify positional order, we calculate the bond orientational
order parameter $\psi_6$, which measures the hexagonal registry of nearest
neighbors~\cite{stein}. $\psi_6$ can be calculated `locally', which does
not consider phase information, or `globally', which allows phase
cancellations. A polycrystal will yield a relatively large value for the local
bond orientational order parameter $\psi_6^l$, even though the global
order parameter $\psi_6^g \sim 1/\sqrt{N_d}$, where $N_d$ is the number
of polycrystalline domains. Eqs.~(\ref{2dglobal}) (global) and
(\ref{2dlocal}) (local) provide expressions for the bond orientational
order parameters in 2D.
\begin{eqnarray}
\label{2dglobal}
\psi_6^{g}&=&\frac{1}{N}\left|\displaystyle\sum_{i=1}^N\frac{1}{n_i}
\displaystyle\sum_{j=1}^{n_i}e^{6\imath\theta_{ij}}\right| \\
\label{2dlocal}
\psi_6^{l}&=&\frac{1}{N} \displaystyle\sum_{i=1}^N\frac{1}{n_i}\left|
\displaystyle\sum_{j=1}^{n_i}e^{6\imath\theta_{ij}}\right|,
\end{eqnarray}
where $\theta_{ij}$ is the angle between a central particle $i$ and
neighbors $j$ and $n_i$ denotes the number of nearest neighbors of
$i$. Two particles are deemed nearest neighbors if their
center-to-center separation $r_{ij} < r_{\min} \sigma_{ij}$.
\begin{figure}
\scalebox{0.45}{\includegraphics{DOS_microphase_bi_2.eps}}
\caption{Density $D(\omega)$ of normal mode frequencies $\omega$ for
$N=1024$ bidisperse frictionless disk packings obtained using
Protocols $1$ and $2$ as a function of the contact number at jamming
onset for $z_J \simeq 4.0$ (black), $4.0 \le z_J \le 4.1$ (red), $4.1
\le z_J \le 4.2$ (green), $4.3 \le z_J \le 4.4$ (blue), and $4.5 \le
z_J \le 4.6$ (violet). The inset shows the same data except that it
focuses on low frequencies $\omega < 1$ and includes power-law fits to
$D(\omega) \sim \omega^{\alpha}$ as dashed lines.}
\label{DOS_bi}
\end{figure}
\begin{figure}
\scalebox{0.45}{\includegraphics{DOS_microphase_mono.eps}}
\caption{Density $D(\omega)$ of normal mode frequencies $\omega$ for
$N=1024$ {\it monodisperse} frictionless disk packings obtained using
Protocol $1$ as a function of the contact number at jamming onset for
$4.1 \le z_J \le 4.2$ (green), $4.5 \le z_J \le 4.6$ (violet), $4.9
\le z_J \le 5.0$ (cyan), $5.4 \le z_J \le 5.5$ (magenta), and $z_J
\simeq 6.0$ (orange). The inset shows the same data except that it
focuses on low frequencies $\omega < 1$ and includes power-law fits to
$D(\omega) \sim \omega^{\alpha}$ as dashed lines.}
\label{DOS}
\end{figure}
The results for the global and local bond orientational parameters
$\psi_6^g$ and $\psi_6^l$ are shown in Fig.~\ref{Q6}. The static
packings obtained from Protocol $1$ possess only local bond
orientational order with $\psi_6^l \approx 0.55$ as found in dense
liquids~\cite{stein}, and $\psi_6^g \sim 1/\sqrt{N}$. Further, there is
no correlation between the packing fraction $\phi_J$ and global or
local bond orientational order. In contrast, for the phase-separated
and partially crystalline packings from Protocol $2$, we find that
there is a strong positive correlation between $\psi_6^l$ and $\phi_J$
and a somewhat weaker correlation between $\psi_6^g$ and $\phi_J$.
The static packings from Protocols $1$ and $2$ have different
structural properties. Those from $1$ are amorphous and possess
similar structural properties even though they exist over a range of
packing fraction. In contrast, there is a positive correlation
between compositional and positional order and packing fraction for
the phase-separated and partially crystalline packings from Protocol
$2$. We will now describe the mechanical properties of the static
packings including the spectrum of normal modes and static shear modulus
as a function of contact number and order.
\paragraph*{Spectrum of Normal Modes}
The spectrum of normal modes provides significant insight into the
structural and mechanical properties of mechanically stable
packings~\cite{longJ}. For example, there is evidence that the
low-frequency region of the spectrum controls the
static shear response of jammed packings~\cite{ellipse}. To calculate the
spectrum, we diagonalize the dynamical matrix of all possible second
derivatives with respect to particle positions evaluated at positions
of the static packing---assuming that no existing contacts break and
no new contacts form~\cite{chapter}. This yields $2N'-2$ nontrivial
eigenvalues $e_i$ after accounting for translational invariance. We
consider here only mechanically stable packings, and thus all $2N' -
2$ of the eigenvalues are nonzero~\cite{foot}.
\begin{figure}
\scalebox{0.4}{\includegraphics{eigenvector_181_413.eps}}
\caption{Eigenvectors corresponding to the modes with frequencies near
the (a) first and (b) second peaks in the density of states
$D(\omega)$ for monodisperse packings with $z_J \simeq 6$ and $\phi_J
\simeq \phi_{\rm xtal}$ for $N=256$. The size of the eigenvector
component for each particle is proportional to the length of the vector
associated with each particle.}
\label{eigenvector}
\end{figure}
\begin{figure}
\scalebox{0.45}{\includegraphics{z_vs_alpha_2.eps}}
\caption{Power-law exponent $\alpha$ for the scaling of the density of
states with frequency in the limit $\omega \rightarrow 0$ ($D(\omega)
\sim \omega^{\alpha}$) as a function of contact number at jamming
onset $z_J$ for bidisperse (circles) and monodisperse (squares)
packings. (The error bars indicate the error in $\alpha$ from
least-squares analysis.) The dashed line is a fit to Eq.~\ref{alpha}
(with $a=0.17$), which interpolates the data between the limiting
values $\alpha = 0$ at $z_J = z_{\rm iso}=4$ and $\alpha=1$ (Debye
behavior) at $z_J=z_{\rm xtal}=6$. The solid line is Eq.~\ref{alpha}
with $a=0$.}
\label{exponent}
\end{figure}
The density $D(\omega)$ of normal mode frequencies $\omega_i =
\sqrt{e_i/N}$, or density of states (DOS), is given by $D(\omega) =
(N(\omega+\delta \omega)-N(\omega))/\delta \omega$, where $N(\omega)$
is the number of modes with frequency less than or equal to $\omega$.
The density of states $D(\omega)$ for packings of bidisperse
frictionless disks is shown in Fig.~\ref{DOS_bi} as a function of the
contact number at jamming onset $z_J$. As in previous studies
\cite{longJ}, we find that for isostatic systems with $z_J \simeq 4$,
$D(\omega)$ possesses a nearly constant regime at low frequencies,
which signals an abundance of low-frequency modes compared to ideal
Debye behavior (where $D(\omega) \sim \omega$ as $\omega \rightarrow
0$) for ideal 2D harmonic solids. For the micro- and macro-phase
separated bidisperse packings generated using Protocol $2$ with $z_J
\gtrsim 4.1$, the density of states develops two other interesting
features. First, $D(\omega)$ develops two strong peaks near $\omega
\simeq 1.0$ and $1.6$ instead of a single broad peak centered near
$\omega \approx 1.4$ for isostatic amorphous systems. (We will see
below that these peaks are associated with crystallization.) Second, we
observe that as $z_J$ increases and the packings become hyperstatic,
the weight in $D(\omega)$ at low frequency ($\omega \lesssim 0.3$)
decreases. As shown in the inset to Fig.~\ref{DOS_bi}, the density of
states scales as a power-law
\begin{equation}
\label{dos}
D(\omega) \sim \omega^{\alpha}
\end{equation}
in the limit $\omega \rightarrow 0$ with a scaling exponent $\alpha$
that varies continuously with contact number $z_J$ as shown in
Fig.~\ref{exponent}. (See Appendix~\ref{system_size} for a discussion
of the system-size dependence of the exponent $\alpha$.) Note,
however, that the plateau in the density of states remains largely
unchanged in the intermediate frequency regime $0.3 \le \omega
\lesssim 1$ over a wide range of $z_J$, which implies that some of the
remarkable features of jamming in isostatic systems also hold for
hyperstatic systems.
\begin{figure}
\scalebox{0.45}{\includegraphics{G_byz_inset.eps}}
\caption{Static shear modulus $G$ versus the deviation in packing
fraction from the jamming onset $\Delta \phi = \phi-\phi_J$ for static
packings at $\langle z_J\rangle = 4.0$ (circles), $4.15$ (diamonds),
$4.35$ (left triangles), and $4.55$ (right triangles). The long dashed
(dot-dashed) line has slope $0.5$ ($0.4$). The inset shows the
power-law scaling exponent $\beta$ for the static shear
modulus ($G \sim (\Delta \phi)^{\beta}$) versus the contact number
$z_J$ at jamming.}
\label{G}
\end{figure}
To test the generality of the results for the density of states, we
also calculated $D(\omega)$ for monodisperse frictionless disk
packings generated using Protocol $1$ as shown in Fig.~\ref{DOS}. The
density of states for monodisperse systems displays similar features
to that for bidisperse systems. 1. A plateau in $D(\omega)$ exists at
low to intermediate frequencies for nearly isostatic systems.
2. Strong distinct peaks are located near $\omega \simeq 1.4$ and
$2.25$ for hyperstatic packings. Eigenvectors that correspond to the
two peak frequencies are visualized in Fig.~\ref{eigenvector}. 3. A
power-law regime $D(\omega) \sim \omega^{\alpha}$ develops in the
$\omega \rightarrow 0$ limit for hyperstatic packings. The exponent
$\alpha$ varies continuously with $z_J$ with a similar functional
dependence to that for bidisperse systems as shown in
Fig.~\ref{exponent}. A notable difference between bidisperse and
monodisperse systems is that a continuous power-law regime in
$D(\omega)$ persists to higher frequencies ($\omega \sim 1$) for
monodisperse compared to bidisperse systems.
The dependence of the scaling exponent $\alpha$ on $z_J$ is displayed
for all bidisperse and monodisperse packings (binned by $z_J$) in
Fig.~\ref{exponent}. We find that $\alpha$ increases monotonically
with $z_J$ and use the suggestive empirical form
\begin{equation}
\label{alpha}
\alpha = (d-1)\frac{z_J-z_{\rm iso}}{z_{\rm xtal}-z_{\rm iso}} +
a (z_J-z_{\rm iso})(z_J-z_{\rm xtal}),
\end{equation}
where $a$ is a fitting parameter, to describe the data between the
limiting values $\alpha=0$ at $z_J = z_{\rm iso}$ and $\alpha = d-1$
(Debye behavior) at $z_J = z_{\rm xtal}$. The continuous increase in
$\alpha$ from $0$ to $1$ as the contact number increases suggests a
different scenario for the behavior of the jamming transition as a
function of $z_J$ and positional order compared to the
first-order-like transition found as the system compacts above random
close packing in simulations of frictional granular
materials~\cite{makse}.
\paragraph*{Static Shear Modulus}
To measure the static linear shear modulus $G$, we slightly deform the
system by applying an infinitesimal simple shear strain $\gamma$
(along the $x$-direction with gradient in the $y$-direction), allowing
the system to relax via energy minimization at fixed strain, and then
measuring the resulting shear stress response, $G=
d\Sigma_{xy}/d\gamma$. In Fig.~\ref{G}, we show the shear modulus
versus the amount of compression $\Delta \phi = \phi - \phi_J$ for
bidisperse packings obtained from Protocols $1$ and $2$ at several
values of $z_J$. We find generally that in the limit $\Delta \phi
\rightarrow 0$ the static shear modulus scales as a power-law with
$\Delta \phi$:
\begin{equation}
\label{geq}
G = G_0 (\Delta \phi)^{\beta},
\end{equation}
where the scaling exponent $\beta$ (and prefactor $G_0$) depend on
$z_J$. As shown in Fig.~\ref{G}, $\beta$ decreases steadily from
$0.5$ to $0.4$ as the contact number $z_J$ at jamming increases. Note
that $\beta=0.5$ for $z_J = z_{\rm iso}$ was obtained in previous work
on isostatic packings~\cite{longJ}. The results in Fig.~\ref{G}
suggest that the critical behavior ({\it e.g.} power-law scaling of
the shear modulus) found in jammed isostatic systems persists when the
jamming onset is hyperstatic. Further studies are required to
determine whether the scaling exponent for the static shear modulus can be
varied over the full range from $0.5$ to $0$.
\begin{figure}
\scalebox{0.4}{\includegraphics{loga_vs_z.eps}}
\caption{The contact number $z_J$ as a function of $a$, where the
condition $r_{ij} \le (1+a)\sigma_{ij}$ determines whether particles
$i$ and $j$ are in contact. The packings shown are $N=1024$, $\phi_J =
0.837$ (circles); $N=1014$, $\phi_J=0.892$ (squares); and $N=2390$,
$\phi_J=0.897$ (diamonds).}
\label{loga}
\end{figure}
\section{Conclusions}
\label{conclusions}
Using computer simulations, we generated a large library of
mechanically stable packings of bidisperse, frictionless disks that
span a wide range of contact number from $z_J=z_{\rm iso}=4$ to
$z_{\rm xtal}=6$ and packing fraction at jamming from $\phi_J \sim
0.84$ to near $\phi_{\rm xtal}$. We find that there is an amorphous,
isostatic branch of packings that spans a finite range in packing
fraction in the large-system limit. Over this range of packing
fraction, these packings are amorphous with no correlation between
bond orientational order or compositional order and $\phi_J$. We also
find a branch of phase-separated and partially crystalline packings
for which the compositional and positional order increase with
$\phi_J$. In addition, we characterize the mechanical properties of
the static packings by measuring the spectrum of normal modes and the
static shear modulus. We find that the mechanical properties of the
packings vary {\it continuously} as the contact number and structural
and compositional order at jamming onset increase from their isostatic
values. In particular, we find that the static shear modulus scales
as a power-law in the amount of compression, $G\sim (\Delta
\phi)^{\beta}$, and that the low-frequency density of states scales as
a power-law in frequency, $D(\omega) \sim \omega^{\alpha}$, and both
$\alpha$ and $\beta$ vary continuously with contact number at jamming
onset. These findings emphasize that jamming behavior in systems with
purely repulsive contact potentials occurs over a range of contact
numbers, not just near $z_J = z_{\rm
iso}$~\cite{hatano,hatano2,otsuki}. In future studies, we will
investigate the relationship between the scaling exponents $\alpha$
and $\beta$, which is likely an important feature of jamming in
hyperstatic systems.
\begin{figure}
\scalebox{0.4}{\includegraphics{phi_z_scatter_collecttypes.eps}}
\caption{Contact number $z_J$ versus packing fraction $\phi_J$ for the
same data in Fig.~\ref{scatter} and an additional set of packings
obtained from thermalizing the configurations in Fig.~\ref{scatter}
with $\phi_J > 0.86$ and then identifying the nearest packing. The
variation in $z_J$ increases with $\phi_J$.}
\label{all}
\end{figure}
\section{Acknowledgments}
We thank the organizers of the Frontiers in Nonequilibrium Physics and
YKIS2009 workshops. We also acknowledge A. Donev, R. Hoy, and
M. Shattuck for helpful conversations. This research was supported by
the National Science Foundation under Grant Nos. CBET-0828359 (LS),
DMS-0835742 (CO, CS), and PHY-0551164. We thank the Kavali Institute
for Theoretical Physics for their hospitality during ``The Physics of
Glasses: Relating Metallic Glasses to Molecular, Polymeric and
Oxide Glasses'' Program. This work also benefited from the
facilities and staff of the Yale University Faculty of Arts and
Sciences High Performance Computing Center and NSF grant
no. CNS-0821132 that partially funded acquisition of the computational
facilities.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,976
|
Alphington Halt railway station was a small station serving the village of Alphington (now a suburb of Exeter) located on the Teign Valley Line, which opened in 1882 and closed in 1961. This diverged from the South Devon Main Line at Exeter and joined the Netwon Abbot to Moretonhampstead line at Heathfield .
History
Alphington Halt had a 100 ft long wooden platform with a flat roofed corrugated shelter located on the eastern side of the single track line with no sidings or passing loop.
Opened by the Great Western Railway in 1928, the station then passed on to the Western Region of British Railways on nationalisation in 1948.
The station was then closed in June 1958 by the British Transport Commission.
The site today
All that remains in the area of the halt are the stone foundations of the bridge that once carried the line over Church Road, and the railway embankment which can be followed as far as the end of Ide Lane, where the remains of the over bridge can still be seen. The line was destroyed beyond this by the building of the A30 dual carriageway. The trackbed re-emerges west of the site of Ide station a little further down the line.
The site has been developed as residential property, modishly called "the halt at alphington".
Notes
References
Station on navigable O.S. map
External links
Disused railway stations in Devon
Former Great Western Railway stations
Railway stations in Great Britain opened in 1928
Railway stations in Great Britain closed in 1958
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,983
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Q: Exporting a variable with multipule values to CSV I'm using PowerShell to try and get a specific field from a CSV, store it as a variable, and output it as another csv. This is mainly because I want to use it as part of a larger script, but I'm having problems...
Import-Csv C:\EmailsListNoBlanks.csv | ForEach-Object{
$Email = $_.Member -split ';'
}
$Email | Out-File C:\EmailListCOMP.csv
However in my CSV I'm only ever getting the last 4 values, whereas I'm expecting a few hundred...
Is there something I'm missing here?
Thanks
Matt
A: I tried something similar to what you are attempting to do and this is what I came up with to have each item on it's own line in the output:
Import-Csv C:\EmailsListNoBlanks.csv | ForEach-Object {
$Email += ($_.Member -split ';') + ("`n")
}
$Email | Out-File C:\EmailListCOMP.csv
The output is not in a csv format, but just a regular text file. The `n adds a newline to the text so that the output is one entry per line.
A: Try this:
Add-Type -AssemblyName System.Collections
$Email = [System.Collections.Generic.List[string]]::new()
Import-Csv C:\EmailsListNoBlanks.csv | ForEach-Object{
[void]$Email.Add( $_.Member -split ';' )
}
$Email | Out-File C:\EmailListCOMP.csv
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,811
|
Mapping the ecological and biophysical character of seabed habitats of the Paraninihi Marine Reserve, Taranaki, New Zealand.
Habitat mapping is important for determining the spatial distribution of biological and physical components of the seabed. Conventional surveying methods, such as diver or drop camera surveys are time consuming and constrained by factors such as depth, water clarity, currents, and weather conditions, which means that is not practical to survey large tracks of sea floor using these methods. Consequently, a substantial proportion of the world's seafloor remains undescribed. In recent years, multibeam sonar (MBES) has revolutionised the way we image, map and understand the marine environment. However, the quantitative characterisation of MBES backscatter imagery for seafloor and habitat mapping remains a developing field. This thesis examines the utility of MBES backscatter imagery as a tool for the characterisation and mapping of biogenic habitats. Pariokariwa Reef, located within Paraninihi Marine Reserve, Northern Taranaki, was chosen as the location for this study because it supports a range of distinct habitats (including sponge gardens of unusually high biomass and diversity) against which to assess our ability to use MBES backscatter imagery to recognise biogenic seabed habitats.
This thesis describes the collection of spatially coincident MBES data (bathymetric and backscatter) within Paraninihi Marine Reserve and outlines techniques used to process and transform this data. Acoustic data was used to generate a predictive habitat map that was linked to the habitat classes derived from observations made on Pariokariwa Reef, over fine spatial scales. Results from the survey, showed MBES successfully produces high resolution bathymetric imagery that revealed the reefs unique morphology. The resolution of the backscatter imagery was fine enough to identify four dominant seabed classes on the reef, but not fine enough to accurately map heterogeneous habitat over small spatial scales. Results from the study suggest that image-based backscatter classification shows promise for the interpretation of MBES backscatter data, for the production of habitat maps. However, this study revealed a new challenge associated with habitat mapping, which is acoustic surveying over complex reef topography. Hence for complex or heterogeneous topographies, MBES data must be generated at a finer resolution in order to acquire the same level of detail that is available in predictive habitat models created from acoustic surveys conducted over flat, homogenous terrain.
I also examined the distribution of biological assemblages over a smaller spatial scale, to that examined using MBES. The purpose of this exercise was to test whether the reefs complex terrain influences biological community composition and distribution. Visual imagery obtained from drop camera and scuba diver surveys, revealed heterogeneous habitat over small spatial scales, across the morphology of the reef. Community composition and distribution significantly changed with reef aspect, with percentage sponge and biogenic reef appearing to be significantly higher over the vertical face of the reef, and within reef overhangs. Percentage silt was highest below the reef, and appears to be a dominant environmental factor influencing the composition and distribution of sponge communities on the Pariokariwa Reef. The findings from this study suggest multibeam sonar can be used as a tool to map biogenic seabed habitat. However, there are challenges associated with acoustic seabed classification across complex terrain, and therefore requires in situ surveys, conducted over smaller spatial scales.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 673
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<template name="afInputEmail_materialize">
<input type="email" value="{{this.value}}" {{atts}}/>
</template>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,902
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Crinia sloanei est une espèce d'amphibiens de la famille des Myobatrachidae.
Répartition
Cette espèce est endémique du Sud-Est de l'Australie. Elle se rencontre du centre de la Nouvelle-Galles du Sud jusqu'au Victoria. Elle est présente jusqu'à d'altitude.
Description
Crinia sloanei mesure en moyenne pour les mâles et pour les femelles.
Étymologie
Son nom d'espèce lui a été donné en l'honneur de Ian F. Sloane de la Savernake Station à Savernake en Nouvelle-Galles du Sud.
Publication originale
Littlejohn, 1958 : A new species of frog of the genus Crinia Tschudi from South Eastern Australia. Proceedings of the Linnean Society of New South Wales, , (texte intégral).
Liens externes
Notes et références
Anoure (nom scientifique)
Myobatrachidae
Faune endémique d'Australie
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,272
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AvesCenter auction information is available through Rich Site Summary (RSS) alerts or feeds, which will allow you to see auction summaries without having to check AvesCenter daily for updates.
With RSS, you can see AvesCenter auctions right on your desktop by installing a free RSS reader. Most RSS readers will check for updates once every hour.
AvesCenter will feed you regular updates on the latest auctions and your favorite sellers.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,232
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The first step is to collect the minimum constituents needed to start the equipment. You should need appliances such as case, power supply or board.
After completing these parts, you is likely to proceed to the assembly of the personal computer to soon enjoy your new acquisition. Fitting to start from the cover, that should be home for the another elements. After open the box you need to unscrew the flank panels and prepare it for installation of the first part: the motherboard (see complete offer at). The plate is mounted in a designated place for it or on a unexampled tray with threaded holes. The constructor provides case with that distance plugs, that have to be screwed into place to cover the holes placed in the mainboard-.
But you don't screwed! However, before professional secures the motherboard, you is likely to want to add some components on it, currently, even when located outside the cover. We will be substantially simpler and more space for movement will rapidly submit the entire PC. So, the board is placed on a flat area and install in the central process unit, memory and CPU cooling system.
To obtain the next part we can use sites such as software house international. How to set up a central processor? At the start you should unlock the clasp CPU socket. Beware to at the same time not to put your palm into the nest, because it is a delicate part and its damage is likely to for attractive deprive us of the opportunity to run the kit, and also mechanical destroy not covered under guarantee. A good central process unit may be purchased, for example, in the software house international (discover more here).
After opening the slot draw note to the processor. In the issue of boards and chips Intel have to at the small indentation on the brink of the laminate. The central processor must be placed in the cradle in such a method that these roaches have entered the guide tabs. This is just possible in specific position CPU, so don't try to fit it with force. After properly setting up the CPU time to complete the procedure, protecting the nest: near them like before it would open.
This is only the beginning of the lodging. Adding additional constituents might significantly turn up the power of the personal computer.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,710
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{"url":"http:\/\/www.my247.com.au\/melbourne\/Highpoint-Hoyts-Cinema.10590","text":"Highpoint - Hoyts Cinema\n\n# Highpoint - Hoyts Cinema\n\nHighpoint City\nInfo\nWhat's On\nMap\nPhotos\nContact\nReviews\ncontact Highpoint - Hoyts Cinema or make a booking\n[03] 9318 6800\nPost Highpoint - Hoyts Cinema to Facebook\nsession times\n\nThursday 14th, May :\n15:45, 21:20\nFriday 15th, May :\n15:45, 21:20\nSaturday 16th, May :\n15:45, 21:20\nSunday 17th, May :\n15:45, 21:20\nMonday 18th, May :\n15:45, 21:20\nTuesday 19th, May :\n15:45, 21:20\nWednesday 20th, May :\n15:45, 21:20\n\nThursday 14th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 18:40, 20:20\nThursday 14th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 18:40, 20:20, 21:20\nFriday 15th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 18:40, 20:20, 21:20\nSaturday 16th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 18:40, 20:20, 21:20\nSunday 17th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 18:40, 20:20, 21:20\nMonday 18th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 18:40, 20:20, 21:20\nTuesday 19th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 18:40, 20:20, 21:20\nWednesday 20th, May :\n10:15, 12:00, 13:00, 14:45, 15:45, 17:30, 18:30, 20:20, 21:20\n\n## 3D The Avengers: Age Of Ultron [CTC]\n\nMonday 27th, April :\n10:30, 13:45, 17:00, 20:15\nTuesday 28th, April :\n10:30, 13:45, 17:00, 20:15\nWednesday 29th, April :\n10:30, 13:45, 17:00, 20:15\n\n## Aerosmith Rocks Donington 2014 [CTC]\n\nFriday 22nd, May :\n21:00\n\n## Avengers-age-of-ultron [CTC]\n\nMonday 27th, April :\n09:40, 10:00, 11:15, 12:00, 12:30, 12:45, 12:50, 13:15, 14:30, 15:15, 15:30, 15:50, 16:00, 16:30, 18:00, 18:20, 18:40, 18:45, 19:15, 19:45, 20:45, 21:10, 21:15, 21:30\nMonday 27th, April :\n10:00, 11:15, 12:00, 12:20, 12:45, 12:50, 13:15, 14:30, 15:10, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 21:00, 21:15, 21:30, 21:50\nMonday 27th, April :\n10:00, 11:15, 12:00, 12:45, 12:50, 13:15, 14:30, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 21:15, 21:30, 21:50\nMonday 27th, April :\n11:15, 12:00, 12:30, 12:45, 13:15, 14:30, 15:15, 15:30, 16:00, 16:30, 18:00, 18:30, 18:45, 19:15, 19:45, 20:45, 21:15, 21:30\nMonday 27th, April :\n11:15, 12:00, 12:45, 13:15, 14:30, 15:15, 16:00, 16:30, 18:00, 18:45, 19:15, 19:45, 20:45, 21:15\nTuesday 28th, April :\n11:15, 12:00, 12:45, 13:15, 14:30, 15:15, 16:00, 16:30, 18:00, 18:45, 19:15, 19:45, 20:45, 21:15\nTuesday 28th, April :\n10:00, 11:15, 12:00, 12:45, 12:50, 13:15, 14:30, 15:10, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 21:15, 21:30, 21:50\nTuesday 28th, April :\n11:15, 12:00, 12:30, 12:45, 13:15, 14:30, 15:15, 15:30, 16:00, 16:30, 18:00, 18:30, 18:45, 19:15, 19:45, 20:45, 21:15, 21:30\nTuesday 28th, April :\n10:00, 11:15, 12:00, 12:20, 12:45, 12:50, 13:15, 14:30, 15:10, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 21:00, 21:15, 21:30, 21:50\nTuesday 28th, April :\n10:00, 11:15, 12:00, 12:10, 12:45, 12:50, 13:15, 14:30, 15:10, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 20:50, 21:15, 21:30\nWednesday 29th, April :\n10:00, 11:15, 12:00, 12:10, 12:45, 12:50, 13:15, 14:30, 15:10, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 20:50, 21:15, 21:30\nWednesday 29th, April :\n10:00, 11:15, 12:00, 12:20, 12:45, 12:50, 13:15, 14:30, 15:10, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 21:00, 21:15, 21:30, 21:50\nWednesday 29th, April :\n11:15, 12:00, 12:30, 12:45, 13:15, 14:30, 15:15, 15:30, 16:00, 16:30, 18:00, 18:30, 18:45, 19:15, 19:45, 20:45, 21:15, 21:30\nWednesday 29th, April :\n10:00, 11:15, 12:00, 12:45, 12:50, 13:15, 14:30, 15:10, 15:15, 15:50, 16:00, 16:30, 18:00, 18:40, 18:45, 19:15, 19:45, 20:45, 21:15, 21:30, 21:50\nWednesday 29th, April :\n11:15, 12:00, 12:45, 13:15, 14:30, 15:15, 16:00, 16:30, 18:00, 18:45, 19:15, 19:45, 20:45, 21:15\n\n## Bee Movie [G]\n\n88 minutes - Comedy \/ Family\nBarry B. Benson (voiced by Jerry Seinfeld) is a young bee eager to explore the world outside of his hive. Whilst discovering New York City, he meets florist Vanessa (Renee Zellweger), and becomes outraged at the selling of honey, leading to a lawsuit with buzz.\nSaturday 16th, May :\n10:30, 12:30, 14:30\nSunday 17th, May :\n10:30, 12:30, 14:30\nTuesday 19th, May :\n10:30, 12:30, 14:30\n\n## Book of Life [CTC]\n\nMonday 27th, April :\n10:00\nTuesday 28th, April :\n10:00\nWednesday 29th, April :\n10:00\n\n## Boychoir [CTC]\n\nMonday 27th, April :\n10:10, 12:30, 14:45, 19:15\nTuesday 28th, April :\n10:10, 12:30, 14:45, 19:15\nWednesday 29th, April :\n10:10, 12:30, 14:45, 19:15\n\n## Cinderella (2015) [PG]\n\n112 minutes - \/\nThe story of \"Cinderella\" follows the fortunes of young Ella whose merchant father remarries following the tragic death of her mother. Keen to support her loving father, Ella welcomes her new stepmother Lady Tremaine and her daughters Anastasia and Drizella into the family home. But, when Ella's father suddenly and unexpectedly passes away, she finds herself at the mercy of a jealous and cruel new family.\nMonday 27th, April :\n10:20, 12:15, 15:10, 17:45, 19:00, 20:15\nTuesday 28th, April :\n10:20, 12:15, 15:10, 17:45, 19:00, 20:15\nWednesday 29th, April :\n10:20, 12:15, 15:10, 17:45, 19:00, 20:15\nSaturday 2nd, May :\n12:15\nSunday 3rd, May :\n12:15\nSaturday 9th, May :\n12:15\nSunday 10th, May :\n12:15\nSaturday 16th, May :\n12:15\nSunday 17th, May :\n12:15\nSaturday 23rd, May :\n12:15\nSunday 24th, May :\n12:15\nSaturday 30th, May :\n12:15\nSunday 31st, May :\n12:15\n\n## Cobain: Montage of Heck [CTC]\n\nThursday 7th, May :\n18:30\nFriday 8th, May :\n21:00\nSaturday 9th, May :\n15:30, 21:00\nSunday 10th, May :\n18:30\n\n## The Duff [M]\n\nMonday 27th, April :\n12:50, 18:15\nTuesday 28th, April :\n12:50, 18:15\nWednesday 29th, April :\n12:50, 18:15\n\n## Fast & Furious 7 [CTC]\n\nAction\nMonday 27th, April :\n10:30, 12:20, 13:30, 15:20, 16:15, 18:20, 19:15, 21:20\nTuesday 28th, April :\n10:30, 12:20, 13:30, 15:20, 16:15, 18:20, 19:15, 21:20\nWednesday 29th, April :\n10:30, 12:20, 13:30, 15:20, 16:15, 18:20, 19:15, 21:20\n\n## Focus (2015) [MA15]\n\n104 minutes - Drama \/ Comedy\nNicky, a seasoned master of misdirection who becomes romantically involved with novice con artist Jess (Margot Robbie). As he?s teaching her the tricks of the trade, she gets too close for comfort and he abruptly breaks it off. Three years later, the former flame - now an accomplished femme fatale - shows up in Buenos Aires in the middle of the high stakes racecar circuit. In the midst of Nicky?s latest, very dangerous scheme, she throws his plans for a loop?and the consummate con man off his game.\nMonday 27th, April :\n21:20\nTuesday 28th, April :\n21:20\nWednesday 29th, April :\n21:20\n\n## Get Hard [MA15]\n\nMonday 27th, April :\n13:20, 17:00, 21:30\nTuesday 28th, April :\n13:20, 17:00, 21:30\nWednesday 29th, April :\n13:20, 17:00, 21:30\n\n## The Gunman [CTC]\n\nMonday 27th, April :\n16:20, 21:30\nTuesday 28th, April :\n16:20, 21:30\nWednesday 29th, April :\n16:20, 21:30\n\n## Home [CTC]\n\nMonday 27th, April :\n10:00, 12:00, 14:00, 16:00\nTuesday 28th, April :\n10:00, 12:00, 14:00, 16:00\nWednesday 29th, April :\n10:00, 12:00, 14:00, 16:00\nSaturday 2nd, May :\n10:00\nSunday 3rd, May :\n10:00\nSaturday 9th, May :\n10:00\nSunday 10th, May :\n10:00\nSaturday 16th, May :\n10:00\nSunday 17th, May :\n10:00\nSaturday 23rd, May :\n10:00\nSunday 24th, May :\n10:00\nSaturday 30th, May :\n10:00\nSunday 31st, May :\n10:00\n\n## Hot Tub Time Machine 2 [MA]\n\nComedy\nFriday 1st, May :\n21:00\nSaturday 2nd, May :\n21:00\n\n## How To Train Your Dragon [PG]\n\n97 minutes - Family\nSet in the mythical world of burly Vikings and wild dragons, and based on the book by Cressida Cowell, the action comedy tells the story of Hiccup, a Viking teenager who doesn?t exactly fit in with his tribe?s longstanding tradition of heroic dragon slayers. Hiccup?s world is turned upside down when he encounters a dragon that challenges he and his fellow Vikings to see the world from an entirely different point of view.\nSaturday 23rd, May :\n10:30, 12:30, 14:30\nSunday 24th, May :\n10:30, 12:30, 14:30\nTuesday 26th, May :\n10:30, 12:30, 14:30\n\n## Insurgent [CTC]\n\nDrama \/\nMonday 27th, April :\n10:15, 13:30, 18:40\nTuesday 28th, April :\n10:15, 13:30, 18:40\nWednesday 29th, April :\n10:15, 13:30, 18:40\n\n## Kung Fu Panda [PG]\n\n88 minutes - Comedy\nPo the Panda (Jack Black), a lowly waiter in a noodle restaurant, is a kung fu fanatic but whose shape doesn't exactly lend itself to kung fu fighting. In fact, Po's defining characteristic appears to be that he is the laziest of all the animals in ancient China. That's a problem because powerful enemies are at the gates, and all hopes have been pinned on a prophesy naming Po as the Chosen One to save the day.\nSaturday 2nd, May :\n10:30, 12:30, 14:30\nSunday 3rd, May :\n10:30, 12:30, 14:30\nTuesday 5th, May :\n10:30, 12:30, 14:30\n\n80 minutes - Comedy \/ Family\nThe mateship between a New York City lion and zebra is put to the test when fate brings them out to the real wilderness.\nSaturday 9th, May :\n10:30, 12:30, 14:30\nSunday 10th, May :\n10:30, 12:30, 14:30\nTuesday 12th, May :\n10:30, 12:30, 14:30\n\n## Paul Blart: Mall Cop 2 [CTC]\n\nComedy\nMonday 27th, April :\n10:15, 12:20, 14:30, 16:30, 18:30, 20:50\nTuesday 28th, April :\n10:15, 12:20, 14:30, 16:30, 18:30, 20:50\nWednesday 29th, April :\n10:15, 12:20, 14:30, 16:30, 18:30, 20:50\n\n## Pitch Perfect 2 [CTC]\n\nComedy \/\nWednesday 6th, May :\n19:00\nThursday 7th, May :\n10:00, 11:00, 11:30, 12:00, 12:45, 13:45, 14:00, 14:45, 15:45, 16:30, 16:45, 17:30, 18:30, 19:00, 19:15, 20:00, 20:30, 21:15, 21:30\nFriday 8th, May :\n10:00, 11:00, 12:00, 12:45, 13:30, 13:45, 14:45, 15:45, 16:00, 16:45, 17:30, 18:30, 19:00, 19:15, 20:00, 20:30, 21:00, 21:15\nSaturday 9th, May :\n10:00, 11:00, 12:00, 12:45, 13:30, 13:45, 14:45, 15:45, 16:00, 16:45, 17:30, 18:30, 19:15, 20:00, 20:30, 21:00, 21:15\nSunday 10th, May :\n10:00, 11:00, 12:00, 12:45, 13:30, 13:45, 14:45, 15:45, 16:00, 16:45, 17:30, 18:30, 19:15, 20:00, 20:30, 21:00, 21:15\nMonday 11th, May :\n10:00, 11:00, 12:00, 12:45, 13:30, 13:45, 14:45, 15:45, 16:00, 16:45, 17:30, 18:30, 19:15, 20:00, 20:30, 21:00, 21:15\nTuesday 12th, May :\n10:00, 11:00, 12:00, 12:45, 13:30, 13:45, 14:45, 15:45, 16:00, 16:45, 17:30, 18:30, 19:15, 20:00, 20:30, 21:00, 21:15\nWednesday 13th, May :\n10:00, 11:00, 12:00, 12:45, 13:30, 13:45, 14:45, 15:45, 16:00, 16:45, 17:30, 18:30, 19:15, 20:00, 20:30, 21:00, 21:15\n\n## Shaun The Sheep Movie [CTC]\n\nMonday 27th, April :\n10:10, 14:30\nTuesday 28th, April :\n10:10, 14:30\nWednesday 29th, April :\n10:10, 14:30\n\n## The Sound of Music 50th Anniversary [G]\n\nSunday 10th, May :\n15:00\n\n## SpongeBob SquarePants: Sponge Out of the Water [G]\n\nMonday 27th, April :\n11:00, 13:00, 15:00\nTuesday 28th, April :\n11:00, 13:00, 15:00\nWednesday 29th, April :\n11:00, 13:00, 15:00\nSaturday 2nd, May :\n14:30\nSunday 3rd, May :\n14:30\nSaturday 9th, May :\n14:30\nSunday 10th, May :\n14:30\nSaturday 16th, May :\n14:30\nSunday 17th, May :\n14:30\nSaturday 23rd, May :\n14:30\nSunday 24th, May :\n14:30\nSaturday 30th, May :\n14:30\nSunday 31st, May :\n14:30\n\n## Spy [TBC]\n\nWednesday 20th, May :\n18:45\n\n## The Age Of Adaline [CTC]\n\nMonday 27th, April :\n11:00, 16:00, 18:30, 21:00\nTuesday 28th, April :\n11:00, 16:00, 18:30, 21:00\nWednesday 29th, April :\n11:00, 16:00, 18:30, 21:00\n\n## The Longest Ride [PG]\n\nDrama\nMonday 27th, April :\n10:30, 15:50, 17:00, 19:45\nTuesday 28th, April :\n10:30, 15:50, 17:00, 19:45\nWednesday 29th, April :\n10:30, 15:50, 17:00, 19:45\n\n## Tinkerbell & The Legend Of The Neverbeast [G]\n\nMonday 27th, April :\n10:20\nTuesday 28th, April :\n10:20\nWednesday 29th, April :\n10:20\n\n## Tomorrowland [CTC]\n\nSci-fi\nMonday 25th, May :\n18:45\n\nmap","date":"2015-04-26 17:11:38","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9999939203262329, \"perplexity\": 11075.032561374976}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246655589.82\/warc\/CC-MAIN-20150417045735-00038-ip-10-235-10-82.ec2.internal.warc.gz\"}"}
| null | null |
Fusão nuclear é o processo no qual dois ou mais núcleos atômicos se juntam e formam um outro núcleo de maior número atômico. A fusão nuclear requer muita energia para acontecer, e geralmente libera muito mais energia do que a que consome. Quando ocorre com elementos mais leves que o ferro e o níquel (que possuem as maiores forças de coesão nuclear de todos os átomos, sendo portanto mais estáveis) ela geralmente libera energia, e com elementos mais pesados ela consome. Até hoje, início do século XXI, ainda não foi encontrada uma forma de controlar comercialmente a fusão nuclear, como acontece com a fissão, embora existam laboratórios de pesquisa que utilizam reatores de fusão nuclear em pesquisas científicas. Um projeto que caminha para a demonstração da viabilidade comercial do uso da fusão nuclear controlada é o ITER.
O principal tipo de fusão que ocorre no interior das estrelas é o de hidrogênio em hélio, onde quatro prótons se fundem em uma partícula alfa (um núcleo de hélio), liberando dois pósitrons, dois neutrinos e energia. Mas, dentro desse processo, ocorrem várias reações individuais, que variam de acordo com a massa da estrela. Para estrelas do tamanho do Sol ou menores, a cadeia próton-próton é a reação dominante. Em estrelas de massa elevada predomina o ciclo CNO.
A fusão de deutério e trítio (isótopos do hidrogênio) pode gerar nêutrons, o que é perigoso e torna o processo menos eficaz. A fusão aneutrônica (sem geração de nêutrons) é possível com o uso de elementos como o hélio-3 (raro na Terra), lítio e boro (abundantes na superfície terrestre), entre outros.
Vale ressaltar que há conservação da energia, e, portanto, pode-se calcular a massa dos quatro prótons e do núcleo de hélio, e subtrair a soma das massas das partículas iniciais daquela do produto desta reação nuclear para calcular a energia produzida.
Utilizando a equação E=mc², pode-se calcular a energia liberada, oriunda da diferença de massas. Uma vez que o valor de c é muito grande (cerca de 3×108 m/s), mesmo uma massa muito pequena corresponde a uma enorme quantidade de energia. Este fato levou muitos engenheiros e cientistas a iniciar projetos para o desenvolvimento de reatores de fusão (Tokamaks) de modo a gerar eletricidade (por exemplo, a fusão de poucos cm³ de deutério, um isótopo de hidrogênio, produziria uma energia equivalente àquela produzida pela queima de 20 toneladas de carvão).
Requisitos para a fusão
Uma substancial barreira de energia deve ser vencida antes que a fusão possa ocorrer. A grandes distâncias, dois núcleos expostos se repelem mutuamente devido à força eletrostática que atua entre seus prótons positivamente carregados. Se os núcleos puderem ser aproximados suficientemente, porém, a barreira eletrostática pode ser sobrepujada pela força nuclear forte a qual é mais poderosa a curta distância do que a repulsão eletromagnética.
Quando uma partícula tal como o próton ou nêutron é adicionado a um núcleo, ele é atraído pelos outros núcleons, mas principalmente por seus vizinhos imediatos devido à força de curto alcance. Os núcleons no interior do núcleo têm mais vizinhos do que aqueles na sua superfície. Desde que núcleos menores têm uma grande razão de superfície para volume, a energia de ligação por núcleon devido à força nuclear forte geralmente aumenta como o aumento do tamanho do núcleo, mas atinge um valor limite que corresponde à vizinhança do núcleon totalmente preenchida.
A força eletrostática, por outro lado, é uma força proporcional ao inverso do quadrado da distância; então, um próton adicionado ao núcleo ira sentir uma repulsão eletrostática de todos os prótons no núcleo. A energia eletrostática por núcleon devido à força eletrostática irá portanto aumentar independentemente do tamanho do núcleo.
O resultado combinado destas duas forças opostas é que a energia de ligação por núcleon geralmente aumenta com o aumento de tamanho do átomo, para elementos até com núcleo do tamanho de ferro e níquel, e diminui para núcleos mais pesados. Eventualmente, a energia de ligação se torna negativa e núcleos muitos pesados não são estáveis. Os quatro núcleos blindados mais compactos, em ordem decrescente de energia de ligação, são 62Ni, 58Fe, 56Fe e 60Ni . Embora o isótopo do Níquel 62Ni seja o mais estável, o isótopo do Ferro 56Fe é uma ordem de magnitude mais comum. Isto é devido em grande parte à grande razão de desintegração do 62Ni no interior de estrelas conduzida pela absorção de fótons.
Uma notável exceção a esta regra geral é o núcleo do hélio-4, cuja energia de ligação é maior que a do lítio, o próximo elemento mais pesado. O princípio de exclusão de Pauli provê um explicação para este comportamento excepcional – isto se dá porque os prótons e nêutrons são férmions, eles não podem coexistir exatamente no mesmo estado. Cada estado energético de um próton ou nêutron em um núcleo pode acomodar uma partícula de spin para abaixo e outra de spin para acima. O Hélio-4 tem uma banda de energia de ligação anormalmente grande porque seu núcleo consiste de dois prótons e dois nêutrons; então todos os núcleons dele podem estar em um estado fundamental. Qualquer núcleon adicional deverá ir para um estado energético alto.
A situação é similar se dois núcleos são colocados juntos. Ao se aproximarem, todos os prótons em um núcleo repelem todos os prótons do outro, até o ponto em que os dois núcleos entrem em contato para que a força nuclear forte domine. Consequentemente, mesmo quando o estado de energia final é mais baixo, há uma grande barreira energética que deve ser ultrapassada primeiro. Na química, este fato é conhecido como energia de ativação. Em física nuclear ele é chamado de barreira de Coulomb.
A barreira de Coulomb é menor para os isótopos do hidrogênio – eles contêm uma única carga positiva em seus núcleos. Um bipróton não é estável, então os nêutrons devem ser envolvidos, de forma a produzir um núcleo de hélio.
Usando combustível deutério-trítio, a barreira de energia resultante é de cerca de 0,1 MeV. Em comparação, a energia necessária para remover um elétron do hidrogênio é 13,6 eV, cerca vezes menos energia. O resultado (intermediário) da fusão é um núcleo instável de 5He, o qual imediatamente ejeta um nêutron com 14,1 MeV. A energia recuperada do núcleo de 4He remanescente é 3,5 MeV, então a energia total liberada é 17,6 MeV. Isto é muitas vezes mais que a barreira de energia a ser transposta.
Se a energia para iniciar a reação vem da aceleração de um núcleo, o processo é chamado de fusão por projétil-alvo; se ambos os núcleos são acelerados, isto é fusão projétil|projétil. Se o núcleo faz parte de um plasma próximo ao equilíbrio térmico, denominamos fusão termonuclear. A temperatura é uma medida da energia cinética média das partículas, então por aquecimento o núcleo deverá ganhar energia e eventualmente transpor a barreira de 0,1 MeV. A conversão das unidade entres elétron-volts e kelvins mostra que esta barreira será transposta quando a temperatura ultrapassar 1 GK, obviamente uma temperatura muito alta.
Há dois fatos que podem diminuir a temperatura necessária. Um é o fato que a temperatura é uma média da energia cinética, implicando que alguns núcleos a esta temperatura poderão já ter uma energia maior que 0,1 MeV, enquanto outros um pouco menos. Estes núcleos na faixa de alta-energia da distribuição de velocidade participam da maioria das reações de fusão. O outro efeito é o tunelamento quântico. O núcleo não precisa sempre ter bastante energia, podendo atravessar, por efeito túnel, a barreira restante. Por esta razão, combustíveis a temperaturas menores podem experimentar eventos de fusão, a uma taxa mais baixa.
A seção transversal da reação σ é uma medida da probabilidade de reação de fusão com uma função da velocidade relativa dos dois núcleos reativos. Se os núcleos têm uma distribuição de velocidade, isto é, uma distribuição térmica com a fusão termonuclear, então eles são úteis para obter uma média sobre a distribuição dos produtos da seção transversal e da velocidade. A taxa de reação (fusão por volume por tempo) é <σv> vezes o produto da densidade dos participantes:
Se um tipo de núcleo está reagindo com si próprio, tal como a reação PP, então o produto pode ser substituído por .
aumenta de praticamente zero a temperatura ambiente para um significativo valor a temperatura de 10 - 100 keV. A estas temperaturas, bem abaixo da energia de ionização típica (13,6 eV no caso do hidrogênio), os reativos da fusão existem um estado de plasma.
O significado de <σv> como uma função da temperatura em um experimento com uma energia de tempo confinamento é determinado pela utilização do critério de Lawson.
Processo de Fusão
O mecanismo de fusão é quase o inverso do mecanismo de fissão nuclear: núcleos leves e rápidos podem colidir, e fundir para formar núcleos mais pesados, sendo que há também uma quantidade considerável de energia liberada nesse processo. Essa energia está associada à dissipação de calor, depende diretamente das massas dos parceiros envolvidos na reação e tem suas propriedades relacionadas com a matéria nuclear, isto é, para que ocorra a fusão, alguns requisitos devem ser satisfeitos pelos parceiros envolvidos no processo:
1) a energia cinética dos núcleos da reação deve ser grande para possibilitar o aumento da probabilidade de penetração na barreira coulombiana; esse processo ocorre em núcleos muito leves, a uma temperatura da ordem de 107K , estando, então, os átomos completamente ionizados, prefigurando um estado de plasma.
2) a densidade de matéria presente nas temperaturas envolvidas na reação de fusão deve ser extremamente alta.
O interior das estrelas, em especial o sol, dispõe de todo cenário propício a esse tipo de reação, a densidade do interior do sol é de cerca de g/cm3 a uma temperatura de 1,5 x 107K. A Figura representa a reação de fusão de hidrogênio em hélio, que ocorre no interior das estrelas e que esteve presente no início da formação do universo, na nucleossíntese primordial
Conforme a temperatura, núcleos mais pesados podem ser formados.
As principais vantagens em relação aos atuais reatores de fissão são:
Combustível de fácil obtenção e em grande quantidade, o deutério pode ser obtido da água do mar e trítio obtido no próprio reator de fusão a partir do lítio, o urânio utilizado na fissão é muito raro e de difícil extração;
A fusão é um processo mais seguro que a fissão, uma vez que a quantidade de combustível empregado é menor, sem liberação descontrolada de energia e as taxas de radiação emitidas são inferiores à taxa de radiação natural que incide na superfície terrestre;
Menor produção de lixo nuclear comparado à fissão, além do que o lixo proveniente da fusão não é matéria prima para fabricação de armas nucleares, como no caso da fissão. Atualmente, a NASA tem investido em pesquisas na construção de reatores nucleares de fusão para gerar energia para foguetes espaciais. Propulsores a fusão seriam mais eficientes e tornariam os foguetes mais velozes, além de propiciar viagens mais longas, uma vez que o combustível (hidrogênio) seria gerado de forma ilimitada no processo.
Fusão em plasma
Em primeiro lugar, recordemos que a colisão de dois núcleos de deutério gera um núcleo de Hélio mais um nêutron e libera uma energia de 5,12 x 10−13 Joules (3,2 Mev). Se esta energia fosse transferida para um grama de água, na forma de calor, a temperatura da água aumentaria de apenas 1,26 x 10−13 °C. Portanto, para se ter um aumento significativo de temperatura da água, gerar vapor e movimentar as turbinas de uma Usina de Energia, necessitamos de um número muito grande de reações de Fusão.
Resta então a questão: Como obter este grande número de reações? A resposta óbvia é: coloque o maior número possível de núcleos de deutério em condições de reação. Muito fácil de responder, mas anos e anos de pesquisa em física de plasma demonstram que é muito difícil fazê-lo.
Para entender as dificuldades vamos tomar, apenas por hipótese, uma certa quantidade de átomos de deutério em estado sólido. Obviamente, um grama de deutério tem um número muito grande de átomos que, se reagissem, forneceriam muita energia. No entanto, os átomos de deutério em estado sólido estão praticamente parados e não têm energia cinética suficiente para vencer a repulsão coulombiana. Portanto, não estão em condições de realizar uma reação de fusão.
Para vencer a repulsão coulombiana deve-se aumentar a energia cinética dos átomos de deutério, o que pode ser feito aquecendo-se o sólido. Ao aumentarmos a temperatura, o sólido sofre uma transição de fase transformando-se primeiramente num líquido e depois num gás. Num gás, uma percentagem grande das partículas tem uma energia cinética próxima da energia cinética média que é proporcional à temperatura:
(onde k é a constante de Boltzmann e T é a temperatura medida em kelvin). Assim, para vencer a repulsão coulombiana, o nosso gás de deutério deve estar a uma temperatura de aproximadamente graus Celsius. (Isto corresponde a uma energia cinética média de 10 keV.)
Esta temperatura elevada traz consigo algumas perguntas. Como aquecer um gás a esta temperatura? Como confinar um gás tão quente? Será que a matéria não se modifica a temperaturas tão altas? As duas primeiras perguntas parecem ter uma natureza tecnológica, no entanto, a sua solução só poderá ser obtida se soubermos mais sobre a terceira indagação cuja natureza científica é evidente. Um primeiro aspecto a ser considerado é que, após uma certa temperatura, um gás usualmente constituído de átomos e moléculas sofre transformações, pois os elétrons são arrancados dos átomos e as moléculas se quebram devido à violência dos choques. Em temperaturas da ordem de a °C não haverá mais átomos e moléculas, mas apenas íons e elétrons viajando e se chocando em velocidades fantásticas.
Estes íons e elétrons não mais se comportarão como um gás, visto que, além das colisões, sentirão os efeitos do campo elétrico e magnético devido às suas cargas e correntes. Isto caracteriza um novo estado da matéria denominado plasma pelos físicos americanos Langmuir e Tonx em 1923.
Portanto, em busca das condições adequadas de confinamento e temperatura para ocorrência de fusões nucleares, nos deparamos naturalmente com este novo estado da matéria que é o plasma. Um estudo das características do plasma vai nos permitir inclusive entender como é possível manter uma certa quantidade de substância confinada a temperaturas tão altas.
Características fundamentais do plasma e suas implicações
Um plasma se caracteriza por ser um gás altamente ionizado, quase neutro e não se encontrar em equilíbrio térmico. A primeira característica (alta ionização) já foi discutida. A quase neutralidade se refere ao fato de que, embora a carga total num plasma (cargas positivas dos íons mais cargas negativas dos elétrons) seja praticamente nula, existem regiões onde se pode ter acúmulos significativos de cargas formando zonas não neutras.
As regiões onde isso ocorre têm dimensões pequenas em comparação com as dimensões totais do plasma. O acúmulo de cargas (positivas ou negativas) vai afetar as colisões entre os íons e elétrons, pois cria pontos de atração e/ou repulsão e estabelece campos de força. Deste modo, o movimento de uma partícula se modificará apenas por choques com contato direto, mas poderão ainda sentir os efeitos da presença de partículas distantes através dos campos de força.
A quase-neutralidade pode ainda gerar movimentos coerentes de um grande número de partículas. Estes movimentos, denominados movimentos coletivos, ocorrem, por exemplo, quando um número grande de íons (cargas positivas) se separa de um número grande de elétrons. Nesta situação, surgem forças atrativas que tendem a restaurar a neutralidade, isto é, aproximam as cargas opostas. Isto causa um movimento oscilatório no qual as cargas opostas se aproximam e se afastam. A aplicação de campos externos pode também gerar movimentos coletivos tais como correntes ou mesmo ondas. Portanto, um plasma difere muito de um gás, pois neste último as partículas só sentem a presença das outras quando sofrem uma colisão. Num plasma as interações de longo alcance geradas pelos campos fazem com que os movimentos de partículas distantes sejam correlacionados. Existem dentro de um plasma dois processos competitivos: de um lado os movimentos coletivos e do outro as colisões.
As colisões tendem a destruir a coerência, isto é, a natureza ordenada dos movimentos coletivos, pois espalham as partículas erraticamente. Num projeto de fusão nuclear em plasma se pretende obter uma solução de compromisso entre os dois processos. Isto é, pretende-se utilizar a coerência dos movimentos coletivos para propiciar um número grande de colisões que gerem fusão. Como os dois processos são antagônicos esta solução de compromisso não é fácil.
Ver também
Energia de fusão
Fissão nuclear
Ligações externas
Instituto Nacional de Pesquisas Espaciais - Laboratório Associado de Plasmas
Instituo de Física da USP - Laboratório de Física de Plasmas
Unicamp, Instituto de Física Gleb Wataghin - Grupo de Física de Plasmas e Fusão Termonuclear Controlada
FURG, Instituto de Matemática, Estatística e Física - Laboratório de Plasma
Instituto Superior Técnico, Centro de Fusão Nuclear - Tokamak ISTTOK
Fusão nuclear
Conceitos fundamentais da física
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Q: JAVA: passing an input or output stream to ITextRenderer (xhtml to pdf converter) I want to convert my XHTML text to a PDF. I converted it to FileOutputStream but I ca'nt find a way to pass it as an input to the ITextRenderer. Is that possible, and how?
the code :
String finalXhtml=xhtmlparser(xmlText);
ByteArrayInputStream finalXhtmlStream = new ByteArrayInputStream(finalXhtml.getBytes());
String HTML_TO_PDF = "ConvertedFile.pdf";
OutputStream os = new FileOutputStream(HTML_TO_PDF);
ITextRenderer renderer = new ITextRenderer();
// renderer.loadDocument(finalXhtmlStream); i can pass a file here can i pass an input or output stream ?
renderer.layout();
renderer.createPDF(os) ;
os.close();
System.out.println("done.");
note: I can pass a file to the ITextRenderer as following:
String File_To_Convert = "report.xhtml";
String url = new File(File_To_Convert).toURI().toURL().toString();
String HTML_TO_PDF = "ConvertedFile.pdf";
OutputStream os = new FileOutputStream(HTML_TO_PDF);
ITextRenderer renderer = new ITextRenderer();
renderer.setDocument(url);
renderer.layout();
renderer.createPDF(os);
os.close();
System.out.println("done.");
please let me know if I have to provide more details.
A: I am using following code to export HTML data to PDF with following code:
renderer.setDocumentFromString(htmls.toString());
renderer.layout();
response.setContentType("application/octet-stream");
response.setHeader("Content-Disposition", "attachment;filename=\"" + fileName + ".pdf\"");
renderer.createPDF(outputStream);
renderer.createPDF(fos);
Now here I am using inline CSS to generate PDF using style but is there any option that I can use setDocumentFromString() function by loading external CSS.
A: I am assuming you are using Flying Saucer. ITextRenderer has a method that does something similar:
public void setDocumentFromString(String content) {
InputSource is = new InputSource(new BufferedReader(new StringReader(content)));
Document dom = XMLResource.load(is).getDocument();
setDocument(dom, null);
}
Adapting your code, what you'd want would look something like this:
String finalXhtml=xhtmlparser(xmlText);
ByteArrayInputStream finalXhtmlStream = new ByteArrayInputStream(finalXhtml.getBytes());
String HTML_TO_PDF = "ConvertedFile.pdf";
OutputStream os = new FileOutputStream(HTML_TO_PDF);
Document document = XMLResource.load(finalXhtmlStream).getDocument();
ITextRenderer renderer = new ITextRenderer();
renderer.setDocument(document, null);
renderer.layout();
renderer.createPDF(os) ;
os.close();
of course you could also do this and skip the inputstream all together:
renderer.setDocumentFromString(finalXhtml);
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,092
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Q: Flutter how to validate any ssn using RegExp I want to validate my ssn using
FilteringTextInputFormatter.allow(RegExp()),
I want it to start with 1 or 2 with 10 length and last 9th numbers any number
,only first one should be 1 or 2
A: TextFormField(
inputFormatters: [
LengthLimitingTextInputFormatter(10),
FilteringTextInputFormatter.allow(RegExp(r'^[12][0-9]*$')),
],
decoration: InputDecoration(
border: InputBorder.none,
labelText: 'PAN Number',
),
)
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 567
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Q: Sqoop incremental load using Informatica BDM I am new to Informatica BDM.I have a use case in which I have to import the data incrementally (100 tables) from RDBMS into Hive on daily basis. Can someone please guide me with the best possible approach to achieve this?
Thanks,
Sumit
A: Hadoop is write onces read many (WORM) approach and the incremental load is not easy stuff. There are following guideline you can follow and validate your current requirement
*
*If the table is a small/mid-size and not having too many records,
better to refresh the entire table
*If the table is too big and incremental load has add/update/delete operation, you can think of staging the delta and perform a join operation to re-create data set.
*For large table and large delta, you can create a version number for all the latest record and each delta may come to a new directory and a view should be created to get the latest version for further processing. This avoid heavy merge operation.
If the delete operation is not coming as change, then you also need to think how to act on it and in such case, you need to get the full refresh.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,304
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The portrait Equestrian Portrait of Elisabeth of France was painted by Velázquez of Elisabeth of France circa 1635, originally for the Hall of Realms, originally a wing of the Buen Retiro Palace in Madrid. It has been in the Prado since the gallery's institution in 1819.
History of the work
Velázquez had been commissioned to paint a series of five equestrian portraits of the royal family
Felipe III and his wife Queen Margaret of Austria,
Felipe IV, his wife Elizabeth of France and their son Baltazar Carlos. This last was smaller than those of the other family members, as it was intended to be hung on a door and therefore viewed from a lower perspective.
This work of Velázquez had significant input from members of his workshop. The technical studies conducted in the Prado Museum under the direction of Carmen Garrido indicated that the five equestrian portraits were painted at the same time and with the same preparation. The idea that Velázquez retouched a painting by an earlier painter to add detail to the queen's clothing and the trappings of the horse has received support from a number of critics; X-rays reveal a painting under the visible one, in which the horse's girth is visible and the queen's clothing is simpler than the existing one. Later, when Velázquez was finishing details in the queen's head and the horse's legs, a more patient painter filled in the meticulous details of the embroidery, thus obscuring details of the painting previously laid down.
Description of the work
The queen is depicted in profile, wearing a jacket with embroidered stars and a gold-embroidered skirt with her arms and initials. The horse is depicted as an overo with a long mane and forelock falling over its face, doing the passage gait; it is facing left in order to provide symmetry with the portrait of Elisabeth's husband, in which work he is facing right.
Elisabeth
Elisabeth of France
1636 paintings
Elisabeth of France
Elisabeth of France
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,338
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Q: Можно ли без костылей узнать результат отработки магического метода __set? К сожалению, __set не возвращает ничего. Остается модифицировать сессию, левую переменную-флаг или плеваться исключениями. Что делают большие дяди в том случае, если присутствует медод __set, но изменять через него запрошенную переменную они не хотят, при этом не хотят и аварийно завершать всю систему?
Мне ближе именно исключения, но народ поговаривает, что они не самый лучший выход для корректировки нормального процесса, а if сюда прикручивается только через костыли.
A: С моей точки зрения, исключения -- естественный способ обработки ошибочных ситуаций в языках, которые достаточно хорошо их поддерживают. (Не знаю, относится ли это к php.) В мануале ошибочная ситуация обрабатывается через trigger_error.
Выброс исключения должен быть частью контракта, и пользователи класса должны быть готовы к такому повороту событий. (Пользу от обработки ошибок исключениями, думаю, вам не нужно объяснять.) Не дело класса думать, хочет ли он завершить обработку аварийно или нет; его дело -- отрапортовать ошибку исключением, а уж окружеющий код пусть решает, ловить это исключение или нет.
Другое дело, что инфраструктура окружающего кода должна соответствовать исключениям, и состояние программы не должно оставаться неопределённым. В C++ для такого традиционно используется идиома RAII, в C# -- using и блок finally.
Мне кажется, установка глобальной переменной при ошибках провоцирует плохой стиль.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,851
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Q: How to generate thumbnail from .eps and .ai files? How can I generate thumbnails from .eps and .ai (and .psd)? files and save them as a picture?
A: For .eps, try IrfanView with its POSTSCRIPT plugin
A: If you are in Windows you can use the "Microsoft Document Image Writer" printer to save any document as a TIFF image File. This file can be opened in any picture/image viewer and converted.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,601
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<?php
namespace Amazon\InstantAccess\Serialization;
use Amazon\InstantAccess\Serialization\Enums\FulfillPurchaseResponseValue;
/**
* Serializable response object used to fulfill a purchase
*
* @see Amazon\InstantAccess\Serialization\Enums\FulfillPurchaseResponseValue For response values
*/
class FulfillPurchaseResponse extends InstantAccessResponse
{
/**
* Set the response content
*
* @param string a string representation of the response
*
* @see Amazon\InstantAccess\Serialization\Enums\FulfillPurchaseResponseValue For response values
*/
public function setResponse($response)
{
if (!FulfillPurchaseResponseValue::isValid($response)) {
throw new \InvalidArgumentException(sprintf('Invalid response value: %s', $response));
}
$this->response = $response;
return $this;
}
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,598
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Firefighters on scene of apartment building fire in St. John's
February 19, 2014 February 19, 2014 John Stamp
Firefighters and police are on the scene of an apartment building fire in the centre of St. John's.
The first call came in shortly after 4:15 p.m. with reports of smoke on the third floor of the building for seniors on Shaw Street.
Flames could also be seen shooting from the rear of the building.
CBC reporter Amy Stoodley said it was difficult to see the building through the smoke and snow, but added she could hear glass breaking.
Some elderly residents of the building were taken to hospital to be treated for smoke inhalation, while others were being escorted to buses for transport to temporary accommodations.
http://www.cbc.ca/news/canada/newfoundland-labrador/firefighters-on-scene-of-apartment-building-fire-in-st-john-s-1.2543757
179 seniors have been evacuated and five have been sent to hospital following 911 calls at 4 p.m. that said the building was on fire.
Superintendent Don Byrne says the fire broke out on the third floor, and there is extensive damage to that part if the building. Thirteen ambulances were on the scene, some from as far away as Clarenville that just happened to be in town today. Metrobuses lined up to take residents to city hall, where they are staying warm and being treated by paramedics.
The extent of injuries of those in hospital is not yet known. RNC are investigating the cause of the fire, but say it's too early to know anything yet.
http://www.vocm.com/newsarticle.asp?mn=2&ID=43608
IAFF Partners With Operation Warm
UPDATE: Residents being transported from scene of apartment fire
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,732
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Program Mariner byl program řízený americkou vesmírnou agenturou NASA, jehož účelem byl průzkum nejbližších planet (Merkuru, Venuše a Marsu) automatickými sondami.
Úkol a plnění programu
Průzkum pomocí nepilotovaných sond měl být prováděn při průletu, případně z oběžné dráhy planet. Vyšší technická úroveň oproti dřívějším programům je umožňovala lépe stabilizovat v prostoru a díky vývoji stále silnějších nosných raket zvyšovat jejich hmotnost a tedy i přístrojové vybavení.
Program uskutečněný v období roku 1962 až 1973 byl splněn dobře. Z deseti vyslaných sond osm dosáhlo cíle a poskytlo Zemi cenné fotografie a další údaje ze všech tří cílových planet. Na získané zkušenosti pak navázaly další sondy Viking a Voyager.
Seznam sond Mariner
Odkazy
Reference
Externí odkazy
Web MEK
Sondy NASA
Mariner
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{
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| 4,863
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Q: Osascript custom function not interpreting command string properly I have this function that I've written:
function alert {
command='display alert '
content="${1} message ${2}"
concat=$command$content
osascript -e "${concat}"
}
When executed like:
alert 'Title' 'Message'
I get the error message:
25:27: syntax error: A end of script can't go after this "my". (-2740)
Why is that the case?
For reference this command works perfectly:
osascript -e 'display alert "Title" message "Message"'
A: You're not putting quotes around the message and title in content. So they're being treated as variable names by the OSAScript interpreter.
function alert {
command='display alert '
content="\"${1}\" message \"${2}\""
concat=$command$content
osascript -e "${concat}"
}
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,640
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Incentive La Marvellous - Shining Star Music Video (Produced by AnnoDominination)!!!
Added by Michael Patrick on December 26, 2018 at 1:45pm
The track 'Shining Star' by Incentive La Marvellous is the featured hit single from the artist's latest EP entitled 'Love Over Lust' insync with his debut music video release as of December 25th 2018!!!
Shining Star was written about a special lady in my life who has become tired of me rehearsing songs about my ex-girlfriends!
Incentive La Marvellous is a local RnB Songwriter from Canada specializing in vocal Hip-Hop Fusion's where melody brings penmanship and music making to it's fullest!!!
https://www.youtube.com/watch?v=54TVMY6Ncgc
https://www.facebook.com/IncentiveLaMarvellous
https://soundcloud.com/incentive-lamarvellous/sets/love-over-lust
https://www.reverbnation.com/incentivelamarvellous
www.oceanfreshproductions.com
Atty Cassandra Daniels Exposes The Child Support…
MY FAVORITE RAP SONGS OF 2021
Suzax feat GB Dix - Accroche-toi Clip Officiel
Rap Insurance (Official Video)
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{
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var dbm = global.dbm || require('db-migrate');
var type = dbm.dataType;
var fs = require('fs');
var path = require('path');
exports.up = function (db, callback) {
var filePath = path.join(__dirname + '/sqls/20161003092733-fix-user-get-target-time-up.sql');
fs.readFile(filePath, { encoding: 'utf-8' }, function (err, data) {
if (err) return callback(err);
console.log('received data: ' + data);
db.runSql(data, function (err) {
if (err) return callback(err);
callback();
});
});
};
exports.down = function (db, callback) {
var filePath = path.join(__dirname + '/sqls/20161003092733-fix-user-get-target-time-down.sql');
fs.readFile(filePath, { encoding: 'utf-8' }, function (err, data) {
if (err) return callback(err);
console.log('received data: ' + data);
db.runSql(data, function (err) {
if (err) return callback(err);
callback();
});
});
};
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"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/www.maths.usyd.edu.au\/ut\/pub-seek.py?au1=Joshi+N&pp=preprint&pgsz=50","text":"# Publication Search Results\n\nMatches for:\n\n\u2022 Author=Joshi N\n\n2. Ramani A, Joshi N, Grammaticos B, Tamizhmani T. Deconstructing an integrable lattice, (2006), preprint\n\n5. Joshi N, Kitaev AV, Treharne PA. On the linearization of the first and second Painlev\u00e9 equations, (2008), preprint\n\n7. Atkinson J, Joshi N. Singular-boundary reductions of type-Q ABS equations, International Mathematics Research Notices, 7 (2013), 1451\u20131481.\n\n8. Atkinson J, Joshi N. Singular-Boundary Reductions of Type-Q ABS Equations, International Mathematics Research Notices, 24 (2012), no. Advance Access, 31 pages.\n\n9. Atkinson J, Joshi N. The Schwarzian variable associated with discrete KdV-type equations, Nonlinearity, 25 (2012), 1851\u20131866.\n\n10. Joshi N, Shi Y. Exact solutions of a q-discrete second Painlev\u00e9 equation from its iso-monodromy deformation problem. II. Hypergeometric solutions, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468 (2012), no. 2146, 3247\u20133264.\n\n11. Joshi N, Shi Y. Exact solutions of a q-discrete second Painlev\u00e9 equation from its iso-monodromy deformation problem: I. Rational solutions, Proceedings of the Royal Society A, 467 (2011), 3443\u20133468.\n\n12. Duistermaat JJ, Joshi N. Okamoto's Space for the First Painlev\u00e9 Equation in Boutroux Coordinates, Archive for Rational Mechanics and Analysis, 202 (2011), no. 3, 707\u2013785.\n\n13. Wood LN, Vu T, Bower M, Brown N, Skalicky J, Donovan D, Loch B, Joshi N, Bloom W. Professional development for teaching in higher education, International Journal of Mathematical Education in Science and Technology, 42 (2011), no. 7, 997\u20131009.\n\n14. Kassotakis P, Joshi N. Integrable Non-QRT Mappings of the Plane, Letters in mathematical physics, 91 (2010), no. 1, 71\u201381. MR2577300\n\n15. Ramani A, Grammaticos B, Joshi N. Second-degree discrete Painlev\u00e9 equations conceal first-degree ones, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 175207 (9pp).\n\n16. Brown N, Bower M, Skalicky J, Wood L, Donovan D, Loch B, Bloom W, Joshi N. A professional development framework for teaching in higher education, Refereed papers from the 33rd HERDSA Annual International Conference, Research and Development in Higher Education: Reshaping Higher Education, HERDSA Conferences, Higher Education Research and Development Society of Australasia, Inc., Australia, (2010), 133 \u2013 143. ISBN 9780908557806\n\n17. Butler S, Joshi N. An inverse scattering transform for the lattice potential KdV equation, Inverse Problems, 26 (2010), 28 pages.\n\n18. Joshi N. Direct 'delay' reductions of the Toda equation, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 022001\u2013022009.\n\n19. Joshi N, Morrison T. Existence and uniqueness of Tronqu\u00e9e solutions of the fourth-order Jimbo-Miwa second Painlev\u00e9 equation, Proceedings of the American Mathematical Society, 137 (2009), no. 6, 2005\u20132014. MR2480282\n\n20. Joshi N, Lafortune S, Ramani A. Hirota bilinear formalism and ultra-discrete singularity analysis, Nonlinearity, 22 (2009), 871\u2013887. MR2486361\n\n21. Joshi N, Spicer P. Direct \"Delay\" Reductions of the Toda Hierarchy, Journal of the Physical Society of Japan, 78 (2009), no. 9, 094006\u20131 to 094006\u20135.\n\n22. Joshi N, Kitaev AV, Treharne PA. On the linearization of the first and second Painlev\u00b4e equations, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 055208. MR2525382\n\n23. Field CM, Joshi N, Nijhoff FW. $$q$$-difference equations of KdV type and Chazy-type second-degree difference equations, Journal of Physics A: Mathematical and Theoretical, 41 (2008), no. 33, 13 pages.\n\n24. Joshi N, Morrison T. New Exact Solutions of Spatially and Temporally Varying Reaction-Diffusion Equations, Analysis and Applications, 6 (2008), no. 4, 371 \u2013 381. MR2459116\n\n25. Hay M, Hietarinta J, Joshi N, Nijhoff F. A Lax pair for a lattice modified KdV equation, reductions to $$q$$-Painlev\u00e9 equations and associated Lax pairs, Journal of Physics A: Mathematical and Theoretical, 40 (2007), no. 2, F61\u2013F73. MR2303490\n\n26. Joshi N, Ormerod C. The general theory of linear difference equations over the max-plus semiring, Studies in Applied Mathematics, 118 (2007), 85\u201397. MR2283569\n\n27. Clarkson PA, Joshi N, Mazzocco M. The Lax Pair for the MKDV Hierarchy, Th'eories asymptotics et equations de Painlev'e, Seminaires et Congres, Societe Mathematique de France, France, (2007), 53\u201364. ISBN 978-2-85629-229-7\n\n28. Miura RM, Ablowitz MJ, Costin O, Joshi N, Kulsrud R, Zabusky NJ. Obituaries: Martin David Kruskal, SIAM News, 40 (2007), no. 3,\n\n29. Joshi N, Kitaev AV, Treharne PA. On the linearization of the Painlev\u00e9 III\u2013VI equations and reductions of the three-wave resonant system, Journal of Mathematical Physics, 48 (2007), no. 10, 103512 (42pp).\n\n30. Joshi N, Lafortune S. Integrable ultra-discrete equations and singularity analysis, Nonlinearity, 19 (2006), 1295\u20131312. MR2230000\n\n31. Ramani A, Joshi N, Grammaticos B, Tamizhmani T. Deconstructing an integrable lattice equation, Journal of Physics. A, 39 (2006), no. 8, L145\u2013L149. MR2209299\n\n32. Gordoa PR, Joshi N, Pickering A. Second and fourth Painlev\u00e9 hierarchies and Jimbo-Miwa linear problems, Journal of Mathematical Physics, 47 (2006), 073504. MR2250304\n\n33. Joshi N, Grammaticos B, Tamizhmani T, Ramani A. From integrable lattices to Non-QRT mappings, Letters in Mathematical Physics, 78 (2006), no. 1, 27\u201337. MR2271126\n\n34. Joshi N. Painlev\u00e9 Equations, Encyclopaedia of Mathematical Physics, 4 (2006), 1\u20135.\n\n35. Joshi N, Kajiwara K, Mazzocco M. Generating function associated with the Hankel determinant formula for the solutions of the Painlev\u00e9 IV equation, Funkcialaj Ekvacioj, 49 (2006), 451\u2013468. MR2297948\n\n36. Joshi N, Pickering A. Generalized Halphen systems, Proceedings of the Royal Society of Edinburgh, 136A (2006), 1287\u20131301. MR2290134\n\n37. Joshi N. Solitons, Encyclopedia of Nonlinear Science, 1 (2005), 849\u2013852.\n\n38. Joshi N. Asymptotics for extended cellular automata, Recent trends in exponential asymptotics, Recent Trends in Exponential Asymptotics, Yoshitsugu Takei (ed.), RIMS Kokyuroku Series, Research Institute for Mathematical Sciences, Kyoto, Japan, (2005), 156\u2013159.\n\n39. Joshi N, Lafortune S. How to detect integrability in cellular automata, Journal of Physics. A. Mathematical and General, 38 (2005), L499\u2013L504. MR2166619\n\n40. Joshi N, Kitaev AV. The Dirichlet boundary value problem for real solutions of the first Painlev\u00e9 equation on segments in non-positive semi-axis, Journal f\u00fcr die Reine und Angewandte Mathematik, 583 (2005), 29\u201386. MR2146852\n\n41. Gordoa PR, Joshi N, Pickering A. B\u00e4cklund transformations for fourth Painlev\u00e9 hierarchies, Journal of Differential Equations, 217 (2005), 124\u2013153. MR2170530\n\n42. Joshi N. The second Painlev\u00e9 hierarchy and the stationary KdV hierarchy, Publications of the Research Institute for Mathematical Sciences, 40 (2004), 1039\u20131061. MR2074710\n\n43. Joshi N, Nijhoff FW, Ormerod C. Lax pairs for ultra-discrete Painlev\u00e9 cellular automata, Journal of Physics. A. Mathematical and General, 37 (2004), L559\u2013L565. MR2098040\n\n44. Kruskal MD, Joshi N, Halburd RG. Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: A Review and Extensions of Tests for the Painlev\u00e9 Property, Integrability of Nonlinear Systems, Lecture Notes in Physics, Springer\/Verlag, Berlin, Heidelberg, (2004), 175\u2013208. ISBN 3-540-20630-2\n\n45. Joshi N, Kajiwara K, Mazzocco M. Generating function associated with the determinant formula for the solutions of the Painlev\u00e9 II equation, Ast\u00e9risque, 297 (2004), 67\u201378. MR2135675\n\n46. Joshi N, Mazzocco M. Existence and uniqueness of tri-tronqu\u00e9e solutions of the second Painlev\u00e9 hierarchy, Nonlinearity, 16 (2003), 427\u2013439. 2004a:34172\n\n47. Maruno KI, Ohta Y, Joshi N. Exact localized solutions of quintic discrete nonlinear Schr\u00f6dinger equation, Physics Letters. A, 311 (2003), 214\u2013220. 2004f:39047\n\n48. Joshi N. Hunting Mathematical Butterflies, Nonlinear Dynamics, From Lasers to Butterflies, World Scientific Lecture Notes in Complex Systems, 1 World Scientific, Singapore, (2003), 77\u2013114. ISBN 981-238-320-4 MR2014643\n\n49. Gordoa PR, Joshi N, Pickering A. A new technique in nonlinear singularity analysis, Publications of the Research Institute for Mathematical Sciences, 39 (2003), 435\u2013449. MR2001184\n\n50. Joshi N. Tritronqu\u00e9e solutions of perturbed first Painlev\u00e9 equations, Theoretical and Mathematical Physics, 137 (2003), 1515\u20131519. MR2057895\n\n Number of matches: \u00a092 Page 1 of 2 Select page: 1\u00a0 2","date":"2013-05-22 09:48:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7658945322036743, \"perplexity\": 7309.704554963229}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368701562534\/warc\/CC-MAIN-20130516105242-00072-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
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Washington pie is a cake (not a pie) found in American cuisine. The earliest known recipes date to the 1850s. It may be an antecedent of Boston cream pie. The earlier Washington pie was a sandwich cake of two yellow sponge layers filled with jam and dusted with powdered sugar, later evolving into the better known chocolate-covered pastry cream filled version. Raspberry jam was a common choice, though any type of jelly could be used. Additional flavorings like kirsch or rosewater could be added. Some modern variations use a vanilla cream filling topped with cherry pie filling and whipped cream. The sugar can be sprinkled using a lace doily to create a decorative pattern.
Slices of Washington pie were sold at stalls at the Northern Liberty Market on Mount Vernon Square until the market was demolished on the orders of Alexander Robey Shepherd in 1872.
Another version, closer to a true pie, was made by soaking pieces of stale and leftover cake with cream, and adding raisins and spices and baking it in pastry crust, always in square pans. It's been described as a "nefarious fraud", but was served at elegant hotels and boarding houses in the Northeastern United States, more commonly than other pies, and it was also sold in quantity by bakers. This version eventually developed a poor reputation during the American Civil War when it started to be sold at coffee stands and other establishments around docks and railroad depots, because "certain bakers, in their efforts to produce great quantities of it, were not so very careful as to what it was composed of. Some bakers got to making it out of stale bread and the like".
Washington pie was served at a Fourth of July celebration in Decatur, Nebraska in 1856. Washington pie was recommended as part of a menu for a Fourth of July party for young ladies in 1901, along with other desserts.
According to a 1934 article published in the Detroit Free Press, the cake layers could be made with cottage pudding instead of traditional sponge cake.
See also
List of cakes
References
American cakes
Sponge cakes
Layer cakes
Independence Day (United States) foods
Historical foods in American cuisine
Foods with jam
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{"url":"https:\/\/uwaterloo.ca\/combinatorics-and-optimization\/events\/tutte-colloquium-matthias-mnich","text":"## University COVID-19 update\n\n### Questions about buildings and services? Visit the list of Modified Services.\n\nPlease note: The University of Waterloo is closed for all events until further notice.\n\n# Tutte Colloquium - Matthias Mnich\n\nFriday, October 19, 2018 \u2014 3:30 PM EDT\n\nTitle:\u00a0New algorithms for maximum disjoint paths based on tree-likeness\n\n Speaker: Matthias Mnich Affiliation: University of Bonn Room: MC 5501\n\nAbstract:\n\nWe study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP\/MaxNDP is currently not well understood; the best known lower bound is $2\u03a9(\u221alog n), assuming$\\mbox{NP}\\subsetneq \\mbox{DTIME}(n^{O(log n)})\\$. This constitutes a significant gap to the best known approximation upper bound of O(\u221an)\u00a0due to Chekuri et al. (2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (1987) introduce the technique of randomized rounding for LPs; their technique gives an O(1)-approximation when edges (or nodes) may be used by O(log n\/log log n)\u00a0paths.\n\nWe strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:\n\n(1) For MaxEDP, we give an O(\u221arlog(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio O(\u221an)\u00a0due to Chekuri et al., as r\u2264n.\n\n2) Further, we show how to route \u03a9(OPT*) pairs with congestion bounded by O(log(kr)\/log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson.\n\n(3) For MaxNDP, we give an algorithm that gives the optimal answer in time (k+r)O(r)n. This is a substantial improvement on the run time of 2krO(r)n, , which can be obtained via an algorithm by Scheffler.\n\nWe complement these positive results by proving that MaxEDP is NP-hard even for r=1, and MaxNDP is W[1]-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless FPT=W[1]and that our approximability results are relevant even for very small constant values of r.\n\nAppeared in Math. Programming, September 2018. Joint work with Joachim Spoerhase (Aalto) and Krzysztof Fleszar (Warsaw).\n\nLocation\nMC - Mathematics & Computer Building\n5501\n200 University Avenue West\n\nWaterloo, ON N2L 3G1\n\n### April 2021\n\nS M T W T F S\n28\n29\n30\n31\n2\n3\n4\n5\n6\n7\n10\n11\n13\n14\n17\n18\n20\n21\n22\n24\n25\n27\n28\n29\n30\n1\n1. 2021 (40)\n1. April (9)\n2. March (13)\n3. February (8)\n4. January (10)\n2. 2020 (119)\n1. December (5)\n2. November (12)\n3. October (12)\n4. September (12)\n5. August (11)\n6. July (17)\n7. June (11)\n8. May (6)\n9. March (11)\n10. February (11)\n11. January (11)\n3. 2019 (167)\n4. 2018 (136)\n5. 2017 (103)\n6. 2016 (137)\n7. 2015 (136)\n8. 2014 (88)\n9. 2013 (48)\n10. 2012 (39)\n11. 2011 (36)\n12. 2010 (40)\n13. 2009 (40)\n14. 2008 (39)\n15. 2007 (15)","date":"2021-04-22 03:54:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5119982361793518, \"perplexity\": 4491.692174537278}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039560245.87\/warc\/CC-MAIN-20210422013104-20210422043104-00250.warc.gz\"}"}
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Supplementary MaterialsPeer Review File 41467_2017_2057_MOESM1_ESM
Supplementary MaterialsPeer Review File 41467_2017_2057_MOESM1_ESM. persist inside the human cells up to one day without affecting cell viability. Phage capsid integrity is lost in lysosomes, and the phage DNA is eventually degraded. We did not detect the entry of phage DNA into the nucleus; however, we speculate that this might occur as a rare event, and propose that this potential mechanism could explain prokaryoteCeukaryote gene flow. Introduction The evolution of cellular life is tightly bound to viruses that use host organisms to complete their life cycle. Bacteriophages, viruses that infect bacteria, are the most numerous replicating entities in the biosphere, with an estimated global population of 1031 phage particles1, 2. Phages play fundamental roles in bacterial ecology and virulence3. Their ability to package DNA fragments of the host genome during phage propagation makes them powerful vehicles for horizontal gene transfer, a dominant process in microbial evolution4. It has been estimated that phages mediate over 1016 gene transfer events each second5. In the face of omnipresent phage-rich environments, animals frequently come into contact with phages. Host mucosal surfaces are densely populated by residential microbial communities that consist largely of bacteria. Within FGF1 this setting, the phage populations are dominating the viral community in the gut6, 7 and have an important contribution to bacterialChost interactions8, 9. Single observations suggest that interdomain genetic exchanges Prasugrel (Effient) from bacteria to eukaryotes have occurred during evolution10C12. Bacterium-to-eukaryote horizontal gene transfer events are suggested to provide novel traits important in conferring advantages for specific niches, such as genes encoding metabolic enzymes13, 14. However, the mechanisms that permit the acquisition of genetic variability via interdomain transfers remain elusive. The cell membrane acts as a barrier between the aqueous cytoplasm and the outside environment, and this efficiently delimits the transfer of molecules, Prasugrel (Effient) such as DNA, across the membrane. Unlike prokaryotes, eukaryotes lack mechanisms for uptake of free DNA from the environment. While it is generally assumed that the enormous reservoir of genetic diversity encompassed by phages is restricted within the borders of the prokaryotic world, evidence is accumulating that gene flow through phages is potentially a horizontal gene transfer pathway between prokaryotes and eukaryotes15C17. In line with this, phage genes have under experimental conditions been integrated into the genome of eukaryotic cells18. Phage genes can also be expressed in eukaryotic cells19C21. While it has been previously shown that phage lambda is capable of transducing mammalian cells20, 21, there is currently no direct evidence demonstrating a specific mechanism by which phages traverse the eukaryotic membrane and enter nonphagocytic cells, and thereby open a door for gene transfer. Here, we show that bacteriophage bound specifically to a mammalian cell receptor can pass the cell membrane Prasugrel (Effient) barrier and be internalized by means of endocytic vesicles. The access to the cell could conceivably provide an entry port for the introduction of foreign genetic material into the cell, even though we did not detect the entry of phage DNA into the cell nucleus. The phageCeukaryotic cell interaction reported here expands the functional capacity of phages and support that phages represent an unexplored factor in the evolution of eukaryotes. Results Binding of bacteriophage to a target on neuroblastoma cells The bacteriophage PK1A2, a member of the family and variant of PK1A, was originally isolated by its ability to bind bacteria containing reduced amounts of its polysaccharide receptor, the K1 polysialic acid capsule22 consisting of 2,8-linked N-acetylneuraminic acid units. The bacterial receptor structure is identical to polysialic acid present on mammalian cells23 and protects the bacterial cell against the immune system during invasive infection24. Compared to the PK1A phage with Prasugrel (Effient) catalytic endosialidase as a tailspike protein, PK1A2 has two amino acid substitutions in the endosialidase that abolish the catalytic activity but still permit polysialic acid binding25. PK1A2 phage is able to recognize and.
← Supplementary MaterialsSupplementary Document → Trafficking of myelin-reactive CD4+ T-cells across the mind endothelium, an essential step in the pathogenesis of multiple sclerosis (MS), is suggested to be an antigen-specific process, yet which cells provide this transmission is unknown
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{"url":"http:\/\/asomef.org.co\/o9gmu2tt\/molar-mass-and-percent-composition-worksheet-d460f8","text":"Find its empirical formula. CJ UVaJ j! For a solution, mass percent equals the mass of an element in one mole of the compound divided by the molar mass of the compound, multiplied by 100%. Mass percent composition is also known percent by weight. 3 mol O=3 \u00d7 16.00 g\/mol= 48.00 g\/mol. Title: Molar Mass worksheet Author: ochs.tf.t Last modified by: Windows User Created Date: 12\/13\/2019 4:13:00 PM 1. Some of the worksheets displayed are Percent composition by mass work, Chemistry computing formula mass work, Percent composition and molecular formula work, Percent composition work ii, Molar mass work, Lwtech learning lab science molar mass, Percent composition work 1, Molar mass practice work. Answer the following questions: 9. Determine the empirical formula of acetic acid. What's the empirica] formula of a molecule containing 65.5% carbon, S. 5% hydrogen, and 29.0% oxygen? 7. Percent Composition Worksheet 1. The percent composition Percent Composition Homework the relative mass of each element in a compound. 2. So for p = 62 * 100 \/ 182 . A compound has an empirical formula of C2H30 and a molar mass of 172 g\/mol. For a solution, mass percent equals the mass of an element in one mole of the compound divided by the molar Worksheet 11.1 KEY Molar Mass Calculations A mole is a standard unit of measurement for amount of a substance. Molecular Mass And Percent Composition mass percent = (mass of solute \/ mass of solution) x 100%. Chemistry: Molar Mass and Percentage Composition KEY Calculate the molar masses and percentage composition of each of the following compounds. a. A similar unit of concentration is molality (m), which is defined as the number of moles of solute per kilogram of solvent, not per liter of solution: $molality\\: =\\: \\frac{moles\\: solute}{kilograms\\: solvent}$ Chemistry: Percentage Composition and Empirical & Molecular Formula. CHEMISTRY COMPUTING FORMULA MASS WORKSHEET Problem Set-up example: Find the formula mass of Ca(NO3)2 Ca: 1 x 40.1 = 40.1 N: 2 x 14.0 = 28.0 O: 6 x 16.0 = 96.0 ____ 10. Solve the following problems. Essential concepts: Molar mass, percent composition. A compound is found to contain 36.5% Na, 25.4% S, and 38.1% O. So Ca = 120*100\/182 . Percent By Mass. 8. Ca3P2. Calculate Percent By Mass And Percent By Volume - Displaying top 8 worksheets found for this concept.. 116.3 g, 20.9% Mg, 24.1% N, 55.0% O 4. A compound has a molar mass of 86 g\/mol and has the percent composition (by mass) of 55.8% C, 37.2% O, and 7.0% H. Determine the empirical formula and the molecular formula. Percent composition and molar mass. Calculate the percent composition of each element for the following compounds. 2. 13. 136.2 g, 29.4% Ca, 23.6% S, 47.0% O 8. Molecular Mass And Percent Composition Mass percent composition describes the relative quantities of elements in a chemical compound. CJ UVaJ j U j\ufffd CJ EH\ufffd\ufffdH*OJ QJ U^J jU\ufffdC This worksheet has 8 problems to solve. Show your work and always include units. C2F2Br2O2. \ufffd\ufffd\ufffd\ufffd \u017e\ufffd \ufffd \ufffd\ufffd\u0462\ufffd\ufffd\u05a2\u0599\ufffd\ufffd\u058d|\ufffd\ufffd\ufffd\ufffdp_\ufffd\ufffd\ufffd j6 CJ EH\ufffd\ufffdH*OJ QJ U^J j1\ufffdC Purpose:\u00a0The percent composition of a compound is the percent, by mass, of each individual element within the compound. Answer the following questions about carbon dioxide, CO 2. Ca(OH) 2 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 3. mol-1\u201d, depending on how your teacher likes to see it written. Solve the following problems. If the molar mass of the compound in problem 1 is 110 grams\/mole, what's the molecular formula? a. Answer sheet included. 142.1 g, 32.4% Na, 22.6% S, 45.0% O 7. Show your work, and always include units where needed. C 3H 3O 2) If the molar mass of the compound in problem 1 is 110 grams\/mole, what\u2019s the molecular formula? CJ EH\ufffd\ufffdH*OJ QJ U^J j\ufffdC Some of the worksheets displayed are Percent composition by mass work, Chemistry computing formula mass work, Percent composition and molecular formula work, Percent composition work ii, Molar mass work, Lwtech learning lab science molar mass, Percent composition work 1, Molar mass practice work. The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. The units of mass are typically grams. b. Nomenclature for Simple Inorganic Coumpounds; Worksheets w\/Solutions. Some of the worksheets displayed are Percent composition by mass work, Chemistry computing formula mass work, Ch 11 ws 3 molarity molality percent solution, Work, Chemistry i work name calculating formula mass, Molar mass work, Percent composition by mass, Percent composition work ii. For more practice with these skills, try the game! A compound has a molar mass of 100 g\/mol and the percent composition (by mass) of 65.45% C, 5.45% H, and 29.09% O. Mg(NO3) 2 8. 12. CJ UVaJ j UCJ H*OJ QJ ^J j CJ H*OJ QJ U^J jO CJ EH\ufffd\ufffdH*OJ QJ U^J $The units of mass are typically grams. Calculate the molar mass for the following compounds SiF 4 KOH 2. Ca3P2 2. You calculate the answer Some of the worksheets for this concept are Percent composition and molecular formula work, Molar mass work, Percent composition and molecular formula work, Formula or molar mass work epub, Chemistry computing formula mass work, Formula work, Empirical and molecular formula work, Student work extra practice questions molar. Mole conversion practice mass to moles 1 step pdf. The molar mass is the sum of the masses of all the atoms in one mole of the compound. formula worksheet 1 h8n and a molar mass of 46 grams per mole answer the of 100 g mole if the percent composition is' 'Moles Molar Mass And Percentage Composition Percent Composition Name Date Show all work for finding the molar mass AND the percent composition requested. Page I of I 'Hour 0)MDlar Mass = The molar mass for question #9 was determined by experiment to be 60.0 g\/mol. ' ( ) * + , 6 7 8 \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd$a$\ufffd \ufffd 8 9 : ; J K L M N O ^ _ a b c q r s t u v ~ \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd 4 \ufffd \ufffd G H \ufffd \ufffd \ufffd \ufffd & * 7 8 \ufffd \ufffd \ufffd \ufffd + , 6 7 \ufffd \ufffd \ufffd \ufffd > 386.2 g, 50.8% Zn, 16.1% P, 33.1 % O 3. C 6H 6O 2 _____ C8H8O2. 2) A compound with an empirical formula of C4H4O and a molar mass of 136 grams per mole. 7. The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. Answer: Molar mass is 22.99 + 35.45 = 58.44 g\/mol Percent composition: %Na = (22.99 \/ 58.44) K 100% = 39.34 % Na 1) What oercent ofMgB\u00bd is magnesium? Molar Mass and Percentage position Worksheet for 9th 12th from Percent Composition Worksheet Answer Key With Work, source: lessonplanet.com. 4O and a molar mass of 136 grams per mole. CJ UVaJ j\ufffd( CJ EH\ufffd\ufffdH*OJ QJ U^J ja&\ufffdC 18. 74.6g, 52.4% K, 47.6% Cl Chemistry: Molar Mass and Percentage Composition KEY Calculate the molar masses and percentage composition of each of the following compounds. Molar Mass And Comp - Displaying top 8 worksheets found for this concept.. Answers: 1. A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. Mass percent is also known as percent by weight or w\/w%. Percent Composition Name Date Show all work for finding the molar mass AND the percent composition requested. (NH4) 2SO4. Mass percent is also known as percent by weight or w\/w%. Percent By Mass - Displaying top 8 worksheets found for this concept.. CaSO4. ' ( ) * + , 4 6 7 > ? Percent Composition Notes and Practice. The molar mass for question #9 was determined by experiment to be 60.0 g\/mol. Percent Composition and Molecular Formula Worksheet I. Percent Composition and Molecular Formula Worksheet I. Chemistry: Molar Mass and Percentage Composition KEY Calculate the molar masses and percentage composition of each of the following compounds. CJ UVaJ j\ufffd EH\ufffd\ufffdUjV\ufffdC Find the formula mass of the following compounds. 19. It is abbreviated as w\/w%. 10. 5. The sum all the mass percentages should add up to 100%. (NH4) 2SO4 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 6. Answer the following questions: 9. C D v w \ufffd \ufffd Show your work and always include units. This Molecular Mass and Percent Composition Worksheet is suitable for 9th - 12th Grade. CJ UVaJ j\ufffd CJ EH\ufffd\ufffdH*OJ QJ U^J js\ufffd\ufffdC 7. 1 mol Fe= 1 x 55.85 g\/mol= 55.85 g\/mol. R S T U Y Z m n o p u v \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd\ufffd\ufffd \ufffd \ufffd\ufffd\ufffd \ufffd\u00b8\u02f8\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdo\ufffd\ufffd\ufffd c\\\ufffd \ufffd j\ufffd# EH\ufffd\ufffdUj\ufffd\ufffd\ufffdC 4) A compound with an empirical formula of C2H8N and a molar mass of 46 grams per mole. Chemistry: Molar Mass and Percentage Composition. KCl. # \ufffd\ufffd\ufffd\ufffd o \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd \ufffd bjbj\ufffd \ufffd \". Showing top 8 worksheets in the category - Mass Percent Chemistry. Show your work and always include units. Zn3(PO4) 2 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 7. \ufffd\ufffd\u0871\ufffd > \ufffd\ufffd ! 3 mol O=3 \u00d7 16.00 g\/mol= 48.00 g\/mol. \" #$ % ) * 4 7 8 ? Find the empirical formula of a compound that is 53.7% iron and 46.3% sulfur. Determine the % by mass \u2026 182.3 g, 66.0% Ca, 34.0% P 5. Some of the worksheets for this concept are molar mass work molar mass work answer key chemistry computing formula mass work ws molar mass chemistry 11 mole conversions molar mass work molar mass \u2026 Mass percent composition is also known percent by weight. A compound has a molar mass of 86 g\/mol and has the percent composition (by mass) of 55.8% C, 37.2% O, and 7.0% H. Determine the empirical formula and the molecular formula. C8H8O2. CaSO4 5. Ca(OH) 2 3. KCl Answers: 1. 1. Ca 3 P 2 g 182.3 P Ca g 62.0 g\/mol 31.0 @ P 2 g 120.3 g\/mol 40.1 @ Ca 3 2 3 Ca 66.0% 100 g 182.3 g 120.3 Ca % P 34.0% 100 g 182.3 g 62.0 P % 2. Percent Composition Notes and Practice. Ca(OH) 2. CJ UVaJ j CJ H*OJ QJ U^J CJ H*OJ QJ ^J CJ OJ QJ ^J CJ OJ QJ ^J jQ EH\ufffd\ufffdUj}\ufffdC 4O and a molar mass of 136 grams per mole. One of the two worksheets is viewable in the Preview so you can see that the questions will be great practice for your students. 3. Ca3P2 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 2. In this chemical compounds worksheet, students calculate the molar masses and the percentage composition for the given compounds. 8. Ca3P2 molar mass is : Ca [ 40 * 3 ] + P [ 31*2] = 182 u . Useful for AQA C2.3. Round atomic masses to the tenth of a decimal place. A compound with an empirical formula of CzHEN and a molar mass of 46 grams per mole. The molar mass for question #9 was determined by experiment to be 60.0 g\/mol. Molar Mass Practice Worksheet Find the molar masses of the following compounds: 1) NaBr 2) PbSO 4 3) Ca(OH) 2 4) Na 3PO 4 5) (NH 4) 2CO 3 6) C 6H 12O 6 7) Fe 3(PO 4) 2 8) (NH 4) 2S 9) Zn(C 2H 3O 2) 2 10) AgF . Example: What percent ofNaCl is sodium? Nicotine is 74.1% carbon, 8.6% hydrogen, and 17.3% nitrogen by mass. Percent Review Worksheet Worksheets for all from Percent Composition Worksheet Answer Key With Work, source: bonlacfoods.com Figure $$\\PageIndex{3}$$: Table salt, NaCl, contains an array of sodium and chloride ions combined \u2026 1. C2F2Br2O2. For instance, the percent composition of oxygen in water is 89%. 2. Percent Review Worksheet Worksheets for all from Percent Composition Worksheet Answer Key With Work, source: bonlacfoods.com CJ UVaJ j\ufffd CJ EH\ufffd\ufffdH*OJ QJ U^J j\ufffd\ufffd\ufffdC The percent composition worksheet. Show your work and always include units. Zn3(PO4) 2. How to Calculate Mass Percent Composition Percent composition can also be used to determine the mass of a certain element that is contained in any mass \u2026 'Molar Mass And Percent Composition Worksheet FREE May 13th, 2018 - Hi Searching For Molar Mass And Percent Composition Worksheet You Are Specifically Here Possibly You Came Via Internet Search Engine After That You Find This Web Site As Well As Chose To See This Website Many Thanks For That''South Pasadena Chemistry chemmybear com 6. \ufffdj \ufffdj \ufffd \ufffd\ufffd \ufffd\ufffd \ufffd\ufffd l \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd ^ ^ ^ ^ D \ufffd $\ufffd \ufffd \ufffd 2 \ufffd \ufffd \ufffd 6 8 8 8 8 8 8$ m \ufffd \ufffd \\ \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \\ _ \ufffd \ufffd \ufffd q _ _ _ \ufffd . CaSO4 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 5. Elements Compounds Mixtures Worksheet; Calculations. If the molar mass of the compound in problem 1 is 110 grams\/mole, what's the molecular formula? percentage Composition : mass of the element in the compund * 100 \/ molar mass of the compound . 3. Percent composition is used to find the percentage of elements in a compound. 1. Answer the following questions about carbon dioxide, CO 2. 12. : \u00b4*. Mg(NO2) 2 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 8. CaCO 3 CH 4 Sn(CO 3) 2 3. 3. Worksheet 11.3 Percent Composition 3. , ; \ufffd \ufffd ; 6 _ \ufffd \ufffd \ufffd \ufffd \ufffd Name: ________________________ Hour: ____ Date: _______________ Chemistry: Molar Mass and Percentage Composition Calculate the molar masses and percentage composition of each of the following compounds. 18. The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. Detailed answer key is included. The molar mass is the sum of the masses of all the atoms in one mole of the compound. CJ UVaJ CJ H*OJ QJ ^J j CJ H*OJ QJ U^J j CJ EH\ufffd\ufffdH*OJ QJ U^J o p q v w \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd A compound with an empirical formula of C 2H 8N and a molar mass of 46 grams per mole. Worksheet 11.1 KEY Molar Mass Calculations A mole is a standard unit of measurement for amount of a substance. Answer: Molar mass is 22.99 + 35.45 = 58.44 g\/mol Percent composition: %Na = (22.99 \/ 58.44) K 100% = 39.34 % Na 1) What oercent ofMgB\u00bd is magnesium? \u00c4 \u00c4 \u00ff\u00ff\u00ff\u00ff 2 2 2 8 j D \u00ae 4 2 \u00ad, \u00e0 \u00e2 \u00e2 \u00e2 \u00e2 \u00e2 \u00bd \u00bd \u00bd P, R, R, R, R, R, R, \ufffd. What is the molecular formula. Its molar mass is \u2026 74.1 g, 54.1% Ca, 43.2% O, 2.7% H 6. chlorite. The mass percent is the mass of an element in a compound expressed as a percentage of the total mass of the compound. 15.03: Solution Concentration - Molality, Mass Percent, ppm and ppb Last updated; Save as PDF Page ID 178209; No headers. Find the empirical formula of a compound that is 53.7% iron and 46.3% sulfur. This worksheet is a good application of molar mass, as students will calculate the mass of the individual atoms of familiar compounds, such as baking soda. O and a molar mass of 136 grams per mole. 182.3 g, 66.0% Ca, 34.0% P 5. CJ UVaJ CJ OJ QJ ^J CJ OJ QJ ^J CJ OJ QJ ^J j\ufffd EH\ufffd\ufffdUj\ufffd\ufffdC Formula mass \/ molecular weight \/ moles \/ % percentage by mass worksheets with range of difficulties. The answers appear after the final question. \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\u057a\ufffd\u02df\ufffd\ufffd\u02eb\u02c2q\ufffd\ufffd \ufffd e^\ufffd \ufffd j\ufffd. Na2SO4 4. Calorimetry Worksheet; Mixed Gas Laws Worksheet; Molar Mass Worksheet; Mole Calculations Worksheet 1; Mole Calculations Worksheet 2; Percent Composition Worksheet Nomenclature Handouts. 7. 2. For a solution, mass percent equals the mass of an element in one mole of the compound divided by the molar mass of the compound, multiplied by 100%. Step 1:Find the molar mass of the compound. 10. 10. The second worksheet has similar questions, with different compounds. CJ UVaJ j U j\ufffd EH\ufffd\ufffdUj\ufffd\ufffd\ufffdC 3) A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. 8. 132.1 g, 21.2% N, 6.1% H, 24.3% S, 48.4% O 2. \u00b2 ?1 \u00b2 R, \u00bd \u00bd \u00bd \u00bd \u00bd R, \u00e2 \u00e2 \u00db g, \u00c5 \u00c5 \u00c5 \u00bd \u201a \u00e2 \u00e2 P, \u00c5 \u00bd P, \u00c5 \u00c5 \u00c6 \\$* \ufffd \u00d8+ \u00e2 \u00ff\u00ff\u00ff\u00ff \u011e >\u00d9\u00b0\u011e \u00ff\u00ff\u00ff\u00ff ? C 3H 3O 2) If the molar mass of the compound in problem 1 is 110 grams\/mole, what\u2019s the molecular formula? Nicotine is 74.1% carbon, 8.6% hydrogen, and 17.3% nitrogen by mass. Calculate the percent composition of carbon for the following compounds. bold atom and ONLY the bold atoms. Some of the worksheets for this concept are Percent composition by mass work, Chemistry computing formula mass work, Ch 11 ws 3 molarity molality percent solution, Work, Chemistry i work name calculating formula mass, Molar mass work, Percent composition by mass, Percent composition \u2026 EH\ufffd\ufffdUjo!\ufffdC Show your work, and always include units where needed. 8. Essential concepts:Molar mass, percent composition. Its molar mass is about 160 g.\/mol. I I. Ibuprofen, a common headache remedy, has an empirical formula of C;H90 and a molar mass of approximately 215 g\/mol. Show your work, and round answers to the ones place. This Molar Mass and Percentage Composition Worksheet is suitable for 9th - 12th Grade. The molar mass for question 9 was determined by experiment to be 600 gmol. Answer the following questions: 9. N 2 O Sr(NO 3) 2 NH 4 NO 3 4. 16 divided by 18 is .89. Purpose: The percent composition of a compound is the percent, by mass, of each individual element within the compound. Determine the empirical formula of acetic acid. Determine the empirical formula of acetic acid. 4) A compound with an empirical formula of C2H8N and a molar mass of 46 grams per mole. I I. Ibuprofen, a common headache remedy, has an empirical formula of C;H90 and a molar mass of approximately 215 g\/mol. Calculate the percent composition of each element for the following compounds. \ufffd \ufffd 6 _ \ufffd 6 _ _ \ufffd \ufffd \ufffd \ufffd 6 \ufffd U\ufffd\ufffd\ufffd\ufffd\ufffd Z ^ \ufffd 6 \ufffd 0 \ufffd CJ UVaJ #\ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd In this molecular mass and percent composition activity, students complete 5 sections where they determine atomic masses of given elements, they determine formula masses for given molecules, they find the molecular mass of 12 molecules and they determine the percent composition of 10 compounds. Percent By Mass - Displaying top 8 worksheets found for this concept.. Chemistry: Percentage Composition and Empirical & Molecular Formula. Percent Composition and Molecular Formula Worksheet Solutions 1) What\u2019s the empirical formula of a molecule containing 65.5% carbon, 5.5% hydrogen, and 29.0% oxygen? 3) A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. A compound with an empirical formula of CzHEN and a molar mass of 46 grams per mole. I use this game instead of a worksheet to build students' confidence and fluency in these problems before we move \u2026 Calculate the molar masses and percentage composition of each of the following compounds. 2) A compound with an empirical formula of C4H4O and a molar mass of 136 grams per mole. Place your final answer in the FORMULA MASS COLUMN. It is abbreviated as w\/w%. In this moles and molecules worksheet, students answer fourteen questions about moles, mass, and the mole and they solve eight problems for molecular formulas, empirical formulas and percent composition. N 2 O Sr(NO 3) 2 NH 4 NO 3 4. 3. Find the formula mass of the following compounds. CHEMISTRY COMPUTING FORMULA MASS WORKSHEET Problem Set-up example: Find the formula mass of Ca(NO3)2 Ca: 1 x 40.1 = 40.1 N: 2 x 14.0 = 28.0 O: 6 x 16.0 = 96.0 ____ Title: Microsoft Word - WS-moles_molar_mass.doc Author: acrosby Created Date: 10\/4\/2007 8:50:46 PM A compound has a molar mass of 100 g\/mol and the percent composition (by mass) of 65.45% C, 5.45% H, and 29.09% O. C 6H 6O 2 _____ Percent composition is the percent by mass of each element found in a compound. This Molar Mass and Percentage Composition Worksheet is suitable for 9th - 12th Grade. Show your work, and round answers to the ones place. The molar mass of water is 18g\/mol, and oxygen has a molar mass of 16g\/mol. Answer the following questions: 9. @ S T U V Z [ n o \ufffd\ufffd\ufffd\ufffd\ufffd\u02b9\ufffd\ufffd \ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\u0586\ufffd\ufffd\ufffdzi\ufffd\ufffd\ufffd\ufffd] j\ufffdC CJ UVaJ j CJ H*OJ QJ U^J !CJ OJ QJ ^J eh r\ufffd \ufffd CJ H*OJ QJ ^J CJ OJ QJ ^J CJ OJ QJ ^J j U j>&. Ca 3 P 2 g 182.3 P Ca g 62.0 g\/mol 31.0 @ P 2 g 120.3 g\/mol 40.1 @ Ca 3 2 3 Ca 66.0% 100 g 182.3 g 120.3 Ca % P 34.0% 100 g 182.3 g 62.0 P % 2. Mg(NO3) 2. (NH4) 2SO4 6. A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. 132.1 g, 21.2% N, 6.1% H, 24.3% S, 48.4% O. Determine the empirical formula of acetic acid. b. Percent Composition by Mass Find the percent composition by mass the . One must know the molar mass of the elements and the compound in order to get percent composition. A compound with an empirical formula of CFBrO and a molar mass of 254.7 grams per mole. Example: What percent of iron (III) hydroxide, Fe(OH) 3, is oxygen? Zn3(PO4) 2 7. What's the empirica] formula of a molecule containing 65.5% carbon, S. 5% hydrogen, and 29.0% oxygen? 'Molar Mass And Percent Composition Worksheet FREE May 13th, 2018 - Hi Searching For Molar Mass And Percent Composition Worksheet You Are Specifically Here Possibly You Came Via Internet Search Engine After That You Find This Web Site As Well As Chose To See This Website Many Thanks For That''South Pasadena Chemistry chemmybear com Example: What percent ofNaCl is sodium? Molar Mass and Percent Composition Card Game Potassium permanganate \u2013 (Oxygen) Calcium. 1. CJ UVaJ j\\, CJ EH\ufffd\ufffdH*OJ QJ U^J jK!\ufffdC Mass percent composition describes the relative quantities of elements in a chemical compound. This Chapter 8 Worksheet-Mass, Mole, Percent Composition Worksheet is suitable for 10th - 12th Grade. CJ UVaJ CJ OJ QJ ^J CJ OJ QJ ^J jX EH\ufffd\ufffdUjf\ufffdC Round atomic masses to the tenth of a decimal place. The molar mass for \u2026 Percent composition and molecular formula worksheet. A compound has an empirical formula of C2H30 and a molar mass of 172 g\/mol. Mole Mass Problems Worksheet Answers Lovely Mole Mass Problems from molar mass worksheet answers , source:thefriendlyghosthunters.net The Molar Worksheets, when used with an example that shows the properties of the molecules in a particular compound, will show the composition of the compound and the answer to the question \u201cWhat is Molar Matter?\u201d that a person may be looking for. Place your final answer in the FORMULA MASS COLUMN. A compound with an empirical formula of C 2H 8N and a molar mass of 46 grams per mole. Determine the % by mass \u2026 10. CaCO 3 CH 4 Sn(CO 3) 2 3. Answer the following questions: 9. 4. Chemistry Worksheet NAME: _____ Mole Conversions and Percent Composition Block: _____ 5. 2. 2. Percent Composition Worksheet 1. Determine the empirical formula of acetic acid. Determine the molar mass of carbon dioxide. Percent Composition and Molecular Formula Worksheet Solutions 1) What\u2019s the empirical formula of a molecule containing 65.5% carbon, 5.5% hydrogen, and 29.0% oxygen? We found some Images about Percent Composition Molar Mass Worksheet: Percent Composition and Molecular Formula Worksheet Molarity Worksheet Chemistry Free Worksheets Library | Download ... 1025 TEST II _ PRACTICE PACK _ 02 Molarity - 0.700 M solution . 1. 19. The sum all the mass percentages should add up to 100%. Showing top 8 worksheets in the category - Calculate Percent By Mass And Percent By Volume. Na2SO4 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 4. Worksheets; Supermarket Science; ... Quiz #2-3 PRACTICE: Molar Masses & Percent Composition For each of the following questions or statements, select the most appropriate response and click its letter: Start . 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Molecule containing 65.5 % carbon, S. 5 % hydrogen, and 53.4 % O of and!","date":"2021-04-12 06:17:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6073899865150452, \"perplexity\": 7239.414331121654}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038066613.21\/warc\/CC-MAIN-20210412053559-20210412083559-00334.warc.gz\"}"}
| null | null |
#include <stdio.h>
#include <stdlib.h> /* exit() */
#include <stdarg.h>
#include <node.h>
#include <node_object_wrap.h>
#include "geos_c.h"
/**
* A convenience for defining repetitive wrappers of GEOS unary
* topology functions which return a new geometry.
*/
#define GEONODE_GEOS_UNARY_TOPOLOGY(cppmethod, jsmethod, geosfn) \
Handle<Value> Geometry::cppmethod(Local<String> name, const AccessorInfo& info) \
{ \
HandleScope scope; \
Geometry *geom = ObjectWrap::Unwrap<Geometry>(info.Holder()); \
GEOSGeometry *geos_geom = geosfn(geom->geos_geom_); \
if (geos_geom == NULL) \
return ThrowException(String::New("couldn't get "#jsmethod)); \
Handle<Object> geometry_obj = WrapNewGEOSGeometry(geos_geom); \
return scope.Close(geometry_obj); \
};
/**
* A convenience for defining repetitive wrappers of GEOS binary
* topology functions which return a new geometry.
*/
#define GEONODE_GEOS_BINARY_TOPOLOGY(cppmethod, jsmethod, geosfn) \
Handle<Value> Geometry::cppmethod(const Arguments& args) \
{ \
HandleScope scope; \
if (args.Length() != 1) \
return ThrowException(String::New("requires other geometry argument")); \
Geometry *geom = ObjectWrap::Unwrap<Geometry>(args.This()); \
Geometry *other = ObjectWrap::Unwrap<Geometry>(args[0]->ToObject()); \
GEOSGeometry *geos_geom = geosfn(geom->geos_geom_, other->geos_geom_); \
if (geos_geom == NULL) \
return ThrowException(String::New("couldn't get "#jsmethod)); \
Handle<Object> geometry_obj = WrapNewGEOSGeometry(geos_geom); \
return scope.Close(geometry_obj); \
};
/**
* A convenience for defining repetitive wrappers of GEOS unary
* predicate functions.
*/
#define GEONODE_GEOS_UNARY_PREDICATE(cppmethod, jsmethod, geosfn) \
Handle<Value> Geometry::cppmethod(const Arguments& args) \
{ \
Geometry *geom = ObjectWrap::Unwrap<Geometry>(args.This()); \
HandleScope scope; \
unsigned char r = geosfn(geom->geos_geom_); \
if (r == 2) \
return ThrowException(String::New(#jsmethod"() failed")); \
return r ? True() : False(); \
};
/**
* A convenience for defining repetitive wrappers of GEOS binary
* predicate functions.
*/
#define GEONODE_GEOS_BINARY_PREDICATE(cppmethod, jsmethod, geosfn) \
Handle<Value> Geometry::cppmethod(const Arguments& args) \
{ \
Geometry *geom = ObjectWrap::Unwrap<Geometry>(args.This()); \
HandleScope scope; \
if (args.Length() != 1) { \
return ThrowException(String::New("other geometry required")); \
} \
Geometry *other = ObjectWrap::Unwrap<Geometry>(args[0]->ToObject()); \
unsigned char r = geosfn(geom->geos_geom_, other->geos_geom_); \
if (r == 2) { \
return ThrowException(String::New(#jsmethod"() failed")); \
} \
return r ? True() : False(); \
};
/**
* A convenience for defining repetitive wrappers of prepared geometry
* GEOS binary predicate functions.
*/
#define GEONODE_GEOS_PREPARED_GEOM_PREDICATE(cppmethod, jsmethod, geosfn) \
Handle<Value> Geometry::cppmethod(const Arguments& args) \
{ \
Geometry *geom = ObjectWrap::Unwrap<Geometry>(args.This()); \
HandleScope scope; \
if (args.Length() != 1) { \
return ThrowException(String::New("other geometry required")); \
} \
Geometry *other = ObjectWrap::Unwrap<Geometry>(args[0]->ToObject()); \
unsigned char r = geosfn(geom->geos_pg_, other->geos_geom_); \
if (r == 2) { \
return ThrowException(String::New(#jsmethod"() failed")); \
} \
return r ? True() : False(); \
};
using namespace v8;
using namespace node;
class Geometry : public ObjectWrap {
public:
GEOSGeometry *geos_geom_;
const GEOSPreparedGeometry *geos_pg_;
Geometry();
Geometry(GEOSGeometry* geom);
Geometry(const char* wkt);
~Geometry();
static void Initialize(Handle<Object> target);
bool FromWKT(const char* wkt);
protected:
static Handle<Value> New(const Arguments& args);
static Handle<Value> FromWKT(const Arguments& args);
static Handle<Value> ToWKT(const Arguments& args);
// GEOS topology operations
static Handle<Value> GetEnvelope(Local<String> name, const AccessorInfo& info);
static Handle<Value> Intersection(const Arguments& args);
static Handle<Value> Buffer(const Arguments& args);
static Handle<Value> GetConvexHull(Local<String> name, const AccessorInfo& info);
static Handle<Value> Difference(const Arguments& args);
static Handle<Value> SymDifference(const Arguments& args);
static Handle<Value> GetBoundary(Local<String> name, const AccessorInfo& info);
static Handle<Value> Union(const Arguments& args);
static Handle<Value> GetPointOnSurface(Local<String> name, const AccessorInfo& info);
static Handle<Value> GetCentroid(Local<String> name, const AccessorInfo& info);
static Handle<Value> Relate(const Arguments& args);
// GEOS unary predicates
static Handle<Value> IsEmpty(const Arguments& args);
static Handle<Value> IsValid(const Arguments& args);
static Handle<Value> IsSimple(const Arguments& args);
static Handle<Value> IsRing(const Arguments& args);
static Handle<Value> HasZ(const Arguments& args);
// GEOS binary predicates
static Handle<Value> Disjoint(const Arguments& args);
static Handle<Value> Touches(const Arguments& args);
static Handle<Value> Intersects(const Arguments& args);
static Handle<Value> Crosses(const Arguments& args);
static Handle<Value> Within(const Arguments& args);
static Handle<Value> Contains(const Arguments& args);
static Handle<Value> ContainsProperly(const Arguments& args);
static Handle<Value> Covers(const Arguments& args);
static Handle<Value> Overlaps(const Arguments& args);
static Handle<Value> Equals(const Arguments& args);
// static Handle<Value> EqualsExact(const Arguments& args); FIXME
// GEOS geometry info
static Handle<Value> GetSRID(Local<String> name, const AccessorInfo& info);
static void SetSRID(Local<String> name, Local<Value> value, const AccessorInfo& info);
// GEOS misc
static Handle<Value> GetType(Local<String> name, const AccessorInfo& info);
static Handle<Value> GetArea(Local<String> name, const AccessorInfo& info);
static Handle<Value> GetLength(Local<String> name, const AccessorInfo& info);
static Handle<Value> Distance(const Arguments& args);
private:
static Persistent<FunctionTemplate> geometry_template_;
static Handle<FunctionTemplate> MakeGeometryTemplate();
static Handle<Object> WrapNewGEOSGeometry(GEOSGeometry *geos_geom);
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,318
|
Kandilli ist die Bezeichnung folgender Orte:
Kandilli (Aşkale), Ortschaft im Landkreis Aşkale der türkischen Provinz Erzurum
Kandilli (Bozüyük), Ortschaft im Landkreis Bozüyük der türkischen Provinz Bilecik
Kandilli (Ereğli), Ortschaft im Landkreis Ereğli der türkischen Provinz Zonguldak
Kandilli (Saimbeyli), Ortschaft im Landkreis Saimbeyli der türkischen Provinz Adana
Kandilli (Tuzluca), Ortschaft im Landkreis Tuzluca der türkischen Provinz Iğdır
Kandilli (Istanbul), ein Stadtteil
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,875
|
\section{Introduction}
\label{Introduction}
Diffusive processes that scale anomalously with time, such that the Mean-Squared Displacement (MSD) of the expanding particle packet is
\begin{equation}
\langle x^2(t)\rangle \sim t^{2H},
\label{HurstDefinition}
\end{equation}
and the Hurst exponent $H\neq1/2$, are widely observed.
This behavior is found both in theoretical models as well as in many experiments, see e.g. \cite{hofling2013anomalous,metzler2014anomalous,metzler2019brownian,oliveira2019anomalous,sabri2020elucidating}. Of-course, if we know the exact underlying process responsible for the dynamics, \eq{H} can be determined exactly and the various features of the system that lead to the deviation from the standard linear scaling of the MSD, expected by the Gaussian Central Limit Theorem (CLT), can be understood. However, when anomalous diffusive scaling is detected in measurements it is not always clear what is responsible for the observed behavior of the system. Imagine, for example, that we obtain an ensemble of data-series describing intra-day trades in financial markets~\cite{bassler2007nonstationary,seemann2012ensemble,chen2017anomalous}, or experimental data obtained from observation of molecules diffusing inside cells, e.g., \cite{tolic2004anomalous,brauchle2010single,xie2008single,weigel2011ergodic,krapf2019spectral,sabri2020elucidating}. Here, the proper characterization of the exact root causes of this phenomenon is very important, since it can have implications on how we understand the underlying functioning of the system.
If, for example, we observe that the MSD grows faster than linearly with time, is this due to temporal correlations in the data that cause random large fluctuations to be followed by similar or even greater ones? Is it the result of a fat-tailed increment distribution, or is it because there is an actual trend of inflation in the system? Our analysis below allows us to give answer to these questions, despite the fact that we cannot completely restore the underlying process just from the data.
To make this more precise: Consider a continuous-time stochastic process \eq{x(t')} defined in the time interval \eq{t'\in[0,t]}. We can choose a number \eq{Q} of observation windows of duration \eq{\Delta=t/Q}, and then, represent this process by a discrete time-series composed of consecutive \emph{increments}, starting at times \eq{\left\{0,\Delta,2\Delta,\ldots,(Q-1)\Delta\right\}}. The increments are $\left\{\delta x_{1},\delta x_{2}, \ldots,\delta_{Q}\right\}=\left\{x(\Delta)-x(0),x(2\Delta)-x(\Delta),\ldots,x(t)-x(t-\Delta)\right\}$. According to the Gaussian CLT, in the limit of large \eq{Q}, if the increments are independent, identically distributed (IID) random variables chosen from a distribution with finite variance, then the MSD will grow linearly with \eq{Q} and thus with time.
Each of the three ways that the CLT can be violated corresponds to a \emph{constitutive effect} that can produce anomalous scaling~\cite{chen2017anomalous}.
For processes with stationary increments, where the probability distribution of \eq{\delta x_j} is independent of time, anomalous diffusive scaling can occur because of long-time increment correlations. This is called the \emph{Joseph effect}~\cite{mandelbrot1968noah,chen2017anomalous,meyer2018anomalous}. A paradigmatic process that exhibits this effect is fractional Brownian motion \cite{lim2002self,chen2017anomalous}. Another cause of anomalous scaling may be that the increment distribution is fat-tailed, in the sense that its second moment is divergent. This is the \emph{Noah effect} \cite{mandelbrot1968noah,chen2017anomalous,meyer2018anomalous}. A L\'evy flight process where the increments are power-law distributed, independent random variables~\cite{shlesinger1986levy,metzler2014anomalous}, but with infinite variance, is one example of a model with this effect.
When the increment distribution is non-stationary, anomalous diffusive scaling can also arise due to the \emph{Moses effect} \cite{chen2017anomalous,meyer2018anomalous}. A paradigmatic model in this case is scaled Brownian motion \cite{jeon2014scaled,thiel2014scaled,safdari2015aging}.
Each of the three effects can appear individually in a system, or in various combinations. Importantly, the three effects can be interconnected with each-other. For example, in \cite{meyer2018anomalous}, it was shown that statistical aging in the process can be associated not only with a Moses, but Noah effect. Among other things, this manuscript will extend our understanding of the coupling of the Moses and Noah effects.
The quantification of the three constitutive effects and the relation between them is given in Sec. \ref{SecThreeExps}.
In this manuscript, we investigate these three constitutive effects in a well studied stochastic process called coupled L\'evy walk~\cite{LevyWalks}. This model is known to have a rich spectrum of statistical behaviors, found by the tuning of a few well defined handles. We explore the emergence of the three effects in different parameter regimes of the model using simulations and methods of time-series analysis of single L\'evy walk trajectories, and compare our findings with analytical results based on the well developed theory for this process. This example shows that the analysis based on the three constitutive effects is a useful tool that can be applied to study other systems as well (see discussion).
In a two-state L\'evy walk \cite{shlesinger1986levy,LevyWalks, froemberg2015asymptotic}, a particle starts at \eq{x=0} at time \eq{t'=0} and then moves in independent steps. Each step has a random duration $\tau$, chosen from a Probability Density Function (PDF) of the form
\EQ{g(\tau)\sim \frac{c}{|\Gamma(-\gamma)|}\tau^{-1-\gamma}}{gTauDefinition}
at long \eq{\tau}, where \eq{c,\gamma>0} are constants.
During each step, the particle travels at a constant velocity \eq{V}, whose magnitude \eq{|V|} can be either $\pm1$ (sometimes referred to as ``genuine L\'evy walk" \cite{LevyWalks}), or a deterministic function of \eq{\tau}, but whose direction is chosen randomly to be either toward the right, along the positive \eq{\hat{x}-}axis (\eq{+}), or left (\eq{-}) along the negative axis. The latter, generalized model, is the case studied in detail in this manuscript (see also e.g., \cite{shlesinger1987levy,akimoto2013distributional,akimoto2014phase,albers2014weak,aghion2018asymptotic}), and the results include also the constant-velocity case. The probability of the direction being to the right or to the left is equal, so the motion is unbiased and the velocity has a symmetric PDF \eq{\phi(V)}.
At time \eq{t'=t}, the process stops. Up to this point, the particle has made \eq{N-1} complete steps, and one, final ``partial" step of duration \EQ{\tau^*=t-\sum_{i=1}^{N-1}\tau_i.}{TotalTime}
The properties of the final step have been shown to have a dramatic affect on the overall behavior of the system \cite{LevyWalks}, as the velocity \eq{V_N} during this step does not necessarily have to be distributed like all its predecessors, see e.g., \cite{froemberg2015asymptotic}. For more on this point, see Sec. \ref{SecModel}.
The number of steps in the process \eq{N \in [1,\infty)}, in the time interval $[0,t]$ is random, and the particle's position at time \eq{t} is given by the sum \eq{x(t)=\sum_{i=0}^{N-1}\chi_i+\chi^*}, where \eq{\chi_i=V_i\tau_i}, and \eq{\chi^*=V_N\tau^*}.
Table \ref{TableNotations} summarizes the main notations we use throughout the paper, by order of their appearance in the main text.
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{c c }
\hline\hline
Notation& Definition\\
[0.5ex] %
\hline
$V,\tau,\chi$&L\'evy walk: step- velocity, duration, displacement\\
$M,L,J,H$&Exponents: Moses, Noah, Joseph, Hurst\\
$\Delta,\delta x$&Time series: increment- duration, size \\
$\mathbf{\mathsf{v}}} % \usepackage{caption$&Time series: Mean velocity during an increment\\
$\alpha,\beta$&Exponents describing the shape of the distribution of $\mathbf{\mathsf{v}}} % \usepackage{caption$\\
$z_\beta$& $\mathbf{\mathsf{v}}} % \usepackage{caption/t^\beta$\\
$v(t)$&Instantaneous velocity of the L\'evy walker at time \eq{t}\\
$\tilde{v}$& $v/t^{\nu-1}$\\
[0.5ex] %
\hline\hline
\end{tabular}
\caption{\footnotesize{The main notations used in this manuscript, by order of their appearance in the main text. Note that the instantaneous velocity $v(t)$ is not always defined, for example in the case of Brownian motion. This does not matter for the general analysis of the three effects, which are defined via $\mathbf{\mathsf{v}}} % \usepackage{caption$, see Sec. \ref{SecThreeExps}. In the example that we use to demonstrate our analysis, namely L\'evy walk, $v(t)$ exists, and $\lim_{\Delta\rightarrow0}\mathbf{\mathsf{v}}} % \usepackage{caption\rightarrow v(t)$, see also Sec. \ref{SecMAndL}.}}
\label{TableNotations}
\end{table}
The structure of the manuscript is as follows: In Sec. \ref{SecThreeExps}, we define the three exponents that quantify the Moses, Noah and Joseph effects. We discuss the relation between them, and their role in determining the scaling shape of the increment PDF. In Sec. \ref{SecModel}, we extend the details on the L\'evy walk model. In Sec. \ref{SecMainResults}, we provide a summary of our main results, obtained from time-series analysis of numerical simulations, and a brief comparison of these results with the theoretical predictions. In Sec. \ref{SecMAndL} we obtain analytic results for the Moses and Noah effects, and in Sec. \ref{JosephExponent} for the Joseph.
We generalize the model in Sec. \ref{Section5}, and the discussion is provided in Sec. \ref{Discussion}.
\section{Story of three exponents:~\eq{M, L} and~\eq{J}}
\label{SecThreeExps}
The complete decomposition of the origin of anomalous diffusion presented in the introduction, was originally derived for discrete-time processes \cite{chen2017anomalous}. In this case, the process starts at \eq{\xi_0=0}, at \eq{n=0}, and evolves in discrete jumps \eq{n=1...N} with duration \eq{\Delta}, until time \eq{t=N\Delta}. The particle's position after \eq{n} steps is denoted \eq{\xi_{n\Delta}}. The Moses effect is quantified by the exponent \eq{M}, given by the median of the sum, of the absolute value of the time-series increments~\cite{chen2017anomalous} $m\left[\sum_{n=1}^{t/\Delta} |\delta \xi_n|\right]\equiv m\left[\sum_{n=1}^{t/\Delta}|\xi_{n\Delta}-\xi_{(n-1)\Delta}|\right]\propto t^{M+1/2}$. Here, \eq{M=1/2} yields a linear relation which is similar to normal diffusion. The Noah effect is defined by the scaling of the median of the sum of square-increments, and quantified by the Latent exponent $L$: $ m\left[\sum_{n=1}^{t/\Delta} (\delta \xi_n)^2\right]\propto t^{2L+2M-1}$. Here again, normal diffusion leads to linear scaling, where $M=L=1/2$. If there is no Moses effect, namely \eq{M=1/2}, the deviation from this scaling is quantified only by the exponent \eq{L}, and it arises if the increment PDF is fat-tailed.
Finally, the Joseph exponent can be defined via the sum over the auto-correlation function \cite{meyer2018anomalous} $\sum_{\Delta'=0}^{\tilde{\Delta}}\langle \delta \xi_n\delta \xi_{n+{\Delta'}}\rangle/\langle (\delta \xi_n)^2\rangle\propto \tilde{\Delta}^{2J-1}$, where \eq{0\leq J\leq 1}. {Here, starting from an arbitrary time point \eq{n}, we sum over a discrete lag time \eq{\Delta'}, up to e.g., \eq{\tilde{\Delta}\sim \OO(t/10)} (\eq{\tilde{\Delta}} is not related to \eq{\Delta}, defined above), and the scaling shape is valid when \eq{\tilde{\Delta},t\gg1}. When \eq{J>1/2}, the correlations decay very slowly with \eq{\tilde{\Delta}}, which leads to a divergent sum when \eq{\tilde{\Delta}\rightarrow\infty}, and superdiffusion (see discussion on ``long-ranged correlations" e.g., in \cite{Beran}). When \eq{J\leq1/2}, the correlation function decays at least as fast as \eq{1/\tilde{\Delta}}, which may lead either to normal diffusion, or in some particular cases to sub-diffusion, see Appen. \ref{AppenHVsMLJ}}.
For a process \eq{x(t)} in continuous time, we divide the time series into \eq{Q} non-overlapping observation windows of duration \eq{\Delta=t/Q} as mentioned in the introduction, and define the average velocity in each time interval
$\mathbf{\mathsf{v}}} % \usepackage{caption(t')\equiv| \delta x_j|/\Delta$, where $\delta x_j=x(j\Delta)-x\left[(j-1)\Delta\right]$, and \eq{(j-1)\Delta<t'<j\Delta}. Fig. \ref{FigCTRWToTSA} illustrates the decomposition of a continuous-time random trajectory, into a time-series of $N$ increments of equal duration $\Delta\ll t$. Now, we can re-write the definition of the Moses effect in terms of the ensemble-time averaged absolute-velocity
(when \eq{\Delta\ll t})
\begin{equation}
\left\langle\overline{|\mathbf{\mathsf{v}}} % \usepackage{caption|}\right\rangle\equiv\left\langle\frac{1}{t-\Delta}\sum_{j=1}^{t/\Delta}\frac{|\delta x_j|}{\Delta}\right\rangle\propto t^{M-1/2}.
\label{MosesDefinitionContinousTime} \end{equation}
We use here the ensemble mean, instead of the median, since it is a more convenient property to study analytically and numerically, hence we assume by this definition that this mean does not diverge.
In the same spirit, the Noah effect is defined via the ensemble-time average of the squared velocity, when \eq{\Delta\ll t}
\begin{align}
&\left\langle\overline{\mathbf{\mathsf{v}}} % \usepackage{caption^2}\right\rangle\equiv\left\langle\frac{1}{t-\Delta}\sum_{j=1}^{t/\Delta} \frac{\left(\delta x_j\right)^2}{\Delta^2}\right\rangle\propto t^{2L+2M-2},
\label{NoahDefinitionContinousTime}
\end{align}
where
\begin{equation}
1/2\leq L\leq 1.
\label{LatentLimits}
\end{equation}
In this definition, one can notice that manifestation of the Noah effect is somewhat different from the case of e.g., a L\'evy flight, since the mean of the squared increments is not divergent. In fact, as we explain in detail below, what leads to $L\neq1/2$ in this case, is that the increment PDF has a regime where its shape is fat-tailed, but this regime has a time-dependent cutoff which is pushed towards $\pm\infty$ as time increases. The resemblance between this observation, and the source of the Noah effect in its original definition on P. $1$, is the reason that we can make the association between the two cases and refer to $L$ in Eq. (\ref{NoahDefinitionContinousTime},\ref{LatentLimits}) throughout this manuscript as the Latent exponent.
The upper bound on $L$, in Eq. \eqref{LatentLimits}, is true because $\langle\overline{\mathbf{\mathsf{v}}} % \usepackage{caption^2}\rangle\leq \langle\overline{|\mathbf{\mathsf{v}}} % \usepackage{caption|}\rangle^2$. Intuitively it means that a tuning of the parameter that leads to a Noah effect beyond $L=1$, would automatically increase the scaling exponent of the first moment and therefore lead to aging and a Moses effect, instead of Noah. The lower bound exists because fat tails of the increment distribution, which are described by a Noah effect, can never lead to a slowing down of the process.
In this work, we will assume that also $\left\langle{|\mathbf{\mathsf{v}}} % \usepackage{caption|}\right\rangle\propto t^{M-1/2}$ and $\left\langle{\mathbf{\mathsf{v}}} % \usepackage{caption^2}\right\rangle\propto t^{2L+2M-2}$. We address the relation between our definitions and the original time-averaged definitions of these effects, which were derived when the ensemble means could be divergent, below (Sec. \ref{Scaling_shapes}). \iffalse, hence in this manuscript we do not address processes such as L\'evy flights, where ensemble-means are infinite for any \eq{t}\fi
Since the ensemble and time averaging procedures are commutative, if we know the first we can immediately obtain the latter via $\left\langle\overline{|\mathbf{\mathsf{v}}} % \usepackage{caption|}\right\rangle\rightarrow\overline{\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle}=(1/t)\int_0^t\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|(t')\rangle\Intd t'$ which yields \eq{=(Const./t)\int_0^t {t'}^{M-1/2}\Intd t'=[1/(M+1/2)]\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle}. Since we can find \eq{\left\langle\overline{\mathbf{\mathsf{v}}} % \usepackage{caption^2}\right\rangle} in a similar way from its ensemble mean, this yields
\begin{align}
\left\langle\overline{|\mathbf{\mathsf{v}}} % \usepackage{caption|}\right\rangle=\frac{\langle |\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle}{M+1/2},\Hquad\mbox{and}\Hquad
\left\langle\overline{\mathbf{\mathsf{v}}} % \usepackage{caption^2}\right\rangle=\frac{\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle}{2L+2M-1}.
\label{EATAVsEA}
\end{align}
Note that Eq. \eqref{EATAVsEA} introduces additional limits on the possible values of $M$ and $L$, for processes with finite $\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle$ and $\langle\mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle$, since the ratio between the time and ensemble averages here has to be positive. These limits are consistent with our results for the L\'evy walk model, in Sec.~\ref{SecMainResults}.
We define the Joseph exponent also in the spirit of the discrete case, via the scaling of the integral
\eq{\int_0^{\tilde{\Delta}} d\Delta^\prime \langle \mathbf{\mathsf{v}}} % \usepackage{caption(t)\mathbf{\mathsf{v}}} % \usepackage{caption(t+\Delta')\rangle/\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle\propto \tilde{\Delta}^{2J-1},}
for large $\tilde{\Delta}$. Here again, \eq{\tilde{\Delta}} should not be confused with \eq{\Delta}, which is the time duration from which we defined \eq{\mathbf{\mathsf{v}}} % \usepackage{caption}.
In this manuscript we will only focus on the case where \EQ{1/2\leq J\leq 1,}{JRange} see Appen. \ref{AppenHVsMLJ} for more explanation. Taking the derivative of the integral with respect to \eq{{\tilde{\Delta}}}, the autocorrelation function is
\begin{equation}
f({\tilde{\Delta}})\equiv\frac{\left\langle \mathbf{\mathsf{v}}} % \usepackage{caption(t)\mathbf{\mathsf{v}}} % \usepackage{caption(t+{\tilde{\Delta}})\right\rangle}{\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2(t)\rangle}\propto {\tilde{\Delta}}^{2J-2},
\label{JosephDefinitionContinousTime}
\end{equation}
at \eq{{\tilde{\Delta}}\gg 1}. For small $\tilde{\Delta}$, we define \eq{f({\tilde{\Delta}})\equiv f_<({\tilde{\Delta}})}, where \eq{f_<} insures that the autocorreletation function is regularized at \eq{{\tilde{\Delta}}\rightarrow0}.
Note that in data analysis there are several known methods to obtain the Joseph exponent without directly calculating the autocorrelation function. These methods have various advantages and disadvantages in practice, see Sec. \ref{JosephExponent} and Appen. \ref{Appenctamsd} and \ref{DFAAndEATMSD}.
We note that by dividing $\delta x_j$ by \eq{\Delta}, and defining the three effects via the mean increment velocity $\mathbf{\mathsf{v}}} % \usepackage{caption$, we did not limit the generality of the definitions at all. The reason is that we did not at this point take the limit \eq{\Delta\rightarrow0}, hence we do not require the instantaneous velocity to be defined. In any process, one can discuss average velocities and increments of a finite-time duration interchangeably.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{FigureForThePaper.png}
\caption {\footnotesize{An example of a L\'evy walk path $x(t')$ (blue) versus time, generated by the model in Sec. \ref{SecModel}. At the total measurement time $t$, the last step is incomplete. Two red dash-dot lines mark the start and end points of one completed L\'evy walk step, whose duration $\tau$ was selected from the PDF Eq. \eqref{MLDistributionOfTaus}, and the step-velocity is $V\sim\tau^{\nu-1}$. Here $\gamma=0.52, \nu=0.5$.
As explained in Sec. \ref{SecThreeExps}, the trajectory is decomposed into a series of consecutive increments $n=1,2,...$, of equal duration $\Delta$, the start and end points of one such increment are marked e.g. by two green dash-dot lines. The size of the average velocity $|\mathbf{\mathsf{v}}} % \usepackage{caption|$ in that increment is also presented.}}
\label{FigCTRWToTSA}
\end{figure}
\subsection{ Relation between $M,L,J$ and $H$}
\label{SecRelationMLJH}
Let \eq{\mathbf{\mathsf{v}}} % \usepackage{caption(0)\equiv0}, using Eq. \eqref{JosephDefinitionContinousTime} and the Green-Kubo relation \cite{meyer2017greenkubo}, the MSD of the process can be written as
\small
\begin{align}
&\langle x^2\rangle=2\int_0^t\Intd {\tilde{\Delta}}\int_0^{t-{\tilde{\Delta}}}\Intd t' \langle \mathbf{\mathsf{v}}} % \usepackage{caption(t')\mathbf{\mathsf{v}}} % \usepackage{caption(t'+{\tilde{\Delta}})\rangle\nonumber\\
&\propto 2\int_1^t\Intd {\tilde{\Delta}}\int_0^{t-{\tilde{\Delta}}}\Intd t' \langle \mathbf{\mathsf{v}}} % \usepackage{caption^2(t')\rangle{\tilde{\Delta}}^{2J-2}\nonumber\\
&+2\int_0^1 \Intd {\tilde{\Delta}} f_<({\tilde{\Delta}})\int_0^t\langle v^2(t')\rangle\Intd t'\nonumber\\
&\propto 2\int_1^t\Intd {\tilde{\Delta}}{\tilde{\Delta}}^{2J-2}\int_0^{t-{\tilde{\Delta}}}\Intd t' {t'}^{2L+2M-2}+c_<t^{2L+2M-1}\nonumber\\
&\propto\frac{2}{2L+2M-1}\int_1^t\Intd {\tilde{\Delta}} {\tilde{\Delta}}^{2J-2}(t-{\tilde{\Delta}})^{2L+2M-1}+c_<t^{2L+2M-1}\nonumber\\
&\underbrace{\sim}_{\footnotesize{\begin{aligned}u&\leftrightarrow{\tilde{\Delta}}/t \\ &t\rightarrow\infty\end{aligned}}}\frac{2t^{2L+2M+2J-2}}{2L+2M-1}\int_0^1\Intd u u^{2J-2}(1-u)^{2L+2M-1}\nonumber\\
& \qquad\qquad \propto t^{2L+2M+2J-2},\qquad\mbox{when}\qquad J>1/2.
\label{ConnectionBetweenExponents1}
\end{align}
\normalsize
In Eq. \eqref{ConnectionBetweenExponents1}, \eq{c_<} is a constant, and in the last step note that since the term \eq{\propto t^{2M+2L-1}} is subdominant with respect to the other when \eq{J>1/2}, we neglected it in the long-time limit. Using Eq. \eqref{HurstDefinition}, this yields
\begin{equation}
H=J+L+M-1.
\label{hjlm}
\end{equation}
The relation in Eq. \eqref{hjlm} was previously shown to hold empirically in a number of models in \cite{chen2017anomalous,meyer2018anomalous}. It was conjectured to be broadly valid, even for systems beyond the case we study here, in particular also when ensemble averages diverge and the Moses and Noah effects are only quantified via their original time-averaged definitions. However, a rigorous derivation in other cases is still needed. For more details see Appen.~\ref{AppenHVsMLJ}.
\subsection{Scaling shapes of the increment distribution}
\label{Scaling_shapes}
Considering ensemble averages allows us to obtain additional insight about the meaning of the Moses and Noah effects. Assume that \eq{\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle} and \eq{\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle} are not divergent. Let $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ be the PDF of finding an increment velocity $\mathbf{\mathsf{v}}} % \usepackage{caption$ at time $t$, given that the process started at at rest at $t = 0$. This increment PDF is said to have a single scaling shape, if for any $x$ and $t$ it can be described by a time-independent function $W(z_\beta)$, such $W(z_{\beta})=t^{\beta}P(\mathbf{\mathsf{v}}} % \usepackage{caption/t^{\beta})$ and $z_{\beta}=\mathbf{\mathsf{v}}} % \usepackage{caption/t^{\beta}$. In our case, we do not restrict $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ to only one such scaling regime, and it can have two different scaling shapes in it bulk and the tails, a situation not uncommon in anomalous diffusion which is associated with multifractality, see e.g., \cite{castiglione1999strong,seuront2014anomalous,rebenshtok2014non,kessler2010infinite,aghion2018asymptotic,grahovac2015asymptotic}. If both the mean of \eq{|\mathbf{\mathsf{v}}} % \usepackage{caption|} and \eq{\mathbf{\mathsf{v}}} % \usepackage{caption^2} are taken from the same scaling regime of \eq{P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)}, then in this regime
\EQ{\lim_{t\rightarrow\infty}t^{\alpha+\beta}P_t(\mathbf{\mathsf{v}}} % \usepackage{caption/t^\beta)\rightarrow W(z_\beta),\quad\mbox{ where}\quad z_\beta=\mathbf{\mathsf{v}}} % \usepackage{caption/t^\beta,}{ICD}
and
\begin{align}
&\qquad\quad L=\alpha/2+1/2,\quad M=\beta-\alpha+1/2\nonumber\\
&(\mbox{equivalently:}\quad \alpha=2L-1,\quad \beta=M+2L-3/2).
\label{ALphaBetaDefinitions}
\end{align}
Notice that since \eq{1/2\leq L\leq1}, Eq. \eqref{LatentLimits}, then \eq{0\leq\alpha\leq1}. The limit function \eq{W(z_\beta)} is responsible for the mean of \eq{|\mathbf{\mathsf{v}}} % \usepackage{caption|} and \eq{\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle} via \EQ{\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|^q\rangle=2\int_{0}^\infty\Intd \mathbf{\mathsf{v}}} % \usepackage{caption|\mathbf{\mathsf{v}}} % \usepackage{caption|^q P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)\underbrace{\approx}_{t\gg1}2t^{q\beta-\alpha}\int_0^\infty \Intd z_\beta|z_{\beta}|^qW(z_\beta),}{ICD1}
for \eq{q=1,2}.
When \eq{M,L} are such that both \eq{\alpha} and \eq{\beta} are zero, the increment PDF has a stationary asymptotic (equilibrium) state. Coincidentally this occurs only when \eq{M=L=1/2}, which as mentioned means that the time-series satisfies at least two of the conditions of the Gaussian CLT. Curiously, \eq{M} can also be half if \eq{\alpha=\beta\neq0}. When \eq{P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)} is non-stationary, we always have a Moses effect. The PDF has a normalized scaling shape, when \eq{\alpha=0} but \eq{\beta\neq0}, namely \eq{L=1/2,M\neq1/2}. This is the onset of a ``pure" Moses effect. Now, the exponent \eq{M} tells us how to re-scale the PDF in-order to find the invariant limit, since \eq{P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)\sim t^{1/2-M}W(\mathbf{\mathsf{v}}} % \usepackage{caption/t^{-1/2+M})}. According to Eqs. (\ref{ICD},\ref{ICD1}), if we define \eq{\langle |z_\beta|^q\rangle_\mathbb{W}\equiv\int_0^\infty |z_\beta|^qW(z_\beta)\Intd z_\beta} for $q=1,2$, then $\langle |v|^q\rangle=2t^{q(1/2-M)}\langle|z_\beta|^q\rangle_\mathbb{W}$. Note that usually, based on intuition taken from Gaussian processes, there is a tendency to vaguely associate the Hurst exponent \eq{H}, with the "self-similarity" property of the process. However in anomalous diffusion that is not necessarily the case; one example is when the MSD is diverging, e.g., in L\'evy flight, another example is the case of multifractality \cite{castiglione1999strong}. In our case, it is \eq{\beta}, not \eq{H}, that may describe this property, from the point of view of the increment PDF.
The onset of a Noah effect means that \eq{\mathbf{\mathsf{v}}} % \usepackage{caption^2} becomes non-integrable with respect to the scaling function which gives the shape of the $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ in the bulk.
In the paradigmatic example for this effect, L\'evy flight \cite{mandelbrot1968noah}, the PDF \eq{P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)} can be e.g., a stationary symmetric L\'evy distribution \eq{l_{\xi,1,0}(\mathbf{\mathsf{v}}} % \usepackage{caption)}, with $0<\xi<2$, defined as the inverse-Laplace transform of \eq{\exp(-|u|^\xi)}, from \eq{u \rightarrow \mathbf{\mathsf{v}}} % \usepackage{caption} \cite{klafter2011first}. In this case, by definition, there is no Moses effect, and the Noah effect rises since $\int_{-\infty}^\infty \mathbf{\mathsf{v}}} % \usepackage{caption^2 l_{\xi,1,0}(\mathbf{\mathsf{v}}} % \usepackage{caption)\Intd\mathbf{\mathsf{v}}} % \usepackage{caption\rightarrow\infty$, though of-course, here it can only be quantified by the original definition of \eq{L}, namely via the time-average of the squared increments of single time-series \cite{mandelbrot1968noah}. If the increment PDF would have e.g., the scaling shape
$P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)\sim t^{-1/\xi}l_{\xi,1,0}(\mathbf{\mathsf{v}}} % \usepackage{caption/t^{1/\xi})$, we would find both a Moses effect, and a Noah effect which is still characterized via the time average.
A more involved scenario that can occur, is when the large fluctuations of the system are reduced such that \eq{\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle} is not strictly infinity, but is increasing with time as in Eq. \eqref{NoahDefinitionContinousTime}, because at its tails the PDF $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ is scaled differently in time with respect to the bulk. Now, the definitions in Eqs. (\ref{MosesDefinitionContinousTime},\ref{NoahDefinitionContinousTime}) are valid. The Noah effect will now appear if the function which describes the asymptotic shape of $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ at the bulk is fat-tailed (in the sense that its variance is infinite), but the mean \eq{\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle} will be given by a second scaling function to which $P_t(v)$ convergence at the tails. If it happens that the mean of \eq{|v|} and \eq{v^2} are obtained from different scaling regimes, then again Eq. \eqref{ICD} and Eq. \eqref{ALphaBetaDefinitions} are not valid, but one can use methods such as estimating fractional moments \cite{rebenshtok2014non,aghion2018asymptotic,grahovac2015asymptotic} to find the various scaling shapes of \eq{P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)}. If both $\langle |\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle$ and $\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle$ correspond to the second scaling function (that describes the large fluctuations), and are proportional to $t^{M-1/2}$ and $t^{2M+2L-2}$ respectively, then Eq. \eqref{ICD} is valid. But in this case, \eq{W(z_\beta)} which denotes these moments might not be normalizable, namely \eq{\int_0^\infty W(z_\beta)\Intd z_\beta\rightarrow\infty}. Here, \eq{\alpha} and the Latent exponent \eq{L} serves as a measure of \textit{how far} the increment PDF is from having a normalized limit shape. When \eq{\alpha>0} and \eq{\beta=0}, equivalently \eq{L>1/2} and \EQ{M=\frac{3}{2}-2L,}{LAndMID} \eq{W(z_\beta)} is an infinite-invariant density, a type of quasi-equilibrium state, see e.g., \cite{aaronson1997introduction,korabel2009pesin,leibovich2019infinite,akimoto2019infinite,aghion2019infinite,sato2019anomalous,aghion2020infinite}. The relation in Eq. \eqref{LAndMID}, if observed in data, can in-fact be used to indicate that the underlying process has an infinite-invariant density in this regime, and it was also observed in the Pommeau-Manneville map \cite{meyer2017infinite}. If \eq{\alpha>0} and \eq{\beta\neq0}, or equivalently \eq{L>1/2} and $M\neq$[Eq. \eqref{LAndMID}], the limit shape of the increment PDF is given by an infinite-covariant density, see e.g., \cite{kessler2010infinite,lutz2013beyond,rebenshtok2014non,holz2015infinite,aghion2017large,wang2019transport,aghion2018asymptotic}.
Note that, in both the invariant and the covariant case, and also in the case when the mean-absolute and mean-squared increments are non-divergent, but they correspond to different scaling regimes of the PDF, a Noah effect cannot appear without a Moses effect.
The different cases for $M,L$ and $\alpha,\beta$ are summarized in Table~\ref{tablelAlphaBeta}.
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{c c c c }
\hline\hline
$\alpha$&$\beta$&$L,M$&\eq{\lim_{t\rightarrow\infty}t^{\alpha+\beta}P(\mathbf{\mathsf{v}}} % \usepackage{caption/t^\beta)}\\
[0.5ex] %
\hline
$0$&$0$&$\frac{1}{2},\frac{1}{2}$&steady-state\\
$0$&$\beta\neq0$&$\frac{1}{2},M>\frac{1}{2}$&normalized scaling limit\\
\iffalse $0$&$\beta\neq0$&$\frac{1}{2},M<\frac{1}{2}$&$t^{-\beta} W(v/t^\beta)$&\textcolor{red}{Normalized scaling limit$?$}\\ \fi
$\alpha>0$&$0$&$L>\frac{1}{2},M<\frac{1}{2}$&infinite-invariant density\\
$\alpha>0$&$\beta\neq0$&$L>\frac{1}{2},(all)$&infinite-covariant density\\
[0.5ex]
\hline\hline
\end{tabular}
\caption{\footnotesize{Summary of the different scaling limit of $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$, that can be found from the Moses $M$ and Latent $L$ exponents, via $\alpha,\beta$ Eqs. (\ref{ICD},\ref{ALphaBetaDefinitions}), if both \eq{\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle} and $\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle$ correspond to the same scaling regimes of the PDF. Note that $\alpha,\beta$ set the restrictions for $M,L$ in the various regimes, not the other way around.
}}
\label{tablelAlphaBeta}
\end{table}
\section{The L\'evy walk model}
\label{SecModel}
As mentioned in the introduction, in this work we analyse a two-state L\'evy walk model. Particularly, here, we consider a continuous range of IID random step velocities, whose distribution is \eq{\phi(V)}. In addition, we assume a nonlinear coupling between the \eq{i}th step duration and the step velocity, namely
\begin{equation}
V_i= \pm\tilde{c}_1\tau_i^{\nu-1},
\label{NonlinearDurationVelocityCoupling}
\end{equation}
where
\begin{equation}
\nu>0.
\label{NuRegime}
\end{equation}
The sign of the step velocity is randomly chosen to be positive or negative with equal probability (the motion is unbiased). The constant \eq{\tilde{c}_1} has units of \eq{distance/(time)^\nu}, but throughout this manuscript we set \eq{\tilde{c}_1=1} for convenience.
Eq. \eqref{NonlinearDurationVelocityCoupling} means that \EQ{\phi(V)=\frac{1}{2}\int_0^\infty\Intd \tau g(\tau)\left[\delta(V-\tau^{\nu-1})+\delta(V+\tau^{\nu-1})\right].}{StepVelocityPDF} Below, in our numerical simulations, we will use a specific example where the IID random step durations are obtained from the distribution
\begin{equation}
g(\tau)=\gamma\tau_0^\gamma\tau^{-1-\gamma}\Theta(\tau\geq\tau_0),
\label{MLDistributionOfTaus}
\end{equation}
though our results are more general (see the discussion, Sec. \ref{Discussion}). Here, \eq{\tau_0>0} can be as small as we wish, and \eq{\Theta(\cdot)\equiv1} when the condition inside the brackets is satisfied and zero otherwise.
For any \eq{g(\tau)} in Eq. \eqref{gTauDefinition}, from Eqs. (\ref{NonlinearDurationVelocityCoupling},\ref{StepVelocityPDF}), when \eq{|V|<1} one finds that \eq{\phi(V)\sim \frac{c}{2(1-\nu)|\Gamma(-\gamma)|}|V|^{-1-\gamma/(\nu-1)}}. For our example, from Eq. \eqref{MLDistributionOfTaus} it follows that the step velocity distribution in the first \eq{N-1} complete steps, when $\nu<1$, is
\begin{equation}
\phi(V)=\frac{\gamma\tau_0^\gamma}{2(1-\nu)}{|V|^{-\frac{\gamma}{\nu-1}-1}}\Theta(|V|\leq\tau_0^{\nu-1}),
\label{StepVelocityPDF1}
\end{equation}
and it has a similar shape but with~$\Theta(|V|\geq\tau_0^{\nu-1})$ replacing the original one when $\nu>1$, hence \eq{c=\gamma \tau_0^\gamma|\Gamma(-\gamma)|}. In this manuscript we focus on the parameter regime
\begin{equation}
0<\gamma<1,
\label{ParameterRegime}
\end{equation}
where \eq{\langle\tau\rangle} is divergent. In various models of non-linearly coupled L\'evy walk, some of them are summarized in the review \cite{LevyWalks}, it was shown that in addition to the various scaling exponents, the statistical properties of the process depend strongly on the treatment given to the last, incomplete, step in the sequence. We choose to correspond with the model studied in \cite{albers2018exact,akimoto2019infinite,bothe2019mean}, where \eq{V_N} is determined from the time interval straddling \eq{t} \cite{wang2018renewal}. With this choice, all the velocities \eq{V_i}, with \eq{i=1..N} are IID, though the duration of the last step is given by Eq. \eqref{TotalTime}. As usual, the displacement at each step (complete and incomplete) is the linear product of the step velocity and its duration.
{\em Instantaneous velocity PDF.}
Akimoto et al. \cite{akimoto2019infinite}, studied the instantaneous velocity PDF \eq{P_t(v)} of the L\'evy walker in the process described above, at time \eq{t\gg1} and the regime where $0<\nu<1$. We can apply their results to our analysis, since in this model we can associate $\mathbf{\mathsf{v}}} % \usepackage{caption$ and $v$ via $v=\lim_{\Delta\rightarrow0}\mathbf{\mathsf{v}}} % \usepackage{caption$, see Sec. \ref{SecMAndL}. The following analytic results are brought from that referenced paper. At long but finite times, $P_t(v)$ assumes different shapes in two separate ranges of \eq{v}: Let \eq{v_c= t^{\nu-1}}, then \cite{akimoto2019infinite}
\begin{equation}
P_t(v)\approx \begin{cases} \frac{t^\gamma}{2(1-\nu)|\Gamma(-\gamma)|\Gamma(1+\gamma)}|v|^{-1-\gamma/(\nu-1)}, & |v|/v_c\leq 1\vspace{5pt}\\
\frac{1-[1-(v/v_c)^{1/(\nu-1)}]^\gamma}{c \Gamma (\gamma+1) }t^\gamma\phi(v), & |v|/v_c>1.
\end{cases}
\label{VelocityPDFSmallV}
\end{equation}
Due to the asymptotic shape of \eq{\phi(v)}, when \eq{v} itself is smaller than unity (regardless of \eq{t}), $P_t(v/t^{\nu-1})$ corresponds in this regime to the scaling function \eq{\sim t^{(\nu-1)}\rho(\tilde{v})}, where \eq{\tilde{v}= v/t^{\nu-1}} and
\begin{equation}
\rho(\tilde{v})\approx \begin{cases} \frac{1}{2(1-\nu)|\Gamma(-\gamma)|\Gamma(1+\gamma)}|\tilde{v}|^{-1-\gamma/(\nu-1)}, & |\tilde{v}|\leq 1\vspace{5pt}\\
\frac{1-[1-(\tilde{v})^{1/(\nu-1)}]^\gamma}{2(1-\nu)|\Gamma(-\gamma)| \Gamma (\gamma+1) }|\tilde{v}|^{-1-\gamma/(\nu-1)}, & |\tilde{v}|>1.
\end{cases}
\label{WzBetaSmallV}
\end{equation}
The scaling function $\rho(\tilde{v})$ is normalized to unity.
On the other hand, at long times $P_t(v)$ has a second scaling shape valid in the region $v>v_c$, since in the limit \eq{t\rightarrow\infty} the support of the region $v/v_c<1$ in Eq. \eqref{VelocityPDFSmallV} goes to zero, and at \eq{v/v_c\gg1}, we can expand \eq{[1-(v/v_c)^{1/(\nu-1)}]^\gamma} as a Taylor series for the small parameter \eq{(v/v_c)^{1/(\nu-1)}}. This yields, to leading order in time,
$
P_t(v)\approx \phi(v)|v|^{\frac{1}{\nu-1}}\frac{t^{\gamma-1}}{c\Gamma(\gamma)}. $
Which means that asymptotically \cite{akimoto2019infinite},
\begin{equation}
\lim_{t\rightarrow\infty}t^{1-\gamma}P_t(v)\rightarrow \mathcal{I}(v),\Hquad\mbox{where}\Hquad \mathcal{I}(v)\equiv\phi(v)|v|^{\frac{1}{\nu-1}}\frac{1}{c\Gamma(\gamma)},
\label{VelocityID}
\end{equation}
and $\phi(v)$ is in Eq. \eqref{StepVelocityPDF1}.
The time-invariant asymptotic limit given by \eq{\mathcal{I}(v)} in Eq. \eqref{VelocityID} is non-integrable around \eq{v=0}, hence it is non-normalizable: \eq{\int_{-\infty}^\infty\mathcal{I}(v)\Intd v
\rightarrow\infty.} As such, this function is the infinite-invariant density of the process \cite{akimoto2019infinite}. Note that when $\nu>1$, the two regimes of the PDF, Eq. \eqref{VelocityPDFSmallV} simply switch places, but their functional shape remains the same.
\begin{figure}
\centering
\includegraphics[width=0.38\textwidth]{Figure1version1.png}
\caption{\footnotesize{Phase diagram of the scaling exponents describing the decomposition of the anomalous diffusion. The three solid lines, separating regions $A$-$B$, $B$-$C$ and $C$-$D$
are respectively:
\eq{\nu=\gamma/2+1/2}, \eq{\nu=\gamma} and \eq{\nu=\gamma/2}. The dashed-line is \eq{\nu=\gamma/2+1}. The results for the three-effect decomposition in the various regimes are discussed in Sec. \ref{SecMainResults}. Region A: $H = \nu$, $J = 1$, $L = 1/2$, $M = \nu-1/2$ (``maximal" Joseph effect, namely the autocorrelation function Eq. \eqref{JosephDefinitionContinousTime} does not decay at large values of $\Delta$, no Noah, $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ has a normalized scaling shape corresponding to $M$). Region B: $H = \nu$, $J = (1+2\nu-\gamma)/2$, $L = 1-\nu+\gamma/2$, $M = \nu-1/2$ (onset of a Noah effect). Region C: $H = \nu$, $J = (1+2\nu-\gamma)/2$, $L = 1- \gamma/2$, $M = \gamma-1/2$ ($P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)\rightarrow\mbox{infinite-invariant density}$). Region D: $H = \gamma/2$, $J = 1/2$, $L = 1- \gamma/2$, $M = \gamma-1/2$ (infinite-invariant density, no Joseph effect). In the $``\infty''$ regime, $H\rightarrow\infty$, and $M,L,J$ are not well defined.
}}
\label{fig2:QgphaseAllTogether}
\end{figure}
\section{Summary of our main results}
\label{SecMainResults}
This summary brings the main results of our analysis of L\'evy walk trajectories generated by the process described in Sec. \ref{SecModel}, and the detailed derivations appear below. For further discussion about the generality of the three-effect decomposition, also see below.
In our simulations, we generated an ensemble of $10^8$ realizations of the process $x(t)$ for different values of $\gamma$ and $\nu$, and observed the increments $\delta x_{j}$ of the paths at different times ranging from $t=10^4$ to $10^8$. We then measured the ensemble averages of $|\delta x_{j}|$, $\delta x_{j}^{2}$ (namely, we used $\mathbf{\mathsf{v}}} % \usepackage{caption$ with observation windows of duration $\Delta=1$), as well as $x^2$, to calculate the values of $M$, $L$ and $H$ respectively. To obtain the value of the exponent $J$, we used a method based on the time-averaged MSD $\delta^2$, as explained in detail in Sec. \ref{JosephExponent} and Appen. \ref{Appenctamsd}. The results of this method correspond to those of a direct measurement of the correlation function, but it is numerically more convenient (see Appen. \ref{Appenctamsd}).
{\em What the data analysis says:} \textbf{Without} relying on prior knowledge about the underlying process, we found that in the range defined by Eqs. (\ref{NuRegime},\ref{ParameterRegime}), the L\'evy walk data exhibits five separate dynamical phases. These phases are summed-up below and in Fig. \ref{fig2:QgphaseAllTogether}. The summation formula, Eq. \eqref{hjlm} is confirmed in all but the ``$\infty$" regime.
\begin{itemize}
\item In regime A, when $\gamma/2+1/2<\nu<\gamma/2+1$: $H = \nu$, $J = 1$, $L = 1/2$, $M = \nu-1/2$. Here, the auto-correlation function does not decay with \eq{\tilde{\Delta}}, in Eq. \eqref{JosephDefinitionContinousTime}, namely the increments are essentially completely correlated. In this situation, we say that the Joseph effect is maximal, since by definition $J$ can never be bigger than its value here. There is no freedom left in the increment distribution for any Noah effect to be present. There can be, however, a Moses effect as the increment distribution does ``age" with time.
The existence of a Moses effect without a Noah effect means that in this regime we expect a single scaling function in the form of $t^{\nu-1}P_t(\mathbf{\mathsf{v}}} % \usepackage{caption/t^{\nu-1})$ to describe the regime of the PDF which gives rise to the first and second moments of $|\mathbf{\mathsf{v}}} % \usepackage{caption|$ (which is therefore no-fat tailed). Our numerics show that this regime extends also to the range $1<\nu<\gamma/2+1$ (and $\gamma<1$).
\item In regime B, $\gamma<\nu<\gamma/2+1/2$: $H = \nu$, $J = (1+2\nu-\gamma)/2$, $L = 1-\nu+\gamma/2$, $M = \nu-1/2$. In this regime all the three effects contribute to the anomalous diffusion.
Here, the Joseph effect is present, but is not maximal, as the auto-correlation function decays as a power-law function of \eq{\tilde{\Delta}}.
This allows for a Noah effect to be present too. Here, the Noah effect means that the scaling shape at the bulk of \eq{P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)} is fat-tailed, in the sense that its second moment is divergent. But the mean of $|\mathbf{\mathsf{v}}} % \usepackage{caption|$ remains unchanged from regime A, so it is expected to still be given by the same scaling regime of the increment PDF as before, namely $\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle$ and $\langle\mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle$ correspond to different regimes of $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$. Accordingly our numerical analysis shows that Eq. \eqref{ICD} is not valid in this case. The Moses effect occurs here in a similar way as it does in regime A, namely also in this regime, the increment PDF is not time-invariant.
\item In regime C, $\gamma/2<\nu<\gamma$: $H = \nu$, $J = (1+2\nu-\gamma)/2$, $L = 1- \gamma/2$, $M = \gamma-1/2$. Still, all three effects contribute to the anomalous diffusion.
Here, just as in regime B, the Joseph effect is present, but is not maximal.
In this regime, the Moses and Noah effects are coupled, with the Moses and the Latent exponents obeying Eq. \eqref{LAndMID}. This suggests that the large fluctuations of the system are described by an infinite-invariant density, Eq. \eqref{ICD} with $\alpha=1-\gamma, \beta=0$.
\item In regime D, \eq{\nu<\gamma/2}: $H = \gamma/2$, $J = 1/2$, $L = 1- \gamma/2$, $M = \gamma-1/2$. Here, $M,L$ remain coupled as in region C. Hence we expect the same infinite-invariant density to be valid in this regime too. Interestingly, now there are no long-range increment correlations and, thus, there is no Joseph effect. At this stage anomalous diffusion occurs due to the non-stationarity of $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ and the fat tails of the scaling-shape describing this PDF at the bulk.
\item When $\nu>\gamma/2+1$, the MSD is divergent. The scaling relations in Eqs. (\ref{NoahDefinitionContinousTime}-\ref{JosephDefinitionContinousTime}) don't hold, and in this regime the decomposition is not valid. We call this the ``$\infty$" regime. See Appen.~\ref{AppendixInfinityRegime}.
\end{itemize}
{\em What we know from the model, in comparison with the data analysis:} When $\gamma,\nu<1$, Eq. \eqref{WzBetaSmallV} and Eq. \eqref{VelocityID} describe two different ways to obtain a time-invariant scaling-shape of the instantaneous velocity PDF $P_t(v)$, the first is valid for small $v$ and the second for large. We can associate this velocity PDF with the distribution of the increment velocity $\mathbf{\mathsf{v}}} % \usepackage{caption$ (see Sec. \ref{SecMAndL}). As expected from the numerics, the analytic results presented in Sec. \ref{SecMAndL} show that the bulk function and the infinite-invariant density describe the shape of the increment PDF in regimes $A$ and $C,D$ respectively, in the range of $v$ which is responsible for the various moments. In regime B, $\langle|v|\rangle,\langle v^2\rangle$, (hence $\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle,\langle \mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle$) are obtained separately from the two scaling regimes. The fact that the Joseph effect, studied in Sec. \ref{JosephExponent}, is ``maximal'' in regime A, matches to the fact that the bulk limit-function describing the PDF is thin-tailed, from the same reason that in regime D it is ``minimal": if the increments are long- (short-) ranged correlated, their size is more (less) predictable from the first step. Therefore, large fluctuations are less (more) possible.
In regime A, when $\nu>1$, it is easy to show that one can find similar results for $\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle$ and $\langle\mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle$ as in the case when $\nu<1$ since as mentioned, the shape of $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ is similar to Eq. \eqref{VelocityPDFSmallV}, but with the two regimes for $v\leq1$ and $v>1$ switching roles. In addition, here $\tau_c^{\nu-1}$ becomes a lower, instead of an upper cutoff for the step velocity PDF in Eq. \eqref{StepVelocityPDF1}. The divergence of the MSD in the ``$\infty$'' regime, was shown analytically in \cite{albers2018exact,bothe2019mean}, further details in Sec. \ref{Section5} and Appen. \ref{AppendixInfinityRegime}.
\section{\eq{M}\& \eq{L}, and how we obtained them}
\label{SecMAndL}
As explained in Sec. \ref{SecThreeExps}, in order to obtain \eq{M} and \eq{L}, we need to examine the temporal behavior of the ensemble-time averages \eq{\left\langle\overline{|\mathbf{\mathsf{v}}} % \usepackage{caption|}\right\rangle} and \eq{\left\langle\overline{\mathbf{\mathsf{v}}} % \usepackage{caption^2}\right\rangle}, where \eq{\mathbf{\mathsf{v}}} % \usepackage{caption} is the mean velocity obtained at increments \eq{\delta x}, whose duration \eq{\Delta} is defined independently from step duration of the underlying L\'evy walk (namely \eq{\Delta\neq\tau}). Choosing \eq{\Delta\ll1}, the mean velocity \eq{\mathbf{\mathsf{v}}} % \usepackage{caption} can be exchanged with the instantaneous velocity \eq{v} of the random walker at various points in time, and then we can replace \eq{\mathbf{\mathsf{v}}} % \usepackage{caption\leftrightarrow v} in Eqs. (\ref{MosesDefinitionContinousTime}-\ref{NoahDefinitionContinousTime}). Accordingly, this means that we can obtain the exponents of the time series from $\left\langle\overline{|v|}\right\rangle$ and $\left\langle\overline{|v|^2}\right\rangle$, where we now use the following definition for the time average of an observable \eq{f}: $\overline{f}=(1/t)\int_0^t f(t')\Intd t'$. We note that here one should use a bit of care, since during an increment of duration \eq{\Delta}, the particle might have ended one step of the underlying random walk, and started another, and in this interval of the motion the mean velocity is different from the instantaneous value before/ after the transition. However we assume that if \eq{\Delta} is small enough, the effect of these occurrences is negligible in the context of the results in this manuscript. This is also confirmed by our numerics.
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{Figure3version3.pdf}
\caption{\footnotesize{Log-log plots for the averages of $|v|$ and $v^2$, as function of time. Red dots and the blue diamonds represent the values of $<|v|>$ and $<v^2>$, obtained from simulated data for different values of $t$, respectively. The solid green and black lines correspond to Eq. \eqref{EA|V|MiddleRegime} and Eq. \eqref{MeansqrvNonIntegrableRegime} respectively. The yellow and the magenta dashed lines represent the leading order terms in these equations, in the long time limit. (a) gives the result in the integrable regime, Sec. \ref{IntegrableRegime}, with $\gamma = 0.5$ and $\nu = 0.875$, (b) the middle regime, Sec. \ref{MiddleRegime}, with $\gamma = 0.5$ and $\nu = 0.625$, and (c) the non-integrable regime, Sec. \ref{NonintegrableRegime}, with $\gamma = 0.5$ and $\nu = 0.375$. The simulation results were generated with $10^8$ realizations and $\tau_c = 0.01$.}}
\label{figEnsembleMoments}
\end{figure*}
\subsection{Three regimes for \eq{M} and \eq{L}}
One can obtain the long-time asymptotic behavior of the ensemble mean of any symmetric observable \eq{\mathcal{O}(v)} in the system, as follows:
\begin{align}
\langle \mathcal{O}(v)\rangle&= 2\int_0^\infty \mathcal{O}(v)P_t(v)\Intd v\nonumber\\
&=2\int_0^{v_c}\mathcal{O}(v)P_t(v)\Intd v+
2\int_{v_c}^{\infty}\mathcal{O}(v)P_t(v)\Intd v.
\label{EnsembleMeanOfFv}
\end{align}
Given Eqs. (\ref{StepVelocityPDF1},\ref{VelocityPDFSmallV}), for the mean of $|v|$, we get
\small
\begin{align}
\langle |v|\rangle&\approx 2\int_0^{v_c}\frac{1}{2(1-\nu)|\Gamma(-\gamma)|\Gamma(1+\gamma)}t^\gamma|v|^{-\frac{\gamma}{\nu-1}}\Intd v\nonumber\\
&+2\int_{v_c}^{{\tau_c}^{\nu-1}}\frac{1-[1-(v/v_c)^{1/(\nu-1)}]^\gamma}{2(1-\nu)|\Gamma(-\gamma)| \Gamma (\gamma+1) }t^\gamma{|v|^{-\frac{\gamma}{\nu-1}}}\Intd v
\nonumber\\
&\approx-\frac{ \Gamma (-\gamma+\nu-1)t^{\nu-1}}{|\Gamma(-\gamma)| \Gamma ({\nu})}+\frac{ {\tau_c}^{\nu-\gamma}t^{\gamma-1}}{|\Gamma(-\gamma)|\Gamma(\gamma) (\gamma-\nu)}.
\label{EA|V|MiddleRegime}
\end{align}
\normalsize
Similarly, for the mean of $v^2$, we get
\small
\begin{align}
\langle v^2\rangle&\approx 2\int_0^{v_c}\frac{t^\gamma}{2(1-\nu)|\Gamma(-\gamma)|\Gamma(1+\gamma)}|v|^{1-\frac{\gamma}{\nu-1}}\Intd v\nonumber\\
&+2\int_{v_c}^{{\tau_c}^{\nu-1}}\frac{1-[1-(v/v_c)^{1/(\nu-1)}]^\gamma}{2(1-\nu)|\Gamma(-\gamma)| \Gamma (\gamma+1) }t^\gamma{|v|^{1-\frac{\gamma}{\nu-1}}}\Intd v
\nonumber\\
&\approx-\frac{ \Gamma (-\gamma+2 \nu-2)t^{2 \nu-2}}{\Gamma (2 \nu-1) \left| \Gamma (-\gamma)\right| }+\frac{{\tau_c}^{-\gamma+2 \nu-1}t^{\gamma-1}}{(\gamma-2 \nu+1) \Gamma (\gamma) \left| \Gamma (-\gamma)\right| }.
\label{MeansqrvNonIntegrableRegime}
\end{align}
\normalsize
To determine the leading behavior of these two means in the long time limit, note that Eqs. (\ref{WzBetaSmallV},\ref{VelocityID}) create a distinction between two different cases, depending on whether $\OO({v})=|v|$ or $v^2$ is integrable with respect to $\rho(v)$, Eq. \eqref{WzBetaSmallV}, or it is integrable with respect to the infinite-invariant density \eq{\mathcal{I}(v)}, Eq. \eqref{VelocityID}. In the first case, the leading order is obtained by first changing variables: $v/t^{\nu-1}\rightarrow\tilde{v}$ $\langle\OO(v)\rangle=2t^{q(\nu-1)}\int_0^{1/t^{\nu-1}}\OO(\tilde{v})\rho(\tilde{v})\Intd \tilde{v}+2t^{q(\nu-1)}\int_{1/t^{\nu-1}}^\infty\OO(\tilde{v})P_t(\tilde{v} t^{\nu-1})\Intd \tilde{v}$, and then in the range $t\gg1$ the first term is $\approx 2t^{q(\nu-1)}\int_0^{\infty}\OO(\tilde{v})\rho(\tilde{v})\Intd \tilde{v}$ and the second term approaches zero since its support vanishes. So, in this case
\small
\begin{equation}
\langle \mathcal{O}(v)\rangle\approx
2t^{q(\nu-1)}\int_0^{\infty}\OO(\tilde{v})\rho(\tilde{v})\Intd \tilde{v}.
\label{EnsembleMeanOfFvWzBeta}
\end{equation}
\normalsize
In the second case, when \eq{\langle \mathcal{O}(v)\rangle_{\mathbb{I}}\equiv\int_0^\infty \mathcal{O}(v)\mathcal{I}(v)\Intd v<\infty}, \eq{\mathcal{O}(v)} is integrable with respect to the infinite-invariant density, the contribution to its mean from the region \eq{v<v_c} can be neglected in Eq. \eqref{EnsembleMeanOfFv} in the limit \eq{t\rightarrow\infty}, to leading order, hence using Eq. \eqref{VelocityID} we get
\begin{equation}
\langle\mathcal{O}(v)\rangle\approx
2\int_{v_c\rightarrow0}^\infty \mathcal{O}(v)P_t(v)\Intd v\underbrace{\rightarrow}_{t\rightarrow\infty} 2t^{\gamma-1}\langle \mathcal{O}(v)\rangle_{\mathbb{I}}.
\label{Eq20ID}
\end{equation}
Notice that in this case, the temporal scaling of \eq{\langle\OO(v)\rangle} is similar for all the integrable observables, since it is determined only by the scaling of the infinite-density.
For \eq{\mathcal{O}(v)=\langle |v|\rangle} and \eq{\langle v^2\rangle} together, there are three regimes of behavior, included within the range \eq{\gamma,\nu<1}: The integrable regime, where both $\langle|v|\rangle$ and $\langle v^2\rangle$ are integrable with respect to \eq{\rho(v)}; The middle regime, where only the mean-absolute velocity is integrable; And the non-integrable regime, where neither observable is integrable (details below).
Figs. \ref{figEnsembleMoments}a-c display simulation results for the temporal behaviour of $\langle|v|\rangle$ and $\langle v^2\rangle$ in the integrable, the middle and the non-integrable regimes, respectively. The simulations match perfectly at long times with the exact expressions in Eqs. (\ref{EA|V|MiddleRegime},\ref{MeansqrvNonIntegrableRegime}), which denote both the leading order behavior of $\langle|v|\rangle$ and $\langle v^2\rangle$ in time, and the next-to-leading order. They also confirm the approach to the leading order asymptotic results, though this approach is slow. The results for the exponents $M$ and $L$ in the various regimes are shown in the lower two panels of Fig. \ref{FigOfFourPhaseDiagrams}. In Fig. \ref{FigICD}, we use the results for these exponents in the three regimes, in order to seek for a time invariant asymptotic shape of $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$, based on Eqs. (\ref{ICD},\ref{ALphaBetaDefinitions}).
\subsection{The integrable regime, \eq{1/2+\gamma/2<\nu<1}}
\label{IntegrableRegime}
In this regime, the leading behavior in time of \eq{\langle|v|\rangle} and $\langle v^2\rangle$ is given by the $\sim t^{\nu-1}$ and $\sim t^{2\nu-2}$ terms in Eq. \eqref{EA|V|MiddleRegime} and Eq. \eqref{MeansqrvNonIntegrableRegime}, respectively. The second term in both equations gives the next-to-leading order behavior. This result agrees with the calculation based on Eq. \eqref{EnsembleMeanOfFvWzBeta}. Similar to the argument in Eq. \eqref{EATAVsEA}, the ensemble-time averages \eq{\langle\overline{|v|}\rangle\propto t^{\nu-1}} and \eq{\langle\overline{v^2}\rangle\propto t^{2\nu-2}}, like their corresponding ensemble averages, and since we associate \eq{v} with \eq{\mathbf{\mathsf{v}}} % \usepackage{caption}, we now obtain the Latent and Moses exponents using Eqs. (\ref{MosesDefinitionContinousTime},\ref{NoahDefinitionContinousTime}):
\begin{equation}
M=\nu-\frac{1}{2},\qquad\mbox{and}\qquad L=\frac{1}{2}.
\label{MosesNoahNonIntegrableRegime}
\end{equation}
Since here both means are obtained from the same scaling limit of $P_t(v)$, we can now associate \eq{\rho(\tilde{v})} in this regime with $W(z_\beta)$, Eq. \eqref{ICD}, and here \eq{z_\beta=\tilde{v}=v/t^{\nu-1}}, so $\beta=\nu-1$ and $\alpha=0$, in agreement with Eqs. (\ref{ALphaBetaDefinitions},\ref{MosesNoahNonIntegrableRegime}).
Fig. \ref{FigICD}a displays the convergence of simulation results of $P_t(v)$ at increasing times, rescaled according to Eq. \eqref{ICD}, as function of $z_\beta$, to the scaling limit Eq. \eqref{WzBetaSmallV}.
Note that the Moses effect originates from the diverging mean duration of the L\'evy walk steps, namely because \eq{\langle\tau\rangle\rightarrow\infty} in \eq{g(\tau)}, Eq. \eqref{MLDistributionOfTaus}, which leads to statistical aging \cite{godreche2001statistics}.
\subsection{Middle regime, \eq{\gamma<\nu<1/2+\gamma/2}}
\label{MiddleRegime}
In this regime, \eq{v^2} is no longer integrable with respect to the scaling function $\rho(v)$. $|v|$, however, still is. Therefore, the leading-order behavior of $\langle|v|\rangle$ and $\langle\overline{|v|}\rangle$, remains proportional to $\sim t^{\nu-1}$, similar to the previous, integrable region. However since $\langle v^2\rangle$ is now integrable with respect to the infinite-density $\mathcal{I}(v)$ instead of $\rho(v)$, its leading behavior is now obtained from Eq. \eqref{Eq20ID}. The result is equal to the term $\propto t^{\gamma-1}$ in Eq. \eqref{MeansqrvNonIntegrableRegime} (and the second term there is now the next-to-leading order behaviour). Therefore, also $\langle\overline{v^2}\rangle\sim t^{\gamma-1}$. Note that in this regime we can obtain the time average of $v^2(t)$ also using arguments based on infinite-ergodic theory \cite{akimoto2019infinite}.
Using Eqs. (\ref{MosesDefinitionContinousTime},\ref{NoahDefinitionContinousTime}), this yields
\begin{equation}
M=\nu-\frac{1}{2},\qquad\mbox{and}\qquad L=\frac{\gamma-2\nu+2}{2}.
\label{MosesNoahMiddleRegime}
\end{equation}
This regime continuously extends the one introduced in Eq. (\ref{MosesNoahNonIntegrableRegime}). The First moment can still be described by the scaling shape of the PDF at the bulk. However, since the second moment of this PDF diverges with respect to $\rho(v)$, here we see for the first time the emergence of a Noah effect, in addition to Moses.
Since the mean of $|v|$ and $v^2$ are obtained from two different scaling regimes of $P_t(v)$, Eqs. (\ref{ICD}-\ref{ICD1}) are not valid, and $\alpha$ and $\beta$ are not defined. Fig. \ref{FigICD}b shows that, if we did not know the model, and try to obtain $\alpha,\beta$ from Eq. \eqref{ALphaBetaDefinitions} in this regime from the data, we would find $\alpha=\gamma-2\nu+1,\beta=\gamma-\nu$, but with this rescaling, $P_t(v)$ does not converge to a time-invariant shape.
\begin{figure*}
\centering
(a)\includegraphics[width=0.32\textwidth]{Mplotv12.pdf}
(b)\includegraphics[width=0.32\textwidth]{Lplotv12.pdf}
(c)\includegraphics[width=0.32\textwidth]{Jplotv12.pdf}
(d)\includegraphics[width=0.32\textwidth]{Hplotv12.pdf}
\caption {\footnotesize{Phase diagrams of the scaling exponents describing the decomposition of the anomalous diffusion of L\'evy walks into three constitutive effects, and their various magnitudes. (a) gives the Moses exponent $M$ that quantifies the Moses effect, (b) gives the Latent exponent $L$ that quantifies the Noah effect, (c) gives the Joseph exponent $J$ that quantifies the Joseph effect, and (d) gives the Hurst exponent $H$. These results were obtained with for $\eta$ = 1 (see Sec. \ref{Section5}). In the ``$\infty$" regime, the Hurst exponent is divergent and the other exponents are not well defined, see Sec. \ref{Section5} and Appen. \ref{AppendixInfinityRegime}}}
\label{FigOfFourPhaseDiagrams}
\end{figure*}
\subsection{The non-integrable regime, \eq{\nu<\gamma}}
\label{NonintegrableRegime}
In this regime neither the first, nor the second moment of $|v|$ are integrable with respect to $\rho(v)$, Eq. \eqref{WzBetaSmallV}. Instead, both the mean velocity, and the mean squared velocity are integrable with respect to the infinite-density. Here, using Eqs. (\ref{StepVelocityPDF1},\ref{VelocityID},\ref{Eq20ID}) we get
$\langle |v|\rangle,\langle \overline{|v|}\rangle\propto t^{\gamma-1}$, as well as $\langle v^2\rangle,\langle \overline{v^2}\rangle\propto t^{\gamma-1}$, so from Eqs. (\ref{MosesDefinitionContinousTime},\ref{NoahDefinitionContinousTime}), we find
\begin{equation}
M=\gamma-\frac{1}{2},\qquad\mbox{and}\qquad L=1-\frac{\gamma}{2}.
\label{MosesNoahNuSmallerThanGamma}
\end{equation}
In this case, we associate $W(z_\beta)$, Eq. \eqref{ICD}, now with the infinite-invariant density \eq{\mathcal{I}(v)}, Eq. \eqref{VelocityID}, and $\alpha=2L-1, \beta=0$, so $z_\beta=v$. The Noah effect tells us that the asymptotic shape of the increment PDF is given by a non-normalizable function, and the relation between $M$ and $L$ here also agrees with Eq. \eqref{LAndMID}, as it should.
Fig. \ref{FigICD}c shows how simulation results of $t^{\alpha}P_t(v)$ at converge increasing times to $\mathcal{I}(v)$, the infinite invariant density.
{As in the other regimes, here the mean duration of the L\'evy walk steps in \eq{g(\tau)}, Eq. \eqref{MLDistributionOfTaus} is divergent, however since $\nu$ is small, the step velocity decays very quickly with the duration. Therefore the step displacement $\chi\sim\tau^\nu$, is almost decoupled from $\tau$.
This implies too things: First, the MSD of the process now mostly depends on how many steps the walker can have between $t'=0$ and $t$, and that is determined only by the value of $\gamma$. So the Hurst exponent in this regime depends only on $\gamma$. Second, by a hand-waving argument we can see why $M$ and $L$ depend only $\gamma$; because if the step displacement depends only on this parameter, the average velocity $\mathbf{\mathsf{v}}} % \usepackage{caption$ in all the time-series increments withing those steps will depend only on this parameter too.
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{Figure3versionfinal.png}
\caption {\footnotesize{Numerical examination of the convergence of the increment PDF $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$ to a time-invariant shape, based on Eqs. (\ref{ICD},\ref{ICD1}) and the quantification of the Moses and Noah effects. From Eq. \eqref{ICD1}: $\alpha = 2L - 1$ and $\beta = M + \alpha - 0.5$. In the Log-log plots (a),(b), and (c), in symbols, we see the rescaled PDF $t^{\alpha+\beta}P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$, obtained from simulation results of $3*10^8$ paths, in regimes A,B and D respectively (in regime C the shape of the PDF is behave similar to the last, at increasing times). The measurements were performed at times $t = 10^4$ (red dots), $10^5$ (green dots), $10^6$ (blue dots) and $10^7$ (black dots). (a) Here we used $\gamma = 0.5$, $\nu = 0.875$, leading to $L = 0.5$, and $M = 0.375$. The figure shows that the simulation results converge at increasing times to the normalized scaling shape given in Eq. \eqref{WzBetaSmallV} (solid mustard line). (b) Here, $\gamma = 0.5$, $\nu = 0.625$, and $L = 0.625$, $M = 0.125$. Attempting to find an asymptotic scaling shape in this regime, which corresponds to Eq. \eqref{ICD}, does not work, since $\langle|\mathbf{\mathsf{v}}} % \usepackage{caption|\rangle$ and $\langle\mathbf{\mathsf{v}}} % \usepackage{caption^2\rangle$ do not correspond to a single scaling regime of $P_t(\mathbf{\mathsf{v}}} % \usepackage{caption)$. (c) Here $\gamma = 0.5$, $\nu = 0.375$, and $L = 0.75$, $M = 0$. $M$ and $L$ here obey the scaling relation in Eq. \eqref{LAndMID}, hence we expect to find that the increment PDF approaches the shape of a non-normalizable infinite-invariant density. This is confirmed by the solid mustard line, that represents Eq. \eqref{VelocityID}. The insets show the same results, but in semi-log plots.}}
\label{FigICD}
\end{figure*}
\section{\eq{J}, and how we obtained it}
\label{JosephExponent}
The Joseph exponent depends on the shape of the auto-correlation function. However, this quantity is difficult to obtain for many systems, analytically and numerically.
In practice, the Joseph effect is often quantified by designated methods, such as the so-called rescaled range statistic (R/S) \cite{Hurst}, wavelet decomposition \cite{WAVELET} or detrended fluctuations analysis \cite{Pen94}. Additional information on the correspondence between our definition of $J$ and the latter method is given in appendix \ref{DFAAndEATMSD}.
Here, for the L\'evy process, we use a measure which is easier to handle analytically; the ensemble averaged time-averaged MSD $\left\langle\overline{\delta^2}\right\rangle$, defined as
\begin{equation}
\langle \overline{\delta^2} \rangle \equiv{} \left\langle{} \frac{1}{t-\Delta} \int_0 ^{t-\Delta}
\left[ x(t_0+\Delta) - x(t_0)\right]^2 \mbox{ d} t_0\right\rangle{}.
\label{EATAMainText}
\end{equation}
Note that Eq. \eqref{EATAMainText} should not be confused with \eq{\langle\overline{\mathbf{\mathsf{v}}} % \usepackage{caption^2}\rangle} in Eq. \eqref{NoahDefinitionContinousTime}, since in the latter the increments are strictly non-overlapping, whereas in \eq{\langle\overline{\delta^2}\rangle} they are.
This quantity is related to the auto-correlation function, via \cite{meyer2017greenkubo}
\begin{equation}
\langle \overline{\delta^2} \rangle \approx \frac{2}{t}\int_0 ^{t}\!\!\mathrm{d}t_0\int_0^{\Delta}\!\! \mathrm{d}t_2\int_0^{t_2}\!\! \mathrm{d}t_1 \langle v(t_1+t_0)v(t_2+t_0) \rangle,
\label{xmxofC}
\end{equation}
when $t\gg \Delta$. The scaling of this function for different types of auto-correlations is discussed in Appen. \ref{Appenctamsd}, where we also show the correspondence between $\langle\overline{\delta^2}\rangle$, the autocorrelation function, and our definition in Eq. \eqref{JosephDefinitionContinousTime}. In all the cases considered in the appendix (even for $J\leq 1/2$), the asymptotic scaling is
\begin{equation}
\langle \overline{\delta^2} \rangle \sim t^{2L+2M-2} \Delta^{2J}.
\label{EATAMSDScaling}
\end{equation}
Our model is described by type (II) in the appendix.
This means that the Joseph exponent is given by the scaling of $\langle \overline{\delta^2} \rangle$ with the lag time $\Delta$ (note, that this observation was already made in \cite{meyer2018anomalous}, however, due to a typo it was read '$t$' instead of '$\Delta$').
To obtain the Joseph exponent in various regimes, we use the results of the calculation of the ensemble-time averaged MSD, obtained for this model in Ref. \cite{albers2018exact}.
Particularly, in that Ref., the scaling of $\langle \overline{\delta^2} \rangle$ with respect to time and \eq{\Delta} was calculated for a general shape of \eq{g(\tau)}, with an asymptotic fall-off as in Eq. \eqref{MLDistributionOfTaus}, at large \eq{\tau}, and it was shown to not depend on the exact behavior at small \eq{\tau}s.
Given this knowledge, the time-averaged MSD (for \eq{0<\left\{\gamma,\nu\right\}<1)} for the L\'evy walk model we study here, has the following scaling \cite{albers2018exact}
\begin{align}
\langle \overline{\delta^2}\rangle\propto\begin{cases}
t^{2\nu-2}\Delta^2,\qquad\qquad \HHquad \gamma/2+1/2<\nu \\
t^{\gamma-1}\Delta^{1+2\nu-\gamma},\qquad \gamma/2<\nu<\gamma/2+1/2\\
t^{\gamma-1}\Delta,\qquad\qquad\quad \nu<\gamma/2
\end{cases}.
\label{TAMSDRadons}
\end{align}
Using Eq. \eqref{TAMSDRadons} and Eq. \eqref{EATAMSDScaling}, we find that \begin{align}
J=\begin{cases}
1,\qquad\qquad\qquad\qquad\quad\quad \gamma/2+1/2<\nu\\
(1+2\nu-\gamma)/2,\qquad\quad\quad\gamma/2<\nu<\gamma/2+1/2\\
1/2,\qquad\qquad\qquad\qquad\quad \nu<\gamma/2
\end{cases}.
\label{jRadons}
\end{align}
Note that since $\gamma<1$, the mean step duration in all these regimes diverges. This implies that the random walker essentially walks in the same direction for almost all of the time $t$, regardless how long it is. In turn, this means the process is correlated in the whole parameter regime that we study. But when $
\nu<\gamma/2$, the average correlations decay rather quickly with $\Delta$ because the step velocity changes only very little with the step duration, hence in this regime we do not see a Joseph effect (the difference between the mean velocity at increments belonging to the same steps of the L\'evy walk, versus increments of other steps, is small). The onset of the effect is above the line $\nu=\gamma/2$. It is maximal when $J=1$, at $\gamma+1<2\nu$.
Fig. \ref{FigOfFourPhaseDiagrams}c shows a phase diagram summarizing the different regimes of the Joseph effect, shown in Eq. \eqref{jRadons}. Fig. \ref{FigOfFourPhaseDiagrams}d shows the different regimes of the Hurst exponents which results from the combined affect of the various effects leading to the anomalous diffusion, and calculated using Eq. \eqref{hjlm}. Our simulation results for several arbitrary samples of values of $\nu$ and $\gamma$ in these regimes agree with the analytic expectation.
\section{A Generalized model}
\label{Section5}
In this section, following \cite{albers2018exact,bothe2019mean} we extend the model displayed above by introducing a new parameter $\eta$. This parameter generalize Eq. \eqref{NonlinearDurationVelocityCoupling} by modifying the relation between the $i$th step velocity $V_i$, its duration $\tau_i$ and the actual time in motion $t'$, as follows
\begin{equation}
V_{\nu,\eta} = \pm\tilde{c}_1 \tau^{\nu-\eta} t'^{\eta-1}.
\end{equation}
Some values of $\eta$ correspond to special cases: The L\'evy walk we studied above corresponds to $\eta=1$, when $\eta=\nu$ we get a Drude-like model \cite{schulz1997anomalous,benkadda1998chaos}, and when $\eta \rightarrow 0$ or $\eta\rightarrow\infty$ we approach either a jump-then-wait type of coupled continuous-time random walk, or a wait-then-jump model, respectively \cite{bothe2019mean}. As we will now show, modifying this parameter changes the onset of the ``$\infty$" regime. Our simulation results suggest that when $\eta$ is within the open range $(0,\infty)$, the behavior of all the effects in regimes A,B,C,D in Fig. \ref{fig2:QgphaseAllTogether} does not change, however the regimes themselves may expand or shrink and disappear.
Let's look again at the PDF {$P_t(x)$}, of the particles' displacement $x$ at time $t$. Here,
\begin{align}
P_t(x) = \int_{-\infty}^{\infty} \Intd x' \int_{0}^{t} \Intd t' A(x',t') r(x-x'|t-t').
\label{maineqP}
\end{align}
where $A(x',t')$ is the joint probability density to land on $x'$ between $x$ and $x+dx$ in a complete step ending at $t'<t$, and $r(x-x'|t-t')$ is the conditional probability density of the displacement in the last, incomplete step given the duration of the walk is $t$.
The following calculation of the MSD for this model is adapted from Ref. \cite{bothe2019mean}.
Let $\hat{f}(k,t)=\int_{-\infty}^\infty f(x,t)\exp(-i kx)\Intd x$, be the Fourier transform of some function $f(x,t)$, from $x\rightarrow k$. Eqs. (\ref{FourierOfP},\ref{FourierOfr}) and Eq. (\ref{FourierOfA}) below, represent the characteristic functions of the probability densities $P,r$ and $A$, respectively. All the functions except for $P$ lack normalization on unity; the zero terms of the expansions are denoted as $r_0(t) = \int r(x|t) dx \neq 1$ and $A_0(t) = \int A(x,t) dx \neq 1$. Let $x_2(t) \equiv \langle x^2(t) \rangle$ be MSD at time $t$, $A_2(\tau) = \int \chi^2A(\chi,\tau) \Intd \chi$ is the marginal second moment of displacement $\chi$ in a single complete step of duration $\tau$ and $r_2(\tau^*) \equiv \int {\chi^*}^2r(\chi^*|\tau^*) \Intd \chi^*$ is the MSD of the displacement $\chi^*$ in the last, incomplete step. The duration $\tau^*$ of the latter is defined in Eq. \eqref{TotalTime}. After Fourier transform, we get
\begin{equation}
\hat{P}(k|t) = 1 - \frac{1}{2} k^2x_2(t) + o(k^2)
\label{FourierOfP}
\end{equation}
\begin{equation}
\hat{r}(k|t) = r_0(t) -\frac{1}{2}k^2r_2(t) + o(k^2)
\label{FourierOfr}
\end{equation}
\begin{equation}
\hat{A}(k|t) = A_0(t) - \frac{1}{2}k^2A_2(t) + o(k^2)
\label{FourierOfA}
\end{equation}
Let $\hat{f}(k,s)=\int_{0}^\infty \hat{f}(k,t)\exp(-st)\Intd t$ be the Laplace transform of $\hat{f}(k,t)$. In Fourier and Laplace space, from Eq. \eqref{maineqP} we obtain
\begin{equation}
\small
\hat{P}(k,s) = A_0(s)r
_0(s) - \frac{k^2}{2}\left[A_0(s)r_2(s) + A_2(s)r_0(s)\right] + o(k^2)
\normalsize
\label{FourierLaplaceOfP}
\end{equation}
On comparing Eq. \eqref{FourierOfP} and Eq. \eqref{FourierLaplaceOfP}, we can now obtain the MSD using
\begin{equation}
\langle x^2(s) \rangle = A_0(s)r_2(s) + A_2(s)r_0(s)
\label{LaplaceOfx}
\end{equation}
Calculating the values on the right-hand side of Eq. \eqref{LaplaceOfx}, and taking the inverse Laplace transform, one can now derive the MSD. Part of this calculation, performed in \cite{bothe2019mean}, was to obtain the second marginal moment of the function $r(x|t)$:
\begin{align}
r_2(t) \simeq \gamma\tilde{c}_1^{2}\tau_0^{\gamma}t^{2\eta} \int_{t}^{\infty} t'^{2(\nu-\eta)-1-\gamma}{d}t'.
\end{align}
From here, we can see that $r_2$, and therefore also $\langle x^2\rangle$, can only obtain a finite value when $\gamma > 2(\nu-\eta)$. This explains the crossover to the ``$\infty$" regime, which occurs when $\nu>\gamma/2+\eta$.
When $\nu<\gamma/2+\eta$ and $\gamma < 1, 2\nu < \gamma$, the MSD is \cite{bothe2019mean}
\begin{align}
\small
\langle x^2(t)\rangle \approx \gamma &\left[ \frac{\Gamma(2\nu+1-\gamma)}{\Gamma(1-\gamma)(2(\nu-\eta)-\gamma)\Gamma(2\nu+1)} t^{2\nu}\right.\nonumber\\
&\left.+ \frac{B(2\nu+1,\gamma-2\nu)}{\Gamma(1-\gamma)\Gamma(1+\gamma)}\tilde{c}_1^{2} t^{\gamma} \right],
\normalsize
\end{align}
where $B(a,b)$ is the Beta-function. This is dominated by the second term, since $2\nu < \gamma$, and therefore $\langle x^2(t) \rangle \propto t^{\gamma}$, which gives the value of the Hurst Exponent as $H=\gamma/2$, similar to what is seen seen in Fig. \ref{FigOfFourPhaseDiagrams}d. When $\nu<\gamma/2+\eta$, but $\gamma <1, 2\nu > \gamma$, the MSD reads \cite{bothe2019mean}
\begin{align}
\small
\langle x^2(t)\rangle \simeq \gamma\frac{\Gamma(2\nu-\gamma)}{\Gamma(2\nu+1)\Gamma(1-\gamma)}\frac{4\nu-2\eta-2\gamma}{2(\nu-\eta)-\gamma} \tilde{c}_1^{2}t^{2\nu},
\normalsize
\end{align}
which gives the value of the Hurst Exponent as $H=\nu$ also similar to Fig. \ref{FigOfFourPhaseDiagrams}(d). The results shown in Fig. \ref{FigOfFourPhaseDiagrams} are for $\eta = 1$, whereas Eqs. (\ref{FourierLaplaceOfP},\ref{LaplaceOfx}) are calculated for any value of $\eta$. This shows that the power-law dependence of the mean squared displacement is independent of the exponent $\eta$ and the particular value of $\eta$ only enters in the prefactors, when $\langle x^2\rangle$ is finite. When $\eta$ is very small, only regime D in Fig. \ref{fig2:QgphaseAllTogether} survives, and beyond it we have the non-scaling ``$\infty$" regime. When $\eta$ is very large, regime A expends higher into the realm of $\nu>1$.
\section{Discussion}
\label{Discussion}
Imagine that you get hold of a ``blind" set of data series, containing the positions of an ensemble of random walkers at various times in the interval \eq{[0,t]}. This data was generated by a L\'evy walk model, but you do not have this prior knowledge. Our analysis allows us to uncover the main features of the hidden process that cause its behavior to scale anomalously with time, despite the fact that we do not know what process generated the data. Elucidating the origins of anomalous diffusion observed in experimental data is crucial in order to understand the underlying functioning of the system, and it is studied therefore these days , e.g., using new advanced methods for single-particle tracing \cite{kepten2011ergodicity,weigel2011ergodic,wang2017three,sabri2020elucidating}. We encourage the verification of our results for example in (but not limited to) future such experiments, in particular e.g. the scaling relation in Eq. \eqref{hjlm} and Eq. \eqref{ICD}, and consequently its application.
In addition to learning about the origins of the anomalous diffusion, one may use the knowledge about the Moses, Noah and Joseph effects in order to try to extrapolate which processes can and cannot be at least good candidates to represent the underlying dynamics. These days, there are many studies which use techniques such as machine learning \cite{munoz2019machine,bo2019measurement,janczura2020machine}, Bayesian statistics \cite{thapa2018bayesian} and more, e.g. \cite{kepten2013improved,thapa2020leveraging}, to try to infer the Hurst exponent or distinguish between various known models such as continuous-time random walk, fractional Brownian motion and others, which lead to anomalous scaling of the MSD, based only on analysis of data obtained from single trajectories.
This issue is even being studied today as part of a multi-group competition to characterize the properties of anomalous diffusion in data, called the ANDI challange \cite{munoz2020andi}.
Though we cannot fully and uniquely restore the underlying dynamics just by discerning the scaling properties of the process from the data, the characterization of anomalous diffusion using three additional exponents $M,L$ and $J$, in addition to the Hurst, does bring additional tools which can be helpful for modeling it. In this sense, this decomposition should also be useful for example for the modeling of diffusion in the membranes of living cells done in \cite{weigel2011ergodic}, where a Moses and a Joseph effect seem to have been observed. Another interesting example is found in \cite{tabei2013intracellular}, where the authors observed intercellular transport of insulin granules in eukrayotic cells, and then used information from the time-averaged MSD (Joseph), and the evolution of the absolute-mean of the increments (Moses) to model it. The authors compared two candidate models to describe their dynamics: fractional Brownian motion and continuous-time random walk, and concluded that non is sufficiently good for a full description of the system. They therefore continued by proposing a different, `hybrid' model based on the previous two \cite{tabei2013intracellular}. Since the first model leads only a Joseph effect, but the second leads to both Moses and Noah, a full three-effect decomposition here, which takes into account also the inherent relation between them, Eq. \eqref{hjlm}, might shed more light on the unified model. Of-course, in any case if one seeks to fully reconstruct the underlying process from the data, a complete knowledge of the entire correlation structure would be required, including two-point, and all the higher order correlations.
The L\'evy walk model that we studied in this paper, is a prototypical example which shows how the three effects analysis can be used for many other processes as well. The results in Fig. \ref{fig2:QgphaseAllTogether} and Fig. \ref{FigOfFourPhaseDiagrams} eventually only depend on two inputs: the shape of the step durations PDF at large $\tau$s, and the coupling between the step durations and the velocity, which can also be translated to the coupling between the step duration and displacement, since the step-displacement $\chi=V\tau$. Therefore, a class of process which can be mapped into a coupled step-duration and step-displacement process, which also includes other processes such as the Pommeau-Manneville map \cite{meyer2018anomalous} and ATTM \cite{massignan2014nonergodic}, will display similar properties as in the various regimes in Fig. \ref{fig2:QgphaseAllTogether} and Fig. \ref{FigOfFourPhaseDiagrams}. These phase diagrams describe their dynamics as well, after change of variables (see e.g., \cite{meyer2018anomalous}).
\\
\textbf{Acknowledgement}
\\
This research is supported by the Max-Planck society (EA, PGM and HK), and in part by the US National Science Foundation, through grant DMR-1507371 (VA and KEB). KEB also thanks the Max-Planck-Institut für Physik komplexer Systeme for its support and hospitality during his visit when this work was initiated.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,249
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Volume 53, Issue suppl_3
Section II: Beta-Cell Therapeutic Targets Other Than ATP-Sensitive K+ Channels| December 01 2004
Beta-Cell-Targeted Expression of a Dominant-Negative Mutant of Hepatocyte Nuclear Factor-1α in Mice: Diabetes Model with β-Cell Dysfunction Partially Rescued by Nonglucose Secretagogues
Maria Sörhede Winzell;
Maria Sörhede Winzell
1Department of Medicine, Lund University, SE-221 84 Lund, Sweden
Giovanni Pacini;
Giovanni Pacini
2Metabolic Unit, Institute of Biomedical Engineering (ISIB), National Research Council (CNR), Padova, Italy
Claes B. Wollheim;
Claes B. Wollheim
3Experimental Diabetology Group, Department of Cell Physiology and Metabolism, University Medical Center, Geneva, Switzerland
Bo Ahrén
Address correspondence and reprint requests to Dr. Bo Ahrén, Department of Medicine, Lund University, B11 BMC, SE-221 84 Lund, Sweden. E-mail: bo.ahren@med.lu.se
Diabetes 2004;53(suppl_3):S92–S96
https://doi.org/10.2337/diabetes.53.suppl_3.S92
Maria Sörhede Winzell, Giovanni Pacini, Claes B. Wollheim, Bo Ahrén; Beta-Cell-Targeted Expression of a Dominant-Negative Mutant of Hepatocyte Nuclear Factor-1α in Mice: Diabetes Model with β-Cell Dysfunction Partially Rescued by Nonglucose Secretagogues. Diabetes 1 December 2004; 53 (suppl_3): S92–S96. https://doi.org/10.2337/diabetes.53.suppl_3.S92
We studied islet function in mice with β-cell-targeted expression of a dominant-negative mutant of hepatocyte nuclear factor (HNF)-1α. At age 2–3 months, anesthetized transgenic and wild-type male mice underwent an intravenous glucose (1 g/kg) tolerance test (IVGTT). It was found that transgenic mice had an abolished insulin response in association with severe glucose intolerance. In other tests, the 5-min insulin response to intravenous arginine was impaired by 79% (P = 0.032) and the 15-min insulin response to gastric glucose was suppressed by 97% (P = 0.006). In islets incubated for 60 min, the insulin response to glucose (3.3–22.2 mmol/l) was impaired by >80% in transgenic mice. In contrast, insulin responses to nonglucose secretagogues were only partially suppressed (to GLP-1 [100 nmol/l] by 40%, to carbachol [1 μmol/l] by 20%, and to palmitate [0.5 mmol/l] by 15%), whereas the response to depolarization by KCl (50 mmol/l) was not reduced. Finally, the IVGTT data insulin sensitivity in transgenic mice was not significantly different from that of wild-type mice. Thus, mice with targeted suppression of β-cell HNF-1α represent a good diabetes model exhibiting severely impaired insulin secretion after glucose with marked glucose intolerance. In contrast, the insulin responses to nonglucose stimuli are not suppressed when the islet insulin content is taken into account.
Maturity-onset diabetes of the young (MODY) is a monogenic form of diabetes representing 2–5% of all cases of type 2 diabetes that usually develops before the age of 25 years and is characterized by β-cell dysfunction (1,2). While the inheritance is autosomal dominant, several genetically different forms of MODY have been described, classified as MODY1-6 (1,3). MODY3, the most common form, is caused by heterozygous mutations in the gene encoding the homeodomain-containing transcription factor hepatocyte nuclear factor (HNF)-1α, which in humans is located on chromosome 12 (4). Patients with MODY3 show a progressively impaired insulin secretion with the risk of subsequent development of diabetes complications if not adequately treated with oral hypoglycemic agents or insulin (1,5–8). Human HNF-1α consists of 631 amino acids and has three functional domains: the dimerization, the DNA-binding, and the transactivation domains. The transcription factor is expressed in liver, kidney, intestines, spleen, and exocrine pancreas besides in β-cells (1,9,10). Mutations responsible for MODY3 have been localized to both the different functional domains and the promoter region of the HNF-1α gene (7,11). In the β-cells, HNF-1α has been shown to regulate expression of glucose transporter-2 (GLUT-2), l-type pyruvate kinase, and insulin (3,10,12,13).
To understand the molecular consequences for β-cell function of mutations in the HNF-1α gene, mice with deletion of the gene (HNF-1α−/−) have been generated (14,15). These mice display severely blunted insulin secretion after challenge with glucose in association with glucose intolerance. In islets, a defective glycolytic signaling has been reported, suggesting targeting of β-cell glucose metabolism (15). However, these general knockout mice also develop severe liver and kidney dysfunction, which is not seen in MODY3 patients. To avoid this complication, mice with β-cell-targeted dominant-negative mutant HNF-1α (RIP-DNHNF-1α) have been generated (16,17). The RIP-DNHNF-1α mice exhibit pronounced hyperglycemia and glucose intolerance in association with severely blunted insulin secretion after glucose challenge, as MODY3 patients, but without any signs of kidney or liver dysfunction. Furthermore, these mice have impaired expression of GLUT-2, reduced β-cell proliferation and abnormal islet cytoarchitecture. They are therefore useful for further studies on the role of HNF-1α in β-cell function and for characterization of the molecular basis of MODY3.
In this study, we have characterized glucose tolerance and β-cell function in RIP-DNHNF-1α mice by performing in vivo challenges with intravenous or gastric glucose as well as intravenous arginine. We also determined the islet content of insulin and conducted islet incubations for studies of in vitro insulin responses to glucose, glucagon-like peptide-1 (GLP-1), the cholinergic agonist carbachol, palmitate, and KCl.
Animals.
As described previously, the dominant-negative (DN) HNF-1α cDNA was inserted into a plasmid under the control of the rat insulin promotor (RIP) for construction of a RIP-DN HNF-1α transgene (16,18). The transgenic mice were generated by pronuclear microinjection of the construct in B6/CBAJ-F1 × B6/CBAJ-F1 zygotes (16). Transgenic and wild-type mice were transported from the animal facility of the University Medical Centre, Geneva, to Taconic A/S, Ry, Denmark. Embryonic transfer was then performed to female C57BL/6J. Transgenic animals were identified by PCR on genomic DNA extracted from tail biopsies using the primer combination 5′-CTGCTAACCATGTTCATGCCT-3′ and 5′-TGAATTGCTGAGCCACCTCTC-3′. Subsequently, mice were moved and housed at the In Vivo Department of the Biomedical Centre, Lund University, and the breeding was continued by mating DN HNF-1α males to wild-type C57BL/6J females purchased from Taconic A/S. Offsprings were then back-crossed to C57BL/6J for at least 10 generations. In the experiments, DN HNF-1α transgenic mice were matched with wild-type littermates. The animals were kept in a 12-h light schedule (lights on at 6:00 a.m.) and given a standard pellet diet (fat 11.4%, carbohydrate 62.8%, protein 25.8% on an energy base, total energy 12.6 kJ/g) and tap water ad libitum. The Lund University Ethic Committee approved the study. Because it was previously demonstrated that the most pronounced diabetes phenotype in this diabetes model was seen in male mice (16,17), the present study was undertaken only in male heterozygous animals.
In vivo experiments.
The in vivo studies were performed in late morning after removal of food from the cages 4 h earlier. The animals were anesthetized with an intraperitoneal injection of midazolam (Dormicum, 0.2 mg/mouse; Hoffman-La-Roche, Basel, Switzerland) as well as a combination of fluanison (0.4 mg/mouse) and fentanyl (Hypnorm, 0.02 mg/mouse; Janssen, Beerse, Belgium). Thirty minutes later, a blood sample was taken from the retrobulbar, intraorbital, capillary plexus in heparinized tubes and d-glucose (1 g/kg; British Drug Houses, Poole, U.K.) or arginine (0.25 g/kg; Sigma) was rapidly injected intravenously. The volume load was 10 μl/g body wt. In one series of experiments, a gastric tube (outer diameter 1.2 mm) was inserted in the anesthetized mice after 16 h of fasting, and glucose (150 mg) was instilled into the stomach. At specific time points after injection or gavage, blood samples (75 μl each) were collected. A total of up to seven samples were taken during each experiment. The removal of this amount of blood has previously been shown not to alter baseline glucose levels in mice (19). Blood was kept in heparinized tubes and immediately centrifuged, whereupon plasma was separated and then stored at −20°C until analysis.
In vitro experiments.
Islets were isolated by standard collagenase digestion (Collagenase P; Roche Diagnostics, Mannheim, Germany) and subsequently handpicked under a stereo microscope. Thereafter, islets were preincubated for 30 min in HEPES balanced salt solution (HBSS; 114 mmol/l NaCl, 4.7 mmol/l KCl, 1.16 mmol/l MgSO4, 20 mmol/l HEPES, 2.5 mmol/l CaCl2, and 0.1% BSA, pH 7.35) containing 3.3 mmol/l glucose. Then, three islets at a time were transferred to a multiwell plate (on ice) containing 200 μl per well of the same buffer in the presence of different concentrations of glucose with or without addition of GLP-1 (100 nmol/l; Peninsula Laboratories, Merseyside, U.K.); carbachol (1 μmol/l; Sigma Chemical, St Louis, MO); palmitate (0.5 mmol/l; Sigma), complexed to 1% fatty acid free bovine serum albumin (ICN Biomedicals, Aurora, OH); or KCl (50 mmol/l; Sigma). When all islets had been transferred, the plate was again placed in an incubator at 37°C; after 60 min, a sample from the buffer was removed for measurement of insulin.
Islet insulin content.
Total islet insulin content was measured in batches of four islets. The islets were frozen and sonicated twice in 100 μl acidic ethanol (0.25 mol/l HCl in 87.5% ethanol). The samples were centrifuged and total insulin was measured in the supernatant.
Assays.
Insulin was determined radioimmunochemically with the use of a guinea pig anti-rat insulin antibody, 125I-labeled human insulin as tracer, and rat insulin as standard (Linco Research, St. Charles, MO). Free and bound radioactivity was separated by use of an anti-IgG (goat anti-guinea pig) antibody (Linco). The sensitivity of the assay is 12 pmol/l and the coefficient of variation (CV) is <3% within assays and <5% between assays. Plasma glucose concentrations were determined with the glucose oxidase technique.
Calculations and statistics.
Data are reported as means ± SEM. From the IVGTT, the acute insulin response (AIR) to intravenous glucose was calculated as the mean of suprabasal 1- and 5-min values, and the area under the insulin curve (AUCinsulin) was calculated using the trapezoid rule for insulin data from 0 to 75 min. The glucose tolerance was quantified from the glucose elimination constant (KG; expressed as percent elimination of glucose per minute) as the reduction in circulating glucose between 1 and 20 min after intravenous administration following logarithmic transformation of the individual plasma glucose values (20). A similar estimation was performed for the total 1- to 75-min glucose disappearance rate (KG[1–75]). This parameter indicates the rate of glucose disappearance during the whole test, when the delayed insulin effect is properly accounted for. The minimal modeling of insulin and glucose data from the IVGTT to assess the insulin sensitivity index (20) was possible only in wild-type mice. It would not make sense in transgenic mice, given the total lack of dynamic insulin after intravenous glucose. Therefore, we estimated insulin sensitivity by using a different approach, based on the general definition of insulin sensitivity following a glucose load (i.e., the glucose disappearance rate at specific insulin levels). In our case, we used the total 75-min glucose disappearance rate divided by AUCinsulin. Pearson's product-moment correlation coefficients were obtained to estimate linear correlation between variables. Statistical comparisons were performed with Student′s unpaired and paired t tests and, when multiple comparisons were performed, with ANOVA.
Body weight and baseline levels of glucose and insulin.
The studies were performed when the mice were 2–3 months of age. Body weight was not different between the two groups (26.8 ± 0.9 g in wild-type mice, n = 13, vs. 26.3 ± 0.8 g in transgenic mice, n = 22). In nonfasted mice, glucose levels were highly elevated in the transgenic mice (14.3 ± 1.1 mmol/l, n = 13) compared with wild-type mice (8.2 ± 0.8 mmol/l, n = 8, P < 0.001). In contrast, the 16-h fasting glucose levels were not different between the groups (5.1 ± 0.5 mmol/l in transgenic mice, n = 7, vs. 5.1 ± 0.4 mmol/l in wild-type mice, n = 7). Nonfasting insulin levels did not differ between the groups (304 ± 50 pmol/l in transgenic mice vs. 290 ± 41 pmol/l in wild-type mice). Similarly, there was no significant difference in fasting insulin between the groups (93 ± 18 pmol/l in transgenic mice vs. 180 ± 61 pmol/l in wild-type mice).
Intravenous glucose.
Following the intravenous administration of glucose (1 /kg), circulating glucose peaked at 1 min and was thereafter eliminated by first-order kinetics. The glucose elimination rate was severely impaired in transgenic mice, resulting in a KG of only 1.1 ± 0.2%/min in transgenic mice versus 2.5 ± 0.2%/min in wild-type mice (P < 0.001). The insulin response to glucose was totally abolished in the transgenic mice; in fact, a slight reduction in insulin levels occurred after 10 and 20 min (Fig. 1). Thus, the AIR, which was 494 ± 78 pmol/l in wild-type mice, was not significantly altered from zero in transgenic mice (−42 ± 25 pmol/l), the difference between the groups being highly significant (P < 0.001). There was a linear correlation between AIR and KG across all animals (r = 0.72, P < 0.001; Fig. 2). Also the total 75-min glucose disappearance rate (KG[1–75]) was markedly suppressed in transgenic mice: 0.64 ± 0.07%/min in transgenic mice versus 1.36 ± 0.11%/min in wild-type mice (P < 0.001). The AUCinsulin was 14.4 ± 1.4 (pmol/l) × min in transgenic mice versus 21.8 ± 2.4 (pmol/l) × min in wild-type mice (P = 0.013). Insulin sensitivity was 0.17 ± 0.02 min−2/(pmol/l) in transgenic mice versus 0.25 ± 0.05 min−2/(pmol/l) in wild-type mice (P = 0.0832).
Intravenous arginine.
Also after the intravenous administration of arginine (0.25 g/kg), wild-type mice responded with a robust increase in insulin levels, which was markedly suppressed in transgenic mice (Fig. 1). The AIR to arginine was only 164 ± 81 pmol/l in transgenic mice versus 770 ± 132 pmol/l in wild-type mice (P = 0.032). This was equivalent to a reduction of arginine-induced increase in AIR by 79 ± 10% in transgenic mice. Circulating glucose did not change significantly after administration of arginine in any of the groups.
Gastric glucose.
After the administration of glucose by a gastric gavage, a marked increase in circulating levels of insulin was observed in wild-type mice, whereas the increase in transgenic mice was only marginal (Fig. 3). Thus, the peak insulin value at 30 min after gastric glucose was 7,298 ± 2087 pmol/l in wild-type mice versus only 306 ± 58 pmol/l in transgenic mice (P = 0.010). The increase in insulin levels observed during the first 15 min after gastric glucose was 3,084 ± 88 pmol/l in wild-type mice versus only 81 ± 9 pmol/l in transgenic mice (P = 0.006). Thus, the increase in insulin levels in transgenic mice, albeit significant (P < 0.001 when calculated using a paired t test), was only 2.6 ± 0.3% of that in wild-type mice (i.e., corresponding to 97% suppression). This was associated with severe glucose intolerance in transgenic mice, resulting in a markedly higher 2-h glucose value in these mice (23.8 ± 1.4 mmol/l) than in the wild-type mice (11.1 ± 1.4 mmol/l; P < 0.001).
Insulin content and secretion in vitro.
The islet insulin content was reduced by 50% in transgenic mice (30 ± 4 ng/islet vs. 60 ± 8 ng/islet in wild-type mice, P < 0.005). Following incubation of freshly isolated islets in different concentrations of glucose (from 3.3 to 22.2 mmol/l), a severe blunting of insulin secretion was observed in transgenic compared with wild-type mice (Fig. 4). However, after incubation with GLP-1 (100 nmol/l), carbachol (1 μmol/l), or palmitate (0.5 mmol/l), the insulin responses were reduced by only ∼40, 20, and 15%, respectively (i.e., transgenic islets retained partial ability to secrete insulin in response to nonglucose secretagogues). KCl (50 mmol/l) completely reinstated insulin secretion from transgenic islets compared with wild-type islets.
This study examined insulin secretion in mice with β-cell-targeted overexpression of a dominant-negative HNF-1α, representing a model of MODY3. The RIP-DN HNF-1α mouse colony was established previously and immunocytochemistry using an antibody against the NH2-terminus of HNF-1α showed marked overexpression in β-cell nuclei in association with abnormal islet cytoarchitecture in association with low pancreatic content of insulin and a reduced expression of GLUT-2 (16). In the present study, we confirm that β-cell-targeted overexpression of a dominant-negative HNF-1α results in nonfasting hyperglycemia, severe glucose intolerance, and markedly suppressed insulin secretory response to glucose both in vivo and in vitro (3,16,17). This is similar to MODY3, confirming the suitability of these mice as an experimental model for the study of this type of diabetes. Interestingly, 16-h fasting glucose levels were preserved in the transgenic mice, which shows that the β-cells respond appropriately to fasting.
This study demonstrates that the insulin responses to intravenous challenges of glucose and arginine as well as to gastric administration of glucose are severely suppressed in the transgenic mice. A difference in degree of suppression was observed in that the insulin response to intravenous glucose was totally abolished, whereas that to gastric glucose was suppressed by ∼97%. The small increase in circulating insulin after gastric glucose remaining in the transgenic mice may be due to activation of insulin secretion by the gut incretins, such as GLP-1. The latter is released by the gastric glucose and stimulates insulin secretion (21). The insulin response to intravenous arginine showed the weakest suppression of these challenges, being attenuated by 79%. This suggests that β-cell stimulation by arginine partially bypasses the defects associated with the expression of the dominant-negative HNF-1α. An impaired insulin response to arginine was previously reported for the perfused pancreas from these mice—arginine-induced secretion was reduced by ∼75% (16).
Previously it has been difficult to isolate islets from these diabetic mice, which was attributed to the disrupted islet structure in the model (16). However, in the laboratory in Lund, isolation was possible and, therefore, we report here for the first time on studies from isolated islets in these mice. It was found that the islet insulin content was reduced by 50% in transgenic mice. We also found that the insulin response to glucose was severely suppressed, which confirms the in vivo results and the previous results in the perfused pancreas (16). In fact, at the high glucose of 22.2 mmol/l for 60 min, insulin secretion was only doubled compared with a 10-fold increase in wild-type islets. In contrast, the insulin responses to GLP-1, carbachol, and palmitate were reduced by only ∼40, 20, and 15%, respectively, in the islets from transgenic mice. This percentage would be even less if taking into account reduced islet insulin content. This suggests that HNF-1α regulates expression of genes, which preferentially are linked to the insulin response to glucose rather than to other secretagogues. Previously, HNF-1α has been shown to control the expression of GLUT-2, which supports such a notion (16,17). In contrast, GLP-1 is more linked to cAMP generation (22) and carbachol to phospholipase C and protein kinase C (23). Overexpression of a dominant-negative HNF-1α in insulin-producing INS-1 cells has suggested that the mechanism underlying the impairment of insulin secretion is defective mitochondrial function. It was demonstrated that ATP generation is attenuated because of downregulation of the tricarboxylic acid cycle enzyme α-keto-glutarate dehydrogenase and upregulation of uncoupling protein 2 (UCP2). Consequently, KATP channel closure and membrane depolarization are not occurring. In turn, the crucial rise in cytosolic calcium due to gating of voltage-sensitive calcium channels is abrogated (18,24). On the other hand, other studies have implicated defective glycolysis rather than mitochondrial dysfunction as a basis for the attenuated glucose-stimulated insulin secretion (15,25). Therefore, more detailed studies are required to establish the importance of HNF-1α for β-cell signaling.
An interesting observation in this study was that the insulin response to palmitate was partially preserved in islets from transgenic mice. Hence, it seems that palmitate rescues the secretion. The mechanism of this rescue remains to be established. Palmitate may serve as a generator of a lipid signal for exocytosis (26,27) and, therefore, a direct stimulation of exocytosis might bypass the perturbations induced by the overexpression of the dominant-negative HNF-1α. This is in agreement with the preserved insulin response to depolarization by KCl, which also has been reported in INS-1 cells overexpressing a dominant-negative HNF-1α (18,24).
The seven-sample IVGTT in mice has previously been shown efficient in estimating insulin sensitivity (19,20). However, due to the severely suppressed insulin response to glucose in transgenic mice, it was not possible to use the minimal model technique for estimating insulin action in this study. Insulin action was nonetheless assessed with another formula that exploits the general definition of insulin sensitivity (i.e., the total glucose disappearance rate during the 75 min after the intravenous glucose load divided by AUCinsulin). In wild-type mice, this measure correlated with insulin sensitivity determined by the minimal model (r = 0.87; P = 0.010). It is worth noting that it is not possible to take into account influences of glucose per se on its own disappearance (glucose effectiveness) in this measure. Nevertheless, this surrogate measure of insulin sensitivity is still a good descriptor of insulin action on glucose disappearance. Insulin sensitivity was not significantly different between the two groups, suggesting that the severe β-cell dysfunction in the transgenic mice does not result in any compensatory change in insulin sensitivity. This is somewhat surprising since an inverse relation between insulin secretion and insulin sensitivity is often seen (20,28). On the other hand, the measure of insulin action used here might be too general. Thus, more detailed measures of insulin sensitivity (such as using the hyperinsulinemic-euglycemic clamp technique) and glucose effectiveness need to be performed in further studies.
Based on the results reported here, we conclude that β-cell-targeted overexpression of a dominant-negative form of HNF-1α in mice results in a good diabetes model exhibiting severe glucose intolerance and absent insulin response to glucose both in vivo and in vitro, consistent with findings in MODY3. In contrast, the insulin responses to nonglucose secretagogues are not inhibited when the islet insulin content is taken into account. This suggests that nonglucose secretagogues partially rescue the β-cell dysfunction in this form of diabetes, perhaps by stimulating exocytosis.
Glucose and insulin levels before and after intravenous administration of glucose (1 g/kg; left panel) or arginine (0.25 g/kg; right panel) in male mice with a β-cell-targeted overexpression of a dominant-negative HNF-1α (transgenic mice; n = 13) and their wild-type counterparts (n = 7). Data are means ± SEM.
Linear relationship (r = 0.72; P < 0.001) between the AIR (i.e., the mean of suprabasal 1 and 5 min insulin values) and KG (i.e., the glucose elimination rate between minute 1 and minute 20) after intravenous administration of glucose (1 g/kg) in male mice with a β-cell-targeted overexpression of a dominant-negative HNF-1α (•, n = 13) and their wild-type counterparts (○, n = 7). Data are means ± SEM.
Glucose and insulin levels before and after gastric administration of glucose (150 mg) in male mice with a β-cell-targeted overexpression of a dominant-negative HNF-1α (transgenic mice; n = 13) and their wild-type counterparts (n = 7). Data are means ± SEM.
Medium insulin concentrations after a 60-min incubation of islets isolated from male mice with a β-cell-targeted overexpression of a dominant-negative HNF-1α and their wild-type counterparts. Upper panel: Islets were incubated in different concentrations of glucose. Lower panel: Islets were incubated with GLP-1 (100 nmol/l), carbachol (1 μmol/l), palmitate (0.5 mmol/l), or KCl (50 mmol/l) in the presence of 11.1 mmol/l glucose. There were 8–24 incubations with three islets in each incubation for each experimental group. Data are means ± SEM.
This article is based on a presentation at a symposium. The symposium and the publication of this article were made possible by an unrestricted educational grant from Servier.
This study was supported by the Swedish Research Council (Grant 6834); The Swedish Diabetes Foundation; Albert Påhlsson Foundation; Region Skåne; the Faculty of Medicine, Lund University (B.A.); and the Swiss National Science Foundation (C.B.W.).
The authors are grateful to Kristina Andersson, Lilian Bengtsson, and Lena Kvist for expert technical assistance.
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The men's single sculls competition at the 1960 Summer Olympics took place at Lake Albano, Italy. The event was held from 30 August until 3 September. There were 13 competitors from 13 nations, with each nation limited to one boat in the event. The event was won by Vyacheslav Ivanov of the Soviet Union, the second man to successfully repeat as Olympic champion (after Australia's Bobby Pearce in 1928 and 1932). It was the third consecutive Soviet victory in the event, with Yuriy Tyukalov winning in 1952 before Ivanov's victories in 1956 and 1960. Ivanov would go on to win again in 1964, becoming the first man to win 3 gold medals in the event. The silver medal went to Achim Hill of the United Team of Germany, the first medal in the men's single sculls for the combined team and the first single sculls medal for any German rower since 1936. Teodor Kocerka of Poland took bronze. Ivanov and Kocerka were the fourth and fifth men to win multiple medals of any colour in the event, with Kocerka previously taking bronze in 1952. It was Kocerka's third straight final in the event, placing fourth between his two bronzes. Australia's three-Games podium streak ended when Stuart Mackenzie fell ill and could not compete.
Background
This was the 13th appearance of the event. Rowing had been on the programme in 1896 but was cancelled due to bad weather. The single sculls has been held every time that rowing has been contested, beginning in 1900.
Three of the 12 single scullers from the 1956 Games returned: gold medalist Vyacheslav Ivanov of the Soviet Union, fourth-place finisher (and 1952 bronze medalist) Teodor Kocerka of Poland, and fifth-place finisher James Hill of New Zealand. The top two scullers in 1960 were Ivanov (defending Olympic champion and two-time reigning European champion) and Stuart Mackenzie of Australia (1956 silver medalist, 1957 and 1958 European champion, reigning British Empire and Commonwealth Games champion, and four-time consecutive Diamond Challenge Sculls winner—he would stretch that streak to 6 in 1961 and 1962). But Mackenzie became ill before the competition and did not compete, leaving Ivanov as the heavy favorite to repeat.
For the first time, no nations made their debut in the event. Great Britain made its 12th appearance, most among nations, having missed only the 1904 Games in St. Louis.
Competition format
This rowing event was a single scull event, meaning that each boat was propelled by a single rower. The "scull" portion means that the rower used two oars, one on each side of the boat. The course used the 2000 metres distance that became the Olympic standard in 1912.
Despite having one more competitor than the 1956 Games, the 1960 format dropped a round. The competition now consisted of only three rounds: semifinals, a repechage, and a final. The six-boat final returned.
Semifinals: Three heats of 4 or 5 boats each. The top boat in each heat advanced to the final, the remaining boats (10 total) went to the repechage.
Repechage: Three heats of 3 or 4 boats each. The winner of each heat rejoined the semifinal winners in the final, with the remaining 7 boats (2nd, 3rd, and 4th in each heat) eliminated.
Final: One heat of 6 boats.
Schedule
All times are Central European Time (UTC+1)
Results
Semifinals
Three heats were held on 30 August. The winner of each heat directly advanced to finals, while the others went to the repechage round.
Semifinal 1
Semifinal 2
Semifinal 3
Repechage
The repechage was held on 31 August. The winner of each repechage heat advanced to the final.
Repechage heat 1
Repechage heat 2
Repechage heat 3
Final
The final was held on 3 September at 16:00.
Results summary
References
External links
Rowing at the 1960 Summer Olympics
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Ant body Ant leg Ant structure under Foldscope#Indiqfoldscopephasel.
Curious listeners exploring microcosmos of biological and non-biological samples, at DST-INSPIRE camp at SASTRA University, on 28th Dec'18.
#Indiqfoldscopephasel Alternaria solani pathogen that causes early blight fungal disease in tomatoes.
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BioWare announced today with a press release that the awaited Star Wars: The Old Republic game update 1.2 will now come early April, instead of March as scheduled. This is as far as the bad news goes.
Game Update 1.2 (now dubbed "Legacy") has been detailed and will include improved UI customization, PvP warzone rankings, guild banks (with up to seven tabs), valuable new in-game items, in game events, appearence customization and a lot of bug fixes and smaller improvements.
Game Update 1.2 will also present the first real iteration of the Legacy system. Players will be able to link their different characters together into a single family tree, giving them the power to unlock and share special Legacy-only abilities. Players will be able to further customize new characters by unlocking different species from any class in their family tree. The Legacy system will also also provide new convenience items for player ships, including an on-board mailbox and Galactic Trade Network terminal. Companion Character affection and moral alignment will also benefit from Legacy bonuses, giving players even more control over their crew mates. .
Flashpoint: Lost Island continues the Kaon Under Siege Flashpoint from Game Update 1.1 – Rise of the Rakghouls. Players must survive a menacing island of mystery on Ord Mantell as they hunt for clues to the Rakghoul virus outbreak that ravaged the Tion Hegemony. But the answers to this riddle may prove fatal!
Operation: Explosive Conflict sends groups of eight or sixteen Imperial or Republic players to a new zone on the planet of Denova, where traitors and mercenaries are selling the rare explosive mineral baradium to the highest bidder. Players will battle through hordes of Droids, mercenaries and deadly creatures all fighting for control of the planet and its valuable resources.
Warzone: Novare Coast pits two teams in an epic battle to control multiple mortar locations and use them to bombard vulnerable enemy bases. Like the Huttball Warzone, Novare Coast can be played Republic vs. Empire or with players of the same faction fighting each other in a thrilling contest of wills to determine each side's greatest champions.
Prior to release, our main priority for Star Wars: The Old Republic was to deliver a high quality game and service, right from day one. Now that we've achieved that, we have shifted our focus to adding more content to the game and improving and refining the experience for our fans. Legacy is our biggest update yet and a great example of the kind of content players can expect for the months and years ahead.
Unrelated to patch 1.2 the "Friends Trial" program will start tomorrow, March the 6th. Current subscribers will be able to invite up to three new players for a free seven days trial (up to level 15) of the game.
Trial members will also receive a limited time offer to purchase the digital version of Star Wars: The Old Republic on Origin.com at a special promotional price.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,301
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We are inching towards the next rate rise. The MPC seem to have gained a bit of confidence in their judgement that Q1 GDP weakness was temporary, sound somewhat reassured on the consumer and are discounting the recent slight dip in wage growth. The shift in the vote to 6–3 from 7–2 in favour of no change partly reflects a response to continued tightness in the labour market and makes it more likely that we will see one rate hike this year — probably in the late summer.
Following the change of tone from the Fed and the ECB last week, it's interesting that the MPC have lowered the threshold for interest rates (to 1.5% from 2%) where they will be prepared to start running down the balance sheet. This is another signal of us moving towards global quantitative tightening in the medium term.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,190
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Q: Executing Mojo failed with `java.lang.NoSuchMethodError` org.codehaus.plexus.util.DirectoryScanner.setFilenameComparator I have created 2 Mojos and recently decided to upgrade all my versions usign the maven version plugin.
Now, an executing one of my Mojos, I get this exception:
An API incompatibility was encountered while executing com.mycompany:spi2jars-maven-plugin:1.0-SNAPSHOT:spi2jars:
java.lang.NoSuchMethodError: 'void org.codehaus.plexus.util.DirectoryScanner.setFilenameComparator(java.util.Comparator)
I know what this errors means, and I retrieve the full stacktrace by running with -X
Caused by: java.lang.NoSuchMethodError: 'void org.codehaus.plexus.util.DirectoryScanner.setFilenameComparator(java.util.Comparator)'
at org.codehaus.plexus.components.io.resources.PlexusIoFileResourceCollection.getResources (PlexusIoFileResourceCollection.java:250)
at org.codehaus.plexus.archiver.AbstractArchiver$1.hasNext (AbstractArchiver.java:560)
at org.codehaus.plexus.archiver.zip.AbstractZipArchiver.createArchiveMain (AbstractZipArchiver.java:221)
at org.codehaus.plexus.archiver.zip.AbstractZipArchiver.execute (AbstractZipArchiver.java:199)
at org.codehaus.plexus.archiver.AbstractArchiver.createArchive (AbstractArchiver.java:1042)
at org.apache.maven.archiver.MavenArchiver.createArchive (MavenArchiver.java:676)
at com.mycompany.Spi2Jars.createJarForLine (Spi2Jars.java:81)
at com.mycompany.Spi2Jars.execute (Spi2Jars.java:47)
So it seems the MavenArchiver is at fault, or maybe a missing jar or incompatible versions.. But how do I check this?
[Update]
I found different versions of the underlying jars using
mvn dependency:tree -Dverbose
For example:
\- org.apache.maven:maven-archiver:jar:3.5.0:compile
[INFO] +- (org.apache.maven:maven-artifact:jar:3.0:compile - omitted for conflict with 3.6.0)
[INFO] +- (org.apache.maven:maven-model:jar:3.0:compile - omitted for conflict with 3.6.3)
[INFO] +- (org.apache.maven:maven-core:jar:3.0:compile - omitted for conflict with 3.6.3)
[INFO] +- (org.apache.maven.shared:maven-shared-utils:jar:3.2.1:compile - omitted for duplicate)
[INFO] +- commons-io:commons-io:jar:2.5:compile
[INFO] +- org.codehaus.plexus:plexus-archiver:jar:4.2.0:compile
[INFO] | +- (org.codehaus.plexus:plexus-utils:jar:3.3.0:compile - omitted for conflict with 3.2.1)
[INFO] | +- org.codehaus.plexus:plexus-io:jar:3.2.0:compile
[INFO] | | +- (org.codehaus.plexus:plexus-utils:jar:3.3.0:compile - omitted for conflict with 3.2.1)
[INFO] | | \- (commons-io:commons-io:jar:2.6:compile - omitted for conflict with 2.5)
[INFO] | +- org.apache.commons:commons-compress:jar:1.19:compile
[INFO] | +- org.iq80.snappy:snappy:jar:0.4:compile
[INFO] | \- org.tukaani:xz:jar:1.8:runtime
[INFO] +- (org.codehaus.plexus:plexus-utils:jar:3.3.0:compile - omitted for conflict with 3.2.1)
[INFO] \- (org.codehaus.plexus:plexus-interpolation:jar:1.25:compile - omitted for conflict with 1.11)
It would be so awesome if the most recent maven plugins played nice together... It seems they don't. How do I resolve this nicely?
A: I had a similar issue (same error message).
I could solve this isse by declaring the following direct dependency in our custom plugin:
<dependency>
<groupId>org.codehaus.plexus</groupId>
<artifactId>plexus-utils</artifactId>
<version>3.3.0</version>
</dependency>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,443
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namespace views {
class View;
} // namespace views
namespace ash {
class UnifiedMediaControlsView;
// Controller class of UnifiedMediaControlsView. Handles events of the view
// and updates the view when receives media session updates.
class ASH_EXPORT UnifiedMediaControlsController
: public media_session::mojom::MediaControllerObserver,
public media_session::mojom::MediaControllerImageObserver {
public:
class Delegate {
public:
virtual ~Delegate() = default;
virtual void ShowMediaControls() = 0;
virtual void OnMediaControlsViewClicked() = 0;
};
explicit UnifiedMediaControlsController(Delegate* deleate);
~UnifiedMediaControlsController() override;
// media_session::mojom::MediaControllerObserver implementations.
void MediaSessionInfoChanged(
media_session::mojom::MediaSessionInfoPtr session_info) override;
void MediaSessionMetadataChanged(
const absl::optional<media_session::MediaMetadata>& metadata) override;
void MediaSessionActionsChanged(
const std::vector<media_session::mojom::MediaSessionAction>& actions)
override;
void MediaSessionChanged(
const absl::optional<base::UnguessableToken>& request_id) override;
void MediaSessionPositionChanged(
const absl::optional<media_session::MediaPosition>& position) override {}
// media_session::mojom::MediaControllerImageObserver implementations.
void MediaControllerImageChanged(
media_session::mojom::MediaSessionImageType type,
const SkBitmap& bitmap) override;
views::View* CreateView();
void OnMediaControlsViewClicked();
// Called from view when media buttons are pressed.
void PerformAction(media_session::mojom::MediaSessionAction action);
void FlushForTesting();
void set_media_controller_for_testing(
mojo::Remote<media_session::mojom::MediaController> controller) {
media_controller_remote_ = std::move(controller);
}
private:
// Update view with pending data if necessary. Called when
// |freeze_session_timer| is fired.
void UpdateSession();
// Update artwork in media controls view.
void UpdateArtwork(const SkBitmap& bitmap, bool should_start_hide_timer);
// Reset all pending data to empty.
void ResetPendingData();
bool ShouldShowMediaControls() const;
void MaybeShowMediaControlsOrEmptyState();
// Weak ptr, owned by view hierarchy.
UnifiedMediaControlsView* media_controls_ = nullptr;
// Delegate for show/hide media controls.
Delegate* const delegate_ = nullptr;
mojo::Remote<media_session::mojom::MediaController> media_controller_remote_;
mojo::Receiver<media_session::mojom::MediaControllerObserver>
observer_receiver_{this};
mojo::Receiver<media_session::mojom::MediaControllerImageObserver>
artwork_observer_receiver_{this};
std::unique_ptr<base::OneShotTimer> freeze_session_timer_ =
std::make_unique<base::OneShotTimer>();
std::unique_ptr<base::OneShotTimer> hide_artwork_timer_ =
std::make_unique<base::OneShotTimer>();
absl::optional<base::UnguessableToken> media_session_id_;
media_session::mojom::MediaSessionInfoPtr session_info_;
media_session::MediaMetadata session_metadata_;
base::flat_set<media_session::mojom::MediaSessionAction> enabled_actions_;
// Pending data to update when |freeze_session_tmier_| fired.
absl::optional<base::UnguessableToken> pending_session_id_;
absl::optional<media_session::mojom::MediaSessionInfoPtr>
pending_session_info_;
absl::optional<media_session::MediaMetadata> pending_metadata_;
absl::optional<base::flat_set<media_session::mojom::MediaSessionAction>>
pending_enabled_actions_;
absl::optional<SkBitmap> pending_artwork_;
};
} // namespace ash
#endif // ASH_SYSTEM_MEDIA_UNIFIED_MEDIA_CONTROLS_CONTROLLER_H_
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,668
|
You are here: Home > News and Insight > Missing evidence 'costs courts dear'
Missing evidence 'costs courts dear'
By Wesleyan
Lawyers are seeing many prosecutions dropped because of missing evidence, paperwork or victims, new figures show.
And some cases are being halted because of the victim's failure to attend a trial, the report finds.
This means that thousands of cases are collapsing before they can go before the courts, according to a member of the London Assembly.
Tony Arbour, who is also a magistrate, has used the Freedom of Information Act to get the data from the Crown Prosecution Service. He calls the results "alarming".
As many as 6,438 of the 73,143 cases scrapped in 2013/14 were down to unavailable exhibits, key statements or further evidence.
In addition, 1,480 cases have been abandoned due to vital specialist evidence being similarly unavailable. Unreceived police files have resulted in a further 1,399 cases being halted.
Mr Arbour says commonsense steps are needed to ensure criminals are brought to justice. He says he is "alarmed" to see the high amount of prosecutions which are not coming to court throughout the country.
Copyright Press Association 2014
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 487
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Campus Conversation Series: "The Origins of the Radical Right and the Crisis of Our Democracy" - 12pm, Tues. 9/26
Sep 5, 2017 2:00 pm by Nicole GottleibCampus Resources / Events
Dear Faculty, Staff, and Students,
Last year, we introduced Campus Conversation series with the theme of What's Going on and Why? The focus was on local, national and international issues that directly affect the lives of those in our community. We heard speakers talk about race and policing, immigration and migration, the election, civil liberties, and the First Amendment. Videos of this series are available on the Campus Conversation page of the Provost's website. This year, rather than following up on the formal presentation the following week, we are lengthening the time of the presentation to 90 minutes and utilizing half of the time for questions and discussion with the audience.
Our first Campus Conversation of this academic year is scheduled for September 26 from noon to 1:30pm in Cardinal Room in Student Center East. Dr. Nancy MacLean of Duke University will provide a lecture and discussion on The Origins of the Radical Right and the Crisis of Our Democracy, based on her new book, Democracy in Chains: The Deep History of the Radical Right's Stealth Plan for America.
I hope you will join me and your fellow students, faculty and staff for this Campus Conversation.
Susan Poser
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,032
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A Citizen's Guide to Addressing West Mayfield Council
October 16, 2019 December 8, 2013 by admin
Take the Right Action
Citizens often come to borough council meetings with a variety of complaints, such as potholes, the condition of their neighbor's yard, barking dogs, or speeding traffic. Frankly, it doesn't matter what the issues are, citizens have the right to address council and to have their voices heard. But what happens after that is another story.
The most commonly heard complaint from citizens is that "council won't do anything" to solve their problems. Of course, there are legitimate reasons why council does not remedy every complaint, solve every issue, or quiet every crisis. Civic issues can be very complicated, fraught with legal, economic, environmental, social, and political challenges. Budget restraints and personnel shortages are frequent reasons for government inaction. Lack of authority is another reason.
Nevertheless, citizens should not be deterred from calling upon their government to do its duty and manage the people's business effectively and fairly.
But how is this done at the grassroots level of local government?
Let's get this straight. Backfence complaining and closed-door grumbling are not going to solve any problem, so the first step is to formally bring an issue before council.
The following is an informative and helpful guide for West Mayfield residents or other citizens who want to engage our local government:
Council meetings are open to the public, meaning that anyone can attend.
The president of council–not the mayor–runs the legally required monthly meetings, which are held on the 2nd Thursday of each month and promptly start at 6:30pm.
The president opens the floor to citizens early in the proceeding, so it is important that citizens be on time. When called upon by the president, citizens may address council on a first come basis. Currently, citizens may address council at the direction of the president, that is, there is no official time limit or other rules in place regulating the citizens' forum. Citizens are free to come and go as they please.
The borough secretary (currently Pat Lansberry) is legally bound to record and maintain official meeting minutes. Although written minutes are available upon request, citizens have the right to record the public meeting via audio or video device.
Citizens addressing council are expected to state their name and address for the record. Citizens are expected to follow typical norms of public speaking, including refraining from using profanity or obscene language, and making libelous and defamatory remarks about others.
Citizens have the right to address council about any issue; however, council's authority to address or act upon an issue is limited by borough ordinances, the Pennsylvania Borough Code, and state statues. Council cannot mediate issues between private parties, nor can it control or regulate private property except as permitted by ordinance or law. However, council does have the authority to create ordinances, as well as amend, repeal or revise existing ordinances as prescribed by law.
Council members cannot act as police officers. When laws are broken, including infractions upon ordinances, the Beaver Falls Police Department should be notified.
As in any representative democracy, it is incumbent upon citizens to convince the majority of their representatives (see below) to support their cause. In West Mayfield Borough, it takes four or more members of council to prevail on the side of any issue—that's the process.
Citizens who can effectively demonstrate popular support for their issue by way of phone calls, letters, petitions, and mass attendance at meetings are more likely to prevail.
A realistic understanding of how local government works can go a long way in preventing despair and frustration among citizens. To learn more about how our local government functions, we encourage every resident to read the borough website and to regularly attend council meetings.
Categories Government
Borough tax office closed Wednesday, Nov. 6, 2013
The West Mayfield real estate tax office at 3628 Matilda St. will be closed Wednesday, November 6, 2013. For more information, contact Tax Collector Kathleen Brewer.
Get Out and Vote in the 2013 Primary!
Nobody will ever deprive the American people of the right to vote except the American people themselves and the only way they could do this is by not voting. ~Franklin D. Roosevelt
West Mayfield residents will have a chance to participate in democracy on Tuesday, May 21st, during the 2013 Pennsylvania Primary Election.
Polls across the state and in the Borough will be open from 7am to 8pm. To learn more about who's running in West Mayfield and where to cast your ballot, visit our Voters Guide.
The Right to Know, or the Right to No?
"Every government record is presumed to be open."
–Terry Mutchler, executive director of the Pennsylvania Office of Open Records
The following article is by Leah Samuel | PublicSource | April 21, 2013
After getting a parking ticket at Pittsburgh International Airport, a driver requested a copy of the Allegheny Police Department's report of the incident. The department didn't respond.
A parent asked the Ligonier Valley School District for documents detailing planned teacher layoffs. The school district said it had no such documents.
A reporter asked Middle Smithfield Township for maps of local sewer lines. The township refused to turn them over, saying that could jeopardize the security of the sewage system.
Eventually, each of these individuals appealed to the Pennsylvania Office of Open Records, which tries to resolve disputes between citizens asking for information and agencies that don't want to give it.
But they should not have had to appeal, said Terry Mutchler, executive director of the office.
"Every government record is presumed to be open," she said, "and the agency has to prove why it shouldn't be."
To read more of this article, please visit the Public Source website:
http://publicsource.org/investigations/right-know-or-right-no
Pancakes/Sausage Breakfast to Support WMVFD
Aiken Refuse Service
2022 West Mayfield Holiday Christmas Party
Chili Bowls & Bake Sale
Categories Select Category Borough News Fire Department News Government Public Safety
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,050
|
{"url":"https:\/\/kodkodgames.gitlab.io\/rotation_matrices\/","text":"# Unlearn rotation matrices as rotations\n\n\u2013 Hey, Markus! What format is this head-rotation representation in?\n\n\u2013 It is a rotation matrix, I answer. Right handed, z forward through the nose and x through the left ear. Our young newly graduated colleague nods his\/her head.\n\nAfter about 10 minutes I hear my name again.\n\n\u2013 Markus\u2026. Eh, what order is it?\n\n\u2013 Oh no! You have opened Wikipedia? Haven\u2019t you? I answer in despair from my desk.\n\nIt happens time to time that a newly graduated engineer (or summer intern) asks me exactly this question. Almost always with the Wikipedia page open at the screen, which I think is horrible (or even worse, some \u201cLearn OpenGL\u201d tutorial).\n\nI steal take a chair to sit down beside the person. This will take a few minutes, we are going to do something that is harder than learning: we are going to unlearn.\n\nIt is interesting, I get no questions, or only very short questions, on Euler angels, Rodriguez rotations and actually only one recurrent question on quaternions. But very often I get questions on rotation matrices. I think it is a bit odd since rotation matrices are very simple in comparison to many other rotation representations. I think a big reason for this is the Wikipedia page. It looks something like this:\n\n$R_x(\\theta) = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & \\cos \\theta & -\\sin \\theta \\\\ 0 & \\sin \\theta & \\cos \\theta \\end{bmatrix}$ $R_y(\\theta) = \\begin{bmatrix} \\cos \\theta & 0 & \\sin \\theta \\\\ 0 & 1 & 0 \\\\ -\\sin \\theta & 0 & \\cos \\theta \\end{bmatrix}$ $R_z(\\theta) = \\begin{bmatrix} \\cos \\theta & -\\sin \\theta & 0 \\\\ \\sin \\theta & \\cos \\theta & 0\\\\ 0 & 0 & 1\\\\ \\end{bmatrix}$ $R = \\begin{bmatrix} \\cos\\alpha\\cos\\beta & \\cos\\alpha\\sin\\beta\\sin\\gamma - \\sin\\alpha\\cos\\gamma & \\cos\\alpha\\sin\\beta\\cos\\gamma + \\sin\\alpha\\sin\\gamma \\\\ \\sin\\alpha\\cos\\beta & \\sin\\alpha\\sin\\beta\\sin\\gamma + \\cos\\alpha\\cos\\gamma & \\sin\\alpha\\sin\\beta\\cos\\gamma - \\cos\\alpha\\sin\\gamma \\\\ -\\sin\\beta & \\cos\\beta\\sin\\gamma & \\cos\\beta\\cos\\gamma \\\\ \\end{bmatrix}$\n\nIt talks about rotations. Rotation around different axes and their relation to Euler angles. This can be a bit confusing when working with for example a head pose. You can of course think about rotation matrices as if the head rotates around different axes in different order, but it becomes kind of hard to interpret:\n\n$\\begin{bmatrix} -0.9987820 & 0.0348782 & -0.0348995 \\\\ 0.0283128 & 0.9844193 & 0.1735424 \\\\ 0.0404086 & 0.1723429 & -0.9842078 \\end{bmatrix}$\n\nSo to interpret this we need to solve the following equation system:\n\n$\\begin{cases} -\\sin(\\beta) & = 0.0404086 \\\\ \\cos(\\beta)\\sin(\\gamma) & = 0.1723429 \\\\ \\cos(\\alpha)\\cos(\\beta) & = -0.9987820 \\end{cases}$\n\nand then we get an \u201cintrinsic rotation whose Tait\u2013Bryan angles are \u03b1, \u03b2, \u03b3, about axes z, y, x\u201d to visualize in our head.\n\nIts sad because I think rotation matrices are one of the easiest representation to interpret.\n\nDon\u2019t think of them as rotations, think of them as a unit vectors of a new coordinate systems.\n\nWe describe where the coordinate system is located related to another coordinate system (where we rotate from), for example from the camera\u2019s coordinate system perspective (z forward, y upwards). The first column of the rotation matrix is the new x-axis expressed in the old coordinate system, the second column is the y-axis and so on. An identity matrix would yield in no rotation since all unit vectors would be the same as the previous coordinate system.\n\n$R = \\begin{bmatrix} X_x & Y_x & Z_x \\\\ X_y & Y_y & Z_y \\\\ X_z & Y_z & Z_z \\end{bmatrix}$\n\nLets go back to the example with the head expressed in the camera coordinate system and assume the head position is atfront of the camera. So by interpret the previous matrix, we can look at the new z-axis:\n\n$Z_{axis} = \\begin{bmatrix} Z_x \\\\ Z_y \\\\ Z_z \\end{bmatrix} = \\begin{bmatrix} -0.0348995 \\\\ 0.1735424 \\\\ -0.9842078 \\end{bmatrix}$\n\n(Remember that z-axis is where the head\u2019s nose is pointing)\n\nWe can quickly see that z-part of the z-axis is almost -1. This means the nose is pointing at the opposite direction as the camera, eg. towards the camera if the person is sitting at front of it.\n\nWe can also se that the persons head is rotated a little bit up (positive y component of the z-axis) and is pointing a little bit to the right of the camera (negative x component).\n\nAnd that\u2019s it! Rotation matrices just describe the unit vectors of a new coordinate system.\n\n\u2013 Hey, Markus! How come this matrix is 4x4?\n\n\/\/ Markus\n\nHN discussion","date":"2020-09-29 22:19:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5975788235664368, \"perplexity\": 839.2333305154426}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600402093104.90\/warc\/CC-MAIN-20200929221433-20200930011433-00775.warc.gz\"}"}
| null | null |
import TranslationsProvider from "language/TranslationsProvider";
import { IModConfig } from "mod/IMod";
import Mod from "mod/Mod";
import Log from "utilities/Log";
export interface IModInfo {
index: number;
config: IModConfig;
state: ModState;
path: string;
folderName: string;
type: ModType;
provides: IModProvides;
instance?: Mod;
initialized?: boolean;
publishedFileId?: string;
steamIDOwner?: string;
lastUpdated?: number;
installDate?: number;
createdDate?: number;
imageOverrides?: IImageOverrides;
customizations?: ICustomizations;
stylesheets?: string[];
languages?: Array<{
path?: string;
instance?: TranslationsProvider;
}>;
log: Log;
loadOrder: number;
}
export declare enum TypeFlag {
Undefined = 1,
Null = 2,
Number = 4,
Boolean = 8,
String = 16,
Object = 32,
Array = 64,
Function = 128,
True = 256,
False = 512
}
export declare enum ModState {
Disabled = 0,
Enabled = 1,
Loaded = 2,
Error = 3,
ChangingState = 4,
Temporary = 5,
LoadedInMultiplayer = 6
}
export declare enum ModType {
Internal = 0,
Local = 1,
Workshop = 2
}
export interface IModProvides {
scripts: boolean;
languages: number;
languageExtensions: number;
stylesheets: number;
imageOverrides: boolean;
customizations: boolean;
}
export interface IImageOverrideDescription {
replace: string;
imagePath?: string;
animated?: boolean;
}
export declare type IImageOverrides = Array<string | IImageOverrideDescription>;
export interface ICustomizations {
hairColors: string[];
skinColors: string[];
hairStyles: Array<string | {
name: string;
path: string;
}>;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 624
|
{"url":"https:\/\/rdrr.io\/cran\/EQUIVNONINF\/man\/bi2wld_ni_del.html","text":"# bi2wld_ni_del: Function to compute corrected nominal levels for the Wald... In EQUIVNONINF: Testing for Equivalence and Noninferiority\n\n## Description\n\nImplementation of the construction described on pp. 183-5 of Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. Second edition.\n\n## Usage\n\n 1 bi2wld_ni_del(N1,N2,EPS,SW,ALPHA,MAXH) \n\n## Arguments\n\n N1 size of Sample 1 N2 size of Sample 2 EPS noninferiority margin to the difference of success probabilities SW width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses ALPHA target significance level MAXH maximum number of interval-halving steps\n\n## Details\n\nThe program computes the largest nominal significance level to be used for determining the critical lower bound to the Wald-type statistic for the problem of testing H:p_1 \u2264 p_2 - \\varepsilon versus K: p_1 < p_2 - \\varepsilon.\n\n## Value\n\n N1 size of Sample 1 N2 size of Sample 2 EPS noninferiority margin to the difference of success probabilities SW width of the search grid for determining the maximum of the rejection probability on the common boundary of the hypotheses ALPHA target significance level MAXH maximum number of interval-halving steps ALPHA0 corrected nominal level SIZE0 size of the critical region of the test at nominal level ALPHA0 SIZE_UNC size of the test at uncorrected nominal level ALPHA ERR_IND indicator taking value 1 when it occurs that the sufficient condition allowing one to restrict the search for the maximum of the rejection probability under the null hypothesis to its boundary, fails to be satisfied; otherwise the indicator retains its default value 0.\n\n## Author(s)\n\nStefan Wellek <[email\u00a0protected]>\nPeter Ziegler <[email\u00a0protected]>\n\n## References\n\nWellek S: Testing statistical hypotheses of equivalence and noninferiority. Second edition. Boca Raton: Chapman & Hall\/CRC Press, 2010, Par. 6.6.3.\n\n## Examples\n\n 1 bi2wld_ni_del(25,25,.10,.01,.05,10) \n\nEQUIVNONINF documentation built on Sept. 19, 2017, 5:06 p.m.","date":"2018-07-21 17:00:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8217312097549438, \"perplexity\": 2661.2093046066348}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676592650.53\/warc\/CC-MAIN-20180721164755-20180721184755-00455.warc.gz\"}"}
| null | null |
Q: React native, state not defined in useSelector after fetch I am trying to load data for my component in useEffect. How can I ensure the state is defined before I go to the next screen?
const [rescueMeContent, setRescueMeContent] = useState({});
useEffect(() => {
async function fetchData() {
const { payload } = await get('xxx');
setRescueMeContent(payload);
}
fetchData();
});
When I first run the app, on the next screen when I try and access the state like this I get an error saying its undefined:
const {
content: {
splashScreen: {
backgroundImage: { url },
buttonText,
informationText,
title: pageTitle,
},
},
} = useSelector((state) => state);
A: useState works in inside only component. You can't reach that from other component. You must use react redux for to use useSelector. You must assign your fetch data to redux state with action. Then you can get that data with useSelector.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,554
|
Tartessiberus cilbanus es la única especie del género Tartessiberus de molusco gasterópodo pulmonado de la familia de los helícidos. Es una especie endémica de la península ibérica.
Descripción
Son similares a los miembros del género Iberus, pero se diferencia de por su apariencia más clara y más globosa, además de una coloración más pálida. La capa delgada y subglobosa y la rádula son similares a las de Allognathus grateloupi, aunque esta última es más globosa, más pequeña y con bandas punteadas (generalmente) más fuertes.
Hábitat y distribución
Solo se conoce en sierra de Grazalema (provincia de Cádiz, Andalucía, España). Habita en zonas de piedra caliza fuertemente karstificadas a gran altura (entre 903 y 1058 m). En periodos secos se refugia en grietas profundas.
Sistemática
La especie fue descrita en febrero de 2021 en base a 4 ejemplares recolectados en la sierra de Grazalema, en el extremo meridional de España continental.
Posición taxonómica
El género Tartessiberus es un miembro de la tribu Allognathini, dada su anatomía genital. Sin embargo, es poco claro si es más próxima a Iberus del sureste de la penı́nsula ibérica o a Allognathus, de las islas Baleares. Su rádula y concha es similar a Allognathus, sin embargo esto podría ser consecuencia de evolución convergente.
Etimología
El nombre del género es en referencia a Iberus, un género de moluscos estrechamente relacionado y, de Tartessos, una antigua civilización del suroeste de la península ibérica. Por otra parte, el nombre específico es en referencia al río Guadalete, concoido en la época romana como Cilbus, al ser en sus inmediaciones donde se descubrió la especie.
Referencias
Bibliografía
Enlaces externos
Helicidae
Géneros monotípicos de moluscos
Moluscos de la España peninsular
Fauna endémica de Mallorca
Animales descritos en 2021
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 789
|
import pytest
from PIL import Image
def image_fixture_factory(path):
@pytest.yield_fixture
def the_fixture():
with Image.open(path) as img:
yield img
return the_fixture
portrait_image_large = image_fixture_factory('tests/images/test-portrait-1200x1600.jpg')
portrait_image = image_fixture_factory('tests/images/test-portrait-450x600.jpg')
landscape_image_large = image_fixture_factory('tests/images/test-landscape-1600x1200.jpg')
landscape_image = image_fixture_factory('tests/images/test-landscape-600x450.jpg')
square_image = image_fixture_factory('tests/images/test-square-600x600.jpg')
tiny_image = image_fixture_factory('test/images/test-tiny-1x1.jpg')
@pytest.fixture()
def models(transactional_db):
from django.apps import apps
dict_ = {M._meta.object_name: M for M in apps.get_models()}
return type(
'models',
(object,),
dict_,
)
@pytest.fixture()
def prime_dimensions_source_image(inmemorystorage, models):
from django.core.files import File
source_image = models.SourceImage()
source_image.original.save(
'tests/images/test-prime-dimensions-599x449.jpg',
File(open('tests/images/test-prime-dimensions-599x449.jpg')),
save=True
)
return source_image
@pytest.fixture(autouse=True)
def _inmemorystorage(settings):
settings.STATICFILES_STORAGE = 'inmemorystorage.storage.InMemoryStorage'
settings.DEFAULT_FILE_STORAGE = 'inmemorystorage.storage.InMemoryStorage'
settings.INMEMORYSTORAGE_PERSIST = True
@pytest.fixture()
def inmemorystorage(_inmemorystorage):
"""
Allows for use of the `InMemoryStorage` backend such that data persists
between different instantiations of the storage backend.
"""
from django.core.files.storage import default_storage
return default_storage
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,640
|
import py.test
from rpython.annotator.model import (
SomeInteger, SomeBool, SomeChar, unionof, SomeImpossibleValue,
UnionError, SomeInstance, SomeSingleFloat)
from rpython.rlib.rarithmetic import r_uint, r_singlefloat
from rpython.rtyper.llannotation import (
SomePtr, annotation_to_lltype, ll_to_annotation)
from rpython.rtyper.lltypesystem import lltype
import rpython.rtyper.rtyper # make sure to import the world
class C(object):
pass
class DummyClassDef:
def __init__(self, cls=C):
self.cls = cls
self.name = cls.__name__
def test_ll_to_annotation():
s_z = ll_to_annotation(lltype.Signed._defl())
s_s = SomeInteger()
s_u = SomeInteger(nonneg=True, unsigned=True)
assert s_z.contains(s_s)
assert not s_z.contains(s_u)
s_uz = ll_to_annotation(lltype.Unsigned._defl())
assert s_uz.contains(s_u)
assert ll_to_annotation(lltype.Bool._defl()).contains(SomeBool())
assert ll_to_annotation(lltype.Char._defl()).contains(SomeChar())
S = lltype.GcStruct('s')
A = lltype.GcArray()
s_p = ll_to_annotation(lltype.malloc(S))
assert isinstance(s_p, SomePtr) and s_p.ll_ptrtype == lltype.Ptr(S)
s_p = ll_to_annotation(lltype.malloc(A, 0))
assert isinstance(s_p, SomePtr) and s_p.ll_ptrtype == lltype.Ptr(A)
def test_annotation_to_lltype():
s_i = SomeInteger()
s_pos = SomeInteger(nonneg=True)
s_1 = SomeInteger(nonneg=True)
s_1.const = 1
s_m1 = SomeInteger(nonneg=False)
s_m1.const = -1
s_u = SomeInteger(nonneg=True, unsigned=True)
s_u1 = SomeInteger(nonneg=True, unsigned=True)
s_u1.const = r_uint(1)
assert annotation_to_lltype(s_i) == lltype.Signed
assert annotation_to_lltype(s_pos) == lltype.Signed
assert annotation_to_lltype(s_1) == lltype.Signed
assert annotation_to_lltype(s_m1) == lltype.Signed
assert annotation_to_lltype(s_u) == lltype.Unsigned
assert annotation_to_lltype(s_u1) == lltype.Unsigned
assert annotation_to_lltype(SomeBool()) == lltype.Bool
assert annotation_to_lltype(SomeChar()) == lltype.Char
PS = lltype.Ptr(lltype.GcStruct('s'))
s_p = SomePtr(ll_ptrtype=PS)
assert annotation_to_lltype(s_p) == PS
si0 = SomeInstance(DummyClassDef(), True)
with py.test.raises(ValueError):
annotation_to_lltype(si0)
s_singlefloat = SomeSingleFloat()
s_singlefloat.const = r_singlefloat(0.0)
assert annotation_to_lltype(s_singlefloat) == lltype.SingleFloat
def test_ll_union():
PS1 = lltype.Ptr(lltype.GcStruct('s'))
PS2 = lltype.Ptr(lltype.GcStruct('s'))
PS3 = lltype.Ptr(lltype.GcStruct('s3'))
PA1 = lltype.Ptr(lltype.GcArray())
PA2 = lltype.Ptr(lltype.GcArray())
assert unionof(SomePtr(PS1), SomePtr(PS1)) == SomePtr(PS1)
assert unionof(SomePtr(PS1), SomePtr(PS2)) == SomePtr(PS2)
assert unionof(SomePtr(PS1), SomePtr(PS2)) == SomePtr(PS1)
assert unionof(SomePtr(PA1), SomePtr(PA1)) == SomePtr(PA1)
assert unionof(SomePtr(PA1), SomePtr(PA2)) == SomePtr(PA2)
assert unionof(SomePtr(PA1), SomePtr(PA2)) == SomePtr(PA1)
assert unionof(SomePtr(PS1), SomeImpossibleValue()) == SomePtr(PS1)
assert unionof(SomeImpossibleValue(), SomePtr(PS1)) == SomePtr(PS1)
with py.test.raises(UnionError):
unionof(SomePtr(PA1), SomePtr(PS1))
with py.test.raises(UnionError):
unionof(SomePtr(PS1), SomePtr(PS3))
with py.test.raises(UnionError):
unionof(SomePtr(PS1), SomeInteger())
with py.test.raises(UnionError):
unionof(SomeInteger(), SomePtr(PS1))
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 543
|
New 2023 Toyota Tacoma Models, Redesign, Release Date
By tamar 27/12/2021 Truck 0 Comments
New 2023 Toyota Tacoma Models, Redesign, Release Date – The 2023 Toyota Tacoma does well in the truck-loving United States, which is no surprise. As a result, it may have something to do with it being possible and fun to operate far from the freeway. With its correct size and not being too heavy to carry along, it's great for moving video games from mobile devices to the midsize space. On the other hand, be careful not to fall victim to the ploy. We don't want anyone else to have access to it. Due to the fact that they're driving on asphalt roads, pioneers are less refined than average drivers. However, the Honda Ridgeline's superior ride quality can't be overstated in these situations. Multiple-pipe engines are another cause of irritation for Toyota owners. It's possible to fall in love with the Tacoma truck if you opt for the V6 and drive on less-traveled roads.
2023 Toyota Tacoma Front View
Combining the cab and cargo bed options for the 2023 Toyota Tacoma is a standout feature, but there are many more options available. The access cab has two doors, while the upgraded cab has a third set of doors. There are honeycomb grilles on the TRD Sport and Out-Road, whereas the more simple grilles are seen on the SR and SR5. It's time to turn the limitation into a sparkling silver membership badge! Additionally, the Toyota TRD Pro boasts the most aggressive Toyota grille in history, with TOYOTA writing, transforming the company's look on low-riding cars.
It can be transformed into a TRD Pro entrance skid meal and hood scoop by using a specific visible. In addition to the basic halogen headlights, everything but the base SR model may be outfitted with fog lights, which are only available on the Limited and higher. There is a Pathway Special Edition and a Nightshade, both of which include a blacked out grille and 16-inch Kevlar-coated wheels.
2023 Toyota Tacoma Interior
You'll need a pickup truck if you want to be one of the few who do, because it has a high-school-style cabin. If you want a car for personal use, don't buy this Tacoma. This product, even though it was made using high-quality materials, does not scream "high school." Only the seats are covered, leaving the dash and entrance solar power panels to stand on their own. Using only natural leather, An electric driver's seat may allow you to enjoy a more comfortable driving position, and the functions are well-placed for ease of use. A lack of options doesn't imply that there aren't many options, but there aren't as many as you may assume either way.
One of the Toyota Tacoma truck's two engines will be discontinued in 2023. Towing the massive Tacoma truck with the 2.7-liter multi-cooking food cooking pot, which has 159 horsepower and 180 lb-ft of torque, can be done, but it will take some time. All four-wheel-drive and four-wheel-drive systems have been tested on this site, which employs a 6-speed automatic transmission. The four-cylinder engine is standard on the SR and SR5, while the V6 is available on a regular basis on higher variants. It's possible to get 278 horsepower and 265 pound-feet of torque by adding 3.5 liters to the more powerful engine, which may be routed to the rear or all four wheels. Only the 4-wheel drivetrain may be accessed by the TRD Pro. As you'll discover in the following paragraphs, a 6-speed automatic transmission is also readily available.
2023 Toyota Tacoma Back View
2023 Toyota Tacoma Release Date and Price
The 2023 Toyota Tacoma is available in eight distinct trims, each with a variety of extras, so you may tailor the price based on your needs! An additional $27,940 is added to this total when using the SR5. The Way Special Edition costs $34,005 instead of $33,060 for the TRD Sports. The more complex TRD Out-Freeway will cost you $34,315. The Nightshade Distinctive Edition costs an additional $1,000, bringing the Limited's MSRP to $38,905. With a starting price of $44,075, the TRD Pro is the most affordable alternative. For example, the 4×4 Limited costs an additional $41,980 when equipped with 4-wheel drive as a standard feature rather than an option. The SR5 goes up to $28,410 and the SR5 goes up to $31,085, when the V6 engine is boosted.
Tags:2023 toyota tacoma access cab, 2023 toyota tacoma access cab 4x4, 2023 toyota tacoma price, 2023 toyota tacoma sr, 2023 toyota tacoma sr5, 2023 toyota tacoma towing capacity, 2023 toyota tacoma trd off road v6, 2023 toyota tacoma trd pro, 2023 toyota tacoma trd sport, build a 2023 toyota tacoma
2023 Toyota Tacoma TRD Pro Colors, Redesign
New 2023 Toyota Tundra Release Date, Price, Specs
New 2022 Toyota Tundra Platinum Electric Interior, Release Date, Specs
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,232
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<head>
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<title>DatapointRequester</title>
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<link rel="stylesheet" type="text/css" href="../../../../doc_stylesheet.css" title="Style">
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<div class="subTitle">com.cosm.client.requester</div>
<h2 title="Class DatapointRequester" class="title">Class DatapointRequester</h2>
</div>
<div class="contentContainer">
<ul class="inheritance">
<li>java.lang.Object</li>
<li>
<ul class="inheritance">
<li>com.cosm.client.requester.DatapointRequester</li>
</ul>
</li>
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<pre>public class <span class="strong">DatapointRequester</span>
extends java.lang.Object</pre>
<div class="block">Class for making requests for datapoint(s) from COSM API</div>
<dl><dt><span class="strong">See Also:</span></dt><dd><code>https://cosm.com/docs/v2/datapoint/</code></dd></dl>
</li>
</ul>
</div>
<div class="summary">
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<li class="blockList">
<!-- ======== CONSTRUCTOR SUMMARY ======== -->
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<h3>Constructor Summary</h3>
<table class="overviewSummary" border="0" cellpadding="3" cellspacing="0" summary="Constructor Summary table, listing constructors, and an explanation">
<caption><span>Constructors</span><span class="tabEnd"> </span></caption>
<tr>
<th class="colOne" scope="col">Constructor and Description</th>
</tr>
<tr class="altColor">
<td class="colOne"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#DatapointRequester()">DatapointRequester</a></strong>()</code> </td>
</tr>
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<caption><span>Methods</span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tr class="altColor">
<td class="colFirst"><code>java.util.Collection<<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a>></code></td>
<td class="colLast"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#create(int, java.lang.String, com.cosm.client.model.Datapoint...)">create</a></strong>(int feedId,
java.lang.String dataStreamId,
<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a>... toCreate)</code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code><a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a></code></td>
<td class="colLast"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#create(int, java.lang.String, com.cosm.client.model.Datapoint)">create</a></strong>(int feedId,
java.lang.String dataStreamId,
<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a> toCreate)</code> </td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>void</code></td>
<td class="colLast"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#delete(int, java.lang.String, java.lang.String)">delete</a></strong>(int feedId,
java.lang.String dataStreamId,
java.lang.String datapointAt)</code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>void</code></td>
<td class="colLast"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#deleteMultiple(int, java.lang.String, java.lang.String)">deleteMultiple</a></strong>(int feedId,
java.lang.String dataStreamId,
java.lang.String startAt)</code> </td>
</tr>
<tr class="altColor">
<td class="colFirst"><code><a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a></code></td>
<td class="colLast"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#get(int, java.lang.String, java.lang.String)">get</a></strong>(int feedId,
java.lang.String dataStreamId,
java.lang.String datapointAt)</code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>java.util.Collection<<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a>></code></td>
<td class="colLast"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#get(int, java.lang.String, java.lang.String, java.lang.String, int)">get</a></strong>(int feedId,
java.lang.String dataStreamId,
java.lang.String startAt,
java.lang.String endAt,
int samplingInterval)</code> </td>
</tr>
<tr class="altColor">
<td class="colFirst"><code><a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a></code></td>
<td class="colLast"><code><strong><a href="../../../../com/cosm/client/requester/DatapointRequester.html#update(int, java.lang.String, com.cosm.client.model.Datapoint)">update</a></strong>(int feedId,
java.lang.String dataStreamId,
<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a> toUpdate)</code> </td>
</tr>
</table>
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<h3>Methods inherited from class java.lang.Object</h3>
<code>clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait</code></li>
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<h3>Constructor Detail</h3>
<a name="DatapointRequester()">
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<h4>DatapointRequester</h4>
<pre>public DatapointRequester()</pre>
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<h3>Method Detail</h3>
<a name="create(int, java.lang.String, com.cosm.client.model.Datapoint)">
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</a>
<ul class="blockList">
<li class="blockList">
<h4>create</h4>
<pre>public <a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a> create(int feedId,
java.lang.String dataStreamId,
<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a> toCreate)
throws <a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></pre>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>feedId</code> - indirect parent of the datapoint</dd><dd><code>dataStreamId</code> - parent of the datapoint</dd><dd><code>toCreate</code> - datapoint to be created over the API</dd>
<dt><span class="strong">Returns:</span></dt><dd>the datapoint that was passed in, on successful operation</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></code> - if failed to create datapoint over the API</dd></dl>
</li>
</ul>
<a name="create(int, java.lang.String, com.cosm.client.model.Datapoint...)">
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<ul class="blockList">
<li class="blockList">
<h4>create</h4>
<pre>public java.util.Collection<<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a>> create(int feedId,
java.lang.String dataStreamId,
<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a>... toCreate)
throws <a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></pre>
<dl><dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></code></dd></dl>
</li>
</ul>
<a name="get(int, java.lang.String, java.lang.String)">
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<h4>get</h4>
<pre>public <a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a> get(int feedId,
java.lang.String dataStreamId,
java.lang.String datapointAt)
throws <a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a>,
<a href="../../../../com/cosm/client/requester/ParseToObjectException.html" title="class in com.cosm.client.requester">ParseToObjectException</a></pre>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>feedId</code> - indirect parent of the datapoint</dd><dd><code>dataStreamId</code> - parent of the datapoint</dd><dd><code>datapointAt</code> - the id of the datapoint to be retrieved</dd>
<dt><span class="strong">Returns:</span></dt><dd>a datapoint object parsed from the json returned from the API</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></code> - if failed to get datapoint over the API</dd>
<dd><code><a href="../../../../com/cosm/client/requester/ParseToObjectException.html" title="class in com.cosm.client.requester">ParseToObjectException</a></code> - if failed to parse the returned json to datapoint</dd></dl>
</li>
</ul>
<a name="get(int, java.lang.String, java.lang.String, java.lang.String, int)">
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<ul class="blockList">
<li class="blockList">
<h4>get</h4>
<pre>public java.util.Collection<<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a>> get(int feedId,
java.lang.String dataStreamId,
java.lang.String startAt,
java.lang.String endAt,
int samplingInterval)
throws <a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a>,
<a href="../../../../com/cosm/client/requester/ParseToObjectException.html" title="class in com.cosm.client.requester">ParseToObjectException</a></pre>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>feedId</code> - indirect parent of the datapoint</dd><dd><code>dataStreamId</code> - parent of the datapoint</dd><dd><code>startAt</code> - </dd><dd><code>endAt</code> - </dd><dd><code>samplingInterval</code> - </dd>
<dt><span class="strong">Returns:</span></dt><dd>a collection of datapoint objects matching the params, parsed
from the json returned from the API</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></code> - if failed to get datapoint over the API</dd>
<dd><code><a href="../../../../com/cosm/client/requester/ParseToObjectException.html" title="class in com.cosm.client.requester">ParseToObjectException</a></code> - if failed to parse the returned json to datapoint</dd></dl>
</li>
</ul>
<a name="update(int, java.lang.String, com.cosm.client.model.Datapoint)">
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</a>
<ul class="blockList">
<li class="blockList">
<h4>update</h4>
<pre>public <a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a> update(int feedId,
java.lang.String dataStreamId,
<a href="../../../../com/cosm/client/model/Datapoint.html" title="class in com.cosm.client.model">Datapoint</a> toUpdate)
throws <a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></pre>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>feedId</code> - indirect parent of the datapoint</dd><dd><code>dataStreamId</code> - parent of the datapoint</dd><dd><code>toUpdate</code> - datapoint to be updated over the API</dd>
<dt><span class="strong">Returns:</span></dt><dd>the datapoint that was passed in, on successful operation</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></code> - if failed to create datapoint over the API</dd></dl>
</li>
</ul>
<a name="delete(int, java.lang.String, java.lang.String)">
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</a>
<ul class="blockList">
<li class="blockList">
<h4>delete</h4>
<pre>public void delete(int feedId,
java.lang.String dataStreamId,
java.lang.String datapointAt)
throws <a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></pre>
<dl><dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></code></dd></dl>
</li>
</ul>
<a name="deleteMultiple(int, java.lang.String, java.lang.String)">
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</a>
<ul class="blockListLast">
<li class="blockList">
<h4>deleteMultiple</h4>
<pre>public void deleteMultiple(int feedId,
java.lang.String dataStreamId,
java.lang.String startAt)
throws <a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></pre>
<dl><dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../com/cosm/client/requester/HttpException.html" title="class in com.cosm.client.requester">HttpException</a></code></dd></dl>
</li>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 36
|
Middle Eastern Cookery
Arto der Haroutunian
Grub Street * London
Published in 2010 by
Grub Street
4 Rainham Close
London
SW11 6SS
Email: food@grubstreet.co.uk
Web: www.grubstreet.co.uk
Text copyright © Arto der Haroutunian 1982, 2008, 2010
Copyright this edition © Grub Street 2010
Design Lizziebdesign
First Published in Great Britain in 1982 by Century Publishing Co. Ltd
A CIP record for this title is available from the British Library
ISBN 978-1-906502-94-2
Digital Edition ISBN 978-1-908117-89-2
All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system, without permission in writing from the publisher.
Printed and bound in Great Britain by MPG, Bodmin, Cornwall
This book is printed on FSC (Forest Stewardship Council) paper
## Acknowledgements
Many thanks are due to all the authors and publishers from whose works I have quoted (see Bibliography), and apologies to those who unintentionally may have been overlooked.
All works from Arabic and French have been edited and translated by myself.
I must also thank all the kind people of North Africa who helped in many a small way in the shaping and writing of this book. Special thanks as well to Odile Thivillier and Rina Srabonian.
## contents
Preface
Introduction
Mezzeh
Churba—soups
Salads
Eggah and kookoo—egg dishes
Pastas, pies and boreks
Kibbehs and kuftas
Yoghurt dishes
Ganachi—cooked vegetables
Dolmas—stuffed vegetables
Pilavs
Kebabs
Fish dishes
Meat dishes
Poultry and game
Firin kebabs and khoreshts
Sauces
Khubz—bread
Torshi—pickles
Desserts and sweet things
Cakes and biscuits
Sweets
Jams and preserves
Ice cream
Khumichk—drinks
Glossary
Bibliography
Footnote
## preface
One of the most heartening memories of my childhood is Sunday lunch, when all the members of our family, as well as guests (mostly young students from the Middle East), sat around our large table and consumed in delight and with gasps of rapture the product of my mother's work. For not only was my mother a remarkable cook but also, being thousands of miles away from home, we were in a vast culinary desert devoid of such familiar vegetables as aubergines and okra; spices such as cumin, sumac and allspice; the honey-soaked, rosewater-scented desserts of our childhood.
We ate. We argued loudly and vociferously. We drank our thick black coffee and nibbled a piece of rahat-lokum, a box of which someone or other had just received from home. Then, thanking the Lord for his bounteous generosity we settled comfortably into our large, Victorian armchairs and, almost in whispers, talked of home, of the sun-drenched streets of Aleppo or Baghdad, the rich souks of Alexandria or the fragrance-inflamed bazaars of Tehran. Someone would then sing 'The song of the immigrant' far away from his village and loved one. Someone else would hungrily describe how to peel and eat a watermelon—'You know, with white goat's cheese, some warm lavash bread with a sprig of fresh tarragon tucked nicely in the centre and all washed down with a glass of cool sous'—all of us lolled in nostalgic euphoria and dreamt of home.
'Home' to us all was the Middle East, not the political entity of today with its strong regional, national and social differences. In those days (the late forties) we were, regardless of our ethnic origins, still Orientals or Levantines, a people who were just waking from centuries of slumber and ignorance, a people who had been mistreated by foreigners, be they Turks, French or English, for their own selfish interests.
In our new environments, temporary for most, permanent for few, we tried hard to emulate the past. Customs were kept, rudiments of our mother tongue were inculcated into our minds and traditions punctiliously adhered to. Lent was strictly kept and for forty days no meats, poultry or fish passed our lips—at least at home. For us children school meals (however tasteless they were) were our salvation. I remember how I relished the school pork chops and steak and kidney pies—especially during Lent.
Over the years changes did take place not only in our domestic lives but up and down the land. More immigrants arrived from India, Pakistan, South East Asia, Cyprus and the West Indies. They too soon settled down, opened their ethnic restaurants, shops and emporiums thus enriching the country with their diverse cultures and adding spice to the eating habits of the natives—particularly important for one such as I who likes his food!
My early childhood was spent in Aleppo, Syria. My father's family originated from Cilicia (Southern Turkey) and my mother's from Armenia. The Aleppo of my childhood was a medium-sized cosmopolitan city soaked in history, rich in commerce and perhaps the most enlightened region of the Levant. In our street lived Armenians, Assyrians, Greeks, Christian Arabs, a well off Turkish family with vast cotton fields and indisputably ugly slanted eyes and not forgetting Jacob the Jew, a carpet dealer and close friend of my father. We all spoke our ethnic tongues, with a little spattering of Arabic. We ate, prayed and lived our lives as had our ancestors for centuries.
This is how my preface should have started: 'The search and collection of authentic recipes from the Middle East by Your's Truly, an exile approaching middle age and in search of his roots.' Well, my roots were my family to whom I turned in earnest, but with some difficulty. For although my family originated in Armenia and Turkey we had, over the years, like the 'sands of desert', scattered all over the east and beyond. So I got in touch with my aunt in Baghdad, my cousins in Kuwait, my sister in Tehran, with other cousins in Egypt, friends of cousins in Cyprus and Ankara, material cousins in Yerevan and Tiblisi and numerous other friends of friends ad infinitum. Finally all those kind people I encountered who had inadvertently 'dropped in' to have a meal in one of our restaurants and who, throughout their meal (and often well after), were subjected to a culinary inquisition of the fiercest kind. My thanks to all those people, with special thanks to the young Saudi Arabian doctor who literally fell into my clutches and had to spend an extra day in a wet and windy Manchester one December until I was satisfied that I had squeezed the last drop of 'culinary blood' from him. But most of all thanks to my mother who was a source of inspiration and infinite information.
The result then is this book, a collection of recipes from all over the Middle East regardless of political and geographical boundaries. By which statement I mean the food of the people of the Middle East: Arabs, Armenians, Assyrians, Azerbaijanians, Copts, Georgians, Kurds, Jews, Lazes, Palestinians, Persians, Turkomans, Turks and all the other minorities who, far too often, are forgotten or ignored by the cataloguers and generalizers of human achievements. For it is my opinion that the true sources of most cultures are best found amongst the indigenous minorities; e.g. the true Egyptian is the Copt, he is not only directly descended from the Ancient Egyptians, but also still retains much of his forefather's culture undiluted by later arrived, desert-oriented Muslim Arabs.
I have also included proverbs, anecdotes, songs and stories of the famous Nasrudin Hodja and Boloz-Mugoush all of which, directly or indirectly, relate to food and all emanating from the rich and varied cultures of the peoples of the Middle East, to whose glorious past and brighter future this book is dedicated.
All the recipes give the right amount of ingredients to feed four people unless I have stated otherwise.
Arto der Haroutunian
## introduction
Geography, history and the people
Ours is the world and all who dwell upon it,
and when we assault, we assault with power.
When kings deal with their peoples unjustly
we refuse to allow injustice among us.
We are called oppressors; we never oppressed yet,
but shortly we shall be starting oppression!
When any body of ours reaches his weaning,
the tyrants fall down before him prostrating.
We have filled the land till it's too strait for us
and we are filling the seas back with our vessels.
So let no man act foolishly against us,
or we shall exceed the folly of the foolhardiest.
Amr ibnel Kulthum—pre-Islamic poet of the sixth century
The greatest single unifying factor in the Middle East is the climate which, throughout the millennia, has imposed a special way of life and has been the mainspring of all social, economic and cultural diversifications. Within the Middle East are the great rivers Nile, Tigris, Euphrates, Arax and Kura. The Nile is fed by the Blue Nile (rising in the Ethiopian Highlands) and the White Nile (rising in the Central African Highlands). The waters of the rivers Tigris and Euphrates mingle some sixty miles before reaching the Persian Gulf. Their junction creates the Shatt-al-Arab waterway which forms the boundary between Iraq and Iran and has been a bone of contention over the centuries. Both rivers rise in the Armenian Highlands and zig-zag their way to the desert lands of Syria and Iraq. The Tigris (the Arrow), so named because of the swiftness of its waters, also receives several tributaries from the Zagros mountains of Iran. The plain here is threaded by numerous tributaries fed from one river to the other creating naturally formed irrigation schemes.
The Arax and Kura rivers rise in the Caucasian Highlands and flow into the Caspian Sea. The Arax borders the Armenian SSR and Turkey, whilst the Kura passes through the Georgian and Azerbaijanian SSR. The waters of the Arax irrigate the rich Ararat plateau.
Other important rivers are the Kizel Irmak (Turkey), Sefid and Kashka (Iran) and the Karun which waters south west Iran. Whilst the smaller rivers Orontes, Litani and the Jordan irrigate Syria, Lebanon, Israel and Trans-Jordan.
With the exception of the Caspian Littoral of Iran and Azerbaijan, 'Pontic' Turkey, and the Caucasian coastline of the Black Sea, rainfall is not only inadequate, but limited to spring and winter. The entire Mediterranean, therefore, stretching from Libya and Egypt, up the Levant to Turkey receives rainfall similar to California and south western Australia.
In the summer, drought is the general rule. Ninety per cent of the entire region is arid desert or semi-desert and whatever forests it once contained have long since vanished. Only five to six per cent of the Middle East is cultivated today and one-fifth of that needs water desperately.
Water is the most important human factor throughout the region. It has been deified throughout the ages and myths have been created around it. To preserve this scant and precious commodity the Ancient Sumerians, Assyrians, Babylonians, Hittites, Urartians and Romans built cisterns and viaducts, the remains of which can still be traced throughout Iraq, Syria and Turkey.
The Persians built qanats—underground conduits—to bring water for hundreds of miles from the mountains to the plains. In parts of Arabia water is so scarce that, rather than waste it, Bedouin women wash their hair in camel urine. The pious Muslim substitutes sand for water as he ritually 'cleanses' himself during prayer time. In ancient Armenia and Iran there existed (and still does) the Water Festival of 'Vartavar'—Burning of Roses. On this day people drench each other with water and the ecclesiastical procession throws rosewater at the congregation. The aim is to invoke the Gods for rain.
Today extensive work is being done to 'green' the land. Huge irrigation schemes such as the building of the Aswan Dam in Egypt, the Euphrates Dam in Syria and Ataturk Dam in Turkey have been established and already the effects of intensive agricultural techniques are seen in a number of highly productive farming areas such as western Lebanon, the Aleppo region of Syria, the Ararat valley in Armenia, the oasis of Damascus and, perhaps the most spectacular of all, the Jewish settlements of Israel, particularly in the Negev.
The position that the Middle East occupies in history is unique for it was here that man first began to cultivate food-plants and to domesticate wild animals. Wheat was cultivated in Jarmo (Kurdistan) 9000 years ago. Gradually peas, lentils and other crops were added to the 'original cuisine' of the inhabitants. With the full adoption of crop and stock raising man gave up nomadism and settlements grew into villages and then towns. As the population increased in the mountain regions people moved down to the plateaux and plains of the Tigris and Euphrates. There they were confronted with hot, semi-arid earth which they had to irrigate with the abundant waters of the two great rivers. The same process undoubtedly occurred around the Nile delta and further afield in the Ganges Basin of India. Towns produced social order, city states, nations and empires with which came literature, law and order.
By about 5000 BC the Egyptians were making wine and bread, rearing animals and cultivating crops. The produce grown at this time was extensive: wheat, barley, spelt, millet, legumes, root crops, melons, olives, grapes, figs and dates and, at a slightly later date, apples, peaches, pomegranates and apricots.
Around 3000 BC the Sumerians built their irrigation canals and city states and evolved a society divided into technological–social classes—nobility, priests, traders, farmers and artisans—divisions that still hold good for most Middle Eastern people today. The Sumerians, a non-Semitic people, were soon joined in Mesopotamia by Semitic tribes coming from Arabia and, in time, a new, mixed people were formed. Further north along the middle of the Euphrates several Semitic races intermingled and created city states. In 2350 BC the entire region was united under the leadership of Sargon, and his empire, in time, was replaced by that of the Amorites (1688 BC) whose most distinguished member was Hammurabi the law giver. The Amorites fell under the rising power of the Indo-European Hittites and Hurrians from the north—the ancestors of the Armenians and most modern Turks. By 1350 BC the Hittites had occupied Syria and wrested Palestine from Egypt and had become the mightiest power in western Asia.
Meanwhile, Egypt had developed its dynastic and political institutions and created one of the seven wonders of the ancient world—the pyramids of Gizeh. The Old Kingdom was followed by the Middle Kingdom (2200 BC) ushered in by the brilliant Twelfth Dynasty when peace, prosperity and overall economic success had made her the most civilized land on earth. Then the Hyksos—the shepherd kings, initially a branch of the Hurrian races—came down from Armenia and on their march, mixing with the Semitic elements, they overran the peaceful empire of the Pharoahs. They introduced the horse, the war chariot became the dominant power until the Pharoah Ahmos succeeded in throwing them out of Egypt and launching her into an imperial stage in history, all of which culminated with Rameses II (1301–1234 BC). He agreed to end all wars and signed a pact with the Hittites which maintained a balance of power and temporary stability. This earliest of international pacts was meant to bring 'Peace and good brotherhood between the contending parties for ever.'
A new Semitic people from the city state of Ashur (Assyrians) were largely responsible for the destruction of the Hittite empire. They were followed by Babylonians whose last king, Balshazzar, was also the last Semitic ruler of the East; henceforth it was the Iranian–Aryan tribes who dominated the canvas of history. Cyrus the Great (550–529 BC) introduced not only a new language, but also a new religion—Zoroastrianism, and the Persian rulers were tolerant of their subjects attempting no imposition of language or religion.
In the spring of 334 BC a twenty-one-year-old Macedonian, at the head of 35,000 fighters, crossed the Hellespont and launched his victorious campaigns all the way to India. Great as those military exploits of Alexander were in themselves 'greater they loom in their cultural consequences. They opened the way to the confrontation, harmonization and final fusion of Greek and Near Eastern ideas and institutions, thereby effecting a pioneering revolution in the world's outlook.'
Alexander died young aged thirty-two and his kingdom was quickly divided between his four ablest generals who henceforth began to quarrel amongst themselves. They all, particularly Seleucus, continued Alexander's policy of planting numerous Greek cities all over his kingdom. In time old soldiers, traders and craftsmen, as well as adventures, followed into the Middle East mixing with the Semitic people and giving birth to a new Hellenistic civilization which, for over a thousand years, remained a dominant feature in the Middle East and was only overtaken by that of Islam in the tenth century AD.
Indeed it is safe to say that in the last 2000 years of Middle Eastern history the two most powerful cultural forces have been those of Hellenism and Islam. Yet Hellenism was primarily an urban feature. The rest of the populace continued to speak their native tongues, whether Egyptian, Persian or Aramaic, and to worship their own deities. Hellenism created an economic uniformity. Transport was improved; the highways were guarded by chains of colonies; inns, resting places and market towns were created and along these routes passed cereals, oil, wine, fruits, minerals, pepper and cinnamon. Saffron came from India via Persia; frankincense and myrrh from Yemen, wheat from Egypt and gold and silver from Armenia. Great cities were founded—Antioch, Iaodicea and, perhaps greatest and most magnificent of them all, Alexandria in Egypt—for generations the cultural capital of the world. 'And then came the Roman wolf.'
In the year 66 BC, with the fall of the mighty Armenian Empire of Tigran II, the Roman general Pompeius took over the command of all the forces in the East and the Roman conquests commenced. The rule of the Romans was a continuation and development of the Hellenistic. The subject races lived in greater security and comparative affluence than their predecessors. Yet in spite of that there were still millions of people who were unhappy with their lot and many of these became ardent followers of the new ideas emanating from Bethlehem. These were swiftly synthesized into religious forms, and arrived in auspicious time in the heart of Rome.
The first nation to be converted to this new religion was Armenia, under Dirtad III in AD 301, and twenty-five years later Constantine raised it to an official state religion in his newly-founded capital Constantinople (Istanbul), strategically situated between Europe and Asia Minor on the Bosphoros. For the next 1100 years Constantinople was to be the focal centre of the entire Middle East militarily, economically and especially spiritually. The temporal triumph of Christianity did not bring the majority of men spiritual freedom or alter their economic lives and this, coupled with ethnic differences and the general resentment of the corrupt bureaucracy ruling from Constantinople, found expression in the dogmatic disputes which in time caused the eventual disintegration of the empire, and paved the way for the rapid ascendancy of Islam.
The Prophet Muhammad was born in AD 570 in Mecca. He began to undergo his religious experience about AD 610. He was vehemently opposed by the merchanttribes of Mecca, among whom he was born, so that in the year 622, with a band of seventy converts, he fled to Medina. There he rapidly made more converts and in a short space of time almost all the tribes of Arabia submitted to his authority. After Muhammad's death (in 632) Abu Bakr became the first Caliph ('successor') and decided to employ the warlike energies of the tribes by invading Syria and Palestine. Within less than a century after the Prophet's death the Arabs had reached the Atlantic in Morocco and the river Oxus in modern Turkistan. In every country the Arabs conquered the great majority of the population, Jewish, Zoroastrian, Pagan or Christian, embraced Islam. One of the most remarkable features of the Arab conquests was the small number of warriors involved. However, Muhammad had authorized the use of women captured in war as concubines: 'The amazing extent of the Arab conquests had enabled them to acquire great numbers of such foreign concubines; Greeks, Persians, Armenians, Egyptians and North African Berbers. Thus a few generations after the conquests the "Arabs" of Syria were ethnologically a different race from the conquerors who had emerged from Arabia after the death of the Prophet.'
Syria became the first seat of the empire, but in the year 750 power was transferred to Baghdad (Iraq). The Arabs reached the height of their glory in the reign of Harun el Rashid (786–809) and his son Al-Mamun (813–833), a period in history which is vividly described in the pages of the Arabian Nights and similar literary works. This was a time of elaborate extravagances enacted at festivals, ceremonial occasions and weddings, and artists, poets and scientists from all corners of the empire were attracted to Baghdad.
Yet slowly a cancer was settling in the heart of the empire which was gradually destroying the very fabric and goals on which it had been created. For the warlike instincts of the nomadic Arab had gradually disappeared. He surrounded himself with luxury which his ancestors could not have dreamt of in the desert and, in the words of Gertrude Bell: 'The ancient ghosts of Babylonian and Assyrian palace intrigue rose from their muddy graves, mighty in evil, to overthrow the soldier Khalif, to strip him of his armour and to tie him hand and foot with silk and gold.'
In the first half of the eleventh century 'a new scourge appeared'; wild Turkish-speaking nomads (the Ghuzz) entered Persia and, under their chiefs (Seljugs), they swept across Persia to Armenia 'massacring, looting and raping as they went.' The Seljugs occupied Baghdad, Syria, Palestine and Egypt and then moved on to Asia Minor where they created the first 'Turkish' empire until its demise some 150 years later under Mongol attacks.
Only Byzantium remained as the bastion of Christianity, but she too was weakened by the loss of Armenia in 1071 at the battle of Manzazkert when Alp Arslan shattered the Byzantine-Armenian forces under the leadership of Emperor Romanus Diogenes. The new emperor, Alexius Comenus, appealed for help to the western Christian powers and the Pope, Urban II, preached his crusade in 1095. Not surprisingly, however, the crusades were a failure, and today only dim reflections of romantic troubadours, chivalric heraldry, sculptured tombstones in old cathedrals and a few Arab words, much maligned and distorted, remain of the 300 years of western Christian presence in the Levant.
Then came the Mongols, burning, pillaging and levelling cities to the ground, only to be followed by far more savage tribes from Central Asia—the Turks. From their countless tribes, in time, the 'Ottoman' branch gradually expanded from their adopted homeland of north western Asia Minor and succeeded, in 1453, in capturing Constantinople. Under Suleyman 'the Magnificent' (1520–68) the Ottoman empire reached its peak and it was now a major naval power. The people of the Ottoman empire that came to rule the Middle East for nearly 400 years formed a heterogeneous complex of religious, linguistic and ethnic groups—Greeks, Slavs, Armenians, Kurds, Arabs, Christians, Muslims, Jews—all artificially held together by the Ottomans. Throughout the life-history of the empire the Turkish element remained a minority, but one that was constantly expanding by deportation, mass murder and forced conversions of other groups. Yet the period of imperial glory did not survive for long, for the sultans, far too often, were interested in matters of the flesh. Dissipation and corruption were rife throughout the court. The seeds of weakness embedded in the Ottoman state and society began to fruit in the late sixteenth century because a government primarily created for warfare rather than welfare and peace lacked the capacity to adapt to change. The subject races had no love for their rulers.
While the Ottoman empire decayed from within the pressures upon it from outside increased. In the seventeenth century the problem was further aggravated by the emergence of two powers, Russia and Austria-Hungary; the opening of a new trade route to Asia via the Cape; the establishment of Dutch and British power in Asia and finally, the constant warfare with neighbouring Persia.
The Russians, ever covetous of 'warm waters', began a long term plan of destroying Ottoman power which began with Peter the Great (1689-1725) and culminated with the Russo-Turkish wars of 1828–9, as a result of which certain Ottoman provinces were surrendered to Imperial Russia. After the Crimean war (1854–1856) more concessions were made to Russia. The sultans tried in vain to reform, or by warfare to subdue, their subjects. Turkey was the 'sick man of Europe' soon to die and disintegrate and no imperial decree could halt that eventuality. The remnants of the empire finally gained their independence after the end of the First World War.
Throughout the centuries the people of the Middle East have remained faithful to their source—the land. Though empires have come and gone they have maintained their deep attachment to their traditions, customs, food and aspirations.
Come with old Khayam, and leave the lot
Of Kaikobad and Kaihostu forget
Let Rustum layabout his as he will,
Or Hattim Tai cry supper—heed them not.
And let us then
along some strips of Herbage strown
That just divides the desert from the sown,
sit to rest
with a loaf of bread beneath the Bough
A flask of wine, a book of verse—and thou
Beside me singing in the wilderness —
And wilderness is Paradise now.
O. Khayyam—Rubaiyat. Trans. E. Fitzgerald from Oriental Caravan.
food in history
After Anu—had created heaven,
Heaven had created—the earth.
The earth had created the rivers,
The rivers had created the canals,
The canals had created the marsh,
And the marsh had created the worm —
The worm went, weeping, before Shamash.
His tears flowing before Eai,
'What will thou give me for my food?
What will thou give me for my sucking?'
'I shall give thee the ripe fig,
And the apricot.'
'Of what use are they to me, the ripe fig
And the apricot?
Lift me up among the teeth
And in the gum cause me to dwell!
The blood of the tooth I will suck,
And of the gum I will gnaw its roots!'
Toothache Incantation from the Ancient Near East.
The raw materials of the earliest civilizations of the Middle East, those of Sumer and Babylon, consisted of barley bread and barley paste accompanied by onions—still highly popular with Iranians and Arabs. Indeed the Prophet Muhammad is reported to have had a great fondness for raw onion, to which the Arab and Persian historians of the Middle Ages attributed aphrodisiac and almost mystical qualities. Also popular were several types of beans, lentils, chickpeas and fish from the nearby rivers and all were washed down with water or, if one could afford it, beer. It was in Sumer that beer—barley beer—was first developed; indeed it is estimated that forty per cent of the barley grown then was used in its manufacture. The Egyptians, who have left us several recipes for beer fermentation, had a famed beverage called hag which was made from 'the red barley of the Nile'. Veal, beef and game were known. However, mutton was the most popular meat—as it still is—and it was either grilled, what are nowadays called 'kebabs', or cooked in large cauldrons with cucumbers, onions and herbs like the guvedge or firin kebabs of today.
Goat and pig meat were also popular, the latter had not yet been rejected by the people of the region. It was, indeed, the Indo-Aryan tribes who (about 2000 BC when they had penetrated the Middle East proper) began a systematic anti-pig campaign since they—cattle rearing people—had a deep-rooted dislike of the pig, an animal that could not be herded, was regarded as filthy due to its eating and living habits and whose meat did not survive long in the hot climate and was therefore dangerous. This aversion towards the pig was later enshrined in the Judeo-Islamic, hence Semitic, ethics.
In the desert regions—and a large portion of the Middle East was and still is desert—the date palm had soon acquired a great importance and was incorporated into the diet of the people. The fruit could be eaten fresh or dried, its juice was made into a syrup for desserts and puddings and into intoxicating drinks such as wine or arak. The Parthians (of the Parni-Aryan tribe), who, from 250 BC until AD 216, were the dominant force in the region, were very fond of the date and date wines, although they were equally partial to the grape wines of Armenia. Hence the Sassanian-Persian expression 'Drunk like a Parthian, stubborn like an Armenian and arrogant as a Roman.'
The first 'raised bread' was most probably discovered in Egypt where, by the twelfth century BC it was commercially produced and sold in specialist bakeries. There were many types of bread, the cheapest and commonest one was called ta—a small, flat bread, the progenitor of the modern pita which is still, appropriately enough, also called khubz Arabi (Arab or Syrian bread). The Egyptians had undoubtedly the most sophisticated and advanced culinary art of their time. In numerous tombs dish after dish of fish, stews, beef ribs, fruits, porridges of some sort or other, as well as figs, dates, cheese, wine and beer in large casks have been found.
Herodotus (484–409 BC) in his Histories attributed the good health of the Egyptians to their custom of purging themselves for three days every month with emetics and clysters, which they did in the belief that all diseases originated from the food a person ate. He was impressed with the wide and rich selection of food available to the public. He notes the date wine, various beers and the numerous fish found in great abundance in the waters of the Nile—some were eaten raw after being dried in the sun or salted, a custom that still prevails in Egypt and the adjacent lands, particularly Lebanon. He noted that the wildlife was rich in small birds such as quails, magpies and swallows which were either stuffed with wheat and roasted, similar to djejij mushwi or amij, or raw after being pickled in brine—a practice still popular in Egypt, Cyprus and Lebanon.
The Greek influence on Middle Eastern eating habits is not hard to see, Alfred Zimmern's caustic remark that the Greeks ate for breakfast 'a kind of porridge' and then for lunch 'another kind of porridge' is not too far from the truth, at least as far as the poor were concerned. These 'porridge' dishes (maza in Greek, puls in Latin) can perhaps better be termed 'dips' and they are still found in such classic Middle Eastern recipes as toureto, fatoush, taramasalata, hummus-bi-tahina and the many vegetable and olive oil-based dips. Incidentally the Arab word for hors d'oeuvre, mezzeh, is undoubtedly derived from the Greek maza. A typical 'porridge' of the time kykeon was made from barley meal and water with aromatic flavours like thyme and mint and it strongly resembles the tarhana or tarkana soup popular in the Balkans and Turkey.
Life under Rome was not much better for the poor, for their basic diet consisted of millet-porridge, coarse bread, olives, figs, beans, cheese and milk. However, the food of the rich was very different, rich and varied. Pickles were imported from Spain, cucumbers from Cappadocia, pomegranates from Libya, wheat—the granary of the empire—from Egypt, wine from Jura, spices from far flung Parthia, India and even China (Chinese cinnamon).
When the Barbarians began to encroach and eventually overran Europe in the fifth century AD the once accessible markets were gone for good. Gaul, which supplied wine and oil; Spain, wheat, fish and pickles; the Black Sea coast, wheat and North Africa, olives and fruits, were all in enemy hands. Then, with the division of the empire into West and East, the glory that was Rome was no more.
It was the turn of Byzantium, straddling Europe and Asia Minor, to rise into prominence; a new Rome, but of Greek origin and oriental in scope and spirit. As the centuries passed the rather complex and often (to us today) obnoxious dishes with their exotic flavours disappeared, to be replaced by a much simpler cuisine primarily made from local ingredients like olive oil, sesame paste, citrus fruits, vegetables and meat—most of which are still retained in the Greek, Turkish and Armenian cuisines. The golden age of the Byzantine cuisine was perhaps the sixth century in the reign of Justinian and Theodore, when many chefs were brought to serve in the court of Constantinople from Persia, Syria, Armenia and even as far away as India.
Until the eighth century Constantinople received, via Alexandria, the rich produce of the Nile valley, Arabia Felix, Ethiopia, India and China. Luxury items such as spices, perfumes, precious stones and raw materials were then exported to Italy, the Byzantine ports of Tripoli, Antioch, Tarsus, Smyrna, Trebizond and Thessalonica did a roaring trade bringing honey and wheat from Bulgaria; fish and salt from Russia; wheat and fruits from North Africa and wine, precious metals and spices from Armenia and Persia.
The Byzantines ate three times a day—breakfast, midday and evening. Hors d'oeuvres were served first and these would have consisted of cheese, olives, cold ham, pork sausages, pickles, fish roe (tarama), artichokes and beans in oil—always olive oil—similar to the plaki dishes of today.
Soup courses would have included vegetable soups, tarhana, i.e. dough- or barley-based, soups, the ever popular tripe soup, a fish or meatball soup with egg and lemon sauce, similar to the avgolemono type soups of today, and the classic kakkavia, a thick fish soup, favoured by the ancient seafarers, which consisted of any small fish—eel, whiting, red or grey mullet, lobster, prawns—olive oil, onions, thyme, parsley, garlic and saffron and sometimes with wine and bay leaves as well. The main course would have consisted of grilled pig's trotters, grilled fish with white sauce. There would be ham, duck, biscuits and various cheeses—particularly the famed Vlach version. Special, often religious, occasions were always celebrated with special meals, the forty days of Lent were broken at midnight on Holy Saturday with a soup made from lamb's innards that was similar to the modern margeritsa. A whole lamb or a suckling pig was spit roasted on Easter Day having been first stuffed with fruits, nuts (pistachios and pine kernels), wheat and raisins all soaked in wine. It is, however, important to remember that the Byzantines—the religious and Greek element at least—were basically vegetarians and 'compelled the people to diets of interminable fish as, for that matter, did the Latin Rite'. Thus the 'true' Byzantine contribution to the cuisine of the Middle East comprised olive oil-based vegetarian dishes and fish dishes—soups, appetizers and stews.
After the death of Muhammad in AD 632 the Arabs poured out of their desert confines into Syria, Egypt and Persia. Very little is known of the economic life of the Umayyad Caliphate. The great part of the Arabs who left Arabia remained Bedouins (nomads). Their presence wrought havoc on the agricultural activities of the settled population, for the overgrazing of goats and camels had a devastating effect on the natural vegetation. The Bedouin knew next to nothing of husbandry and agriculture. Their needs were little, their cuisine very poor. Milk and milk products from their flocks and herds formed (and still does) the mainstay of their diet. Milk was drunk warm straight after milking or thickened in a goat's skin —yoghurt was not introduced until after the conquest of Persia. Sometimes unleavened bread was made by grinding the grain in a stone quern, kneading it with a little water and then throwing it from hand to hand until it became round and thin. It was then baked on a convex metal dish (saj) heated on dried camel dung or, when available, dead tree branches (see khubz saj). Very little meat was eaten—maybe only a few times a year on festive occasions such as a wedding celebration, the birth of a son, the arrival of important guests or during the Holy weeks of Ramadan. There was no rice initially. This was introduced at a later date via Persia. There were some fruits and a great deal was made with dates. The Arabians had a penchant for locusts either roasted, ground up and stored to add flavour to future meals or grilled on twigs—like kebabs. To the Bedouin, whatever the desert offered was good fare such as large monitor lizards, gazelles, bustards and dead camels as well as his goats and sheep. Yet in a short period of time the conquering Arabs were themselves living the luxurious life of the inhabitants of the newly acquired territories.
From the late eighth to the tenth centuries 'Arab' civilization was at its peak. The empire was rich in wheat—the granaries of Syria and Egypt produced crops which so far exceeded their wants that considerable quantities were exported to other regions. The Persians introduced rice, since the 'Mawali' from Huzistan and the Caspian regions had long been accustomed to rice and there was a growing demand for it in the cities.
Dates were grown all over Iraq and southern Persia, while Syria and Upper Mesopotamia produced various fruits and nuts—walnuts, hazelnuts, and pistachios. The finest apples came from Lebanon and Palestine, figs and grapes from Syria, plums from Transjordan, apricots and melons from Armenia and oranges and lemons from Oman, Syria and Egypt.
Under the Abbasid rule there was created an enormous economic market where the supply and transport of materials, goods and foodstuffs from quite distant regions was centred. Saffron from Isfahan and Hamadhan, red wine from Armenia, olive oil from Syria and Tunisia, sugar from the Yemen—all could be found in the bazaars of Baghdad. Tradesmen, bankers and adventurers from all the corners of the empire converged on the capital which, according to Benjamin of Tudelo, the fourteenth-century traveller-historian, was, next to Constantinople, the other great city: 'Merchants came to it from every land and save for Baghdad there is no city in the world to compare with it.' A vast metropolis was created out of a once small oasis where different nationalities lived in their quarters, Jews, Armenians, Christians, Persians, etc. The splendour and sensual gratifications competing to satisfy the appetites of the rich are well illustrated in the Tales of 1001 Nights. Food was abundant—at least to those privileged few who could retire to their inner sanctums (kaah), sit on richly decorated mattresses and cushions, listen to the finest singers, musicians and poets and dine off gold plates. Several manuals on the art of cooking were written, three of which have survived. The historian Al-Mesudi in his Muruj-al-dhahab (Meadows of Gold) portrays a typical symposium at the court of Caliph Al-Mustakfi who proclaimed to his friends, 'It is my desire that we should assemble on such and such a day and converse together about the different varieties of food, and the poetry which has been composed on the subject.' On the prescribed day one of the members of the circle said, 'Commander of the faithful, I have some verse by Ibn-al-Mu'tazz in which the poet describes a tray containing bowls of kamakh,' and he proceeded to recite. Others followed, each in turn praising dishes such as tardina, medira, sanbusaj, buran sauce and gateif. A rice pudding called aruzza was described by the poet Muhammad ibn-al-Wazir of Damascus in the following words:
O glorious aruzza! what a boon,
Thou cook as lovely as high heaven's noon!
Purer than snow thou hath been furrowed twice
By handiwork of wind and frosted ice...
Whilst sugar sprinkled upon every side
Flashes and gleams, like light personified.
One culinary influence on Baghdad was the Sasanian. The Sasanians received most of their culinary tradition from the Parthians, and the duality of Good and Evil, the internecine warfare between light and dark, was strongly reflected in Parthian–Persian food. The khoresht dishes of Iran are the product of this philosophy. Fruits—apples, plums, damsons, apricots and cherries—are cooked with chillis, nuts, vinegar, sour grapes and meats. Bitter limes or sumac powder are added to vegetables, chicken and ducks, e.g. khoreshte-hoo (lamb with peas and prunes), khoresht eghooreh (meat with sour grapes, sugar and spices) etc. This use of fruits, nuts, wine, honey, vinegar, as an ensemble was, of course, also popular with the Romans, but in the centuries that followed it virtually disappeared from the European-Byzantine cuisine. Today it only appears in the tajin (stew) dishes of North Africa—once part of the Greco-Roman world, in the khoreshts of Iran and the stews of the Caucasus.
There were other influences that infiltrated Baghdad. Kebab-type meats from Armenia, cakes from Egypt, cous-cous (semolina rice) dishes from North Africa, 'Armenian' fat bread topped with nuts, olive bread from Antioch, 'Frankish' roast lamb—similar to the rostos of Cyprus and Greece, meghmuma from India—a dish of aubergines, mutton, onions and spices—the ancestor of mussaka, meat or chicken pieces cooked in milk or yoghurt from the Caucasus, Persia and India. A typical example of the latter was medira—meat, cut into small pieces including the sheep's tail, and cooked with curd. The meat—mutton or chicken—was placed in a saucepan, a little salt was added as well as water and it was brought to the boil. When cooked a large peeled onion and leeks were added as well as coriander, cumin, mastic and ground cinnamon. All this was transferred to a bowl to which curdled milk or yoghurt was added with a little lemon juice and fresh mint. Such was the fame of this dish that the poet wrote:
Medira on the festive tray
Is like the moon in full array;
Upon the board it gleams in light
Like sunshine banishing the night...
Upon a platter it is brought
Of onyx, in Tehama wrought...
Medira cannot rivalled be
To heal the sick man's malady...
'Tis as delicious as 'tis good–
A very miracle of food.
All this lavishness bewildered and quenched the gastronomic desires of the once 'hungry' Arabs of Baghdad and they set about systematically developing this mêleé of cuisines which, by the grace of Allah! was bequeathed to them. With the spread of Islam the food of the Caliphs penetrated all corners of the empire and beyond into Europe via Spain, Armenian Cilicia and Sicily.
There has been little change in the basic diet of the people of the Middle East since the days of the Caliphs, except for the addition of new spices and vegetables which arrived after the discovery of the New World, potato, tomato, maize, red pepper, green pepper, hot chillis, French beans, peanuts, vanilla and others, the incorporation of which has given Middle Eastern cooking its uniqueness.
The advent of the Mongols and the numerous Turkish speaking tribes from Central Asia (eleventh to fourteenth centuries) added very little to the existing repertoire. The nomadic cuisine was unique for its poverty. They ate no bread, rice or grains, fruit and very few vegetables except the occasional bulbs of the red tulip, some wild carrots and onions, etc. From March until October they depended exclusively on their flocks and herds. They lived off mutton and horse meat. 'Kazy'-type sausages made of horse flesh, or roast hump of camel, stewed feet or braised paunch were great delicacies. In leather sacks they carried a little rice flour with which they prepared dumplings which were then boiled in water with a little salt mantu. These nomads who conquered Asia Minor, Persia and, for a period, most of the Middle East, depended on two basic items—blood and milk. According to Marco Polo when on the move each Mongol had a string of eighteen horses and mares and they travelled 'without provisions and without making a fire, living only on the blood of their horses; for every rider pierces a vein of his horse and drinks the blood.' About half a pint a day of blood could be drunk from each horse without damaging the health of the animals. Milk and milk derivatives—curds, soured milk, cheese and a fermented drink made from mare's milk called kumiss (mare's milk contains four times as much vitamin C as ordinary cow's milk)—was drunk fresh or dried under the sun. This is how Marco Polo describes this drying process. 'First they bring the milk to the boil... they skim off the cream that floats on the surface... then they stand the milk in the sun and leave it to dry. When they are going on an expedition they take about 10 lbs of this milk and every morning they take out about half a pound of it and put it in a small leather jacket shaped like a gourd, with as much water as they please then, while they ride, the milk in the flask dissolves into a fluid which they drink. And this is their breakfast.' Kumiss, which has been described as a kind of desert champagne, has aquired a certain mystique. It was brewed in early spring—with the birth of the first foal which was a time of celebration. It was served at weddings, funerals and religious festivals. The beverage was prepared by adding a 'starter'—the process is similar to making yoghurt—of old kumiss to fresh mare's milk, then churning steadily for an hour or more until the liquid had reached the exact peak of alcoholic acidity. A bag of this brew always hung outside the nomad's tent—yurt—and a charming custom existed whereby anyone who passed within arms length would agitate the bag just to keep the kumiss fresh! Kumiss today is still prepared in parts of Turkey, but it is far more popular in the Northern Caucasus and the Central Asian republics.
When the Ottomans took over the leadership of the moribund Arab Caliphate the pendulum once again swung towards the West. The Arab-Persian cuisine was relegated to a secondary position and the Byzantine-Armenian cuisine came to the fore. A quick glance at 'Turkish' food will immediately verify this fact. The persistent use of olive oil—still almost unknown in Central Asia—the many varied fish dishes, all bearing Greek names, the numerous Anatolian–Caucasian kebabs—still unknown in the original homelands of the Turks where everything grilled, and that means mutton, is called shashlyk—a Russian terminology. Finally the sweet pastries such as baklava (from the Armenian baki-halva, meaning Lent sweet) or kataif (from the Arabic ataif) are completely unknown in Central Asia even today.
Yet as it was with the Arabs in Baghdad so it became with the Ottomans in Constantinople, in a short period of time the conquering races were themselves living the luxurious life of the inhabitants of the newly acquired territories!
When Suleiman I came to the throne in 1520 Constantinople had once again become 'the world's most beautiful city' and, on the occasion of his marriage to Roxelana (a Russian slave captured by raiders in Galicia), free breakfast was served to all, i.e. bread and olives for the poor, cheese, bread, fruit and rose petal jam for the better off: 'the streets of Constantinople were jammed for the celebration... mounds of fruit—Mardin plums, Azerbaijan pears, Smyrna grapes, Temesvir prunes... hadgis cooked on portable stoves... the sherbert sellers did a roaring trade pouring the sherbert through a lump of snow stuck on the end of the vessel's spout; sherberts of lemon juice and snow flavoured with honey, amber and musk; others of water lilies, together with sherbert made of violets and honey, there were squares of rice-jelly sprinkled with rosewater or fruit soup with ice floating in it.' All this a mere seventy years after the conquest!
The poor, as ever, continued to hack a miserable living as the rich wallowed in luxuries 'beyond belief', and it befell to Kaygusuz Abdel the fifteenth century mystic poet to petition the Lord concerning this injustice on behalf of the starving masses—whose lot has not improved much since the day he penned these immortal words.
Lord I humbly beg of You, hear my reverend request,
These are words straight from the heart, they are not spoken in jest.
First, a hundred thousand loaves, also fifty thousand pies,
One hundred sixty thousand buns, profusely buttered on both sides.
A thousand piglets should suffice, if added to a thousand sows,
With sixty of their young, some fifty thousand water buffaloes.
Ten thousand cows, a thousand oxen for a mustard stew,
The trotters separately served in vinegar, with garlic too.
A thousand sheep in casserole, an equal sum of goats at most,
But fifty thousand lambs and kids to grill upon the spit, or roast.
Innumerable chickens, ducks, and in the same proportion, geese,
Some to make succulent kebabs, and others to be fried in grease.
Pray let there be dish after dish of pigeons and of tender quail,
Partridge and pheasant caught in nets, arriving in an endless file.
Fifty thousand pots of rice, and saffron puddings are inferred,
A thousand pots of porridge, the butter with a drumstick stirred.
Soups with pleasant flavouring, meatballs gently made, I beg,
Ducklings, and on trays of brass, sweetmeats made of starch and egg.
Fifty thousand pasties and the same amount of baklava,
Honey and almond cakes galore, and countless plates of fresh okra.
Helva fit for conquerors, served on trays and heaped in bowls,
For eager fingers to scoop up, making quite enormous holes.
Forty thousand, fifty thousand pecks of apricot and cherry,
Apple, pear and vintage grape, will be enough to make us merry.
Social and religious influences
The ways of life of the people of the Middle East vary from the completely westernized to the traditional peasant—Bedouin, unchanged for generations. The most European regions are Israel, the Caucasus and Lebanon where the majority of people live under completely westernized conditions and, not surprisingly, they are mostly non-Muslim.
Israel, a new nation, was created from the remnants of the second great genocide of our century. Its people have mostly come from Russia, Germany, USA and other western countries. These Askhanazim (Western Jews) are all, bar religion, Europeans; while the Sephardim (Oriental Jews) are, bar religion, indistinguishable from their neighbours—the Arabs, amongst whom they lived (prior to the birth of the State of Israel in 1948) in peace and harmony for centuries. Israel is, in essence, a western state on the periphery of the Muslim-Arab world and intrinsically so too is Lebanon, a small country that has been a haven for generations for minorities (Christian or heretic Muslims) but which, since 1975, has been undergoing serious political surgery. The majority of the Lebanese are 'Christian' of one denomination or another!
The people of the Caucasian Republics of Armenia and Georgia are Christians—belonging to the Eastern Orthodox branch. The republic of Armenia today is only a tenth of historic Armenia which stretched from the Mediterranean to the Caspian Sea. Like the Jews the Armenians have suffered throughout the ages. A large portion of the population was forcibly converted to Islam or deported. The climax of these misdeeds culminated with the first genocide of our century (1915–20) when some two million people, out of a population of four million, were slaughtered and the survivors scattered all over the Middle East and beyond. The Georgians, an intelligent and highly gifted people, are mostly Christian. Muslim Georgians—Lazes—emigrated to the Ottoman empire in the seventeenth century.
Azerbaijanians are Muslims and are related to the Turks, particularly to those living in Iran.
Cypriots are Christians with a Muslim Turkish minority.
All these lands are on the borders of the heartland of Islam, for Islam is the dominant religion with 150 million followers throughout the Middle East, and since it is not merely a religion but more a way of life there is a strong social homogeneity throughout the region; and centuries of Muslim domination have influenced socially and culturally such non-Muslim people as Greeks, Armenians, Copts, Jews and Zoroastrians.
Outside the highly industrialized and westernized regions of the Middle East there lie, for want of a better word, the semi-westernized regions. These, in essence, are the large cities such as Cairo, Tehran, Damascus, Ankara, Istanbul, etc. where the official and professional classes and the rich merchants live under semi-European conditions. Most of these people live in houses or apartment blocks. They wear European clothing, drive cars and their homes are furnished with tables, chairs, beds, dressing tables, modern kitchens, refrigerators, radios and often televisions and cassette players. In a typical house the front door usually leads to the reception room which is commonly used by men. To get to the domestic quarters, i.e. kitchen and bedrooms, one has to use a separate door leading from the hall and this is usually barred to men. When guests arrive they will sit down first in the men's room, but after a short time the ladies are expected to 'visit' domestic quarters where the females of the house sojourn. This segregation of the sexes has almost been eradicated amongst the westernized classes where the women not only do not cover their faces, but are seen in public wearing the 'latest' styles of clothing. Female emancipation, although not yet as advanced as in Europe, has certainly arrived in the Middle East.
A generation ago the streets of Alexandria, Baghdad or Damascus were filled with women clad in black, today this phenomenon is a rarity and is only practised among the elderly, or women living in small towns or villages.
There have been attempts at social and cultural reforms—Ataturk in Turkey, Reza Shah in Iran and Nasser in Egypt—but the slow, traditional and extremely conservative mind of the average devout Muslim has built barriers which only time—a lot of it—and education will dismantle.
In the villages, or in the tents of the nomads, there will often be no furniture. The guest room, mandarah, is spread with coloured rugs or carpets and with quilts, mattresses and cushions laid around the walls for people to sit on. Usually no chambers are furnished as bedrooms. The bed, during the day, is rolled up and placed to one side or, when one is available, in a small adjoining room called a khazneh which is used as a bedroom in winter. During the summer many people in the towns and villages sleep on the flat roof tops. The centre of the men's room, which is usually regarded as a reception room, is kept empty. The guests sit around with their backs to the walls.
All Middle Easterners are very hospitable. The unwritten rule is to please one's family, acquaintances, guests or hosts. It is a great honour to be a guest, but a greater honour to be host. When an unexpected guest arrives a space is immediately created for him at the head of the table and coffee or sherbet is offered. He must never refuse, to do so is taken as an offence.
When food is served it is brought either on a large dish or in numerous small dishes and placed on the ground in the centre of the room. The guest is then invited to join the family. He must refuse—at least three times—but give in upon a great deal of cajolling and entreating. Even when he has previously dined he is not expected to refuse, but to make a gesture by tasting a little of this and that. Before he sits down at the table the guest will wash his hands with soap and water in a copper basin called a tisht or, at least, have some water poured over his right hand. He is then offered a napkin. He must never refuse dishes that have been sampled by others present at dinner, to do so will give great offence. He must comment on the delicacy of the aroma emanating from the meal, pay little compliments such as on the tenderness of the meat or the thinness of the housewife's kibbeh or the sweetness of the baklava, etc.
Each person bares his right arm to the elbow—before he begins to eat, he says Bismillah—in the name of God. This is generally said in a low, but audible voice; and by the master of the house first. It is considered both as a grace and as an invitation to any person to partake of the meal... The master of the house first begins to eat, the guests or others immediately follow his example. Neither knives nor forks are used; the thumb and two fingers of the right hand serve instead of those instruments; but spoons are used for soups, or rice or other things that cannot easily be taken without... to pick out a delicate morsel and hand it to a friend is esteemed polite... Each person breaks off a small piece of bread, dips it in the dish, and then conveys it to his mouth, together with a small portion of the meat or other contents of the dish. The piece of bread is generally doubled together, so as to enclose the morsel of meat.
The above lines from Edward William Lane's brilliant book Manners and Customs of the Modern Egyptians are as true today as when they first appeared in 1836. In the intervening 170 years very little has changed in the social and cultural attitudes of the average Middle Easterner. Perhaps he does not sit on the floor, but uses a table and chairs. He may even use knives and forks, have table napkins, several plates instead of one, but the traditional Middle Eastern manners and rules of etiquette still remain.
At the end of a meal one must lick one's fingers as a sign of great satisfaction and say 'el hamdullah'—praised be the Lord, and get up to wash one's hands and mouth. This brief description of the table manners of the Muslims also applies to the other nationalities, whether Christians or Jews. A Christian will say a few words of thanks to the Lord before dinner and will end by thanking the Lord for what he has just received. He will then, in turn, thank his host and the lady of the house 'May your hand remain young and fresh,' or 'May God be bounteous and generous, and increase one to a thousand,' or 'May your table always be plentiful.'
During the meal people will converse in pleasantaries, but avoid discussing politics or business. The meal is a family affair even where there are several guests and the focus of attention will inevitably be on them.
There is always found an air of solemnity and dignity at a Muslim table. However, up in the Caucasus amongst the lush hills and valleys, there is a much lighter and more boistrous ambiance. When Armenians or Georgians are dining they show a liveliness and geniality not common amongst Arabs, Turks and Persians. Immediately the guests and hosts are seated a tamada—chairman—is democratically elected and he cannot refuse! It is his function to keep order and act as an arbiter between the diners. To be elected as a tamada is a great honour for it signifies a person of respect, wit and erudition. He will thank all for their presence and, one by one, introduce each guest in a light-hearted manner, reciting certain anecdotes from their past, often lavishing them with praise and, after each introduction, a toast will be drunk in honour of that particular guest. The atmosphere during dinner is always happy and gay. The tamada may ask someone to recite a poem, or plead with someone else to sing an old folk song—he cannot be refused—for during the dinner the tamada is king. A great deal of wine is consumed during such dinners for Armenians and Georgians, being of the Christian faith are, unlike the Muslims, under no religious censure. The meal will end with a few words from the tamada thanking the lady of the house for all her work, her wonderful cooking, expressing the appreciation of it by all and, turning to the assembled diners he will conclude:
It is time gentlemen we departed
For we have wined and dined in content;
the landlady's task for tomorrow
Has increased similarly in content.
A few words must be said about the dietary laws of the Middle Easterners. The three major religions Judaism, Christianity and Islam were the product of the desert where they were first seeded and nurtured. Today the predominant influence is that of Islam, often called a 'Christian heresy' and indeed, one can go further back and regard Christianity as a 'Judaic heresy'. These religions have a common source—the nomadic people of the Middle East, their way of life, their culture, beliefs, superstitions, fears and dreams shaped in metamorphical and surrealist colours and forms. With the development of a religion comes the 'commandments'—the law and order, the 'do' and 'don't' acts. And from amongst these social and ethical codes emerges the accepted norms of good conduct in life and the hereafter.
The dietary laws of the Semitic races—Arabs and Jews—are virtually the same. The slight differences, which have arisen over the centuries, are due more to the political and economic situation of the people at any one given time than to any purely religious uplifting or development.
Since religion has been the dominant factor in keeping the Jewish people together, this very old religion has evolved codes of behaviour that touch every aspect of a person's social life. The third book of Moses—Leviticus—is mostly devoted to health, cleanliness and social niceties as well as the permitted or forbidden diets of a devout Jew.
Whatsoever parteth the hoof, and is cloven footed, and cheweth the cud, among the beasts, that shall ye eat. Nevertheless shall ye not eat of them that chew the cud, or of them that divide the hoof: as the camel, because he cheweth the cud, but divideth not the hoof; he is unclean unto you.... And whatsoever hath no fins and scales ye may not eat. (Leviticus 11, 3–12)
Moreover ye shall eat no manner of blood, whether it be of fowl or of beast in any of your dwellings. (Leviticus 7, 26–27)
In short, a practising Jew is not permitted to eat pork, shellfish and, according to some authorities, turbot.
The prophet Muhammad deals lengthily in the Koran with food:
Believers, be true to your obligations. It is lawful for you to eat the flesh of all beasts other than that which hereby announced to you. Game is forbidden while you are on a pilgrimage. You are forbidden carrion, blood, and the flesh of swine. You are forbidden the flesh of slaughtered animals, and of those beaten or gored to death... You are forbidden the flesh dedicated to any other than Allah; also of animals sacrificed to idols. (Koran. Al-Maida 53–54)
Also permitted are 'the game of the sea'—which includes shellfish. All fruits and vegetables are permitted, as well as dairy produce. There are slight differences between Jewish and Muslim dietary laws and the more significant are:
a) | Muslims do eat camel meat as well as shellfish;
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b) | 'Thou shalt not seethe a kid in its mother's milk' (Deuteronomy 14–21) does not generally apply to the Muslims;
c) | Muslims are forbidden to drink alcoholic or fermented drinks. This does not really apply to Jews.
As for the Christians—Greeks, Copts, Armenians, Georgians, Maronites, Assyrians, Nestorians, etc.—they are permitted to eat and drink whatever they like—except during Lent. All the eastern Christians partook the forty days of Lent in earnest. The Greeks, Armenians, Georgians and Copts—the majority of Christians—fasted most rigorously as Paul Ricaut observed as far back as 1679.
For as the eastern people have always been more abstemious in their diet, and less addicted to excess in their tables and ordinary banquets than the western or northern nations, so by this custom of living they support more easily the severe institution of their Lents; who in the time of their feasts are not so free in their eating and dancing, as we are in our time of abstinence and fasting; for that which we call a collation, or lenten-table, will serve an Armenian for an Easter dinner.
The Christians observe 'the great Lent' before Easter, when they do not eat any fish, meat, butter, eggs, etc, but live on olive oil-based dishes, or tahina-based dishes 'the smell of which is sufficient to overcome a tender stomach'. The other fasts were on the Feast of our Lady's Assumption (early August), Feast of Pentecost and Epiphany (late December).
As for the Muslims they are forbidden to eat, drink or smoke from first dawn to sunset during the month of Ramadan, i.e. in the month in which the Koran was revealed to Muhammed. (The great feast falls on the 10th of the month of Dhuel Hijja—the day of pilgrimage outside Mecca.) The lesser Muslim feasts come at the end of Ramadan.
For Jews the Sabbath is a day of rest when all work is forbidden, which means all food must be prepared the day before. The most important feast is the Passover, when bread is banished from the home to be replaced by matzo—unleavened bread. The passover meals vary from the rest of the year when most vegetables—save peas and beans—are used. The 'uniqueness' of the Jewish cuisine comes to itself in the Passover period for the housewives were obliged to create different and tasty dishes to break the matzo monotony.
Today these dietary laws, as well as the fasts and feasts, are observed in varying degrees of laxity by the Middle Easterners, but generally pork is most uncommon, except in the Caucasus, Greece, Cyprus and parts of Lebanon. Wine is also uncommon, as are all alcoholic beverages, except with the Christians who, incidentally, produce all the wines available locally.
The long lasting influence of Islam has affected the social and, indirectly, the religious attitudes of non-Muslims who have unwittingly adopted many of the latters' customs and integrated them into their Christian beliefs.
To all those who followed the path of righteousness the Prophet promised that they:
shall recline on jewelled couches face to face,
and there shall wait on them immortal youths
with bowls and ewers and a cup of purest wine
that will neither pain their heads nor take away
their reasons,
with fruits of their own choice
And flesh of fowls that they relish,
And theirs shall be the dark-eyed houris,
Chaste as hidden pearls; a querdon
For their deeds.
(Koran. Al-wagia 56–15)
general features of the cuisine
Good wholesome food makes a good and healthy man.
Kurdish saying.
A major characteristic of the Middle Eastern cuisine is its overall simplicity; simple to prepare and simple to digest. Yet behind this apparent simplicity there lies thousands of years of experimentation with raw materials, utensils, cooking techniques and a subtle and intuitive understanding of climatic requirements.
The Middle East is rich in vegetables and fruits of exceptional quality—mainly due to the mineral rich soil and abundance of sunshine—and when these gifts of nature are better utilized with the aid of modern agricultural skills, as in the Caucasus and Israel; the produce is on a par with the finest available in the world. There is hardly any vegetable or fruit that does not grow in the region and the few exceptions are mainly those that prefer a tropical habitat such as pineapple, guava, breadfruit, calalu, soursop, etc.
Present on any Iranian table will be a large bowl of fresh vegetables which are eaten raw, 'Cucumbers are so good that Iranians eat them like fruit; once when I was working in a library, a young woman kindly peeled one and gave it to me as though it were an apple or orange.'
Vegetables and fruits are often dried for winter use, e.g. okra, aubergines, peppers, tomatoes, grapes, apricots, plums, peaches, etc. For instance, aubergines of a medium size are washed, their stems removed, their flesh scooped out, washed again thoroughly and finally threaded on strings and hung to dry under the sun. Vegetables dried in this manner make excellent dolmas in winter and indeed some people prefer a dried vegetable to a fresh one for their dolma.
As a rule vegetables are never boiled, except in Israel and parts of the Caucasus where European influences, e.g. Polish and Russian are at their strongest. The people of Asia, in general fry their vegetables in oils, fats or butter.
Olive oil is used for dishes which are to be eaten cold and, since a substantial portion of all Middle Eastern food comprises cold dishes—mezzehs, salads, fried vegetables and some fish and meat dishes—olive oil is the most popular cooking ingredient, second only to samna.
Samna is usually made from buffalo's milk, which has been melted over boiling water and clarified. Samna is similar to ghee (see Glossary), so popular in Indian cooking, whence it most probably originated. It appeared with the advent of the Indo-Iranian tribes into the Middle East for, even today, the Iranians, Kurds, Afghans and the Caucasians use very little olive oil in their cooking, but prefer samna which has a strong individual flavour and a little of it will go a long way. The samna produced in Hama (Syria) and Isfahan (Iran) are perhaps the most famous in the region. In recent years increasing use is made of European-type butters and margarines, whilst olive oil is often substituted by corn oil, groundnut oil and particularly by sunflower seed oil.
During the Middle Ages people made much use of alya; the rendered fat from a sheep's tail; and sesame seed oil—tahini. There exists in the Middle East a remarkable breed of sheep with a lean body, but a tail of pure fat which can sometimes weigh up to 20–30 lbs and has to be carried on specially constructed crutches! The thrifty nomad naturally made use of this fat alya, and almost all the Turkish, Syrian, Iraqi and Bedouin dishes, even just a generation or two ago, were fried or cooked with it. Almost every ancient manuscript suggested the use of this fat, 'dissolve tail and remove unwanted sediment, add meat into this oil and fry...' They also highly recommended tahini, which is still very popular in Egypt, particularly amongst the Copts; as well as throughout the Mediterranean coastline, and with the Armenians. In recent years Israelis, who seem to have developed a great liking for this oil, have incorporated it into many interesting new dishes.
If to the Chinese meat means pork, then to the Middle Easterner it implies lamb, although in recent years other animal flesh such as beef and veal are increasingly being eaten. Lamb is still the main meal of the region. The little pork that is used, mainly by Christians and 'liberated' Israelis who call the meat 'white beef', is usually grilled.
As a general rule people fry their meat and vegetables before adding water and spices when preparing soups and stews. This method of cooking gives a richer flavour and helps the meat retain its own juices by sealing the pores. This method of cooking is not restricted solely to the Middle East. It appears in Indo-Pakistani cuisine as well as that of North Africa and the Balkans.
Meat dishes have always been the food of the rich and even today the greatest honour a Bedouin can bestow on his guests will be the slaughter of a lamb—a tradition that precedes the story of Abraham and Isaac.
A great deal is made of minced meat. It is fried and used in soups and stews, mixed with chopped vegetables for kebabs and turned into sausages that can be stored for use in winter.
Middle Eastern food is spicy, but not hot. The exceptions are some dishes from the Yemen, Anatolia and those of North African origin. The finest exponents in the art of using herbs and spices are the Caucasians and Iranians. The latters' cuisine has much in common with those of the Indian subcontinent. Yet to the uninformed most Iranian food appears bland. In fact it is subtle and the Iranians have created a refined cuisine where spices and herbs—usually fresh—are used in just the right quantities and combinations to create the required flavours. They make great use of saffron—once so highly prized by the Greeks and Romans—and turmeric, also known as 'oriental saffron'. The people of the Gulf States, Iraq and Iran make the most use of turmeric, while Afghans, Iranians and, to a lesser extent the Turks and Caucasians, prefer saffron.
If it is the criterion in the West for a 'town' to have a 'Woolworth'-type store, a post office, a social security centre and a Chinese restaurant—or a take-away at least, then an equivalent 'town' in the Middle East should have a public bath, a mosque or two (or a church or synagogue, depending on the country, naturally), and a souk (covered market) with at least one attarine (spice street) where scores of small shops are filled to the brim with boxes, sacks and jars of every spice and herb imaginable.
In Turkey the most popular spices and herbs are sweet basil, dill and marjoram. In Armenia they are mint, tarragon and sumac. The Syrians love coriander, cinnamon, caraway and allspice; while Egyptians go for sesame, cumin, tamarind and thyme. The Gulf States prefer cloves, cumin, chillis, coriander and cress. The Kurds have a liking for fenugreek and garlic chives as do the Iranians who also make much use of nutmeg and cardamom.
Nuts and fruits—fresh and dried—are also used to create some very unusual flavours. The habit of mixing them with vegetables and animal flesh is an old one, but it still continues in the Caucasian and Iranian cuisines.
Nuts are used in sauces, soups, pilavs, stuffings and meat dishes as well as desserts and pastries. Turks and Iranians make a great deal of almonds; Armenians and Georgians prefer walnuts, Syrians and Lebanese prefer pine kernels and pistachios while the Lazes of Turkey and the Iraqis have a penchant for hazelnuts.
A Middle Eastern ingredient that has in recent years been popularized in the West, is yoghurt.
Bread, cheese and yoghurt is still probably the basic diet of the majority of the people of Turkey, Iran and Kurdistan. Yoghurt on its own, or as an integral part of a meal, is at its best in the Iranian, Armenian and Turkish cuisines which between them have hundreds of such dishes.
Another staple ingredient is rice, with which pilav dishes are prepared—see Glossary. Before the introduction into the region of rice there was burghul—cracked wheat—which is used in soups, pilavs, as a stuffing and for the kibbeh dishes. The past masters of this ingredient are the Armenians, although it is also popular in Syria and Lebanon.
Kebabs, perhaps the most original (and in some respects the oldest) method of cooking, also began in the Middle East. It was on the mountain ranges of the Caucasus that the art of cutting meat into small portions and marinating it in oil flavoured with spices and herbs was first developed.
The finest kebab dishes are found in Anatolia, Armenia and the Caucasus where people in the villages still cook their meals in the charcoal burning clay ovens which are sunk into the ground and called tonir or tandir—similar to the tandoori ovens of Northern India. In the rest of the Middle East people in the past, and still today, take their food to be cooked by the furunji (professional bakers). These furuns are an integral part of the Middle Eastern scene. All races and creeds intermingle and the women spend hours gossiping while the children devour freshly baked breads and cakes. A typical furun is on two levels. At the higher street end a salesman sits selling the freshly baked breads, cakes and savouries. A side entrance leads to the lower level and here the baker and his assistants are busy cooking both the shop's and the customer's food. He does this by placing the casserole dishes on a long wooden paddle which he carefully pushes into the centre of the oven—traditionally heated with wood, but more often today with gas. The entire oven is covered with stone so that the food is equally cooked all over. This method of cooking is slow. Several times the baker retrieves the dishes to check their progress and when done the casseroles, or breads, are counted, paid for and hurriedly taken home to be consumed.
All this smacks of the Middle Ages, not surprisingly, for there is much in the psyche of the Middle Easterner that impels him to keep in touch with his origins. Perhaps it is because of the unchanging desert, or the rigid religious doctrines that still prevail; the strong family ties with the strict code of honour, dignity, respect, virtue and tradition.
This deep deference to the past is perhaps one of the prime reasons for the unwavering attachment to the ancestral dishes, and the almost reverential treatment of their continuity into the future. 'I am what I am for I eat what I eat' said Boloz Mugush one day when confronted by an arrogant and ill-tempered Turkoman. 'My great grandmother passed it to my grandmother who in turn passed it to my mother, who passed it on to her daughters, the secret of enjoyment, happiness and satisfaction, that is why I am what I am. You are what you are for you do not know how to eat.'
## mezzeh
Mezzeh is the food of the traveller, whether on a picnic, pilgrimage or long journey. Mezzeh is a large selection of little things and the ideal places to acquaint oneself with this 'way of life' are—or rather were before civil war intervened—the mountain resorts of Zahleh, Bikfayah, and Jounieh in Lebanon. In small open-air restaurants literally hundreds of dishes were arranged on tables containing, amongst others, most of the recipes in this chapter and many such as salads, kibbehs, kebabs and pickles from other chapters; for strictly speaking there is no such thing as a mezzeh and yet everything can be called mezzeh.
Mezzeh is a way of life for everyone from the Bedouins in a desert oasis sharing their tit-bits to the large family sitting around the table sharing a little of this, a little of that, something cold, something warm, a plate of chopped fried liver left from yesterday, a few olives, cubed cheese, hot or cold minced meat balls, etc. Everything available in the kitchen can be part of the mezzeh—and herein lies its beauty, for a little of this and that usually mounts up to create a rich and sumptuous table with a variety of dazzling colours and contrasting textures and flavours.
Mezzeh makes an ideal buffet table and is particularly suitable to the Western way of life as it has been for centuries in the East.
Once the proprietor of a well-known restaurant in Zahleh, Lebanon boasted that he could, in a matter of a few hours, lay a mezzeh table containing over 200 items. His challenge was accepted. Three hours later, when our party arrived, the proprietor apologized that due to unforseen circumstances he had lost his bet. The mezzeh table—a magnificent sight stretching from one end of the open-air balcony to the other (over fifteen metres) contained 'only 175 items'.
He had proved his point.
muhammarah pomegranate and walnut dip
A Syrian speciality also popular with Armenians who call it garmeroug. This is a hot, piquant dip which makes an excellent appetizer and this Aleppan saying vouches for its 'acridity': 'Only the devil can cope with our Aysha's muhammarah, and he only comes this way once a lifetime—thank the Lord.'
It is a must on any mezzeh table, but is also a good accompaniment for all kinds of cooked meats and kebabs.
2 tablespoons red chilli pepper
150 ml/¼ pint olive oil
25 g/1 oz stale, dry breadcrumbs
1 tablespoon pomegranate juice (if available) or 2 tablespoons lemon juice
175 g/6 oz walnuts, ground
1 teaspoon ground cumin
1 teaspoon allspice
salt to taste
Garnish chopped parsley
Moisten the chilli pepper with 2 tablespoons of water in a bowl. Add all the remaining ingredients and mix thoroughly until well blended. Spoon into a small bowl and garnish with a little chopped parsley. Chill until ready to serve.
hab-el-jose walnut balls
Easy to make, attractive and very tasty they are one of the specialities of Antioch of Crusader fame. Once it was the capital of a Christian kingdom, a seat of learning and one of the great cities of antiquity. Today it is a shabby little town in southern Turkey—such is the destiny of men and mice!
Antioch was built in the year 300 BC by Seleucus the Great who:
... consulted with the priest Amphion for the choice of a site for the new capital that he wished to give to his immense Empire...an eagle flew rapidly to the summit of Mount Silpius and let fall there the viscera of victims slain on the altar. The divine will was made manifest... and yet no city in the world has suffered more or oftener from the cataclysms of nature, famine, floods, earthquakes and plague; or from the lust and cruelty of man who has betrayed, burnt, pillaged and utterly destroyed it.
(Syria As It Is)
150 g/5 oz walnuts, ground
50 g/2 oz breadcrumbs
½ teaspoon cumin
tahina cream
½ teaspoon cayenne pepper
salt to taste
olive oil to grease fingers
50 g/2 oz sesame seeds
Garnish pinch paprika
In a bowl mix together the walnuts, breadcrumbs and cumin. Add sufficient tahina to form a soft paste. Add the cayenne pepper and salt to taste. Now grease your fingers with the oil and break the paste into small pieces. Shape into walnut-sized balls. Pour the sesame seeds on to a plate and roll each walnut ball in them. Arrange on a large plate and stick a cocktail stick in each. Sprinkle with the paprika.
avocado im egozim avocado with walnuts
Here is a new recipe from a new land—Israel—and yet see how Middle Eastern it really is! Avocados were first introduced by Jewish settlers early this century in Palestine and over the years many exciting dishes have been created—or so the creators thought. In reality they are nearly all variations on age-old recipes, all tried and tested by time.
A refreshing and delightful first course. Serve it with pita bread.
1 large, ripe avocado, stoned and peeled
2 tablespoons lemon juice
1 small onion, finely chopped
3 pickled cucumbers, thinly sliced
1 stick celery, chopped
75 g/3 oz walnuts, quartered
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon ground cumin
Garnish
½ small red pepper, thinly sliced
25 g/1 oz black olives, stoned
Cube the avocado flesh, place in a bowl and sprinkle with the lemon juice. Add the onion, cucumbers, celery and walnuts. Season with the salt, pepper and cumin, toss and chill for 30 minutes. Before serving garnish with the strips of red pepper and the black olives.
sumpoogi aghtsan aubergine and pepper salad
An Armenian speciality, this is a cold, spicy salad of cooked aubergines, with onion, peppers, garlic and spices.
There are several variations of this salad not only in the Middle East, but in Greece, Bulgaria and even in Romania where aubergines are not all that popular. This particular recipe is from Tarsus—the birthplace of St. Paul.
For some years Hoca Nasrettin was the Cadi (judge) of a large agricultural region in Anatolia and one day he was called upon to settle a quarrel between two neighbouring farmers who had fought in the court of the Mosque. One had prayed for rain to help grow his aubergines, the other wished Allah to keep the weather fine for the threshing. Hoca Nasrettin listened to one first and said 'But my dear man, you were perfectly right.' Then he listened to the other and replied 'But my dear man, you were perfectly right.' His wife heard him pronounce judgement, flushed red with anger and shouted 'How can a just Cadi agree with both parties to a suit? Are you crazy?' The Hoca smiled and said 'You also, dear wife, are perfectly right.'
4 medium aubergines
1 green pepper, seeded and thinly sliced
1 small onion, thinly sliced
4 tomatoes, thinly sliced
2 cloves garlic, crushed
1 teaspoon salt
½ teaspoon chilli pepper
2 teaspoons ground cumin
4–6 tablespoons olive oil
juice 2 lemons
2 tablespoons parsley, chopped
Pierce each aubergine 2 or 3 times with a sharp knife. Place them whole in a hot oven and bake until they are soft when poked with a finger (about 20–30 minutes). Remove from the oven and leave until cool enough to handle. Peel off the skin, scraping off and retaining any flesh which may come away with it. Chop the flesh and put in a large bowl. Add the green pepper, onion, tomatoes and garlic. In a small bowl mix the spices with the olive oil and lemon juice. Pour the dressing over the vegetables and mix thoroughly. Stir in the parsley, taste and adjust seasoning if necessary.
Serves 4–6 people.
hunkar beyendi aubergine with cheese
A classic aubergine dip from Turkey. This version is with grated cheese. There is one from Istanbul which excludes the cheese, but has very finely chopped green pepper, parsley and onion. Serve on its own as a mezzeh or as an accompaniment to kebab and lamb dishes. 'The king likes it' is the literal meaning of this dish.
2 aubergines
1 tablespoon lemon juice
25 g/1 oz butter
2 level tablespoons flour
6 tablespoons hot milk
3 tablespoons grated cheese, Cheddar, feta or mozzarella
1 teaspoon salt
½ teaspoon chilli pepper
Make 2 or 3 slits in each aubergine and cook them over charcoal, under the grill or in a hot oven until the skins are black and the flesh soft when poked with a finger. Remove from the heat and, when cool enough to handle, peel off the skins reserving any flesh that gets stripped off with the skin. Put the flesh in a saucepan and mash with a fork. Add the lemon juice and simmer over a low heat for five minutes, stirring frequently. Meanwhile, in another saucepan melt the butter, add the flour and cook until it turns golden brown, stirring constantly. Stir this into the aubergines and then gradually stir in the milk until the mixture is creamy. Add the cheese, salt and chilli pepper and cook for 2 more minutes until the mixture forms a thick purée.
mutabbal aubergine dip with tahina
A Syrian-Lebanese classic. The aubergine flesh is grilled then finely chopped and mixed with tahina and spices. It is served with pita bread or, more traditionally, khubuz-saj. Of all aubergine hors d'oeuvres this is the one most regularly served in hotels and restaurants throughout the Arab world.
I have also included two other aubergine dips. One, salat chatzilim, is from Israel and substitutes mayonnaise for the tahina. The other is Armenian khentzorov sumpoog and combines aubergine with apple. This combination of fruit and vegetable is typical of Caucasian cooking.
3 large aubergines
3 cloves garlic, crushed
1 teaspoon salt
50–75 ml/2–3 fl oz tahina paste
juice 2 lemons
1 teaspoon chilli pepper
1 teaspoon ground cumin
1 tablespoon olive oil
Garnish
2 tablespoons parsley, chopped
a few black olives
Make 2 or 3 slits in each aubergine then cook over charcoal, under a hot grill or in a hot oven until the skins are black and the flesh feels soft when poked with a finger. When cool enough to handle peel off the skin, scraping off and reserving any flesh which comes away with it. Put the flesh into a large bowl and mash with a fork. Add the garlic and salt and continue to mash or pound the mixture until it is reduced to a pulp. Add the tahina, lemon juice and chilli pepper and stir thoroughly. Spoon the mixture on to a large plate, smooth it over and sprinkle with the cumin. Pour the olive oil over the top and garnish with the parsley and black olives.
salat chatzilim
2 aubergines
50 ml/2 fl oz mayonnaise
2 hard-boiled eggs, finely chopped
1 tablespoon parsley, finely chopped
2 tablespoons onion, finely chopped
1 large clove garlic, crushed
2 tablespoons olive oil
2 tablespoons lemon juice
1 teaspoon salt
½ teaspoon black pepper
Garnish
1 green pepper, thinly sliced
1 lemon, cut into wedges
3 tomatoes, quartered
2 spring onions, sliced
Make 2 or 3 slits in each aubergine and cook over charcoal, under a hot grill or in a hot oven until the skins are black and flesh soft when poked with a finger. When cool enough to handle peel off the skin, scraping and reserving any flesh which comes away with it. Chop the flesh and then mash until smooth. Place the aubergine purée in a salad bowl, add the remaining ingredients except the garnish and mix thoroughly. Chill the mixture for 2–3 hours.
To serve arrange the salad over the centre of a large plate. Form a pattern over it with the strips of green pepper and arrange the lemon wedges, tomato pieces and spring onions decoratively around the edge of the plate.
khentzorov sumpoog
450 g/1 lb aubergines
3 tablespoons vegetable oil
2 medium onions
1 tablespoon lemon juice
1 teaspoon sugar
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon dillweed
2 eating apples
1 tablespoon sumac powder (optional)
Make 2 or 3 slits in each aubergine and cook over charcoal, under a hot grill or in a hot oven until the skins are black and the flesh soft when poked with a finger. When cool enough to handle peel off the skin, scraping off and reserving any flesh which comes away with it. Chop the flesh. Heat 2 tablespoons of the oil in a large frying pan, add the aubergine flesh and sauté for 2–3 minutes, stirring constantly.
Remove the flesh with a slotted spoon and drain on kitchen paper, then place in a large bowl.
Meanwhile, slice 1 onion and finely chop the other and set both aside. Add the remaining oil to the frying pan, add the sliced onion and sauté until soft and golden. Add the fried onion to the aubergine flesh together with the lemon juice, sugar, salt, black pepper and dillweed. Peel, core and finely chop the apples and add to the bowl together with the chopped onion. Mix thoroughly, sprinkle with the sumac powder, and chill before serving.
imam bayildi
aubergines stuffed with peppers, tomatoes and onions
You might as well expect tears from the dead as a decent meal from an Imam.
Turkish proverb.
One of the great classics of the 'Ottoman' cuisine—I have quoted the name Ottoman as I know there exists a great deal of controversy about the origin of this dish. The Greeks vehemently claim it to be theirs while any self-respecting Kurd or Armenian will be offended if it is called anything but theirs. There are very few dishes in the Middle East that have stirred up as much controversy as the dish that made the Imam faint. Why did this poor man collapse on to his patterned divan? Some say that he lost consciousness because his wife had used too much expensive olive oil in its preparation, others, more cynical, innuendoed to the effect that the Imam ate so much of the dish (free, naturally) that he just made it home before passing out on the now famed patterned couch.
There was a rumour circulating in Akflenir (where Hoca Nasrettin lived) that he was once invited to the house of the local Papaz (Christian priest) and that the latter's wife prepared this magnificent aubergine dish stuffed with peppers, onions, tomatoes and spices all cooked in the finest grade of olive oil. There was apparently nothing else but this dish—since it was the Christian Lent and no meat, poultry or fish could be consumed. Well, the Hoca ate and ate and ate and went home and complained that Allah was unfair since he gave opportunity for the Gavours (Christians) to be able to prepare such magnificent dishes, due to their religious laws.
'Hoca effendi, do you want to change your religion for a mere dish?' asked his wife. 'Allah forbid,' said the Hoca, 'but the mere thought of that dish would make many a faint heart succumb to the temptation. However, wife, I have cajoled from the Papaz the recipe. Here it is. Make some for supper.'
Here is the selfsame recipe, with not quite so much olive oil!
4 medium aubergines, washed and dried, leave the stalks on
6 tablespoons olive oil
2 onions, thinly sliced
2 green peppers, seeded and thinly sliced
2 fat cloves garlic, coarsely chopped
2 ripe tomatoes, sliced
3 tablespoons tomato purée
2 teaspoons salt
½ teaspoon cayenne pepper
1 teaspoon allspice
2 tablespoons parsley, chopped
12 tablespoons cooking oil
450 ml/¾ pint water
Garnish parsley, chopped
Make a slit about 5 cm/2 in long down each aubergine. Salt the insides and leave for 15 minutes.
Meanwhile, heat the olive oil in a large saucepan. Add the onions, green peppers and garlic and fry gently until the onion is soft but not brown. Add the sliced tomatoes, tomato purée, salt, cayenne pepper and allspice and cook for 5 more minutes, stirring occasionally. Stir in the chopped parsley, remove from the heat and set aside.
Rinse out the aubergines under cold running water and then pat dry.
Heat the cooking oil in a frying pan, add the aubergines and fry gently, turning several times, until the flesh begins to soften. Take care not to spoil the shape. Remove the aubergines from the pan with a slotted spoon and place, side by side, in an ovenproof dish, slits uppermost. Carefully prise open the slits and spoon some of the onion mixture into each aubergine. Add the water to any remaining onion mixture, stir and pour over the aubergines. Bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 1 hour.
Remove and set aside to cool. Transfer to a serving dish and chill until ready to serve. Garnish with some chopped parsley.
hummus-bi-tahini chickpeas with tahina
A classic of the Syrian-Lebanese cuisine which is also very popular in Israel, and Jordan. This purée of chickpeas, tahina, garlic and cumin is served in all the hotels and restaurants. There are several slight variations. The recipe below is the one my mother brought over with her from Syria.
I suggest you reserve a tablespoon of the unpuréed chickpeas and when the dish is to be served use them to decorate the surface. A must on the mezzeh table, serve with pita, lavash or kubuz saj.
450 g/1 lb chickpeas, soaked overnight in cold water
3 cloves garlic, peeled
300 ml/½ pint tahina
1 teaspoon chilli pepper
3 teaspoons salt
2 teaspoons ground cumin
juice 2 lemons
Garnish a little red pepper, cumin, olive oil, lemon juice and chopped parsley
Rinse the chickpeas under cold running water and place in a large saucepan ¾ filled with cold water. Bring to the boil then lower the heat and simmer until the chickpeas are tender. Remove any scum which appears on the surface and add more water if necessary. Drain the chickpeas into a sieve and wash thoroughly under cold running water. Retain a few of the chickpeas to use as a garnish. Using a liquidizer reduce the rest to a thick paste or purée. You will need to add a little water to facilitate the blending, but take care not to add too much or the purée will become too thin. While liquidizing the chickpeas add the cloves of garlic—this will ensure they are properly ground and distributed.
Empty the purée into a large bowl, add the tahina, chilli pepper, salt, cumin and lemon juice and mix very thoroughly. Taste and adjust seasoning to your own liking.
To serve—use either individual bowls or a large serving dish. Smooth the hummus with the back of a soup spoon from the centre out so that there is a slight hollow in the centre. Decorate in a star pattern with alternating dribbles of red pepper and cumin. Pour a little olive oil and lemon juice into the centre and then garnish with a little chopped parsley and the whole chickpeas. Serves 8–10 people.
nvig chickpeas with spinach
An Armenian dish served during the forty days of Lent when traditionally no meat, poultry or fish dishes were permitted.
Makes an excellent appetizer or a side dish with roasts and kebabs or an excellent light lunch or supper dish with pickles, salads, breads and yoghurt.
100 g/4 oz chickpeas, soaked overnight in cold water. You can use a 425 g/15 oz tin of cooked chickpeas
450 g/1 lb fresh spinach, washed thoroughly and chopped. If you use frozen spinach buy leaf spinach and chop it. The ready chopped variety is too fine
300 ml/½ pint water
4 tablespoons tomato purée
50 g/2 oz butter
1 teaspoon salt
1 teaspoon sugar
½ teaspoon black pepper
1 tablespoon cumin
If you are using tinned chickpeas drain them and follow the recipe from the next paragraph. Wash the chickpeas and place in a saucepan half filled with water. Bring to the boil then lower heat and simmer for 45 minutes to 1 hour or until the chickpeas are tender. Remove any scum and add more water if necessary. Strain and leave until cool enough to handle. Remove the skins from the chickpeas. The easiest way to do this is to hold a pea between thumb and forefinger and squeeze the pea from the skin.
Put the spinach and chickpeas into a saucepan, add the water, tomato purée, butter, salt, sugar, black pepper and cumin. Stir the mixture, bring to the boil then simmer for 30–40 minutes or until the spinach is tender and the water has evaporated.
Chill before serving.
falafel chickpea rissoles
One of Egypt's classic dishes inherited from the days of the Pharaohs. Falafel is the equivalent of fish and chips in Britain or hamburgers in the New World and, like the latter, it has spread beyond its native land and is today equally popular in Syria, Lebanon and Israel where it is sold in small take-aways as a sandwich with chopped salad, pickles and a little tahina sauce all enclosed inside a hot pita bread.
The recipe below is an Egyptian one using chickpeas only, however there are slight variations. In Lebanon dried broad beans are popular and are often used in 50/50 proportions with chickpeas. The breadcrumbs are eliminated. Falafel is better known as tameya amongst the Copts of Egypt (who are the direct descendants of the Pharaohs). They prepare this dish during Lent and distribute it to non-Coptic friends as an act of penance.
Sometimes the chickpeas are substituted altogether with ful nabed—dried white broad beans. If you would like to try tameya with broad beans buy them ready-skinned and soak for twenty-four hours (see Glossary for further instructions) and then use the following recipe.
A dry falafel 'ready-mix' is now sold in many Middle Eastern shops. There are many brands, but I have found that one from Alexandria called 'St George' makes the most successful rissoles. However, it does need 'spicing up' a little and I suggest you add ½ teaspoon cumin, 1 tablespoon finely chopped parsley, 1 clove garlic crushed and 1 teaspoon coriander with the water when you mix it.
450 g/1 lb chickpeas soaked overnight in cold water and then cooked in 75 ml/3 fl oz water
1 egg, lightly beaten
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon turmeric
2 tablespoons coriander leaves or parsley, finely chopped
½ teaspoon ground coriander
½ teaspoon cayenne pepper
1 clove garlic, crushed
1 tablespoon tahina paste or olive oil
50 g/2 oz fresh white breadcrumbs
50 g/2 oz flour
sufficient oil for deep frying
Pass the chickpeas twice through a mincer and place in a large bowl. Add all the remaining ingredients except the flour and frying oil. Knead the ingredients until the mixture is soft but firm. Form into 2.5 cm/1 in balls and then flatten slightly between the palms of your hands. Coat with the flour.
Heat the oil and when hot add the rissoles, a few at a time, and fry for about 3 minutes or until they are evenly browned. Remove with a slotted spoon, drain on kitchen paper and serve hot with pita bread and salad.
Serves 8–10 people.
tahinov tzaghgagaghamp
Cauliflower with tahina
A speciality from Antioch (Turkey). This dish appears in both the Syrian and Turkish cuisines. It is simple and appetizing and makes a clever use of cauliflower and tahina. Cauliflowers are widely used in the Middle East. They are often fried in oil or, as in Egypt, boiled, drained, dipped in beaten eggs and breadcrumbs and then fried in olive oil (see section on Salads).
1 head of cauliflower, about 700 g/1½ lbs
150 ml/¼ pint tahina paste
2 cloves garlic, crushed
150 ml/¼ pint cold water
1 teaspoon salt
2 tablespoons parsley, finely chopped
juice of 1 lemon
Break the cauliflower into florets and rinse.
Half fill a large saucepan with lightly salted water and bring to the boil. Add the cauliflower and simmer until just tender.
Meanwhile, pour the tahina into a bowl. Add the garlic, lemon juice and half the water. Using a fork mix thoroughly adding more water, a little at a time, until the mixture has the consistency of thick mayonnaise. Taste and adjust the seasoning if necessary.
Drain the cauliflower, pat dry and chop into smaller pieces. Place in a serving dish, pour the tahina mixture over and mix thoroughly. Sprinkle with the parsley and serve hot or cold.
tahiniyeh garlic and tahina dip
This dip is also known as tarator-bi-tahina. It is usually served with hot bread, pickles and olives.
150 ml/¼ pint tahina paste
juice 2 lemons
300 ml/½ pint milk
2 cloves garlic, crushed
1 tablespoon parsley, finely chopped
1 teaspoon salt
½ teaspoon chilli pepper
50 g/2 oz white breadcrumbs
Garnish 1 tablespoon parsley, finely chopped, ½ teaspoon cumin
Pour the tahina into a bowl and stir in the lemon juice. The mixture will become very thick. Slowly add the milk, stirring until you have a mixture of a thick creamy consistency. Add the garlic, parsley, salt and chilli pepper. Taste and adjust the seasoning if necessary. Add the breadcrumbs and mix thoroughly. Pour the mixture into a serving bowl and sprinkle with the parsley and cumin.
This dip will keep for several days in a refrigerator if covered. If you find it thickens then stir in a little more milk.
kadoo pish gaza Courgette dip
A simple recipe from Iran which is traditionally served at breakfast time, or as a starter, with thin, flat bread.
3 medium courgettes
1 tablespoon vinegar
2 teaspoons salt
3 tablespoons vegetable oil
1 onion, grated
225 g/8 oz tomatoes, blanched, skinned and chopped
juice ½ lemon
½ teaspoon black pepper
½ teaspoon paprika
Garnish 2 tablespoons parsley or fresh tarragon leaves, finely chopped
Cut 0.6 cm/¼ in from the head and tail of each courgette and then cut into 0.6 cm/¼ in rounds. Place the slices into a bowl of cold water, add the vinegar and 1 teaspoon of the salt and set aside for 30 minutes.
Meanwhile, heat the oil in a saucepan and fry the onion for a few minutes until soft and golden brown. Drain the courgettes and add to the onion. Add the tomatoes, remaining salt, lemon juice, pepper and paprika and about 225 ml/8 fl oz of water. Cover and simmer for about 30 minutes or until the courgettes are tender. Reduce this mixture to a purée either through using a liquidizer or a potato masher. Pour the dip into a shallow bowl and sprinkle with the garnish.
el–Ful egyptian brown beans
What do you expect from the children of El-Ful?
Arab insult about Egyptians.
One of the national dishes of Egypt, but also popular in adjacent territories. It is often eaten for breakfast—and I well remember my childhood when for a halfpenny I used to get a bowl of ful and a crust of flat bread at the 'Ful shop' near my school. Naturally one had to queue, sometimes up to fifteen minutes, then gobble the food and depart quickly.
If a host is to honour a guest he orders a bowl of el-ful from the nearest shop and to refuse to eat would be tantamount to an insult.
Adored and worshipped by the Egyptians (rich and poor) el-full is brown broad beans cooked slowly in a special pot that tapers to a narrow neck (idra) which helps to stop over-evaporation by permitting the steam to condense on the sloping sides and drop back into the pot.
You can buy tinned Egyptian and Cypriot el-ful—also known as ful-medames from most continental and Middle Eastern stores. Or you can buy the dry beans and follow the simple recipe below.
The Egyptians and Sudanese like to add hard-boiled eggs (hamine) to the dish to give it substance. While the Alexandrians like to smother their ful in a sauce of tomatoes flavoured with garlic (see section on sauces). Always serve el-ful with pita bread.
700 g/1 ½ lbs Egyptian broad beans, soaked overnight and then drained
3 cloves garlic, crushed
2 tablespoons olive oil
juice 2 lemons
1 teaspoon salt
½ teaspoon black pepper
4 hard-boiled eggs, shelled
2 tablespoons parsley, finely chopped
Put the beans into an ovenproof casserole and cover with water. Bring to the boil, then place in an oven preheated to 120°C, 250°F, gas mark ½ and bake for 4–7 hours, depending on the quality of the beans. At the end of the cooking time the beans should be soft but not broken up. Drain the cooking liquid from the beans and discard it. Stir the garlic, olive oil, lemon juice, salt and pepper into the beans. Spoon the mixture into 4 soup bowls, place a hard-boiled egg in the centre of each one. Sprinkle the parsley thickly over the top and serve immediately.
engouyzov lupia green beans in a walnut sauce
The Caucasians have a great penchant for walnuts. The famed cerkez tavugu and the various satsivi sauces are all made with the large and flavoursome walnuts of the region.
This is an Armenian recipe, but it is equally popular with the Abkhazians and particularly with the Georgians—famed for their nimble folk dancers and that tyrant of a leader, Joseph Stalin.
The dish can be served hot or cold, but I prefer the latter.
450 g/1 lb french beans, fresh or frozen, trimmed and cut into 5 cm/2 in pieces
75 g/3 oz walnuts
2 cloves garlic, chopped
1 small onion, finely chopped
3 tablespoons chopped coriander
leaves, if available
1 teaspoon ground coriander
2 tablespoons olive oil
2 tablespoons wine vinegar
1 tablespoon lemon juice
2 teaspoons paprika
1–2 teaspoons salt to taste
a little chicken stock
1 tablespoon parsley, finely chopped
Garnish pinch cayenne pepper
Half fill a large saucepan with lightly salted water and bring to the boil. Add the beans and cook briskly for 8–10 minutes by which time the beans should be cooked but still crisp. Drain and leave to cool.
In a blender or mortar crush the walnuts and garlic to a paste. Empty the paste into a large bowl and add the onion, coriander, oil, vinegar, lemon juice, paprika and salt and mix well. The mixture needs to have a thick, creamy consistency and so if you think it is too dry then stir in a little chicken stock. Taste and adjust seasoning if necessary. Add the beans and stir until they are coated with the sauce. Lightly mix in the parsley and then pile into a serving bowl. Sprinkle with a little cayenne pepper.
hulba fenugreek dip
A speciality of Yemen, this is a hot, spicy dip made from fenugreek, chillies, onion, garlic, lentils and rice. Sometimes lamb or chicken meat is finely chopped and added to the dip to make it more substantial. If you prefer you can eliminate the rice and make up the quantities with lentils, or vice versa.
Please note this is a hot dish—no exaggeration—particularly if you can find the hot chillies that a Yemeni would appreciate.
4 tablespoons ground fenugreek
225 ml/8 fl oz water
4 hot chillies
1 teaspoon salt
1 large tomato, peeled and finely chopped
2 spring onions, finely chopped, including heads
2 cloves garlic, crushed
½ teaspoon black pepper
¼ teaspoon ground cardamom
¼ teaspoon turmeric
75 g/3 oz boiled lentils
75 g/3 oz boiled rice
1 teaspoon coriander leaves, chopped
2 tablespoons ghee (clarified butter)
about 150 ml/¼ pint stock
Garnish ½ teaspoon powdered saffron
Place the fenugreek in a bowl, add the water and leave to soak for 5–6 hours. Carefully pour off the water then beat the fenugreek with a fork until frothy.
Cut the chillies in half, remove and discard seeds and stalks then chop the flesh very finely. Add the chillies to the fenugreek together with the salt, tomato, spring onions, garlic, black pepper, cardamom and turmeric and mix well. Now add the cooked lentils and rice and mix thoroughly with a wooden spoon. Transfer the mixture to a saucepan and stir in the coriander leaves, ghee and stock. Cook over a low heat, stirring occasionally, until the mixture is thick. Add a little more water if necessary. Pour the hulba into a large bowl, decorate with ½ teaspoon saffron and serve with flat bread as a mezzeh dip.
fasoulya piyazi turkish bean salad
There are several variations of this bean salad. The recipe below makes use of green peppers, onion and olives, but in Syria the olives are omitted and a finely chopped tomato is added. In western Turkey two hard-boiled eggs (each cut into eight lengthways) are added with the tomato.
You can serve this dish as an appetizer, as a meal on its own with warm bread, or as a side dish with kebabs.
225 g/8 oz haricot beans, soaked in cold water overnight
1 clove garlic, crushed
2 tablespoons olive oil
1 small onion, thinly sliced
1 small green pepper, thinly sliced
salt and pepper
1 tablespoon parsley, chopped
25 g/1 oz black olives, stoned and halved
juice 1 lemon
Drain the beans and put them into a large saucepan. Cover with water, add ½ teaspoon salt and bring to the boil. Lower the heat and simmer until the beans are tender. Add more water if necessary. Strain the beans and place in a bowl. Add the garlic, olive oil, onion, green pepper and salt and pepper to taste. Stir well and set aside to cool. Taste and adjust the seasoning if necessary. Sprinkle with the parsley and olives and squeeze the lemon juice over the top.
besarah broad bean dip
This recipe is from Egypt and is similar to other vegetable dips of the region. It makes use of dried broad beans (ful nabed) which are available from most Middle Eastern shops. However, normally these beans still have their skins on. To remove these soak the beans in cold water for forty-eight hours, changing the water two or three times. Remove the skins.
Egyptians often add one to two teaspoons of melokhia to this dish to give it a green tint, but Libyans do not do this. Also in Egypt this dish is always served with a garnish of an onion sauce called taleyeh (see below).
Serve hot in individual bowls with bread of your choice.
350 g/12 oz dried broad beans soaked and skinned
900 ml/1½ pints water
1½ teaspoons salt
½ teaspoon black pepper
2 teaspoons dried mint
1 teaspoon melokhia (optional)
Garnish
Olive oil, chopped onion, spring onions, radishes and lemon wedges
Taleyeh
1 large onion
4–5 tablespoons olive oil
1 clove garlic, finely chopped
First prepare the taleyeh by halving the onion lengthways and then slicing thinly to give the pieces a semi-circular shape.
Heat the oil in a small saucepan, add the onion and fry until golden. Add the chopped garlic and fry for a further 1–2 minutes. Remove from the heat and set aside.
Place the soaked and skinned beans in a saucepan, cover with the water, bring to the boil then lower the heat and simmer for 1½ hours by which time the beans should be soft. Add a little more water if the beans are likely to boil dry. Blend the beans in a liquidizer or pass through a sieve and return the purée to the saucepan. Add the salt, pepper, mint and melokhia if available and cook over a low heat until the mixture is thick and bubbling. Transfer to bowls and garnish each with a little of the taleyeh.
Serve the other garnishes separately so that people can help themselves according to their taste.
tabouleh burghul and vegetable salad
Made with cracked wheat tabouleh is a mixture of burghul and vegetables, and has an 'earthy' flavour. Popular in Syria and Jordan it is perhaps at its best in Lebanon where it almost reaches a sublime stage when prepared by a Maronite housewife whose skills in mixing the ingredients of this dish cannot really be surpassed. She piles the tabouleh on to a large plate like a pyramid and decorates it with tomatoes, black olives, small pickling-type cucumbers, sprigs of parsley, strips of red and green pepper and pomegranate seeds. Always part of the mezzeh table, tabouleh has several variations. The recipe below is a standard one from Lebanon.
75 g/3 oz fine burghul
1 cucumber, peeled and finely chopped
4 tomatoes, finely chopped
1 green pepper, seeded and finely chopped
½ onion, finely chopped
4 tablespoons parsley, finely chopped
2 tablespoons dried mint or fresh mint, finely chopped
1 teaspoon salt
juice 2–3 lemons
4 tablespoons olive oil
To serve 1 lettuce, preferably Cos, washed
Rinse the burghul in a large bowl several times until the water you pour away is clean. Squeeze out any excess water. Put the chopped vegetables, parsley and mint into the bowl and mix thoroughly with the burghul. Stir in the salt, lemon juice and olive oil. Mix well together, leave for 15 minutes and then taste and adjust seasoning if necessary. Arrange lettuce leaves around the edge of a plate and pile the salad into the centre.
The ideal way to eat this is to make a parcel of tabouleh by folding a little of it up in a lettuce leaf or a pita bread.
each armenian burghul salad
If tabouleh is the pride of the Maronite Christian Lebanese, each is the pride of the Cilician Armenians. This is a speciality from the region of Gavour Daglari (Christian Mountains) once the stronghold of crusading nobles and their marauding armies.
Each is an appetizer made with burghul and vegetables incorporating pomegranate juice (the Armenian touch) and cumin. It can be eaten with pita bread or, like tabouleh, with lettuce leaves. I regard this as superior to tabouleh.
150 g/5 oz burghul
6 tablespoons parsley, finely chopped
1 onion, finely chopped
2 spring onions, finely chopped
2 tomatoes, finely chopped
1 teaspoon salt
½ teaspoon cumin
½ teaspoon cayenne pepper
1 tablespoon tomato purée
1½ tablespoons pomegranate juice or about 3 tablespoons lemon juice
4 tablespoons olive oil
Garnish Cos lettuce leaves
Rinse the burghul in a large bowl and squeeze out any excess water. Add all the remaining ingredients and mix well. Set aside for 15 minutes then taste and adjust seasoning if necessary. Arrange the lettuce leaves around the edge of a serving plate and pile the salad into the centre.
To eat make a parcel of the each by folding a little of it up in a lettuce leaf or in pita bread.
bazerghen burghul and walnut salad
Some years back an Assyrian friend of mine was incensed when I described bazerghen as a Syrian dish. 'It's ours,' he protested indignantly. 'It's been enough to lose one's lands, but it is the ultimate in "massacres" to steal and monopolize even our food. Bazerghen is ours!'
So, the recipe below is most definitely of ancient origin—Assyrian in fact. Although it is also very popular with the Kurds of Syria and Iraq and the Syrian peasants in general.
Serve it as an appetizer with pita, lavash or any other flat bread and lettuce leaves and home-made pickles.
100 g/4 oz fine burghul
4 tablespoons olive oil
½ onion, finely chopped
½ teaspoon oregano
½ teaspoon coriander
½ teaspoon allspice
1 teaspoon salt
½ teaspoon black pepper
3 tablespoons parsley, finely chopped
2 tablespoons walnuts, finely chopped
½ teaspoon cumin
½ teaspoon cayenne pepper
50 g/2 oz tomato purée diluted in 2–3 tablespoons water
Garnish
black olives
lettuce leaves
3–4 radishes, thinly sliced
Place the burghul in a bowl and rinse several times with cold water until the water you pour off is clean. Soak the burghul in water for 5 minutes and then drain and squeeze out excess water.
Meanwhile, heat the oil in a small saucepan and fry the onion for a few minutes until soft and transparent.
Place the burghul in a bowl and add the onion and oil. Add all the remaining ingredients and mix very thoroughly. Place in the refrigerator to chill. Before serving spoon the salad on to a bed of lettuce leaves and garnish with the olives and radishes.
houm miss raw meat with burghul
There are two versions of this classic appetizer made with raw meat and cracked wheat. One is popular amongst Armenians, the other amongst the Lebanese. In houm miss the meat/burghul proportions are about equal and the mixture is divided into small patties and eaten dipped into a hot, cooked meat and nuts mixture. The Lebanese version kibbeh naya has twice as much meat as burghul and is usually eaten with olive oil and bread or lettuce leaves.
Like many Middle Eastern dishes the origin of this one is lost in the mists of time. On the one hand it has certain resemblances to the famed steak tartare which Marco Polo first encountered on his travels in China. On the other hand it resembles a great many Assyrian raw fish and meat dishes still popular in scattered regions of the Middle East. What, however, is certain is that it cannot be of Arab origin since raw meat or fish is forbidden to pious Muslims as the meat still retains blood. It is strictly a Levantine speciality and particularly popular with the Christian Lebanese and Syrians who are of Greek, Armenian and Crusader origin. It is best made just before you are ready to eat it as it will dry fairly quickly.
100 g/4 oz fine burghul
100 g/4 oz raw very lean lamb, minced twice
½ tablespoon onion, very finely chopped
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon chilli pepper
Accompaniment
50 g/2 oz minced lamb
½ tablespoon onion, chopped
3–4 walnuts, chopped
½ teaspoon salt
pinch black pepper
½ tablespoon parsley, finely chopped
Garnish pine kernels or split almonds
½ tablespoon parsley, finely chopped
First prepare the accompaniment by placing the minced lamb and onion in a small pan and frying over a low heat, stirring frequently for 10 minutes. Add the walnuts, salt and pepper and continue to fry until the meat is cooked. Stir in the parsley, remove from the heat and set aside.
Wash the burghul in a bowl until the water you pour off is clear. Empty the burghul on to a baking sheet and knead for 5 minutes, dampening your hands with warm water occasionally. Add the lean lamb, onion, salt and peppers and knead into the burghul for 5–10 minutes, dampening your hands occasionally.
Get your serving dish ready and fill a small bowl with water. Wet your hands, take a piece of the burghul mixture about the size of a walnut and squeeze it in the palm of your hand to make a boat shape. Use up all the mixture in this way and arrange the patties around the edge of the plate. Stick one pine kernel or split almond into the top of each piece. Heat the cooked meat through and empty it into the middle of the plate and sprinkle with the parsley. To eat dip one end of each patty into the hot meat.
kibbeh naya
50 g/2 oz fine burghul
1 level teaspoon salt
½ teaspoon black pepper
100 g/4 oz very lean lamb, minced twice
1 tablespoon onion, very finely chopped
1 tablespoon olive oil
pinch chilli pepper
1 teaspoon pine kernels
Accompaniment
bowlful of Cos lettuce leaves
1 onion, quartered
Wash the burghul in a large bowl until the water you pour off is clean. Empty the burghul on to a large plate or baking sheet, season with the salt and black pepper and knead for 5 minutes, wetting your hands if the mixture sticks to them. Add the minced lamb and chopped onion and knead for a further 5–10 minutes until the mixture is smooth. Keep wetting your hands if it makes the kneading easier. Spread the kibbeh mixture over a large plate and press until smooth forming a slight depression in the middle. Pour the olive oil into the centre and sprinkle a little chilli pepper all over the surface. Sprinkle the pine kernels over the top and serve immediately accompanied by the lettuce and onion.
banri aghtsan cheese and tomato salad with spices
This is a family recipe which is superb in its simplicity. It is ideal as an appetizer with warm bread.
It is traditionally made with feta cheese. You can make your own for real authenticity. However this salad is equally successful made with white Stilton or even cottage cheese.
225 g/8 oz feta cheese, or white Stilton or cottage cheese
2 tablespoons onion, finely chopped
2 large tomatoes, finely chopped
2 tablespoons olive oil
1 tablespoon lemon juice
1 tablespoon ground allspice
1 tablespoon dried thyme or 2 tablespoons fresh thyme
½ tablespoon black pepper
Garnish
1 tablespoon parsley, finely chopped
lettuce leaves
Rinse the feta under cold water and cut the cheese in 0.6 cm/¼ in cubes and place in a large bowl. Add all the remaining ingredients and mix well with a fork. At this stage taste the salad. Feta cheese is often quite salty and it may not be necessary to add any salt, but if you are using another kind of white cheese you may need it.
Decorate a serving dish with a few lettuce leaves and pile the cheese salad into the centre. Sprinkle with the parsley and serve.
mortadella sausages with eggs cooked in wine
The only thing this sausage has in common with the Genoese version is its name—a relic of bygone ages of profitable commerce when shiploads of Venetian and Genoese merchants traversed the Middle East, sealing commercial deals with Christian, Muslim, enemy and friend, and carrying to Europe the gold, silver, silks and spices of the east.
Very little has remained in the Middle East of these glorious days: a few ruined castles scattered along the Mediterranean coastline; some words in the native languages; and a few dishes. Mortadella is one of them. However, through the ages it has changed and localized—in this case 'Armenianized', for mortadella (sometimes called gololig) is considered a 'classic' Armenian appetizer. It consists of boiled eggs wrapped in spiced lean meat, shaped into sausages and cooked in wine.
This recipe makes three fair-sized sausages each of which serves four people, and which will keep in the refrigerator for a week. If you wish to freeze the sausages then simply prepare the recipe without the hard-boiled egg filling.
1.35 kg/3 lb lean lamb
50 g/2 oz fresh breadcrumbs
2 eggs, beaten
3 teaspoons salt
2 teaspoons black pepper
2 teaspoons cinnamon
2 teaspoons allspice
6 cloves garlic, crushed
6 boiled eggs, shelled
300 ml/½ pint red wine
Remove all the fat and gristle from the meat and mince twice.
Put the meat on to a large, clean surface. Make a well in the centre, add the breadcrumbs, beaten eggs, spices and garlic, and mix until well blended. Knead the mixture until the texture is smooth. Divide the mixture into 3 balls and shape each one into a rectangle 15 × 7.5 cm/6 × 3 in. Hollow out the middle slightly and build up the sides of each rectangle. Place 2 hard-boiled eggs, end to end, down the middle of each rectangle. Ease the meat up over the eggs and seal so that the eggs are completely enclosed. Roll them gently between your palms until they have a smooth sausage shape and then tie each one up like a parcel with string.
Empty the wine into a saucepan, then add sufficient water to cover the parcels. Bring to the boil then simmer for ¾–1 hour. Remove sausages and leave to cool, then serve cut into thin slices arranged on a bed of lettuce leaves and garnished with olives, radishes and spring onions.
yershig string sausage with garlic and spices
An Armenian speciality which is also popular in Turkey where it is called soujuk.
Yershig is a spicy sausage that is fried or grilled and eaten on its own as an appetizer or part of a mezzeh table. It is also often fried with mushrooms, tomatoes and eggs to form an omelette.
In the old days the villagers made yershig towards the end of the summer and kept them for the harsh winter days ahead when it was difficult to get out and when fresh milk, meat and vegetables were impossible to find. It is laborious to make, but you can prepare enough to last for months as it will freeze well.
There are all kinds of superstitions based on the qualities of this dish as well as several folk songs.
2.75 kg/6 lb minced lamb
4 teaspoons crushed garlic
4–5 teaspoons salt
2 teaspoons each black and chilli pepper
2 teaspoons ground cumin
4 teaspoons ground allspice
4 teaspoons nutmeg
Put all the ingredients into a very large bowl and knead until everything is well blended. Taste and adjust seasoning to your own liking. Cover the mixture and leave in the refrigerator overnight.
Bags
If you are going to store the mixture in bags then
a | Cut 12 oblongs, 10×15 cm/4×6 in, out of a light cotton material e.g. muslin. Make the oblongs up into 6 bags by sewing up 3 sides of each bag.
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b | Divide the mixture into 6 portions and spoon one into each bag.
c | Pass a rolling pin over each bag so that the mixture is distributed evenly and is about 2.5 cm/1 in thick.
d | Leave about 5 cm/2 in empty at the open end, fold this over and fasten with a few stitches in a thick thread.
e | Leave about 30 cm/1 ft of thread so that the bags can be hung up to dry.
f | Hang the bags up in a cool place, e.g. larder for 2–3 weeks.
g | Store in a refrigerator if you are going to use them fairly quickly or else store in a deep freeze.
Skins
This is the ideal way to make yershig. The prepared meat is put into intestines and divided into the required size. Intestines can be purchased from most butchers if you order them in advance and they are usually already cleaned and prepared. Put them into water for about three hours before you need them. This softens them and makes them easier to handle. To put the mixture in the intestines you need a large plastic funnel with a nozzle width of about 2.5 cm/1 in.
a | Fit one end of an intestine over the nozzle and gently work the whole of the intestine on to the nozzle.
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b | Force the meat down through the funnel into the intestine. As the intestine fills up it will slip off the nozzle.
c | When the whole intestine is full run it lightly through one hand to distribute the meat evenly. Set aside.
d | Continue in this way until you have used up all the meat mixture.
e | To make into sausages—fold one intestine in half and then tie or knot at certain intervals to give you sausages of the length you require.
f | Hang up to dry and store in same way as yershig in bags.
To serve
Peel off the cotton material or intestine and cut the yershig into 2.5 cm/1 in pieces. Heat some butter, ghee or cooking oil in a frying pan, add the pieces of yershig and fry for 5–10 minutes, turning occasionally. Remove with a slotted spoon.
Serve sprinkled with a little chopped parsley and garnish with lemon wedges.
aboukht/basturma dried beef in fenugreek
A classic of the Armenian cuisine, this is a speciality of the regions of Van and Kaissery. Aboukht is salted beef, dried under the sun and cured with a hot fenugreek paste (chaimen). Unfortunately good aboukht is not easily available outside an Armenian home although pale imitations can sometimes be purchased from a few Middle Eastern or Greek stores. Therefore I suggest you try making your own. It makes an excellent appetizer when sliced thinly and eaten with bread, olives and pickles. It also makes a tasty omelette, aboukhty tzoo.
1 rib of beef with bones removed
Paste
3 tablespoons ground fenugreek
3 tablespoons paprika
½ tablespoon salt
½ tablespoon ground black pepper
½ tablespoon cumin
½ tablespoon allspice
¼ tablespoon cayenne pepper
3 cloves garlic, crushed
Cut the meat into slices 2.5 cm/1 in thick, 7.5 cm/3 in wide and any length you like. Put a thick piece of string through one end of each piece of meat and tie into loops. Immerse the pieces of meat in a large pan of brine and leave for a week at least.
Remove the meat, wash under cold running water and then leave in a pan of cold water for about 1 hour and then hang by the loops over the pan to drain for 1–2 hours. Lay a piece of muslin on a flat surface and arrange the pieces of meat on it side by side. Put another piece of muslin over the top. Place a board across the meat and lay a heavy weight on top to squeeze the juices from the meat. Change the cloths when they become saturated. Most of the moisture will have been extracted at the end of 3 days.
Hang up the meat again in a cool dry place where there is a good movement of air for 1–2 weeks until the meat is quite dry.
Put all the ingredients for the paste in a bowl and, adding a little water at a time, mix to a thick, smooth paste. Put the paste in a large bowl, add the pieces of meat, turn to coat with the paste and leave for 1–2 weeks. When removed each piece of meat should have a thick coating of paste. Hang for another week. It is then ready to use.
It will keep for several months in a cool dry place if you wrap it well. It will also freeze, but you must wrap it several times or else its strong flavour will permeate other food. If it becomes too dry then soak it in the chaimen (paste) until it softens enough to slice without breaking.
To serve slice the aboukht very thinly. It is generally eaten as part of a larger hors d'oeuvre selection, or in bread as a sandwich.
herissah chicken with wheat
On the feast of St Mary people go to church, slaughter a lamb or chicken, cook herissah and feed 'the multitude'—relations, friends and passers-by as well as strangers who are on a pilgrimage.
Although now part of the Christian Church's festive rituals, herissah predates Christianity by millenia and was connected with the sacrificial slaughtering festivities of Zoroaster and perhaps even older to the times of Abraham and Isaac.
It is traditionally made with whole-grain gorghod which is soaked overnight in water and then drained. Skinless grain is virtually unobtainable outside Armenia and Turkey therefore I suggest you substitute pearl barley or a coarse burghul. Lamb or chicken meat is used. The recipe below is for chicken. There are, of course, several variations of herissah in other Middle Eastern countries. In Turkey it is known as keskek and in Syria and Lebanon as herisa, but they all descend from the same origin—the high mountains of Anatolia where Urartians sacrificed animals to their Gods and were wise enough not to throw them to the vultures, but cooked them mixed with wheat and spices.
175 g/6 oz whole grain, skinless wheat which you might be able to find in a Middle Eastern store or 175 g/6 oz pearl barley or 350 g/12 oz coarse burghul
900–1350 g/2–3 lb chicken, cut
into serving pieces
salt and black pepper to paste
1 teaspoon cinnamon
50 g/2 oz butter
1 teaspoon paprika
2 teaspoons ground cumin
Put the chicken pieces in a large saucepan, cover with water, season with salt and pepper and simmer until the flesh is tender. Remove the chicken from the stock and leave to cool, remove the flesh and discard the bones. Shred the meat as finely as possible and return to the stock.
Add the wheat, pearl barley or burghul and cinnamon to the stock and simmer for about 30 minutes or until the grain is tender. While it is cooking beat constantly with a wooden spoon until the mixture has the consistency of smooth porridge.
Melt the butter in a small pan and stir in the paprika.
Spoon the herissah into individual soup bowls. Before serving spoon a tablespoon of the butter mixture into the centre of each bowl. Sprinkle a little of the cumin powder over each portion and serve immediately. Serve with lavash or pita bread and some home-made pickles, fresh tarragon leaves and spring onions.
çerkez tavugu circassian chicken
A great Georgian speciality—almost everything with a walnut sauce hails from the Caucasus including most of the so-called Turkish tarator sauces. This dish was probably introduced into Turkey by the beautiful Circassian girls who were bought for the whiteness of their skin, their fair hair and nimble fingers and who, for centuries, adorned the harems of the sultans and the rich.
This dish can be served cold as an appetizer (as here) or warm as a main course.
1.35 kg/3 lb chicken, cut into 4 pieces
1 onion, coarsely, chopped
1 carrot, peeled and cut into rings
175 g/6 oz walnuts
2 thick slices of white bread
1 teaspoon salt
½ teaspoon black pepper
Garnish
1 tablespoon olive oil
1 teaspoon paprika
1 tablespoon parsley and/or tarragon, finely chopped
Place the chicken pieces in a large saucepan, cover with water, add the onion, carrot and a pinch of salt and bring to the boil. Lower the heat and simmer for about 45 minutes or until the chicken is tender. Transfer the chicken pieces to a plate and when cool enough to handle strip the flesh from the bones.
Return the bones to the stock and boil until the stock is reduced.
Cut the chicken flesh into strips about 5 cm/2 in long and 1.2 cm/½ in thick.
Grind the walnuts and bread in an electric blender and empty into a saucepan. Slowly stir in some of the stock until you have a smooth paste. Season with the salt and pepper. If you find the sauce is too thin then simmer over a low heat until it thickens.
Put the paprika and oil in a small bowl and set aside.
Pile the shredded chicken into the centre of a serving plate and spoon the sauce over the top. Set aside to cool. Just before serving dribble the oil-paprika mixture over the chicken and sprinkle with the parsley and/or tarragon.
arnavut çigeri albanian fried liver
The Ottoman empire comprised many races and nationalities. It included Albanians who played a major part in her success. They were reputed to be honest—a rare distinction in any time—brave and fair. They produced some of the best known soldiers and administrators of the empire thus originating the expression: 'Live in a vilayet (district) where the vali (governor) is an Albanian and you may live to see old age.'
This is one of the best hors d'oeuvres from the time of the Ottomans and was introduced by the mountain people of Albania.
It is simple and tasty.
1 small onion, thinly sliced or 5 spring onions, chopped
450 g/1 lb lamb's liver, you can use calf's liver instead
2 tablespoons flour
2 tablespoons paprika
50 ml/2 fl oz olive oil
1 clove garlic, crushed
Garnish 2 tablespoons parsley, finely chopped
Arrange the onion on a serving plate.
Wash the liver, pat dry and cut into small pieces about 2.5 cm/1 in square, removing skin or tough pieces.
Place the flour in a small bowl, add 1 teaspoon of the paprika and mix. Add the liver pieces to the flour and toss until well coated. Heat the oil in a frying pan, add the liver pieces and fry for 2–3 minutes, turning once or twice. Do not overcook. The pieces of meat should still be pink and juicy inside. Remove with a slotted spoon and arrange on the onion.
Pour off all but about 3 tablespoons of the oil from the pan. Add the remaining paprika and the garlic and cook for 1 minute, stirring all the time. Pour this paprika-oil mixture over the liver and set aside to cool. Garnish with parsley and serve.
lsannat mtabbili lamb's tongue salad
A speciality from Syria, Lebanon and Palestine. It has, in recent years, become very popular in seaside restaurants always accompanied by a glass of arak and bread.
Often a lemon juice and olive oil dressing is poured over the salad, but in Lebanon and Palestine a tahineyeh sauce similar to the tahineyeh dip, but using water instead of milk to mix, is used as dressing. I prefer to sprinkle the salad with 1 teaspoon of cumin and juice of 1 lemon.
6 small lamb's tongues
1 onion, peeled and quartered
1 clove garlic
1 stick celery, washed and cut into 12 cm/½ in slices
1 clove
2 bay leaves
1 teaspoon salt
3–4 peppercorns, crushed
1 tablespoon lemon juice
2 tablespoons olive oil or 3–4 tablespoons of tahina dressing (see recipe for tahineyeh)
Garnish
2 tablespoons parsley, finely chopped
wedges of lemon
½ teaspoon paprika
Wash the tongues under cold running water. Place them in a large saucepan and add sufficient water to cover. Add the onion, garlic, celery, clove, bay leaves, salt and crushed peppercorns. Bring to the boil, lower the heat, cover and simmer for about 1 hour or until the tongues are tender. Leave to cool and then peel each tongue by carefully cutting one edge and removing the skin with your fingers. With a sharp knife cut the tongues into slices about 3 mm/⅛ in thick. Arrange the slices decoratively over a plate and chill for about 1 hour.
Mix the lemon juice and oil in a small bowl and pour over the slices or pour over 3–4 tablespoons of the tahina dressing. Garnish with the parsley, paprika and lemon wedges. Serves 4–6 people.
taramasalata fish roe dip
Taramasalata is a Greek and Cypriot speciality. It is the roe of the grey and red mullets which are found in abundance in the waters of the eastern Mediterranean.
Outside Greece, Turkey and Cyprus fresh tarama roe is not readily available, but it can be purchased in jars from most Middle Eastern shops. Tarama has a strong flavour and this is broken down with the addition of bread. Cypriots sometimes substitute the bread with mashed potato—a custom I dislike for the taste is not that of the traditional tarama. You can use carp or red caviar's roe instead.
Taramasalata is also popular on the Aegean coast of Turkey.
4 slices white bread, trimmed of crusts
100 g/4 oz tarama
4 tablespoons cold water
2 teaspoons onion, finely chopped
juice 1½ lemons
100 ml/4 fl oz olive oil
Garnish
1 tablespoon parsley, finely chopped
radishes, cucumber, celery
black olives
Soak the bread in water and then squeeze dry. Place in a shallow bowl, preferably a wooden one. Gradually add the tarama and crush the eggs in with a mortar. Little by little add the cold water. Add the onion and the lemon juice and keep crushing with a mortar or transfer to a blender. Add the oil, a little at a time, until the mixture is smooth and has a light, pinkish colour. Transfer to a serving dish and refrigerate.
Garnish with the parsley and black olives and serve with radishes, etc. and warm pita bread.
barbunya bilakisi fish plaki
This dish is Greek-Turkish (Byzantine) as most plaki dishes are. In Turkey barbunya (red mullet) is usually used, but since this fish is not always easily available in this country I suggest you use halibut, cod, hake or any other firm-fleshed white fish.
This Turkish recipe incorporates potatoes which the Greeks often omit. The latter often like to add extra flavour with oregano and bay leaves.
4 small pieces white fish, e.g. halibut, cod, hake or mullet
4 medium carrots, peeled and thinly sliced
6 sticks celery, cut into 2.5 cm/1 in long pieces
1 small onion, thinly sliced
1 green pepper, seeded and cut into 8 pieces
2 large potatoes, peeled and cut into 1.2 cm/½ in thick slices
6 tablespoons olive oil
3 cloves garlic, crushed
2 tablespoons tomato purée
1 teaspoon salt
½ teaspoon cayenne pepper
900 ml/1½ pints water
Garnish
1 tablespoon parsley, chopped
lemon wedges
Place all the vegetables in a large colander and wash thoroughly, then drain. Heat the olive oil in a large saucepan, add the vegetables, stir well, cover and then cook for about 30 minutes, shaking the pan occasionally.
Meanwhile, brush the base of an ovenproof dish with olive oil and arrange the washed and dried fish in it. After 30 minutes remove the vegetables from the saucepan with a slotted spoon and arrange over the fish.
Return the saucepan to the fire and heat up any oil that remains and add one more tablespoon of oil. Add the garlic and fry for 1 minute. Now add the tomato purée, salt and pepper, stir well and cook gently for 3 minutes. Stir in the water, raise the heat and bring to the boil. Pour this sauce over the vegetables, place in an oven preheated to 200°C, 400°F, gas mark 6 and cook for about 45 minutes or until the carrots are tender. Remove from the oven and set aside to cool.
Arrange the pieces of fish on a serving dish and spoon the vegetables and sauce over the top. Sprinkle with the parsley and serve with the lemon wedges.
midya litsk stuffed mussels served cold
One of the most sophisticated dishes from Armenia popularized by the nineteenth-century chef-hotelier Tokatlian of Istanbul. The mussels are cleaned and then stuffed with rice, pine kernels, currants and spices.
about 30 mussels
4–6 tablespoons olive oil
2 medium onions, finely chopped
75 g/3 oz rice, washed thoroughly
50 g/2 oz pine kernels or walnuts, coarsely chopped
50 g/2 oz currants
1 tablespoon parsley, chopped
1 heaped teaspoon salt
1 level teaspoon allspice
½ teaspoon chilli pepper
Garnish lemon wedges
Put the mussels to soak in a large saucepan filled with salted water. Discard any mussels with open or broken shells or any that float to the surface.
To prepare the filling, first heat 4–6 tablespoons of olive oil in a saucepan, add the onions and fry until soft. Add the rice and fry for about 3–5 minutes, stirring frequently to prevent sticking. Stir in all the remaining ingredients and cook for a further 5–10 minutes. Taste and adjust seasoning if necessary. Remove from the heat.
Scrub and wash each mussel shell thoroughly. Force open each mussel with a sharp knife. If you find them difficult to open put them in a very thick-bottomed saucepan, cover, put over a low heat and steam for a few minutes—they should then begin to open. Cut off the beard—the fibrous bits that keep the mussel attached to its shell—but do not remove the flesh. Leave in slightly salted water until ready to use.
To fill take one mussel at a time and put a teaspoon of the filling inside—do not pack too tightly as the rice will swell when cooked. If you loosen the joint a little the shell should stay closed, but if not tie up with cotton. Pack tightly into a saucepan, cover with an inverted plate to stop them moving while cooking and pour in enough water to cover. Bring to the boil then lower the heat and simmer for 1–1½ hours. Drain off the water, allow to cool and arrange on a serving dish with lemon wedges.
Serves 4–6 people.
tunig mackerel in olive oil
This dish is better known in Turkey as uskumru pilakisi. The Armenians' favourite fish is ishkan (salmon trout) found only in Lake Sevan. The Turks undoubtedly favour barbunya (red mullet), but both nations agree that mackerel comes a good second. It is salted, pickled (lakerda), used in soups, fried, grilled and prepared in the following way in olive oil and wine. This is a marvellous starter.
4 medium sized mackerel
300 ml/½ pint olive oil
4 onions, thinly sliced
2 carrots, peeled and thinly sliced
2 cloves garlic, halved
½ teaspoon chilli pepper
1 tablespoon tomato purée
1 glass white wine (or extra stock or water)
600 ml/1 pint fish stock or water
salt to taste
Garnish
1 tablespoon parsley, chopped
lemon wedges
Scale and clean the insides of the fish, but do not cut off heads or tails.
Heat half the oil in a large frying pan, add the onions and fry until soft. Add the carrots and garlic and fry for a further 10 minutes. Remove from the heat and stir in the rest of the ingredients. Bring to the boil, lower the heat, cover and simmer for 15 minutes. Arrange the mackerel in the sauce, cover and cook for a further 20 minutes or until the mackerel are tender. Turn off the heat and leave to cool. Place the mackerel on a serving dish, pour the sauce over them and sprinkle with the parsley. Serve with the lemon wedges.
Sardalya tavasi fried sardines and other small fish
All along the Mediterranean coastline small fish restaurants serve sardines or other small fish such as sprats or whitebait. They all occur in abundance in the Black Sea and Sea of Marmara.
The fish are simply cleaned, tossed in flour and fried in oil. All they need then is a few drops of lemon juice squeezed over them and they are ready to eat.
sardalya sarmasi
This is another clever Turkish method of cooking sardines or other small fish. Each one is cleaned, wrapped in a vine leaf, brushed all over with oil and then deep fried in sizzling oil, a few at a time, for about 3 minutes until crisp on all sides.
Serve with lemon juice and pickles.
Both these methods are popular with Turks, Syrians, and Cypriots while fried small fish (blehat samk) are much loved by Egyptians, Libyans and the Lebanese who eat them—heads, tails and all!
nkhaat mtabbli brain salad
A great delicacy throughout the Middle East, brains (lamb or calf) are prepared as salads, as here, in omelettes or are fried as in the following recipe. There are many regional variations. This is a typical one from Palestine-Jordan.
450 g/1 lb lamb or calf brains
2 tablespoons vinegar
3½ teaspoons salt
1 clove garlic
½ teaspoon black pepper
juice 1 lemon
4 tablespoons olive oil
Garnish 1 tablespoon parsley, finely chopped
Place the brains in a bowl, add enough cold water to cover together with the vinegar and 2 teaspoons of salt. Leave for 20–30 minutes and then drain.
Place the brains in a saucepan with enough cold water to cover and 1 teaspoon of salt and bring just to the boil. Lower the heat and simmer for 15 minutes or until the brains are tender. Drain and when cool enough to handle peel off the skin and any veins. Cut the brains into bite-sized pieces and place in a salad bowl.
In a small bowl mash the garlic with the remaining salt and the pepper. Add the lemon juice and oil and mix well. Pour the dressing over the brains covering all the pieces. Do not toss. Sprinkle with the parsley and serve.
erebouni lamb's brain fritters
The origins of most ancient cities are lost in legends and myths and it can only be a miracle which has kept the foundation date of one of the oldest of them all—Erebouni—carved in Urartian script on stone and dated 782 BC. It was built by slaves for the mighty King Argishti I.
This Armenian recipe does not claim such longevity, it is simply named in honour of the fortress-capital which lies a few miles away from the modern capital Erevan.
about 450 g/1 lb lamb's brains
2 tablespoons vinegar
salt
2 egg yolks
1 tablespoon fresh dill, finely chopped or ½ tablespoon dried dillweed
50 g/2 oz grated cheese, e.g. kasseri, kashkaval or Cheddar
oil for frying
Garnish lemon wedges
Place the brains in a large bowl, add enough cold water to cover together with the vinegar and 2 teaspoons of salt and leave to soak for 20–30 minutes. Drain and remove any skin and veins. Place the brains in a saucepan with enough cold water to cover, add a teaspoon of salt and bring just to the boil. Lower the heat and simmer for about 15 minutes or until tender. Drain, place the brains in a bowl and mash with a fork. When the brains are cool, add the egg yolks, dill and cheese and mix to a smooth paste. Taste the mixture and add a little salt if necessary.
Put enough oil in a large frying pan to cover the bottom by about 0.6 cm/¼ in and heat. Take tablespoons of the mixture and place in the oil. The mixture will spread a little so do not put them too close together. When set and golden on the undersides turn and cook the other sides until golden. Remove and drain on kitchen paper. Keep warm while you cook the remaining mixture in the same way.
Serve garnished with the lemon wedges.
aloo-chap spicy potato dip
A recipe from the Abadan region of Iran for those who like hot, spicy food. Adjust the quantity of chilli pepper to suit your particular taste. This dip is usually accompanied with a glass of ice cold vodka. It can be eaten either warm or cold.
450 g/1 lb potatoes, peeled and boiled until tender
100 ml/4 fl oz milk
3 tablespoons onion, finely chopped
2 tablespoons parsley, finely chopped
1 teaspoon salt
1 teaspoon chilli pepper (or more)
½ teaspoon black pepper
Garnish
1 tablespoon parsley, finely chopped
pinch cumin
pinch paprika
Put the cooked potatoes in a large bowl with the milk and mash until completely smooth. Add the remaining ingredients and mix until they are all well blended.
To serve place in a shallow bowl, smooth over the surface and garnish attractively with the parsley, paprika and cumin.
## churba–soups
There are no consommé-type soups in the Middle East for their soups are often eaten as a meal with bread.
The repertoire is rich and varied, ranging from simple vegetable soups to meat and vegetable soups. Pulses—lentils, chickpeas, beans and peas as well as fruits and nuts are incorporated. Also pomegranate seeds, tripe, yoghurt, soured cream, rice, barley, wheat and burghul.
Often dried bread—toasted or fried in a little butter—is added to give more substance. The soups are garnished with hard-boiled eggs or parsley, tarragon or basil and when unexpected guests drop in—as they often do in the East—the hostess immediately adds some more stock so that the soup will go further and honour will be preserved.
To drink 'soup from the same spoon' is the greatest honour a Kurd can bestow a guest—be he friend or stranger and to 'share a bowl of soup' to most Middle Easterners is synonymous with the American Indians 'smoking the pipe of peace'.
vartabedi chorba lentil soup
And Esau said to Jacob, Feed me, I pray thee, with that same red pottage;
for I am faint...
And Jacob said, Sell me this day thy birthright.
And Esau said, Behold, I am at the point to die: and what profit shall this
birthright do to me?
And Jacob said, Swear to me this day; and he sware unto him:
and he sold his birthright unto Jacob.
Then Jacob gave Esau bread and pottage of lentils;...
thus Easu despised his birthright. (Genesis 25, 30–34)
Lentil soup is perhaps, next to the yoghurt-based soups, the most popular soup in the Middle East. Red, green or brown lentils can be used and there are literally hundreds of recipes. To try and do them justice I have included below two of the more famed ones. The first recipe is the simplest. It is called 'the young priest's soup'. This version is with macaroni, but often it comes with rice or with a combination of the two.
175 g/6 oz brown or red lentils, rinsed
1.8 litres/3 pints stock or water
100 g/4 oz macaroni, cut into 2.5 cm/1 in pieces
50 g/2 oz butter or oil
1 onion, finely chopped
salt and chilli pepper to taste
Place the lentils in a large saucepan, add the stock or water and bring to the boil. Lower the heat and simmer until the lentils are nearly tender. You may need to add more stock or water, especially if using brown lentils which take longer to cook. Add the pieces of macaroni and continue simmering. You may need to add a little more water again. When both the lentils and macaroni are cooked heat the butter or oil in a small saucepan, add the onion and fry until golden. Pour into the soup, season with salt and chilli pepper to taste and serve.
Serves 4–6 people.
shreet ads majroosh lentil soup with cumin
Popular throughout the Arab lands, Israel and Turkey. Arabs often break bread into small pieces and add to the soup to give it body. You can make croutons to serve in it by cutting a thick slice of bread into 1.2 cm/½ in cubes and frying in hot oil.
450 g/1 lb ads majroosh—red lentils, rinsed
1.8 litres/3 pints stock or water
1 onion, quartered
1 tomato, quartered
1 stick celery with leaves, chopped
1 clove garlic, coarsely chopped
50 g/2 oz butter
1 tablespoon onion, chopped
2 teaspoons ground cumin
1 teaspoon salt
¼ teaspoon black pepper
Garnish lemon wedges, croutons
Place the stock or water in a large saucepan and bring to the boil. Add the lentils, onion, tomato, celery and garlic and stir. Reduce the heat and simmer for 30–45 minutes or until the lentils are tender. The length of time will depend on the quality of the lentils.
Meanwhile in a small pan melt half the butter and fry the chopped onion until golden. Remove from the heat.
Purée the soup in a liquidizer or by rubbing through a sieve with the back of a wooden spoon and discarding any remaining bits of vegetables. Return the soup to the saucepan and cook for a further 5 minutes, stirring all the time. Add the cumin, salt and pepper. Just before serving stir in the remaining butter. If you like a light soup add a little more water, otherwise simmer for a few more minutes.
Serve in individual bowls topped with a few freshly fried croutons and sprinkled with the fried onion. Serve the lemon wedges separately.
bezelye çorbasi split pea soup
A Turkish recipe from Anatolia which is old, tested and tasty. It is traditionally served with a bowl of natural yoghurt. A simpler version, which is an Arab favourite, omits the spinach and replaces the carrot with a finely chopped stalk of celery.
225 g/8 oz split peas, soaked overnight in cold water
25 g/1 oz butter
1 onion, thinly sliced
1 carrot, chopped
2 bay leaves
1 teaspoon salt
½ teaspoon black pepper
225 g/8 oz spinach, finely chopped
a little milk
Garnish
2 tablespoons fresh mint, finely chopped or 2 teaspoons dried mint
1 tablespoon paprika
Drain the split peas. Half fill a large saucepan with water or stock, add the split peas, bring to the boil.
Melt the butter in a small saucepan, add the onion and fry until soft.
Add the onion and butter to the split peas together with the carrot, bay leaves, salt and pepper. Simmer until the split peas are almost tender. Add the chopped spinach, mix well and continue to simmer. Add a little more water or stock if necessary. When all the ingredients are tender remove the bay leaves and purée the soup in a liquidizer. Return the soup to the saucepan and thin to the required consistency with a little milk. Stir in half the paprika and bring to the boil. Taste and adjust the seasoning if necessary. Serve in individual bowls sprinkled with the remaining paprika and the mint.
ab-gusht-e-bademjan aubergine and lentil soup
Bademjaneh Bam avaf nadoreh. Nothing in heaven or earth can stop the growth of Bam's aubergines. (Bam being a town in Iran famed for its aubergines.)
There are several variations of this Iranian aubergine soup. Some incorporate chunks of meat cut into 5 cm/2 in pieces. However, the recipe below is simple fare which is served with hot bread and a bowl of natural yoghurt. Turmeric gives the soup a light golden hue.
The Armenian soup sumpoogi abour incorporates 250 g/9 oz chickpeas and ½ teaspoon each of the following: thyme, mint, marjoram and cumin, but omits the lentils, turmeric, cinnamon and tomato purée. The chickpeas are boiled until tender and then half are added to the soup and the remaining are puréed and added to the soup just before serving to thicken it. It is garnished with chopped tarragon or parsley and a sprinkling of cumin.
2 small aubergines, peeled and sliced
50 g/2 oz butter
1 onion, thinly sliced
100 g/4 oz brown lentils, rinsed
1 teaspoon salt
½ teaspoon black pepper
1 tablespoon tomato purée
1 teaspoon turmeric
½ teaspoon cinnamon
1.8 litres/3 pints water
Arrange the aubergine slices on a large plate, sprinkle with salt and set aside for 30 minutes.
Meanwhile melt half the butter in a large saucepan and sauté the onion until soft and golden. Add the remaining butter.
Rinse the aubergine slices, dry with kitchen paper and add to the saucepan. Fry for a few minutes until lightly browned all over. Add all the remaining ingredients, stir well and bring to the boil. Cover the pan, lower the heat and simmer for 30–45 minutes or until the lentils are tender. Add a little more water if the soup becomes too thick.
Serves 4–6 people.
spanak çorbasi spinach soup
A recipe from eastern Turkey from the region of Kars famous for its kebabs and brigands.
25 g/1 oz butter
1 onion, thinly sliced
2 sticks rhubarb, sliced
1 stick celery, thinly sliced
1.8 litres/3 pints water
1½ teaspoons salt
450g/1 lb fresh spinach or 225 g/8 oz frozen leaf spinach
2 eggs, beaten
150 ml/¼ pint smetana (soured cream)
juice 1 lemon
½ teaspoon black pepper
Melt the butter in a saucepan, add the onion and fry until soft. Add the rhubarb and celery and a small amount of water. Cover and cook over a low heat until tender. Add the water and the salt and bring to a quick boil. If using fresh spinach wash very thoroughly and squeeze out excess water. If using frozen spinach then thaw and squeeze out the water. Chop the spinach and add it to the soup. If the spinach is fresh cook for 15 minutes, if frozen cook for 5 minutes.
Make a sauce of the beaten eggs, smetana and lemon juice with a little of the hot stock. Remove the soup from the heat and stir in the sauce until well blended. Add the black pepper and a little more salt if it is needed. Serve immediately.
Serves 4–6 people.
havuç çorbasi carrot soup
A wonderful soup from Anatolia with a creamy texture and delicate flavour.
50 g/2 oz butter
450 g/1 lb carrots, peeled and chopped into small pieces
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon sugar
½ teaspoon dillweed
1.2 litres/2 pints water
scant 1 tablespoon flour
75 ml/3 fl oz milk
2 egg yolks
Garnish 2 tablespoons walnuts, finely chopped
Melt half the butter in a saucepan, add the carrots and sauté for a few minutes until well coated with butter. Add about 300 ml/½ pint water, together with the salt, pepper, sugar and dillweed. Bring to the boil and simmer until the carrots are very soft. Drain the carrots retaining a little of the liquid to help purée them if necessary. Place the carrots in a liquidizer and blend to a smooth purée.
Bring 1.2 litres/2 pints of water to the boil in a large saucepan, add the carrot purée, stir well and continue simmering.
Meanwhile melt the remaining butter in a small saucepan, remove from the heat and stir in the flour. Slowly add the milk and stir until smooth. Return to the heat and stir constantly until the mixture thickens. Remove from the heat, add the egg yolks and stir until well blended and smooth. Add some of the hot soup, a little at a time, and stir constantly until it is thin and well blended. Return this mixture to the soup, stir and heat through, but do not boil or it will curdle. Serve in individual bowls sprinkled with the chopped walnuts.
tarkana burghul and yoghurt soup
One of the very few ancient recipes that has come down to us virtually intact. The nomadic tribes of Asia made tarkana from pellet-shaped pieces of dough, made from flour, salt, eggs and water, which were left to dry in the sun. They were cooked in salted water and eaten either as thick soup, or in stews and casseroles made with meat, poultry or wild game.
Tarkana is made in this way in Hungary, Romania and the Balkans. In the Middle East however—Turkey, Armenia and Syria only—burghul and yoghurt are used to make it. This is wholesome, earthy soup which is a favourite in winter. You can buy tarkana ready made from Greek and Armenian shops. The Syrians and Lebanese call it kishk. However, as always where possible, I suggest you make your own. It is quite simple to make.
Tarkana
12.5 g/½ oz yeast
250 ml/8 fl oz warm water
225 g/8 oz plain flour, sifted
½ teaspoon salt
225 g/8 oz large burghul
300 ml/½ pint yoghurt
Dissolve the yeast in 3 tablespoons of the water in a large bowl. Mix in the rest of the water. Add the flour, salt, burghul and yoghurt and mix with a wooden spoon until well blended. Cover with a clean tea towel and leave to rest overnight.
The following day form the dough into walnut-sized pieces, then flatten between your palms to 0.3 cm/⅛ in thickness. Arrange on a baking sheet.
Traditionally these tarkanas are dried under the sun, but a warm oven will do just as well. Place in a warm oven, about 160°C, 325°F, gas mark 3, and when dried on one side turn over with a spatula and dry thoroughly.
Break into smaller pieces and store in airtight jars. This quantity will make enough tarkana for 6 separate meals.
Tarkana soup
about 1.2 litres/2 pints water or stock
50 g/2 oz tarkana
25 g/1 oz butter
1 large onion, finely chopped
1½ teaspoons dried mint
½ teaspoon salt
¼ teaspoon paprika
150 ml/¼ pint yoghurt
Bring the water or stock to the boil in a large saucepan. Add the tarkana, lower the heat and simmer for about 30 minutes or until the tarkana is soft.
Meanwhile, melt the butter in a small saucepan, add the onion and fry until soft and golden. Add the mint, salt and paprika and mix well. Remove and keep warm.
When the tarkana is done add the mint-mixture, mix well and simmer for a further 5 minutes. Remove from the heat and just before serving stir in the yoghurt little by little. Serve immediately.
churba kavkaski vegetable and rice soup
A soup from the Caucasus which is rich and wholesome with a creamy-pink colour. It is similar to many southern Russian soups, but it has a definite Middle Eastern flavour.
25 g/1 oz butter or ghee
1 onion, finely chopped
175 g/6 oz white cabbage, shredded
1 beetroot, peeled and diced
2 sticks celery, diced
2 carrots, peeled and diced
3 tomatoes, blanched, peeled and sliced
1.8 litres/3 pints water
1½ teaspoons salt
½ teaspoon black pepper
175 g/6 oz long grain rice, washed thoroughly under cold water
2 egg yolks
200 ml/1/3 pint milk
Melt the butter or ghee in a large saucepan, add the onion and fry until golden. Lower the heat, add the remaining vegetables and fry gently for 5 minutes. Add the water, salt and pepper and bring to the boil. Lower the heat and simmer for 15 minutes. Add the rice, cover and continue to simmer for about 20 minutes or until the rice and carrots are tender. Turn off the heat, remove the lid and leave to cool.
Beat the egg yolks and milk together in a small bowl and very slowly stir into the soup. Do make sure that the soup is really cool or else it will curdle. When ready to serve re-heat, but do not allow to boil. Taste, adjust seasoning if necessary and serve immediately. Serves 4–6 people.
gololig rice, tomatoes and meatballs soup
A filling Armenian soup of rice, tomatoes and meatballs flavoured with tarragon.
There are many such soups in the Caucasus, some having chunks of meat or pieces of chicken or even fishballs.
225 g/8 oz lamb, minced twice if possible as this will make it easier to knead
1 large onion, finely chopped
1 tablespoon parsley, finely chopped
1 egg, beaten
salt and pepper to taste
1.8 litres/3 pints stock or water
2 tablespoons tomato purée
1 heaped teaspoon dried tarragon
50 g/2 oz long grain rice, washed
12.5 g/½ oz butter
Put the meat into a large mixing bowl, add half the onion, the parsley, egg and ½ a teaspoon each of the salt and black pepper. Knead the mixture until well blended and smooth. It will help if you dampen your hands with cold water from time to time. Break off small pieces of the mixture and roll between your palms to make small balls about 2 cm/¾ in in diameter.
Bring the stock or water to the boil in a large saucepan and season it with tomato purée, tarragon and salt and pepper to taste. Gently add the meatballs and simmer for 20 minutes. Stir in the rice and cook for a further 10–15 minutes or until the rice is tender.
Just before serving melt the butter in a small saucepan, add the remaining onion and fry until golden. Stir into the soup and serve immediately. Serves 4–6 people.
melokhia melokhia leaf soup
A classic Egyptian soup, pre-dating the pyramids, that was the food of the peasants fellehine—but today is the pride and joy of the middle class Egyptians.
Melokhia (see Glossary) is usually eaten twice daily and each family has its 'authentic' recipe.
It is rather difficult to purchase fresh melokhia outside Egypt, Libya and Cyprus, but dried versions can be bought from some Middle Eastern stores.
Stock
a chicken or a rabbit or a knuckle of beef or veal
1 onion, quartered
2 tomatoes, blanched, peeled and quartered
1 clove garlic
salt and pepper to taste
Soup
50 g/2 oz dried melokhia leaves or 450 g/1 lb fresh leaves if you can find them
3 cloves garlic
2 tablespoons butter or oil
1 tablespoon cayenne pepper
1 tablespoon ground coriander
Salt and pepper to taste
First make the stock by placing all the ingredients in a large saucepan, covering well with water and simmering for 2–3 hours. Add more water from time to time when necessary. You will need about 1 litre/2–3 pints of stock for the soup. Remove any scum which may form and adjust the seasoning at the end of the cooking time. Strain the stock into a large saucepan.
Crush the dried melokhia leaves. If they are not brittle enough then dry them out in a warm oven for a few minutes. Place them in a bowl and moisten with a little hot water until they double in bulk. Add them to the stock and simmer for 20–30 minutes.
Make the garlic sauce—taklia—by crushing the garlic with a little salt and frying in the butter or oil. When it turns brown add the cayenne pepper and coriander and stir to a smooth paste. Add this to the soup, cover and simmer for a further 2 minutes. It is important to stir occasionally and not to overcook as the leaves will otherwise sink to the bottom.
Finally check the seasoning and adjust accordingly. Serves 4–6 people.
churba-bi-banadora tomato and onion soup
A typical Arab soup—this one is from Syria-Lebanon. There are many variations. Sometimes the soup is prepared without meat, but traditionally chunks of lamb, or goat, are added to give substance. The Syrian peasants also add pieces of dry bread to make a really hearty meal.
40 g/1½ oz butter or ghee
2 onions, thinly sliced
450 g/1 lb lamb, cut into 3.5–5 cm/1½–2 in cubes
2 cloves garlic, finely chopped
900 g/2 lbs tomatoes, blanched and chopped
1 tablespoon tomato purée
1.8 litres/3 pints water
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon dillweed
Garnish 2 tablespoons parsley, finely chopped
Melt the butter or ghee in a large, deep saucepan. Add the onions and cook for about 5 minutes or until soft, but do not overcook. Add the meat and garlic, stir well, cover the pan and leave to simmer for 15 minutes. Add the chopped tomatoes, stir well and continue cooking. Dilute the tomato purée in a little water and add to the pan. Stir well then add the remaining water, the salt, pepper and dillweed and bring to the boil. Lower the heat, cover the pan and simmer for about 1 hour or until the meat is very tender. Taste and adjust seasoning if necessary. Serve immediately with a garnish of a little chopped parsley.
Serves 4–6 people.
marak avocado im batzal
Onion and avocado soup
An Israeli soup making use of avocados which the kibbutzim first introduced into the region in the early years of this century. Avocados have become big business, are imported in large quantities and have inspired Israeli chefs to exploit this pear-shaped fruit with its distinctive flavour.
4 tablespoons oil
1 onion, chopped
2 spring onions, chopped
2 cloves garlic, crushed
1.2 litres/2 pints chicken stock
salt to taste
½ teaspoon black pepper
¼ teaspoon nutmeg
1 ripe avocado, peeled and mashed
1 teaspoon lemon juice
grated rind 1 lemon
1 egg yolk
Garnish 1 tablespoon parsley, finely chopped
Heat the oil in a large saucepan, add the onion, spring onions and garlic and sauté for about 5 minutes, stirring frequently, until the onion is golden. Add the stock, salt, pepper and nutmeg and bring to the boil. Lower the heat and simmer for 30 minutes, stirring occasionally. Mash the lemon juice and rind into the avocado and add it to the soup. Mix it in well and simmer for a further 5 minutes.
Beat the egg yolk in a small bowl, add a few tablespoons of the soup and mix well. Add the egg mixture to the soup, stir well and simmer for a further 5 minutes. Serve garnished with the chopped parsley.
mantabour dumpling soup
One day the villagers were gathered under the old walnut tree dining on bread, cheese, fresh vegetables and some fruit. A large cauldron of mantabour adorned the dinner table. Boloz Mugush passed by on his donkey.
'Welcome brother Mugush,' the villagers called. 'Just the man to give an answer to an intriguing problem.'
He got off his donkey and joined the villagers.
'Problems?' he enquired. Just then his eyes caught sight of the cauldron, his nostrils inhaled the rich aroma of the soup. He felt a pang in his stomach. He had not eaten for over twelve hours.
'Life, brother,' said the village elder, 'death and immortality.'
'Life is a bowl of mantabour. Some like it, some don't, some want it cool, others warm, some with tomato, others prefer yoghurt. Life and death can be found in a bowl of wholesome mantabour. As for eternity it is mere extension. When I have finished eating a bowl of this soup, what is there left? Only the memory of a wonderful experience.'
His eyes were now fixed on the bubbling cauldron.
'Eternity, I'll tell you about eternity. The Mongols came in their hordes, burning, raping, destroying all. Where are they now? Gone to eternity. What have they left behind?'
'Nothing.' Someone shouted.
'Wrong,' he interjected, 'Manti; thank God for that.' Saying this he got up and helped himself to a bowl of hot mantabour.
'This then, my friend, is immortality.'
Manti then, is of Mongolian origin (mantu in Korean and Chinese), and variations of this dish exist throughout Europe and Asia under different names; manti in Turkish, dyushbara in Azerbaijanian, mantabour in Armenian, vareniky in Ukrainian and pelmeny in Russian. The finest manti dishes are prepared by the Turkomans of Anatolia and the Armenians.
There are many variations. I have chosen two; one with a tomato sauce and the other with a yoghurt sauce.
Dough
225 g/8 oz plain flour
salt
1 egg
cold water
Filling
225 g/8 oz minced lamb
1 onion, chopped
1 tablespoon parsley, chopped
1 egg
1½ teaspoons salt
1 teaspoon black pepper
Tomato sauce
1.8 litres/3 pints water or stock
50 g/2 oz butter
6 tomatoes, blanched, peeled and chopped
4 cloves garlic, crushed
1 teaspoon basil
2 tablespoons tomato purée
salt and black pepper
Sift the flour and salt into a mixing bowl, break an egg into the centre and begin to knead. Adding a little water at a time, continue kneading until you have a dough which comes away from the sides of the bowl and the fingers easily. Continue kneading for a few minutes and then set aside.
In another bowl mix together the ingredients for the filling.
Flour a large working surface. Divide the dough in two and roll out one ball until it is as thin as possible. With a circular cutter 2.5–3.5 cm/1–1½ in in diameter, cut out as many circles of pastry as possible. Gather up the scraps, roll them out and cut more circles. Repeat with the other ball of dough. Place a small ball of the meat mixture in the centre of each circle. Dampen the edge of the circle and then pinch the edges up to make 4 corners which trap the meat inside but do not hide it completely from view. Continue until you have used up all the ingredients. You have now made the manti.
In a large saucepan bring to the boil about 1.8 litres/3 pints of lightly salted water or stock. Put the manti in gently and simmer for 30 minutes.
Meanwhile, melt the butter in a small saucepan, add the tomatoes and cook for 2–3 minutes. Add the garlic, basil, tomato purée and salt and pepper to taste. Stir this tomato mixture into the soup and simmer for a further 10 minutes. Taste to check the seasoning and add a little more water if the sauce is too thick. Serves 4–6 people.
Yoghurt sauce
1.2 litres/2 pints stock
1 teaspoon salt
225 g/8 oz labna (drained yoghurt)–see recipe p. 376
1 clove garlic, crushed
2–3 teaspoons dried mint
Follow the instructions above up to the third paragraph, but use stock for boiling instead of water. When ready to serve bring the stock to the boil in a large saucepan. Carefully transfer the manti to the stock and simmer for 5 minutes. Remove from the heat, add the labna and garlic and stir gently for about 3 minutes. Do not boil the soup or it will curdle. Rub the mint to a powder between the palms of your hands and stir into the soup. Serve immediately.
eshkaneh onion soup
A classic of the Iranian cuisine, this soup is a speciality of the region of Shiraz. It is an unusual soup with a sweet and sour flavour, traditionally eaten with bread and a bowl of fresh herbs.
If you cannot find limes then use lemons as suggested.
60 g/2½ oz butter
3 onions, thinly sliced
2 tablespoons flour
1.8 litres/3 pints water
1½ teaspoons salt
1 teaspoon black pepper
1 teaspoon turmeric
juice 2 lemons or 1 lime
2 tablespoons sugar
1 tablespoon dried mint
½ teaspoon cinnamon
2 eggs
Melt the butter in a large saucepan, add the sliced onions and fry for several minutes until soft and golden.
Place the flour in a small bowl and dissolve in a few tablespoons of water. Stir this mixture into the fried onions. Now add the water and stir well. Add the salt, pepper and turmeric, stir and simmer for about 30 minutes. Now stir in the lemon or lime juice and the sugar and simmer for a further 10 minutes. Just before removing from the heat stir in the mint and cinnamon. When ready to serve break the eggs into a bowl, beat thoroughly, stir into the soup and serve immediately.
Serves 4–6 people.
ashe-e-joe barley soup
A staple diet of Anatolian villagers, barley soup has many variations. This recipe is from Iranian Azerbaijan and is a rather richer version. Often meat is added; about 350 g/12 oz cut into 5 cm/2 in chunks; as well as kidney beans, lentils and vegetables such as spinach, leeks, fresh coriander, etc. Yoghurt is an essential part of this soup.
100 g/4 oz pearl barley
40 g/1½ oz butter
1 large onion, finely chopped
100 g/4 oz chickpeas, soaked overnight in cold water
1.2 litres/2 pints stock
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon turmeric
100 g/4 oz fresh dillweed or spinach, washed and finely chopped
300 ml/½ pint yoghurt
Garnish
1 tablespoon dried mint sautéed in 1 tablespoon butter
Place the barley in a bowl, cover with water and leave to soak for 1 hour.
Melt the butter in a small saucepan, add the onion and fry until the onion is soft and turning golden.
Drain the chickpeas and pearl barley and place in a large saucepan with the stock and bring to the boil. Remove any scum that appears on the surface. Add the salt, pepper and turmeric, stir well, lower the heat and simmer for 30 minutes. Add the dillweed or spinach and any other chopped vegetables of your choice and simmer for a further 30 minutes or until the chickpeas are tender.
Put the yoghurt in a bowl and, adding a few tablespoons of the hot soup at a time, stir until the yoghurt has been well diluted. Pour the soup into a large tureen and stir in the yoghurt mixture.
Quickly sauté the mint in the butter and then pour over the top of the soup. Serve immediately.
ashe-e-anar pomegranate soup
A locked garden is my sister, my bride
a closed spring, a sealed fountain.
Your branches are a pomegranate orchard
with all precious fruit, henna and roses
saffron and spikenard, cassia, cinnamon
with frankincense trees, myrrh and aloes,
all perfect spices...
Eat, my friend, drink —
lover, be drunk with love.
The Song of Songs
This soup is popular throughout the Caucasus and northern Iran. It has recently been introduced into Israel by immigrants from the USSR.
You can buy pomegranate syrup, dibs ruman, from good Middle Eastern stores or you can prepare your own as suggested in the Glossary section.
The people on the Caspian coastline often add small meatballs, while the Armenians prefer small gololig made of burghul and meat—see section on kibbi.
1 tablespoon butter
1 onion, chopped
4 spring onions, thinly sliced
75 g/3 oz long grain rice, washed thoroughly under cold running water
1½ teaspoons salt
½ teaspoon black pepper
1 teaspoon dried oregano
4 tablespoons parsley, finely chopped
100 g/4 oz fresh spinach washed thoroughly, squeezed dry and chopped
1.8 litres/3 pints water and 2 tablespoons pomegranate syrup or 1.5 litres/2½ pints water and about 300 ml/½ pint pomegranate juice and 2 tablespoons sugar
Garnish
1 tablespoon dried mint
½ teaspoon white pepper
½ teaspoon cinnamon
Melt the butter in a large saucepan, add the onion and fry until soft and turning golden. Add the rice, salt and pepper and fry gently for a few minutes. Stir in the oregano, parsley and spinach. Add the water and pomegranate syrup or juice and bring to the boil. Lower the heat and simmer for 20–25 minutes or until the rice and spinach are tender. Remove the soup from the heat.
Mix the mint, cinnamon and white pepper together and sprinkle over the soup. Taste for seasoning. The soup should have a pungent sweet-sour flavour.
Serves 4–6 people.
avgolemono soupa egg and lemon soup
Popular throughout Turkey and the Arab speaking countries, this is a Greek speciality—hence its Greek name. In Arabic it is called beid bi lemoun and in Turkish terbiyeli çorba.
This soup does not reheat well so make just the amount you need and prepare it just before serving.
Although in Greece fish or meat stocks are used, amongst Turks and Arabs chicken stock is the norm and I prefer the latter.
1.5 litres/2½ pints stock, chicken, meat or fish
50 g/2 oz long grain rice, washed thoroughly under cold running water or 50 g/2 oz any type of small soup noodles
½ teaspoon salt
3 eggs, separated
juice 2 lemons
¼ teaspoon white pepper
Garnish 2 tablespoons parsley or chives, finely chopped
Bring the stock to the boil in a large saucepan. Add the rice or noodles and salt, lower the heat and simmer for 15–20 minutes or until tender.
Meanwhile, in a large bowl beat the egg whites until stiff. Add the egg yolks and continue beating until creamy. Gradually add the lemon juice, beating constantly until thick and frothy. Add a few ladlesful of the hot soup to the eggs and stir vigorously.
Remove the soup from the heat and gradually pour the egg mixture back into the saucepan making sure that you beat constantly or else the soup will curdle. Beat for 1–2 minutes, season with the pepper and serve immediately garnished with the parsley or chives.
NB Once the rice or noodles are cooked it is important that the soup is not brought to the boil again.
churba-ful-sudani peanut soup
An Egyptian soup of African origin. It has a rich earthy flavour and is popular amongst the felehine.
450 g/1 lb fresh, shelled peanuts
600 ml/1 pint milk
600 ml/1 pint stock
salt and pepper to taste
Garnish 4 tablespoons of double cream or 2 tablespoons melted ghee
Spread the peanuts over a baking tray and place in an oven preheated to 190°C, 375°F, gas mark 5. Roast for about 15 minutes or until the skins can be easily removed. The length of time will depend on the freshness of the peanuts. Leave to cool and then rub off the skins by squeezing the nuts between thumb and forefinger. Grind the nuts in a blender or pass through a mincer. Pour the powdered nuts into a large saucepan and add the milk, little by little, stirring constantly. Stir in the stock and bring to the boil. Season to taste with the salt and pepper. Simmer for about 10 minutes, stirring frequently. Add a little more water or stock if the soup is too thick for your taste.
Serve in individual bowls topped with a little cream or melted ghee.
madzounabour yoghurt soup
'No one will call his madzoun black.'—No one will accept his mistakes.
The Middle Eastern repertoire is particularly rich in yoghurt-based soups; and I have been reliably informed that there are more such soups than the entire range offered by Messrs Heinz! Consequently it has been difficult to choose between rivalling recipes. I have categorized my choice thus:
a) Plain yoghurt soups
b) Soups with rice, noodles, barley, etc.
c) Yoghurt with meat
Always use fresh natural yoghurt and make sure it is stabilized. (See Glossary, for the preparation of yoghurt and on how to stabilize it.)
600 ml/1 pint yoghurt
1 egg
600 ml/1 pint water
1 teaspoon salt
¾ teaspoon black pepper
50 g/2 oz butter
1 small onion, finely chopped
2 teaspoons mint, dried and crushed
Garnish
2 thick slices bread cut into 1.2 cm/½ in cubes
cooking oil
Put the yoghurt into a saucepan. Add the egg to the yoghurt and mix well with a wooden spoon. Put on a low heat and stir continuously until the yoghurt is just begining to boil. Add the water, season with the salt and pepper and return to the low heat.
Meanwhile, in a small saucepan melt the butter, add the onion and mint and cook until the onion is soft, but not brown. Pour into the soup, bring to the boil and simmer very gently for a few minutes.
Heat a little cooking oil in a small saucepan and when it is very hot add the cubes of bread and fry until golden and crisp. Remove from the fat and put into a bowl.
To serve put the soup into individual bowls and add the croutons at the last moment.
tutmaj yoghurt soup with noodles
Any small pasta is suitable although, ideally, the noodles should be made at home as they still are in most villages in Turkey and Iran.
Traditionally chortan—dried powdered yoghurt—was used to make most yoghurt soups, but fresh natural yoghurt, preferably home-made, is perfectly suitable.
900 ml/1½ pints yoghurt
2 egg yolks
450 ml/¾ pint water
100–125 g/4–5 oz noodles
1 teaspoon salt
½ teaspoon pepper
50 g/2 oz butter
1 onion, finely chopped
2 tablespoons dried mint, crushed
Place the yoghurt and egg yolks in a large saucepan and mix well with a wooden spoon. Bring slowly to the boil over a low heat, stirring constantly. Stir in the water, noodles, salt and pepper and bring to the boil. Lower the heat and simmer for 8–10 minutes until the pasta is just cooked.
Meanwhile, melt the butter in a small saucepan, add the onion and mint, and fry until the onions are soft. Pour the onion mixture into the soup and stir well. Serve immediately.
NB Another soup—gololigi-tutmaj—has the same ingredients as above with the addition of 225 g/8 oz minced meat seasoned with 1 teaspoon salt, ½ teaspoon black pepper and ½ teaspoon paprika, shaped into small marble-sized balls and fried in butter, then added to the soup to give it substance.
dovga yoghurt soup with meatballs
An Azerbaijanian soup from the Caucasus which is rich and full of flavour. Wherever possible use fresh herbs.
225 g/8 oz minced meat (lamb or beef)
1 onion, finely chopped
salt and black pepper to taste
900 ml/1½ pints yoghurt
1 tablespoon flour
1.2 litres/2 pints stock or water
25 g/1 oz long grain rice, washed thoroughly under cold, running water
50 g/2 oz chickpeas, soaked overnight in cold water, cooked in water until just tender and drained
100 g/4 oz spinach, chopped
3 tablespoons parsley, finely chopped
2 spring onions, finely chopped
3 tablespoons fresh dill, chopped or 1 tablespoon dried dillweed
In a large bowl mix the meat, onion, salt and pepper. Knead until well blended and smooth. Make small walnut-sized balls and put to one side.
Pour the yoghurt into a large saucepan.
Put the flour into a small bowl and blend to a smooth paste with a little of the stock or water. Stir this into the yoghurt and then stir in the remaining stock or water. Season with a little salt and pepper. Add the meatballs and rice and, on a low heat, simmer for 10–12 minutes, stirring gently and very frequently. Add the cooked chickpeas and the spinach and simmer for a further 10–12 minutes until the rice is tender and the meat is cooked. Add the parsley, onion and dill and cook for a further 5 minutes. Serve immediately.
NB An Armenian soup—gololig tzavarov—incorporates 50 g/2 oz coarse burghul and omits the rice and chickpeas.
Serves 4–6 people.
ashe-e-reshteh vegetable and noodle soup
A thick wholesome soup from Kurdistan that is a meal in itself. You can omit the spinach and leek and incorporate potatoes cut into large cubes, or you can increase the quantity of noodles to 75 or 100 g/3 or 4 oz.
Here is a story about Nasrudin Hodja and his deep knowledge of Kurdish. Hearing that a man wanted to learn the Kurdish language, Nasrudin offered to teach him. Nasrudin's own knowledge of Kurdish was limited to a few words.
'We shall start with the word for "hot soup"' said the Mulla. 'In Kurdish, this is Aash.'
'I don't understand, Mulla. How would you say "cold soup"?
'You never say "cold soup". The Kurds like their soup hot.'
25 g/1 oz butter
1 onion, sliced
1 tablespoon chickpeas, soaked overnight in cold water
1 tablespoon dried haricot beans, soaked overnight in cold water
1 tablespoon brown lentils
¼ teaspoon turmeric
2½ litres/4 pints stock
1 tablespoon salt
few grindings black pepper
2 tablespoons parsley, chopped
100 g/4 oz leek, chopped and washed
100 g/4 oz spinach, chopped and washed
50 g/2 oz egg noodles
1 tablespoon flour mixed to a paste with 1–2 tablespoons water
300 ml/½ pint yoghurt
Garnish
25 g/1 oz butter
1 tablespoon chopped fresh mint or 1 teaspoon dried mint
Melt the butter in a large saucepan, add the onion and fry until soft and golden. Drain the chickpeas and beans and add to the pan together with the lentils, turmeric, stock, salt and pepper. Bring to the boil, lower the heat, cover and simmer for 1 hour or until the chickpeas and beans are tender. Add more water if necessary. Add the parsley, leek and spinach and simmer for 20 minutes. Stir in the noodles and cook for a further 10–12 minutes until tender.
Spoon a few tablespoons of the soup into the paste and then stir the paste slowly into the soup. Remove the soup from the heat and stir in the yoghurt. Just before serving melt the butter in a small pan and fry the mint for a few minutes. Pour it over the soup and serve.
Serves 4–6 people.
dugun çorbasi wedding soup
Soon after his return from the mosque, the bridegroom leaves his friends in a lower apartment enjoying their pipes and coffee and sherbet. If the bridegroom is a youth or young man, it is considered proper that he, as well as the bride, should exhibit some degree of bashfulness; the bride has a shawl thrown over her head; and the bridegroom must give her a present of money which is called 'the price of the uncovering of the face', before he attempts to remove this, which she does not allow him to do without some apparent reluctance, if not violent resistance, in order to show her maidenly modesty... the bridegroom now, in most cases, sees the face of his bride for the first time. Having satisfied his curiosity respecting her personal charms, he calls to the women (who generally collect at the door, where they wait in anxious suspense) to raise their cries of joy; and the shrill sounds make known to the persons below and in the neighbourhood, and often, responded to by other women, spread still further the news, that he has acknowledged himself satisfied with his bride. (From Modern Egyptians.)
He then descends to join his friends and no doubt eat heartily a bowl of the wedding soup—and, who knows, more often than not he probably needed it!
50 g/2 oz butter
1 onion, thinly sliced
450 g/1 lb lean leg of lamb, cut into 2.5 cm/1 in pieces (450 g/1 lb of lean beef can be substituted)
1.8 litres/3 pints water
2–3 teaspoons salt
½ teaspoon black pepper
2 eggs
1 tablespoon lemon juice
2 tablespoons yoghurt
Melt the butter in a large saucepan; add the onion and sauté until golden. Add the meat and sauté for 5–8 minutes or until nicely browned. Add the water and simmer for half an hour or until the meat is tender—removing any scum which may appear on the surface. Add the salt and pepper and mix well.
In a small bowl beat together the eggs, lemon juice and yoghurt. Stir a few tablespoons of the stock into the yoghurt mixture, mixing well between each one. Pour the yoghurt mixture into the soup, taste and adjust seasoning if necessary. Serve immediately.
Serves 4–6 people.
midya çorbasi mussel soup
A classic from the days when the Ottoman rulers paraded up and down the Sea of Marmara in their gilded craft, embraced by their ageing entourage of followers, slaves, eunuchs and concubines. They no doubt stopped at one of the many small restaurants on the Bosphorous to eat a bowl of mussel soup followed by a kebab of red mullet, pike or bass.
1.5 litres /2½ pints mussels
1.5 litres/2½ pints water
2 cloves garlic, chopped
1 large onion, coarsely chopped
2 tablespoons flour
4 tablespoons milk
½ teaspoon black pepper
½ teaspoon allspice
Garnish 2 tablespoons parsley, finely chopped
Scrape the mussels, transfer to a colander and wash thoroughly under cold running water. Place the mussels in a large saucepan with the water, garlic and onion and bring to the boil. Simmer until the mussels open up and then remove them with a slotted spoon and reserve. Strain the liquid in the pan through a fine sieve into another pan.
Now remove the mussel shells with a sharp knife and set the mussels aside.
Return the soup to the heat and bring to the boil.
Place the flour in a small bowl and mix to a smooth paste with the milk. Stir in a few tablespoons of the soup to make a thin paste and then stir into the soup. Add the mussels, stir gently and cook for 3–4 minutes. Season with the black pepper and allspice, stir and remove from the heat. Transfer to a soup tureen, garnish with the parsley and serve immediately.
kharcho chicken soup with walnuts
A soup from Georgia with a fascinating combination of chicken, walnuts and sour plums.
1–1.5 kg/2–3 lb chicken, cut into serving pieces
1.8 litres/3 pints water
1 teaspoon salt
1 large onion, finely chopped
3 large tomatoes, blanched, peeled,
seeded and mashed
175 g/6 oz walnuts, coarsely chopped
6–8 sour plums, thinly sliced
2 cloves garlic, crushed 1 teaspoon cinnamon ½ teaspoon black pepper 2 bay leaves
Garnish 3–4 tablespoons coriander leaves, fresh dill or parsley, finely chopped
Put the chicken pieces in a large saucepan with the water and salt. Bring to the boil and remove any scum which appears on the surface. Add the onion, lower the heat, cover and simmer for about 1 hour or until the chicken is tender. Remove the pieces of chicken and, when cool enough to handle, remove the flesh and cut into small pieces. Return the chicken to the pan and add the tomatoes, walnuts, plums, garlic, cinnamon, pepper and bay leaves. Bring to the boil and cook, uncovered, for a further 10–15 minutes. Serve garnished with the herbs of your choice.
Serves 4–6 people.
shusha gololig stuffed meatball soup
A classic of the Armenian cuisine from the region of Susha in Kharapak. Instead of each meatball being stuffed with a whole hard-boiled egg, some people like to chop the eggs and mix with 1 tablespoon chopped pistachio nuts (or walnuts), 2 tablespoons fried minced onion and ½ teaspoon salt. This is then formed into 4 balls which are then enclosed in the minced meat mixture. You can use large burghul (cracked wheat) instead of rice.
Meatballs
450 g/1 lb lamb or beef, minced twice
1 teaspoon salt
1 small onion, finely chopped
2 teaspoons plain flour
1 tablespoon milk
1 tablespoon brandy
1 small egg, beaten
½ teaspoon black pepper
2 tablespoons parsley, finely chopped
4 hard-boiled eggs, shelled
Soup
3 tablespoons ghee
1 small onion, finely chopped
1.8 litres/3 pints stock, lamb or beef
1 tablespoon salt
75 g/3 oz long grain rice, washed under cold running water
3 tablespoons fresh tarragon, finely chopped or 2 teaspoons dried tarragon
Garnish 1 tablespoon sumac powder (optional)
Place the minced meat and salt in a large bowl and knead. Add the chopped onion, flour, milk, brandy, egg, pepper and parsley and knead for 5–10 minutes until the mixture is well blended and smooth. Cover and refrigerate for 30 minutes.
Remove the bowl from the refrigerator and divide the meat into 4 equal parts. Keeping your hand damp, roll each portion of meat into a ball and then hollow out each ball. Place a hard-boiled egg in each hollow and build the meat up around them until the eggs are completely and evenly enclosed. With dampened hands smooth off the surfaces and form into slightly oval shapes. Refrigerate the meatballs while you prepare the soup.
Melt the ghee in a small pan, add the onion and fry until golden.
Bring the stock to the boil in a large saucepan, add the salt and rice, lower the heat and simmer for 10 minutes. Add the meatballs and cook for a further 10–12 minutes. Add the onion mixture and the tarragon to the soup and simmer for a further 5 minutes. Just before serving halve the meatballs and serve in individual bowls with a little sumac sprinkled over the top.
Serves 4–6 people.
## salads
To Middle Easterners a salad ranges from hummus-bi-tahina, taramasalata and similar puréed dishes through the numerous boiled and fried vegetable salads to the fresh vegetables dressed in olive oil, lemon juice, yoghurt and other combinations. Indeed, all the dishes under the headings 'Mezzeh' and 'Vegetables', together with those in this section, would be regarded as salads.
Colour, texture and presentation are paramount in the mind of the housewife when she prepares her salads. Unusual combinations of fruit and vegetables; nuts and vegetables; as well as cold meats such as tongue, liver and meatballs; are served prior to, or traditionally with, warm meat, poultry, kebabs or stews.
Talmudic, Koranic and the native folklore traditions recommend the use of certain vegetables for medicinal, sexual and psychological cures; 'Radishes are good for fever.... which may be caused by eating hot bread' or 'Garlic destroys parasites in the entrails' are amongst many beliefs still held by most Middle Easterners.
The four basic dressing ingredients are: olive oil—believed to possess almost supernatural healing qualities; lemon juice; wine-vinegar and yoghurt—the supposed cure for all intestinal maladies.
olive oil and lemon dressing
Although by far the most popular, this dressing of Greek origin—latho-lemono—is almost unknown throughout Iran, Iraq, the Gulf States and parts of the Caucasus where vegetables are eaten raw sprinkled with a little salt—aghtsan.
100 ml/4 fl oz olive oil
4–5 tablespoons fresh lemon juice
2 teaspoons parsley, finely chopped, or oregano or mint or dillweed
1 clove garlic, crushed (optional)
1 teaspoon salt
½ teaspoon black pepper
Mix the ingredients together in a bowl with a fork before pouring over the vegetables.
You can keep this dressing in a screw-top bottle or jar and refrigerate for further use. Before using remember to shake well.
You can increase or decrease the proportions according to your personal taste. It is excellent with all fresh or boiled vegetables.
oil and vinegar dressing
Popular in Cyprus, Greece, Turkey, Syria and Lebanon.
The vinegar should be wine vinegar.
Prepare as for the oil and lemon dressing above.
Cypriots often add ½ teaspoon or more of dry mustard—lathoxitho.
In Turkey this dressing—sirkeli salatasi—is served with any raw vegetable, as well as dried beans and beetroot salads.
yoghurt dressing
Yoghurt is a must with many dishes, particularly in Turkey, Armenia and Iran. Fresh natural yoghurt is an ideal accompaniment, but it is often mixed with herbs and spices.
One of the most popular dressings is a mixture of garlic and yoghurt—sughtorov madzoon.
300 ml/½ pint yoghurt
1 clove garlic, crushed
½ teaspoon salt
½ teaspoon dried mint
1 spring onion, finely chopped (optional)
Pour the yoghurt into a bowl. Mix the garlic and salt together, add to the yoghurt and mix well. Sprinkle the dried mint and onion over the top. Serve with all types of fried vegetables and fresh vegetables of your choice.
domates salatasi tomato salad
Popular throughout the Middle East this salad, in its simplest form, contains only tomatoes, onions or spring onions and chopped parsley with an olive oil-lemon juice dressing.
The recipe below is from Turkey and is richer, very attractive and appears with all roasts, kebabs and stews.
4 tomatoes, sliced
1 cucumber, thinly sliced
juice 1 lemon
2 tablespoons olive oil
1 tablespoon parsley, finely chopped
1 teaspoon mint, finely chopped or ½ teaspoon dried mint
½ teaspoon salt
¼ teaspoon black pepper
a few black olives
Garnish pinch cumin
On a large plate arrange the tomato and cucumber slices decoratively. In a cup mix the lemon juice, oil, parsley, mint, salt and pepper. Pour this dressing over the tomatoes and cucumbers and chill for 1–2 hours. Just before serving arrange some black olives on the plate. Sprinkle with the cumin and serve.
kefit aghtzan mixed salad with tahina
The literal meaning of this salad is 'of your choice or taste'—suggesting that there are no set rules—except the tahiniyeh dressing.
Any vegetable available is suitable, e.g. radishes, mushrooms, etc. However the recipe below is a typical one from Armenia.
1 green pepper, seeded and with white pith removed
2 tomatoes
½ cucumber
1 cooked potato, peeled
2 sticks celery
2 carrots, peeled and grated
handful of black olives, stoned
2 hard-boiled eggs
3–4 tablespoons tahiniyeh dressing (see below)
Garnish 1 tablespoon parsley, finely chopped
Tahiniyeh dressing
150 ml/¼ pint tahina paste
juice 2–3 lemons
approximately 300 ml/½ pint milk or water or a mixture of the two
2 cloves garlic, crushed
1 teaspoon salt
½ teaspoon chilli pepper
1 heaped teaspoon ground cumin
Prepare this dressing first by pouring the tahina into a mixing bowl, adding the lemon juice and stirring until the mixture thickens. Slowly stir in the milk and/or water until a thick, creamy texture is obtained. Season with the garlic, salt, pepper and cumin. This will make about 450 ml/ ¾ pint and it will keep in the refrigerator for at least a week.
Cut the green pepper, tomatoes, cucumber, potato and celery into small pieces and put into a large mixing bowl. Add the grated carrots and black olives. Shell the eggs, chop them coarsely and stir carefully into the vegetables. Spoon in the tahiniyeh dressing and gently mix everything together. Pile into a serving dish and sprinkle with the chopped parsley.
garmir gaghamp cabbage salad
A Caucasian favourite where the cabbage is first pickled with beetroot, giving it a crimson colour.
As well as the recipe below this pickled cabbage can be thinly shredded and served with grated carrot with an olive oil-lemon juice dressing, and in the recipe that follows clever use is made of apples and grapes.
Delicious with all kinds of meat and poultry dishes.
Pickling
900 g/2 lb white cabbage
900 g/2 lb beetroot
leaves of 5–6 stalks celery
10 sprigs parsley
450 ml/¾ pint red wine vinegar
2 teaspoons paprika
Wash the cabbage and remove the outer leaves. Put it into a large saucepan, cover with water, bring to the boil and simmer for about 30 minutes. Remove the cabbage from the water and set aside to drain and leave until cool enough to handle. Pull back the outer leaves gently and carefully open and separate the inner ones, but without detaching them from their base. Put the cabbage into a deep casserole.
Peel the beetroot, cut into 2.5 cm/1 in pieces and add to the cabbage. Add the celery leaves, parsley, vinegar and paprika. Add sufficient water to cover the cabbage by about 7.5 cm/3 in. Cover and leave for approximately 1 week.
At the end of the week remove the cabbage, which should now be deep red in colour and have a marvellously piquant flavour. This pickled cabbage is known as garmir gaghamp and it is often served as a salad by itself simply put on a plate and cut into quarters.
gaghampi aghtzan cabbage salad with chives
900 g/2 lb pickled white cabbage
1 onion, peeled and thinly sliced
2 apples, peeled, thinly sliced and tossed in 2 teaspoons lemon juice
50 g/2 oz grapes, halved and seeded
100 g/4 oz black or green olives, stoned
2 tablespoons olive oil
salt and black pepper
Garnish pinch of basil, 1 teaspoon paprika
Shred the cabbage and put into a large salad bowl. Add the onion, apples, grapes and olives. Stir in the olive oil and salt and pepper to taste. Place in the refrigerator for 2–4 hours. When serving arrange the vegetables in a pyramid shape and sprinkle with the basil and paprika.
Another Armenian variation is to put the shredded cabbage in a salad bowl with 50 g/2 oz finely chopped walnuts, 1 tablespoon sesame seeds, 1 clove garlic, finely chopped, 1 teaspoon salt and 1 teaspoon sumac.
Mix thoroughly and chill for at least 2 hours before serving.
salat benoosach hakibbutz kibbutz salad
This salad is a by-product of the kibbutzim of Israel which are an experiment in practical communism—as against theoretical waffle!—that seems still, after several decades, to attract considerable notice and adherents.
It is an array of vegetable salads that also include Cos lettuce and homegrown herbs, e.g. basil, tarragon, mint, parsley, etc.
Also included on the table would be hard-boiled eggs, pickled herrings, etc—in short anything and everything that the kibbutz produces.
The basic idea is to mix and season the vegetables according to one's own taste. Quantities are not included as this type of salad is best suited to large groups, e.g. buffets, parties and picnics.
fennel, raisins, celery seeds, lemon juice, olive oil
Grate the fennel; add the raisins and season with the celery seeds, lemon juice and olive oil.
tangerines, mayonnaise, cinnamon, chopped hazelnuts
Divide the tangerines into segments. Mix the mayonnaise with the cinnamon and hazelnuts. Dress the tangerines with this mixture.
cucumber, yogurt, dried mint
Slice the cucumber, add the yoghurt and sprinkle with the dried mint.
leeks, black olives, radishes (sliced), lime juice, olive oil, chopped tarragon
Wash and slice the leeks into a bowl, add all the remaining ingredients and mix well.
tomatoes, chopped parsley, olive oil, sumac powder
Slice the tomatoes and sprinkle with the parsley, olive oil and sumac.
Onions (sliced into rings), pimentos (chopped), olive oil and lemon juice, salt, oregano
Arrange the onion rings on a plate. Mix the remaining ingredients together and pour over the onions.
Serve each salad in a separate bowl.
Garnish them with extras, e.g. lettuce, herbs, spring onions, radishes, etc.
fattoush bread salad
The Egyptians 'shew a great respect for bread, as the staff of life—the name they give to it is eysh which literally signifies Life—and they on no account suffer the smallest portion of it to be wasted if they can avoid it. I have often observed an Egyptian take up a small piece of bread which had by accident fallen in the street or road and, after putting it before his lips and forehead three times, place it on one side in order that a dog might eat it, rather than let it remain to be trodden under foot.' (From Modern Egyptians.)
This respect for bread is general throughout the region, since for centuries people's existence chiefly depended on this 'life'. Indeed the Armenian expression hatz oudel—to eat bread—is most apt for it implied 'to dine' and bread was the paramount ingredient. As this Syrian salad shows, not a morsel was wasted, for when bread dried they made fattoush with it. This is a wonderful salad with an unusual texture. Prepare it in advance and chill, but do not stir in the bread until just before serving or it will lose its crispness.
1 large cucumber, chopped
1 lettuce heart, shredded
5 tomatoes, chopped
10 spring onions, chopped
1 small green pepper, chopped
1 tablespoon fresh coriander leaves, chopped
1 tablespoon parsley, finely chopped
1 tablespoon fresh mint, finely chopped
1 clove garlic, crushed
6 tablespoons olive oil
juice 2 lemons
½ teaspoon salt
¼ teaspoon black pepper
5 thin slices bread, lightly toasted and cut into small cubes
Place all the ingredients, except the bread, in a large mixing bowl. Toss the salad so that all the vegetables are coated with the oil and lemon juice. Chill until ready to serve and then stir in the cubes of bread and serve immediately.
adas salatasi lentil salad
A popular peasant salad. This is a Kurdish recipe from Diarbekir, Turkey. The Kurdish cuisine varies from region to region since, as a nation, they are spread between Turkey, Iraq and Iran. However, their food is, in general, perhaps nearer to the Armenians with whom they seem to have had a love-hate relationship for generations.
5 tablespoons oil
1 onion, finely chopped
225 g/8 oz lentils, washed
juice 1 lemon
1 clove garlic, crushed
600 ml/1 pint water
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon paprika
½ teaspoon cumin
Garnish 3 tablespoons parsley or tarragon, finely chopped
Heat the oil in a saucepan, add the onion and fry until soft. Add the lentils, lemon juice, garlic and water and bring to the boil. Lower the heat and simmer for about 45 minutes or until the lentils are tender. Add a little more water if necessary. Do not let the lentils get too soft. About 5 minutes before serving strain off any remaining water and add the salt, pepper, paprika and cumin and mix well. Transfer to a serving dish and garnish with the parsley or tarragon. Serve cold with all kinds of roast and grilled meats.
fasulya piyazi white bean salad
A popular salad throughout Greece, Turkey and Armenia.
There are several variations. The simpler versions may only include, apart from the beans, stoned and halved olives and a sliced tomato.
The recipe below is from Istanbul and is usually served cold, although it can also be eaten warm. If it is to be eaten cold then cook in olive oil, if warm then cook in butter.
Serve with cold meats and fish dishes.
225 g/8 oz haricot beans, soaked overnight in cold water
3 tablespoons oil (or butter if it is to be eaten warm)
1 large onion, thinly sliced
5 tomatoes, blanched, peeled and chopped
2 cloves garlic, crushed
½ teaspoon dried basil
3 tablespoons parsley, finely chopped
1 teaspoon salt
¼ teaspoon black pepper
¼ teaspoon allspice
Garnish
1 tablespoon parsley, finely chopped, lemon wedges, 2 hard-boiled eggs, quartered
Bring a large saucepan half-filled with lightly salted water to the boil, add the drained beans and simmer until soft. The time will depend on the quality and age of the beans and so add a little more water if necessary.
Heat the oil or butter in a large saucepan, add the onion and fry until soft and golden. Add the tomatoes, garlic, basil, parsley, salt, black pepper and allspice and mix thoroughly. Cook over a low heat for 5–10 minutes, stirring frequently.
Strain the beans and place in a serving dish. Retain a little of the cooking water. Pour the tomato mixture over the beans and mix well. If you think the mixture is too dry then you can stir in a little of the bean water. If this dish is to be eaten cold then refrigerate for 2–3 hours. Serve garnished with the parsley, eggs and lemon wedges.
salad-e-esfanaj Spinach and nut salad
A Kurdish-Iranian salad, popular in the Caucasus. For this recipe only fresh spinach is suitable. Ready salted pistachios are also recommended.
450 g/1 lb fresh spinach
½ cucumber, peeled, quartered and thinly sliced
6–8 black olives, stoned and thinly sliced
½ medium onion, thinly sliced
3 spring onions, thinly sliced
3 tablespoons parsley, finely chopped
3–4 tablespoons fresh tarragon, finely chopped
4–5 radishes, thinly sliced
2 tablespoons pistachios, chopped
4 tablespoons olive oil
juice 1 lemon
1½ teaspoons salt
1 teaspoon paprika
½ teaspoon black pepper
Garnish
½ teaspoon sweet basil
½ teaspoon marjoram
Wash the spinach thoroughly under cold running water, remove the stems and dry the leaves. Cut leaves into small pieces. In a bowl combine the spinach, cucumber, olives, onion, spring onions, parsley, tarragon and radishes. Add the pistachio nuts.
In a separate bowl mix together the oil, lemon juice, salt, paprika and black pepper. Pour this mixture over the salad and toss well. Refrigerate for 1–2 hours. Just before serving garnish with the basil and marjoram.
Azerbaijanian variation
Wash, drain and thinly shred 450 g/1 lb fresh spinach and place in a salad bowl with 150 ml/¼ pint double cream, 1 teaspoon salt, ½ teaspoon black pepper, juice 1½ lemons and ½ teaspoon dillweed. Shell and slice 2 hard-boiled eggs, remove the yolks, chop and sprinkle over the salad. Decorate with the rings of egg whites and sprinkle with 1 teaspoon paprika.
chirov aghtzan fruit and vegetable salad
A brilliant recipe from the Caucasus. Fresh fruits mix with vegetables to make a salad that goes well with poultry, pork and veal.
1 apple, peeled, cored and thinly sliced
1 quince, peeled, cored and thinly sliced
1 hard pear, peeled, cored and thinly sliced
1 ring of pineapple, chopped (optional)
¼ cucumber, peeled and thinly sliced
2 spring onions, thinly sliced
3 tablespoons olive oil
1 tablespoon raisins, soaked for 30 minutes in water beforehand
Garnish
5-6 lettuce leaves, washed and dried
1 tablespoon sumac
Mix all the ingredients together in a large bowl. Arrange the lettuce leaves around the edge of a large plate. Pile the salad into the middle of the plate. Sprinkle the sumac over the top and serve immediately.
asbourag aghtzan asparagus salad
He who boils asparagus and then fries them in fat, and then pours upon them the yolks of eggs with pounded condiments, and eats every day of this dish, will grow very strong—and find it a stimulant for his amorous desires. (From The Perfumed Garden.)
Serve with veal, poultry, pork and roast meats.
1 lb fresh asparagus
1 teaspoon salt
1 teaspoon dried dillweed
2 hard-boiled eggs, shelled and chopped
Dressing sughtorov–madzoun (see recipe)
Garnish sprinkling of paprika
Cut off the tough and coarse ends of the asparagus and discard. Wash, clean and dry with a paper towel. Cut into 2.5 cm/1 in pieces. Half-fill a large saucepan with water, bring to the boil and add the salt and the asparagus pieces. Cook for 10-20 minutes or until tender. Drain in a colander and set aside to cool. Place the pieces in a salad bowl, add the chopped eggs, dillweed and the yoghurt dressing and mix well. Serve cold, garnished with a sprinkling of paprika.
enginar artichokes in oil
The Yahoudi [Jew] has as much in common with Jerusalem as has the artichokes—i.e. both are foreigners to the place. (Palestinian saying.)
This, however, is neither a Jewish nor a Palestinian recipe. It hails from Turkey and is similar to the Greek aginares me anitho, although it was no doubt around before any of the above mentioned people were even heard of. It is often served as an appetizer, but also makes a good accompaniment to chicken, turkey, veal and game dishes.
4 globe artichokes
juice 2 lemons
12.5 g/½ oz flour
150 ml/¼ pint olive oil
1 tablespoon sugar
½ teaspoon salt
¼ teaspoon black pepper
2 cardamom seeds
4 black peppercorns
1 carrot, peeled and diced
8 spring onions, chopped
Garnish
1 tablespoon parsley, finely chopped
1 tablespoon fresh dill or 1 teaspoon dried dillweed
Wash the artichokes, place in a large saucepan of boiling water and simmer for 15 minutes. Remove from the water and, when cool enough to handle, hold each one by the stalk and remove the coarse outer leaves.
In a large saucepan bring 1.2 litres/2 pints of water to the boil together with the juice of 1 of the lemons.
Place the flour in a small bowl, add 4–5 tablespoons of the water and mix to a thin paste and then stir this into the pan of water.
Slice away the top of the artichoke leaves, pull out the prickly leaves and then use a teaspoon to scrape out the hairy choke. Place the artichokes immediately in the pan of water, stalks uppermost. Add all the remaining ingredients including the rest of the lemon juice. If the bases of the artichokes are not covered, add a little of the water in which they were first boiled. Bring to the boil, cover pan, lower heat and simmer for about 1 hour. Remove from the heat and leave to cool. Remove the artichokes and arrange in a serving dish. Pour over half the juice in which they were cooked and sprinkle with the parsley and dill.
salatit shawander beetroot salad
Popular in all countries beetroot is served with a variety of dressings including olive oil-lemon juice, tahina and yoghurt.
This goes well with all meat, poultry and fish dishes.
700 g/1½ lb beetroot, washed
1 tablespoon salt
water
Place the beetroot in a large saucepan, add the salt and sufficient water to cover by about 5 cm/2 in. Bring to the boil and then lower the heat and simmer for 45–60 minutes or until tender. Strain into a colander and leave until cool enough to handle. Rub off the skins and then slice or cube the beetroot into a bowl.
You can now make use of an olive oil-lemon juice dressing (see recipe) or prepare a favourite Egyptian salad called shawander tahiniyeh, where the beetroot is tossed in a tahiniyeh dressing.
To prepare the tahiniyeh dressing see the recipe and then make half the quantity suggested there. Put the cubed or sliced cooked beetroot in a bowl and chill. When ready to serve pour the dressing over the beetroot, toss well and serve immediately.
borani chogondar
This recipe from Iran is a yoghurt dressing. It uses labna (drained yoghurt), but there is no reason why you cannot use ordinary yoghurt.
300 ml/½ pint yoghurt or labna
½ teaspoon salt
¼ teaspoon black pepper
2 tablespoons lemon juice or vinegar
Garnish 1 tablespoon fresh mint, finely chopped, or 1 teaspoon dried mint
Put the cubed or sliced cooked beetroot in a bowl and chill until ready to use. Add the yoghurt, salt, and pepper to the bowl and mix well then stir in the lemon juice. Pile into a serving dish, garnish with the reserved beetroot and the mint and serve immediately.
mshat arnabeet cauliflower cheese
In the Middle East cauliflower is usually first lightly cooked in salted water and then served with one of the following dressings:
(a) olive oil-lemon juice
(b) tahiniyeh
(c) sughtorov-madzoon
This recipe, however, from Palestine and Jordan, makes use of cheese and nutmeg which gives a beautiful fragrance to an otherwise rather bland vegetable.
Makes an ideal accompaniment to meat and poultry dishes.
1 cauliflower
225 g/8 oz grated cheese, Cheddar is very suitable
2 tablespoons flour
1 egg
150 ml/¼ pint milk
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon grated nutmeg
oil for frying
Garnish
2 tablespoons parsley, finely chopped
1 lemon cut into wedges
Wash the cauliflower, break into florets and dry with kitchen paper.
In a large bowl mix the cheese, flour, egg and milk together to form a thick batter. Season with the salt, pepper and nutmeg.
Heat some oil in a large frying pan. Now dip the florets, one at a time, into the cheese batter and then fry gently until soft and golden. Drain on kitchen paper. Arrange the cauliflower cheese on a serving plate, sprinkle with the parsley and garnish with the lemon wedges.
bakla broad bean salad
Give me a woman juicy as a mulberry,
Broad hipped and soft as a broad bean
And throw all those okra-shaped ones
To the devil.
Kurdish wisdom
If young broad beans can be found then they are often cooked in their pods, as in this recipe. They are delicious with an olive oil and lemon dressing. However Turks, Kurds, Assyrians and particularly the Iranians like to prepare the beans in the following manner and serve them with a yoghurt dressing.
900 g/2 lb young broad bean pods, washed and stringed
juice 1 lemon
1 teaspoon salt
4–5 spring onions, cut into 1.2 cm/½ in pieces
2 tablespoons fresh mint, chopped, or 1 tablespoon dried mint
2 tablespoons fresh dillweed, chopped, or 1 tablespoon dried dillweed
½ teaspoon allspice
350 ml/12 fl oz water
100 ml/4 fl oz oil
1 tablespoon parsley, finely chopped
Sauce sughtorov-madzoon dressing—see recipe
Put the washed beans in a large bowl, add the lemon juice and salt, toss and leave to rest for 15 minutes.
Meanwhile, line the bottom of a saucepan with vine leaves or cabbage leaves. Cover the leaves with a layer of beans. Sprinkle the spring onions, mint and dillweed over the beans and then arrange the remaining beans over the top. Add the allspice, water and oil, cover and cook for 1–1½ hours or until the beans are tender. Remove from the heat and leave to cool. Transfer to a serving dish and sprinkle with the parsley. To serve prepare the yoghurt sauce and pour over the beans.
Serves 4–6 people.
brass aghtzan leek with yoghurt sauce
Leek is a remedy for snake bite.—Tosefta Shabbeth.
This salad makes a fine accompaniment for fish and chicken dishes.
8 leeks
600 ml/1 pint water
juice 1 lemon
10 or more peppercorns
1 teaspoon salt
4 coriander seeds
3 spring onions, finely chopped, including green heads
3 sprigs parsley
Yoghurt-mustard dressing
300 ml/½ pint natural yoghurt
3 egg yolks
2 teaspoons lemon juice
1 teaspoon salt
½ teaspoon black pepper
¾ teaspoon fennel seeds
1 teaspoon sumac powder
1½ teaspoons Dijon mustard
Garnish 2 tablespoons parsley, chopped
Cut the roots and most of the green tops off the leeks and remove any coarse outer leaves. Wash carefully under cold running water to remove all the grit and sand between the layers.
Prepare a stock by bringing 600 ml/1 pint of water to the boil in a saucepan. Add the lemon juice, peppercorns, salt, coriander seeds, chopped onions and the sprigs of parsley and simmer for about 10 minutes. Arrange the leeks in a large frying pan or flameproof dish and pour the stock over the top. Cover and simmer gently for 20–30 minutes or until the leeks are tender. Switch off the heat and leave to cool.
Meanwhile prepare the yoghurt-mustard sauce. Place the yoghurt, egg yolks and lemon juice in a bowl and beat thoroughly. Place the bowl over a pan of simmering water and cook the sauce for 12–15 minutes, stirring frequently until the sauce is thick. Add the remaining ingredients, mix well and remove from the heat.
Remove the leeks and drain on kitchen paper. Arrange them on a large plate and pour the yoghurt-mustard sauce over the top. Sprinkle with the parsley and serve.
NB A variation popular amongst Arabs and Turks is to cut the leeks into 5 cm/2 in pieces and cook in lightly salted water until tender. Drain and leave to cool. Place in a serving dish, pour over an olive oil-lemon juice dressing and toss.
Sprinkle with 1 tablespoon chopped parsley or mint.
kaleh joosh date and walnut salad
Dates act as a laxative.—Gittin.
This Iranian dish can be served either as an appetizer or as a salad accompaniment to poultry and fish dishes. It has a delicate flavour and very attractive appearance, and is traditionally made with kashk—liquid whey, but yoghurt is an excellent substitute.
50 g/2 oz butter
1 large onion, finely chopped
1 tablespoon flour
300 ml/½ pint fresh natural yogurt
2 teaspoons dried mint
2 cloves garlic, finely chopped
½ teaspoon saffron
1 teaspoon warm water
8–10 stoned dates, thinly sliced lengthwise
3 tablespoons walnuts, chopped
Melt half the butter in a pan, add the onion and fry until soft, stirring frequently. Sprinkle the flour over the onions and mix well. Add the yoghurt, stir and bring the mixture almost to the boil. Do not actually boil or the mixture will curdle. Remove the pan from the heat and transfer the mixture to a shallow serving dish.
Melt the remaining butter, add the mint and garlic and mix well. Pour this mixture over the yoghurt. Dissolve the saffron in the warm water and sprinkle over the yoghurt mixture. Now sprinkle the sliced dates and chopped walnuts over the mixture and serve.
jajig yoghurt and cucumber salad
Mint to the quantity of three eggs, one of cumin and one of sesame are good for angina pectoris.—Gittin.
A classic of the Middle Eastern cuisine, this salad, with several variations, appears everywhere from the Balkans down through to India.
The first recipe—the standard one—hails from Armenia, the second is Turkish from the region of Manisa—the ancient Greek city of Magnesia ad Sipylum.
These salads can be eaten with everything. They are particularly good with all roasts, kebabs, pilavs, kibbeh dishes and dolmas.
600 ml/1 pint yoghurt
½ teaspoon salt
1 clove garlic, crushed
1 cucumber, peeled, quartered lengthways and finely chopped
2 tablespoons fresh mint, finely chopped, or 2 teaspoons dried mint
1 little red pepper as a garnish
Place the yoghurt in a mixing bowl and stir in the salt, garlic, cucumber and mint. Place in a refrigerator to chill. Pour into 1 bowl or individual bowls, sprinkle with a little red pepper and serve.
biberli jajig
6 small, green, hot peppers
600 ml/1 pint yoghurt
1 teaspoon salt
1 clove garlic, crushed
3 tablespoons parsley, finely chopped
1 clove garlic, thinly sliced
2 tablespoons olive oil
Grill the peppers for about 4 minutes, turning once. Leave to cool then peel off the skins and remove the seeds. Cut them into small pieces about 0.6 cm/¼ in square.
Pour the yoghurt into a large bowl, add the salt and crushed garlic and stir for several minutes. Add the chopped peppers, parsley and sliced garlic and mix again. Serve in individual bowls with a little olive oil poured over the top.
In another variation the Iranians make this salad with labna (drained yoghurt) to which, as well as cucumber and mint, they add 1 large grated onion, 3–4 grated radishes, 50 g/2 oz chopped walnuts, 1 teaspoon dried dillweed and 50 g/2 oz seedless raisins.
The ingredients are mixed thoroughly then shaped (use an ice cream scoop if you have one) into balls and arranged on lettuce leaves.
kebabi peyvaz kebab salad
The relish with which Arabs and Iranians enjoy onions, which they often eat as though munching an apple, cannot just be explained by the fact that they, as good Muslims, wish to emulate the Prophet Muhammad who is reputed to have had a great fondness for this sad vegetable. A further reason must be their belief in the aphrodisiac qualities that this down to earth bulb of the lily family was claimed by the Ancients to possess. No wonder the children of Israel, while fleeing from Egypt, cried out bitterly when they were deprived of the famed Egyptian onions which they preferred to manna! Yet Hippocrates—that Greek hypocrite—declared them bad for the body.
I prefer the opinion of Shayhk Nefzawi who eulogizes the onion's excellent virtues in The Perfumed Garden:
Take one part of the juice pressed out of pounded onions and mix it with two parts of purified honey. Heat the mixture over a fire until the onion juice has disappeared and the honey only remains... this beverage is to be partaken of during winter and on going to bed. Only a small quantity is to be taken and only for one day. The member of him who has drunk of it will not give him much rest during the night!
Now I know why the Israelis were crying in the desert!
The recipe is a classic, simple and a must with all kinds of kebabs.
2 large onions, thinly sliced
1 bunch fresh parsley or tarragon, finely chopped
2–3 tablespoons sumac powder
Mix all the ingredients together in a large salad bowl. Spread this salad over a serving plate and lay the kebabs over the top. When serving the kebabs give a little of this salad with each one.
leninakani aghtzan vegetable kebab salad
The Caucasians—like the Californians and the Australians—love to eat out and they create excuses to have picnic parties where they barbecue whole pigs and lambs as well as the kebabs for which they are famed.
Almost always the accompaniment to these kebabs are jajig, kebabi peyvaz and this vegetable salad from Leninakan.
It is also excellent with all kinds of roasts.
1 aubergine, cut in ½ lengthways and then into 2.5–3.5 cm /1–1½ in cubes
1 green pepper, seeded and quartered
2 tomatoes
8 mushrooms
2 tablespoons onion, finely chopped
2 tablespoons fresh tarragon or parsley, finely chopped
2 tablespoons olive oil
2 tablespoons lemon juice
salt and black pepper
Thread the vegetables on to 1 or 2 skewers. Grill, preferably over charcoal, but an electric or gas grill will do, until the vegetables are cooked. Skin and quarter the tomatoes. Combine all the vegetables in a salad bowl and add the onion, tarragon or parsley. Stir in the olive oil, lemon juice and salt and pepper to taste. Mix well and serve either warm or cold.
## eggah and kookoo–egg dishes
Egg and egg-based dishes are very popular throughout the Middle East. The Ancient Egyptians evolved special hatcheries—maamal el-faroog, which were operated by using artifical heat, i.e. dung, where eggs were placed in small ovens, a heat of 100–103°F, 38–9°C was raised and by the twentieth day the eggs were hatched and the chickens distributed amongst the peasants who had first supplied the eggs. Eggs, coloured and hard-boiled, or, hard-boiled and then deep fried, are sold by street vendors or in small 'sandwich bars' with little cornets of paper filled with salt, or salt and cumin, to dip them in.
On festive occasions—Easter for the Christians and the birth of Muhammad for the Muslims—the eggs are coloured and, amongst the Greeks and Armenians, decorated in patterns and religious images, similar to the Russian Easter eggs.
Eggs are also used in stews or encased in minced meats or made into eggah or kookoo type omelettes which a traveller in Persia in 1824 described as 'a large omelette about 5 cm/2 in thick'. These dishes are not omelettes in the Western sense of the word for they are usually thick and heavy with various kinds of herbs, vegetables, meat or chicken and are almost meals in themselves—they are often eaten as such with yoghurt, bread and salads.
Eggah (Arabic) or kookoo (Persian) can be eaten hot or cold and they are often cut into small segments and served as appetizers or on a buffet table.
Small eggah are cooked in specially made copper or tin pans similar to 'rock-cake' tins while larger ones are prepared in heavy frying pans usually with lids. These dishes can be cooked either on top of the cooker or in the oven. The choice of oil varies. The Syrians and Armenians prefer olive oil, especially if they are to be eaten cold, while the Iranians make their kookoos in ghee or butter. The kookoo dishes of Iran form an integral part of the cuisine and they are exceptionally rich and varied in the choice of ingredients and their subtle blending. This type of omelette originated in Northern Iran from whence it spread to the Arab lands and the Caucasus.
beid hamine coloured boiled eggs
These eggs are boiled in water with onion skins to give the shells a light brown colour. My mother used to add 'Turkish' ground coffee to give the eggs a darker colour. To make them red she used to add some cut beetroot to the water. Nowadays, with the advent of food dyes, only one's artistic preferences limit the choice of colour.
On Easter Sunday Christians have a basketful of coloured eggs with which they play a game called 'Find the champion'. One person takes an egg and holds it in one hand covering as much as possible leaving only a little of one end showing between thumb and forefinger. Another person picks up an egg and, using one end of it, tries to crack the shell of the other egg. Whosoever's egg first cracks at both ends loses and the winner goes on challenging all comers until the champion is found.
The eggs are peeled and eaten dipped in salt and pepper or salt and cumin or in a mixture of salt, cumin, coriander and cinnamon, with an accompaniment of bread, salads and pickled vegetables. There are various semi-religious connotations for the colouring of eggs and they all stem from the pre-Christian rituals of re-birth and regeneration of Ancient Egypt and Mesopotamia.
12–15 eggs
water
2–3 tablespoons oil
skins of several onions—these will give a reddish-brown colour. If you want a darker colour use a little ground coffee. Or use food dyes—the quantity you use depends on the strength of colour you want
To serve salt, cumin
Half fill a large saucepan with cold water and carefully arrange the eggs in the water. Add the onion skins—and coffee if you are using it or the food dye. Add the oil. This will slow down the rate of evaporation of the water. Bring gently to the boil and then simmer over a very low heat for 5–6 hours, keeping an eye on the water level and adding more if necessary. Remove from the heat and leave to cool. Serve in their shells so that the colour can be admired.
betzaim eggs in wine
'Lady', insisted the corner grocer, 'these are the best eggs we've had for months.' 'Then keep them,' snapped the customer. 'Who needs eggs you've had for that long?'
This Israeli recipe where the eggs are poached in a wine sauce makes a tasty breakfast or snack.
200 ml/8 fl oz dry white wine
200 ml/8 fl oz water
½ teaspoon salt
pinch white pepper
4 teaspoons flour
6 eggs
6 slices toast
1 teaspoon dried mint
1 teaspoon mixed herbs
½ teaspoon paprika
Pour the wine and water into a large saucepan and season with the salt and pepper. Bring to the boil.
Put the flour into a bowl, add a few tablespoons of the stock and mix to a smooth paste. Stir this into the stock and simmer until it thickens, stirring constantly. Break the eggs carefully into the sauce and cook gently until the eggs have set. Remove with a slotted spoon and place each egg on a round of toast. Sprinkle lightly with the mint, herbs and paprika and serve. Serves 6 people.
betza sabra egg sabra
A recipe from Israel dedicated to the Sabras—Israeli-born youth who derive their names from the prickly cactus growing wild in the desert.
Sabras—incidentally this is also the Arab name for this fruit—have a very thick, greenish skin, but are juicy and sweet inside. Thus are reputed to be the youth of Israel (the Jew not the Palestinian!).
This recipe needs care as it is a little elaborate, but it is quite simple in essence and is well worth the effort. Serve as an appetizer or a savoury.
4 eggs, hard-boiled
2 tablespoons tahiniyeh (see recipe)
1 egg, well beaten
½ teaspoon Worcestershire sauce
4–5 cream cracker biscuits, crushed very finely
oil for frying
4 lettuce leaves
Garnish
black olives
tomato slices
gherkins
½ teaspoon paprika
Shell the eggs and cut each one in half lengthways. Remove the yolks. Place the yolks in a bowl with the tahina tarator and mash until smooth. Divide this mixture into 4 and put each part back between two of the egg whites. Reshape into 4 whole eggs.
In a small bowl mix the beaten egg with the Worcestershire sauce. Carefully dip the stuffed eggs into the egg-sauce mixture. Roll each egg in the biscuit crumbs until covered generously.
Heat some oil in a small saucepan. Place the eggs very gently in the pan, two at a time, and deep fry until golden, turning very carefully. Serve on individual plates with each egg bedded in a lettuce leaf. Garnish with the olives, tomato slices and gherkins and sprinkle with a little paprika.
çilbir eggs on toast with yoghurt
A classic dish which, in the context of the Middle Eastern cuisine, means old, traditional, of peasant origin and simple. This savoury snack fits all these requirements. It has that particular Anatolian touch—yoghurt.
40 g/1½ oz butter
6 eggs
6 large rounds of toast
300 ml/½ pint yoghurt
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon cumin
25 g/1 oz melted butter
1 teaspoon paprika
Melt the butter in a large frying pan. Break the eggs gently into the pan and cook until firm.
Meanwhile, arrange the rounds of toast on a large serving plate. Place 1 egg on top of each round of toast. Beat the yoghurt with the salt, pepper and cumin and spoon over the eggs. Mix the melted butter and paprika together and dribble over the eggs. Serve immediately.
Serves 6 people.
yumurtali inçir eggs with figs
An omelette of figs!—not really so unusual. The Middle East is full of such delightful recipes, particularly the Turkish-Armenian cuisine where, aside from figs, cherries, apples and quinces are treated in this way with the addition of herbs and nuts.
A particularly good omelette tzirani tzvadzegh makes use of dried apricots, thinly sliced and cooked with eggs, ½ teaspoon cinnamon, 1 tablespoon finely chopped pistachio nuts and 1 tablespoon seedless raisins.
For the recipe below, from Turkey, use fresh or dried figs and accompany with salads, bread and pickles.
8 eggs
4 tablespoons single cream or milk
½ teaspoon salt
60 g/2½ oz butter
175 g/6 oz fresh or dried figs, thinly sliced or chopped
Break the eggs into a bowl and beat lightly. Add the cream and salt and stir.
Melt the butter in a frying pan. Add the chopped or sliced figs and fry for 4–5 minutes, stirring occasionally. Pour the egg mixture over the figs and cook gently until set. Serve immediately.
tzvazegh small omelettes with mint
Traditionally served at Easter these small omelettes can be eaten hot or cold. I prefer the latter, especially when they are sandwiched in the pouch of a warm pita bread.
They make a good appetizer and look attractive on a mezzeh or buffet table. This is a family recipe, but there are many other fine variations. My mother had a special pan for making them, but I have used a 'rock-cake' tin and it is perfectly satisfactory.
They will keep in the refrigerator 2–3 days—if permitted to do so!
3 teacups parsley, finely chopped
1½ teacups onion, finely chopped
2 tablespoons dried crushed mint
6 eggs, beaten
1 tablespoon flour
3 cloves garlic, crushed
2 teaspoons salt
1½ teaspoons black pepper
cooking oil
Place all the ingredients, except the oil, in a large bowl and beat with a fork until well blended.
One third fill each compartment of the cake tin with oil and place over a moderate heat. When the oil is hot put one heaped soup spoonful of the mixture into each compartment and cook gently until the mixture is completely set and golden underneath. Turn each small omelette over and cook until the other side is also golden. When they are ready remove to a serving dish.
If necessary add a little more oil to each compartment and continue cooking the mixture until it is finished.
If preferred the mixture can be cooked as one large omelette in a frying pan.
mirza ghassemi eggs and aubergines
A speciality of Gilan in northern Iran on the Caspian coast. It is named after a great eighteenth century poet and philosopher, Mira Ghassemi. Serve with hot flat bread, pickles and salad.
2 medium aubergines
50 g/2 oz butter
6 cloves garlic, finely chopped
1 large onion, finely chopped
1 teaspoon turmeric
1 teaspoon salt
½ teaspoon black pepper
1 tomato, blanched, peeled and finely chopped
4 eggs, well beaten
Make 2 or 3 slits in each aubergine and cook in a hot oven until the skins are black and the flesh soft when poked with a finger. When cool enough to handle peel off the skin, scraping off and reserving any flesh which comes away with it. Put the flesh into a bowl and mash with a fork.
Melt the butter in a large frying pan. Add the garlic and onion and fry for a few minutes until the onion is soft. Add the turmeric and mashed aubergine and stir thoroughly. Add the salt, pepper and tomato and cook over a low heat for about 5 minutes. Pour the beaten eggs over the mixture. Stir lightly from time to time and when the eggs are just set remove from the heat and serve immediately.
eggah-bi-qarnabit cauliflower eggah
Hit the egg with a stone and it goes to the devil.
Hit the stone with the egg and again it goes to the devil.
A Syrian recipe popular throughout the Arab countries, this makes a fine quick lunch. Serve it with salad, bread, pickles and perhaps some cold meats.
40 g/1½ oz butter
1 small onion, finely chopped
about 225 g/8 oz cauliflower florets, broken into very small pieces
2 spring onions, finely chopped
3 tablespoons parsley, finely chopped
4 eggs
150 ml/¼ pint milk
1 teaspoon salt
¼ teaspoon black pepper
pinch nutmeg
Garnish ½ teaspoon paprika
Melt half the butter in a pan and add the onion, cauliflower and spring onions. Fry for a few minutes until the onion softens. Remove from the heat and stir in the parsley.
In a bowl beat the eggs and milk together and then stir in the salt, pepper and nutmeg. Stir the cauliflower mixture into the eggs.
Melt the remaining butter in an ovenproof dish and pour the mixture into the dish. Place in the centre of an oven preheated to 190°C, 375°F, gas mark 5 and bake for about 30 minutes or until nicely browned. Remove from the oven, sprinkle with the paprika and serve.
eggah-bi-djadj wa rishta
chicken and macaroni eggah
A rich and sumptious meal when served with fresh salads. You can use macaroni, spaghetti or tagliatelle, although traditionally, of course, the housewives prepared their own rishta (noodles) which was a long and laborious job.
350 g/12 oz macaroni, etc.
350 g/12 oz cooked chicken meat, cut into small pieces
2.5 litres/4 pints chicken stock
4 large eggs
3 cardamom pods, cracked or a pinch of ground cardamom
1 teaspoon salt
½ teaspoon black pepper
¼ teaspoon allspice
2 tablespoons parsley, finely chopped
40 g/1½ oz butter
1 green or red pepper, thinly sliced
Garnish
25 g/1 oz butter
1 tablespoon paprika
2 large eggs
Bring the chicken stock to the boil in a large saucepan, add the macaroni, stir and boil briskly for 5 minutes. Pour into a colander and leave to drain.
Meanwhile, break the eggs into a large bowl and whisk with a fork. Add the chicken pieces, cardamom, macaroni, salt, pepper, allspice and parsley and mix well.
Melt the butter in a large frying pan, add the green or red pepper and fry until just soft. Add the egg mixture and move the pan from side to side to spread evenly. Cook over a low heat for about 30 minutes or until set.
Meanwhile, melt remaining butter for garnish in a small frying pan. Break the 2 eggs into a small bowl, whisk, pour into the pan and cook until set.
Slide the omelette on to the eggah, sprinkle liberally with the paprika and serve straight from the pan.
kookoo-ye-sabzi herb kookoo
This, the most famed kookoo, is prepared with fresh herbs and is traditionally served on New Year's Day—Norouz. The abundance of fresh herbs symbolizes the coming years' fruitfulness.
All kinds of herbs—some unfortunately only found in Iran—can be used so do experiment. As well as those mentioned below try chervil, tarragon, etc.
40 g/1½ oz butter
2 lettuce leaves, finely chopped
6 spring onions, finely chopped
2 leeks, washed thoroughly and finely chopped
6 tablespoons parsley, finely chopped
3 tablespoons coriander, finely chopped
6 eggs
3 tablespoons spinach, finely chopped, optional
½ teaspoon turmeric
¼ teaspoon cinnamon
½ teaspoon dillweed
1 teaspoon salt
¼ teaspoon black pepper
3 tablespoons walnuts, chopped
2 tablespoons raisins or sultanas
Melt half the butter in a large frying pan. Add the lettuce, spring onions, leeks, parsley, coriander and spinach and fry for 5 minutes, stirring regularly.
Meanwhile, break the eggs into a bowl. Add the remaining ingredients except the butter and mix well.
Use the remaining butter to grease the bottom and sides of an ovenproof casserole dish. Pour in the egg mixture, add the fried vegetables and mix well. Bake in the centre of an oven preheated to 180°C, 350°F, gas mark 4 for about 45 minutes or until set and lightly golden. Remove, cut into wedges and serve hot or cold.
kookoo-ye-loobia bean kookoo
Kookoo dishes are usually cooked in the oven, this recipe, however, is often cooked on top of the stove. Serve with rice, salad and pickles.
350 g/12 oz string beans, washed, drained and cut into 1.2 cm/½ in pieces
50 g/2 oz butter
1 small onion, finely chopped
½ teaspoon saffron
6 eggs
1 teaspoon salt
¼ teaspoon black pepper
Place the beans in a large saucepan with some lightly salted water and bring to the boil. Simmer until just tender and then drain.
Meanwhile, melt half the butter in a large frying pan, add the onion and sauté until golden. Stir in the saffron and the beans and fry for 2–3 minutes.
Break the eggs into a mixing bowl, add the salt and pepper and beat thoroughly. Add the onions and beans and mix well.
EITHER—melt the remaining butter in an ovenproof dish, add the egg mixture and bake in an oven preheated to 180°C, 350°F, gas mark 4 for 30–45 minutes until firm and golden;
OR—melt the remaining butter in a large frying pan, add the egg mixture and cook over a low heat. When the centre is almost firm, very carefully place a large plate over the pan and invert, dropping the omelette on to the plate.
Very carefully slide the omelette back into the pan and cook for another 3–4 minutes.
Cut the kookoo into wedges and serve.
kookoo-ye-tareh ba gerdoo
leek and walnut kookoo
This tasty kookoo of leeks and walnuts makes a good snack or appetizer and a substantial meal when served with yoghurt, salad and bread.
50 g/2 oz butter
350 g/¾ lb leeks, washed carefully and finely chopped
5 tablespoons walnuts, finely chopped
1 tablespoon flour
½ tablespoon turmeric
1 teaspoon salt
¼ teaspoon black pepper
6 eggs, well beaten
2 tablespoons parsley, finely chopped
Melt half of the butter in a frying pan. Remove from the heat and stir in the leeks, walnuts, flour, turmeric, salt and pepper. Now add the beaten eggs and parsley and stir thoroughly.
Melt the remaining butter in a baking or casserole dish and pour in the egg mixture. Spread the mixture evenly with the back of a spoon. Place in the centre of an oven preheated to 180°C, 350°F, gas mark 4 and bake for about 45 minutes or until the kookoo is set and golden. Remove from the oven, cut into squares and serve with yoghurt, salad, etc.
kookoo-ye-sibzamini potato kookoo
There are several variations of this popular dish where, as well as potatoes, other vegetables such as onions, tarragon and spinach are included. On the Caspian Sea coast mushrooms are incorporated. While in Iraq eggeh-bi-batata is prepared with thinly sliced potatoes, onions and tomatoes which are fried in ghee for 15 minutes and then added to an egg mixture seasoned with I teaspoon salt, ½ teaspoon turmeric, ½ teaspoon black pepper and ½ teaspoon caraway. The whole is then cooked in a frying pan, over a low heat, for 30–40 minutes or until well set.
This recipe is from Shiraz, Iran.
2 large potatoes
50 g/2 oz butter
6 eggs, well beaten
4 spring onions, finely chopped
2 tablespoons parsley, finely chopped
1 teaspoon dried dillweed
1 teaspoon salt
½ teaspoon black pepper
Garnish 1 teaspoon paprika
Peel and boil the potatoes in lightly salted water until tender. Strain and mash the potatoes. Add half of the butter, the beaten eggs, onions, parsley, dill, salt and pepper and beat until smooth.
Melt the remaining butter in a baking dish and swirl it around to coat the sides. Pour the potato mixture into the dish, smooth it over evenly, place in an oven preheated to 180°C, 350°F, gas mark 4 and bake for about 45 minutes or until a light golden colour. Sprinkle with the paprika and serve warm.
## pastas, pies and boreks
There is a charmingly innocent scene in a film about the adventures of Marco Polo—the thirteenth century Venetian merchant-traveller—where he naively asks a Chinese man what the thin reeds in his hand are.
'Spagget,' answers the man, 'you eat it.'
Marco Polo immediately takes a handful and adds it to his other 'finds' from the mysterious East which he later, we are told, introduces into Europe.
This, as usual when dealing with things Eastern, is far from the truth. For, had Marco Polo stopped at one of the many eating houses on his visits to Ajaccio (Cilician Armenia) or Constantinople or Persia, he would have been served several spagget-type dishes. Indeed pastas were known and used by the Ancients, as well as the Romans whose legions often, for days, depended on them between battles.
Pasta recipes appear in medieval Arab, Persian and Armenian recipes—usually under the name of rishta or arshta—a Persian word meaning thread.
Today in the villages of Turkey, the Greek islands, Armenia and most Arab-speaking countries, rishta is still home-made and I have included a few recipes using this type of pasta. However, commercially produced pastas of all kinds are becoming increasingly popular.
On a basic recipe for rishta, which makes long thin threads of dough similar to flat pastas like tagliatelle.
rishta
450 g/1 lb plain flour
1 teaspoon salt
2 eggs, beaten
5–6 tablespoons water
Sift the flour and salt into a large bowl. Make a well in the centre and add the beaten eggs and 4 tablespoons of the water. Mix well and knead until the dough is firm. Add a little more water if necessary. Knead for about 10 minutes.
Lightly flour a working top and divide the dough into 3–4 portions. Roll each portion out as thinly as possible, working from the centre. When all the sheets are rolled as thinly as possible leave them to rest for 45 minutes. Carefully roll each sheet up tightly—like a Swiss roll—and then cut them into 0.6 cm/¼ in, or less, slices. Unroll the threads and spread them out on the floured surface for at least 10 minutes. To serve, bring some lightly salted water to the boil in a large saucepan, add the rishta and simmer for about 5–6 minutes. The exact amount of time will depend on the thickness. Drain and use as required.
baki mussaka spaghetti with aubergines
For this recipe you can make your own rishta or use spaghetti, marcaroni or tagliatelle.
This dish is traditionally prepared during the forty days of Lent. Sometimes it is served as an hors d'oeuvre, but it is at its best as a savoury meal with salad and pickles.
2 medium aubergines, sliced
350 g/12 oz spaghetti or marcaroni
4 tablespoons oil
1 onion, finely chopped
1 clove garlic, crushed
1 green pepper, finely chopped
2 tomatoes, blanched, peeled and chopped
2 tablespoons tomato purée
1 teaspoon salt
½ teaspoon cinnamon
½ teaspoon cayenne pepper
225 g/8 oz grated cheese, e.g. haloumi,
Gruyère, Parmesan or Cheddar butter
Arrange the aubergine slices on a large plate, sprinkle with salt and set aside for 30 minutes. Rinse under cold running water and dry with kitchen paper.
Meanwhile, bring a large saucepan half filled with lightly salted water to the boil, add the pasta and boil for a few minutes. Drain into a colander, rinse under warm water and set aside.
Heat the oil in a large saucepan, add the onion and fry until soft. Add the garlic and continue to fry until the onion is golden. Add the aubergine slices and fry for a few minutes until lightly coloured all over. Add a little more oil if necessary. Now add the green pepper, tomatoes, tomato purée, salt, cinnamon and cayenne pepper. Lower the heat and simmer until the aubergines are soft. Add just a little water if the mixture is very dry and turn gently so that the slices don't break up.
Butter a large baking dish and arrange half of the pasta over the bottom. Arrange the aubergine mixture over the pasta and then arrange the remaining pasta evenly over the aubergine filling. Sprinkle the cheese evenly over the top and place in the centre of an oven preheated to 190°C, 375°F, gas mark 5 for about 30 minutes or until the surface is golden. Remove from the oven, cut into squares and serve warm.
banirov arshda pasta with cheese
Here is a story about a niggard of Isfahan who loved cheese.
A merchant who had lately died at Isfahan and left a large sum of money, was a great niggard; that for many years he denied himself and his son, a young boy, every support except a crust of coarse bread. He was, however, one day tempted by the description a friend gave of the flavour of a cheese to buy a small piece; but before he got home he began to reproach himself with extravagance and, instead of eating the cheese, he put it into a bottle and contented himself, and obliged his son to do the same, with rubbing the crust against the bottle and enjoying the cheese in imagination. One day he returned home later than usual and found his son eating his crust and rubbing it against the door.
'What are you about, you fool?'
'It's dinner time father, you have the key so I could not open the door—I was rubbing my bread against it because I could not get to the bottle.'
'Can you not go without cheese one day, you luxurious little rascal?
You'll never be rich!'
(Sketches of Persia, J. Murray)
An Armenian favourite which makes a fine starter or lunch when served with salads and pickles.
1 egg
5 tablespoons parsley, finely chopped
½ teaspoon salt
½ teaspoon black pepper
1 clove garlic, crushed
100 g/4 oz butter, melted
350 g/12 oz grated cheese, e.g. haloumi, kashkaval or Cheddar
225 g/8 oz marcaroni
Mix the egg, parsley, salt, pepper, garlic and 2 tablespoons of the melted butter together in a bowl. Add the grated cheese, mix well and set aside.
Bring a large saucepan half filled with lightly salted water to the boil. Add the macaroni and cook for about 10 minutes or until just tender. Strain in a colander, return to the saucepan and stir in 2 tablespoons of the butter.
Lightly grease an ovenproof dish about 22.5 cm/9 in in diameter and spread half the macaroni over the bottom. Spread two thirds of the cheese mixture over the macaroni and top with the remaining macaroni. Spread the remaining cheese over the top and sprinkle with the rest of the butter. Place in the centre of an oven preheated to 200°C, 400°F, gas mark 6 and bake for 20–30 minutes. Cut into squares and serve hot.
chrod borek water pastry
A classic from Anatolia, both Turks and Armenians claim this as their speciality. However, since borek-type pastries are a speciality of both nations and since, for centuries, they were the backbone of the Ottoman Empire I have come to a Solomonian decision. Chrod borek is fifty per cent Turkish and fifty per cent Armenian—the Turkish half the 'water' and the Armenian half the 'pastry'!
The pastry is dipped in boiling water, then immediately into cold water to give it that crisp effect.
This recipe is a family one based on my great grandmother's.
Some people include meat, chicken or spinach fillings with this pastry, but I prefer it as it is.
Pastry
6 eggs
1 teaspoon salt
2 teaspoons cooking oil
450 g/1 lb plain flour
175 g/6 oz butter, melted
Filling
450 g/1 lb cooking cheese, grated
3 tablespoons parsley, chopped
salt and pepper to taste
Beat the eggs together in a large mixing bowl and stir in the salt and oil. Add the flour and mix until you have a soft dough. Knead for several minutes until the dough is smooth. Divide the dough into about 12 portions and roll each into a ball. Grease a baking tray, place the balls on it set well apart from each other, cover with a teatowel and leave to rest in a cool place overnight. Roll each ball out thinly to the shape of the baking tray. Set them aside.
In a very large saucepan boil up about 7½ litres/12 pints of water with 1 tablespoon salt.
Fill another large pan with cold water.
Dip each sheet of dough into the boiling water and hold for about 30 seconds. Remove and dip immediately into the cold water. Dry on a teatowel and set aside.
Make the filling by mixing together in a bowl the cheese and parsley with salt and pepper to taste.
Place 2 sheets of the dough in the baking tray. Pour 1–2 tablespoons of the melted butter over the second sheet. Continue buttering every second sheet until you have used up half of the sheets. Spread the filling over the 6th sheet. Continue adding the sheets and buttering every second one until they are all used up. Pour any remaining butter over the last sheet. Cook in an oven preheated to 200°C, 400°F, gas mark 6 for about 30 minutes. Remove from the oven and leave to rest for 5 minutes. If you like it soft then cover for 5 minutes, but if you prefer it crunchy then leave it uncovered.
Cut into 7.5 cm/3 in squares and serve warm.
kyurza meat filled pastries
A speciality from Azerbaijan similar to the numerous dumpling-type pastries which originated with the Mongols. Indeed, Azerbaijanians are part ethnically and part in culture the descendants of the Mongol and Tartar nomads who traversed this important transit area in search of pastures green.
Kyurza-type dishes are found from the Balkans (they are called majcomboc in Hungary, wutzerlin in Romania) through to Afghanistan (aushag) to Central Asia (chebureki) and finally to China (jui pao).
These pastries are boiled in salted water and served with yoghurt.
Dough—see recipe for rishta.
Filling
4 tablespoons ghee
225g/8 oz minced lamb
1 onion, finely chopped
1 teaspoon salt
½ teaspoon black pepper
Garnish
1 teaspoon cinnamon
2 tablespoons parsley or tarragon, finely chopped
To serve natural yoghurt
Prepare the rishta dough, divide into 2 balls, cover with a lightly dampened cloth and set aside.
Melt the ghee in a saucepan, add the meat and onion and fry, stirring frequently, until the onion is soft and the meat browned. Season with the salt and pepper and set aside.
Sprinkle a work top with some flour, take one of the balls of dough and roll out as thinly as possible. With a 7.5 cm/3 in pastry cutter cut out as many circles of dough as possible. Repeat with the other ball of dough.
Place about 1½ teaspoons of the meat mixture in the lower half of each circle. Moisten the edges with cold water and fold the top half over to make a half-moon shape. Seal the edges with your fingertips or with the prongs of a fork. Repeat until all the ingredients have been used up.
Half fill a large saucepan with lightly salted water and bring to the boil. Drop 6–8 pastries into the water and simmer for 8–10 minutes or until the pastries rise to the surface. Remove with a slotted spoon, drain on kitchen paper, arrange on a serving dish and keep warm in the oven. Cook the remaining pastries in the same way.
If there is any meat filling left over heat it through and spoon over the pastries. Sprinkle with the cinnamon and chopped parsley or tarragon and serve with a bowl of yoghurt.
NB Kaloyrka is a Cypriot dish similar to kyurza. It has 2–3 tablespoons freshly chopped parsley added to the filling. After they have been cooked as above they are served topped by hot saltsa tomato sauce (see recipe). Grated cheese, e.g. Cheddar, haloumi or Parmesan is often served with it.
mante small, boat-shaped pastries
'What a lucky person. Throw him into the sea, he'll soon float like a mante.'
Mante also appears in the chapter on soups—see mantabour, while here it features as small, meat-filled pies—versatile little pastries! This recipe is a family one and it looks very attractive.
Dough
225 g/8 oz plain flour
½ teaspoon salt
50 ml/2 fl oz oil
about 100 ml/4 fl oz cold water
Filling
1 tablespoon oil
1 onion, finely chopped
2 tablespoons parsley, finely chopped
1 tablespoon tarragon or mint, finely chopped, or 1 teaspoon dried tarragon or mint
a little melted ghee
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon allspice
½ teaspoon cumin
450 g/1 lb minced lamb
150–200 ml/6–8 fl oz stock
Sift the flour and salt into a mixing bowl and make a well in the centre. Add the oil and gradually add the water and mix to a soft dough. Knead for 5–10 minutes until the dough is smooth then shape into a ball, cover with a damp cloth and leave to rest for 30–40 minutes.
Meanwhile, to prepare the filling heat the oil in a saucepan, add the onion and fry until soft. Add the parsley, tarragon or mint, salt, pepper, allspice and cumin, stir well and cook for a further 2 minutes. Put this mixture into a bowl, add the minced meat and mix well.
Lightly flour a work top and roll out the pastry as thinly as possible. Cut into 5 cm/2 in squares. Place a teaspoon of the filling in the centre of each square, fold up the 2 opposite sides and pinch at the 2 ends to seal firmly. This should leave the meat exposed at the top and the pastry shaped like a small boat.
Grease a baking dish and pack the mante in tightly, meat sides uppermost. Brush all over the tops with the melted butter.
Bake in an oven preheated to 180°C, 350°F, gas mark 4 for about 45 minutes.
Heat the stock until boiling, pour all over the mante and return the dish to the oven for a further 15 minutes. Serve with yoghurt, salads and pickles.
borek
There are many large pie-type dishes in the Middle East with different fillings, e.g. kottopitta—chicken pie with filo pastry, tyropitta—with a cheese filling; tagine min laham—meat pie and pirasapide—leek pie.
Nevertheless, the housewives have lavished their attention and artistic flair on the smaller savoury pies and pastries.
They are found throughout the region, but it is in modern Turkey where the finest examples are found. This is why I have used the Turkish word borek as a general heading. These borek come in many shapes—triangles, fingers, half moons and small parcels, as well as in individual pots.
The usual doughs are the following: filo, flaky and shortcrust. Some boreks have their own doughs, a few of which I have included.
Shapes vary from region to region as do fillings, but generally the most popular ones are—cheese, meat, spinach and chicken.
Borek can be fried, boiled or baked. They are served as appetizers or savouries; can be eaten hot or cold and are often accompanied by salads and pickles.
There are many borek-type recipes in Arab, Turkish and Armenian manuscripts, one of the oldest is sanbusak which is found in Masudi's 'Meadows of Gold'. Another is oughi-dobrag—brains in a bag—which is found in an eleventh-century Armenian manuscript and the recipe for which I have given.
borek hamuru pastry for borek
This is one of the traditional doughs used for making borek. It does take a little time to make, but is not difficult and the borek will be soft and flaky.
You can, however, substitute commercial puff pastry with very satisfactory results.
450 g/1 lb plain flour, sifted
1 teaspoon salt
200 ml/8 fl oz cold water
1 teaspoon lemon juice
50 g/2 oz butter, melted and clarified
225 g/8 oz butter, chilled
Mix the flour and salt together in a large bowl. Make a well in the centre, add the water and lemon juice and, with a wooden spoon, mix thoroughly. Add the melted butter and knead with your hands for 10 minutes. Shape the dough into one large ball, cover with a damp cloth and leave for 30 minutes.
Lightly flour a work top. Roll the dough out until about 0.6 cm/¼ in thick. Now put the block of butter in the middle of the dough. Fold the pastry back over the fat so that it is completely hidden. With a rolling pin, well floured, flatten the dough to 1.2 cm/½ in thickness. Fold the dough in half and refrigerate for 10 minutes. Return to the floured work top and roll out until 0.6 cm/¼ in thick. Fold in half and refrigerate for a further 10 minutes. Keeping the working top well-floured roll the dough out once more as thinly as possible. Now cut into the desired size and shape for the borek.
savoury fillings–cheese
The quantity of cheese fillings below, and all other savoury fillings given, are for pastry made with 450 g/1 lb flour, etc. So increase or decrease depending on the quantity of borek you want.
450 g/1 lb grated cheese, e.g. haloumi, feta, Cheddar, Gruyère, or a mixture of these
2 eggs, beaten
½ teaspoon black pepper
salt to taste–amount will depend on the saltiness of the cheese
Mix together, taste and adjust seasoning if necessary.
450 g/1 lb crumbly white feta cheese or Stilton, Munster or even cottage cheese
4 tablespoons parsley, finely chopped
½ teaspoon black pepper
1 small clove garlic, crushed
salt to taste–feta is a salty cheese and if using it taste before adding
Crumble cheese into a bowl with a fork, add remaining ingredients and mix well.
herb fillings
450 g/1 lb grated cheese, e.g. feta, Cheddar, Gruyère, etc.
2 tablespoons parsley, finely chopped
1 tablespoon dill, chopped
1 tablespoon mint, chopped
½ teaspoon black pepper
¼ teaspoon allspice
salt to taste
Mix together and taste before adding the salt.
egg and cheese
450 g/1 lb grated cheese, e.g. Cheddar, feta, etc.
2 eggs, beaten
10 tablespoons milk
3 tablespoons oregano
½ teaspoon white pepper
salt to taste
Mix all the ingredients together and taste before adding any salt.
savoury fillings–meat
a
2 tablespoons ghee or butter
1 onion, finely chopped
450 g/1 lb minced lamb or beef
2 tablespoons pine kernels or chopped walnuts
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon allspice
Melt the butter in a pan, add the onion and fry until soft. Add the meat and fry until browned, stirring frequently. Add the nuts, salt, pepper and allspice, stir well and cook for 15–20 minutes or until the meat is tender. If the mixture becomes dry moisten with a few tablespoons water. Remove from the heat and leave to cool.
b
2 tablespoons ghee or oil
1 large onion, finely chopped
450 g/1 lb minced meat
6 tablespoons parsley, finely chopped
1 teaspoon ground cinnamon
½ teaspoon sugar
1 teaspoon salt
½ teaspoon black pepper
Heat the ghee or oil, add the onion and fry until soft. Add the meat and fry until browned, stirring frequently. Add the remaining ingredients, stir well and cook for 15–20 minutes or until the meat is tender. If the mixture is too dry moisten with a few tablespoons of water and continue cooking. Set aside to cool.
cerkez puf boregi circassian meat borek
The dough used here is the Middle Eastern version of puff pastry. The borek are deep fried. Serve with yoghurt.
Filling
25 g/1 oz butter
1 small onion, finely chopped
450 g/1 lb lamb, minced twice
3 tablespoons parsley, finely chopped
1 teaspoon salt
½ teaspoon black pepper
Dough
450 g/1 lb plain flour
1 teaspoon salt
2 egg yolks
1 tablespoon yoghurt
4 tablespoons olive oil
½ teaspoon lemon juice
100 ml/4 fl oz milk
50–75 g/2–3 oz butter, melted
oil for frying
First prepare the filling by melting the butter in a saucepan. Add the onion and fry until soft. Add the minced meat, parsley, salt and pepper, mix well and continue frying for 15–20 minutes until the meat is cooked. Remove from the heat and leave to cool while you prepare the dough.
Sift the flour and salt into a large bowl. Make a well in the centre and add the egg yolks, yoghurt, olive oil and lemon juice. Mix the ingredients with a wooden spoon. Now knead the mixture for about 10 minutes, adding a little of the milk at a time, until you have a soft, smooth dough. Shape the dough into a large round ball and dust with flour. Sprinkle a work surface with flour and, breaking off lumps of dough about the size of a walnut, roll each out as thinly as possible into a circle about 20 cm/8 in in diameter. Continue until you have used up all the dough. Brush the upper surface of one circle of dough with the melted butter, place another circle of dough on top of it and brush its upper surface with fat. Continue stacking the circles of dough in this way, brushing each upper surface with the fat until they form one pile.
Liberally sprinkle the work surface and the rolling pin with flour and roll out the pile of dough circles until you have one large sheet of dough which is very thin. Cover with a clean cloth and leave to rest for 30 minutes. Cut the pastry into 10 cm/4 in squares. Place 2 teaspoons of the filling in the centre of each square.
Either (a) fold into a triangle, pinching the edges with your thumbs or a fork or (b) spread the filling in a ridge about 1.2 cm/½ in from one edge then fold the edge over it and turn in the 2 sides and roll into a sausage shape.
Continue until you have used all the dough and all the filling.
Heat sufficient oil in a pan to deep fry the borek and fry, a few at a time, until golden. Remove with a slotted spoon, drain on kitchen paper, arrange on a serving plate and keep warm while you cook the remaining borek. Serve hot.
kutab caucasian meat borek
There are several variations of this dish. This recipe is from Baku on the Caspian Sea coast.
Dough see recipe for borek hamuru
Filling
450 g/1 lb minced lamb
1 onion, finely chopped
2 tablespoons freshly squeezed pomegranate juice (if available) or 1 tablespoon lemon juice
1 teaspoon cinnamon
½ teaspoon basil
1 teaspoon salt
¼ teaspoon black pepper
Frying 8–10 tablespoons ghee
First prepare the dough. Then place all the filling ingredients in a large bowl and knead well. If you dampen your hands it will make it easier.
Sprinkle a work top with flour and roll out the pastry until 0.3 cm/⅛ in thick. Cut the dough into 7.5 cm/3 in circles. Put 1½ teaspoons of the filling in the centre of one half of each circle. Dampen the edges with cold water and fold over to make a half moon shape. Seal the edges with your fingers or the prongs of a fork.
Heat the fat in a large pan, add the borek, a few at a time, and fry gently until one side is golden. Turn with a slotted spoon and cook the other side in the same way. Do not cook too quickly or the filling will not be cooked through.
With a slotted spoon transfer to a serving dish and keep warm while you cook the remaining borek in the same way.
Serve warm, sprinkled with sumac and accompanied by yoghurt, salads and pickles.
Savoury fillings–spinach
a
450 g/1 lb fresh spinach or 225 g/8 oz frozen leaf spinach
8 tablespoons oil
1 small onion, finely chopped
1 teaspoon salt
½ teaspoon black pepper
juice 1 lemon
If using fresh spinach trim stems, wash thoroughly several times in cold water and drain. If using frozen spinach then allow it to thaw.
Half fill a large saucepan with lightly salted water and bring to the boil. Add the spinach and simmer for 8–10 minutes and then strain into a colander. When cool enough to handle squeeze as much moisture as possible from the spinach and then chop. Heat the oil in a large pan, add the onion and fry until soft. Add the spinach, salt and pepper and cook over a low heat for about 5 minutes, stirring occasionally. Stir in the lemon juice and set aside to cool.
b
450 g/1 lb fresh spinach or 225 g/8 oz frozen spinach
1 tablespoon ghee
110 g/4 oz grated cheese, e.g. Cheddar, Gruyère, etc.
1 egg, beaten
½ teaspoon black pepper
salt to taste
Wash the fresh spinach thoroughly and drain or thaw the frozen spinach. Chop the leaves finely and place in a saucepan with the ghee. Cover and cook over a low heat until the leaves are tender, stirring occasionally. Add the cheese, egg, pepper and salt to taste, mix well and set aside to cool.
c
450 g/1 lb fresh spinach or 225 g/8 oz frozen spinach
2 tablespoons oil
1 medium onion, finely chopped
2 tablespoons pine kernels or chopped walnuts
2 tablespoons raisins
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon allspice
Prepare spinach as described under the first filling. Heat the oil in a pan, add the onion and fry until soft. Add the remaining ingredients, mix well and set aside to cool.
NB Sometimes the nuts are fried in oil before being added to the spinach.
tahinov-spanaki borek spinach and tahina borek
This is an Armenian speciality made during Lent, using tahina paste.
Dough see recipe for borek hamuru, or use a large packet of puff pastry
Filling
350 g/12 oz fresh spinach or about 175 g/6 oz frozen leaf spinach
3 tablespoons ghee or oil
2 onions, finely chopped
40 g/1½ oz long grain rice, washed
100 ml/4 fl oz water
25 g/1 oz sesame seeds
6–7 tablespoons parsley, finely chopped
2 tablespoons walnuts, finely chopped
2 tablespoons tomato purée
175 g/6 oz tahina paste
¼ teaspoon black pepper
½ teaspoon allspice
1 teaspoon cayenne pepper
2 teaspoons salt
To glaze milk or beaten egg
Prepare the dough as described. If using fresh spinach trim stems, wash thoroughly several times in cold water and drain. If using frozen spinach then allow it to thaw. Half fill a large saucepan with lightly salted water and bring to the boil. Add the spinach and simmer for 8–10 minutes and then strain into a colander. When cool enough to handle squeeze as much moisture as possible from the spinach and then chop.
Heat the ghee or oil in a pan, add the onion and fry until soft, remove with a slotted spoon and reserve.
Place the rice in a small pan with the water, bring to the boil, then simmer until the water has been absorbed.
Place the sesame seeds on a piece of foil and toast under a hot grill until golden.
Place all the filling ingredients in a large bowl and mix thoroughly.
Sprinkle a work top with flour and roll out the pastry to 0.3 cm/⅛ in thickness. Using a pastry cutter cut the pastry into 7.5 cm/3 in rounds. Put 1 teaspoon of the filling in the lower half of each circle. Moisten the edges with cold water and fold over to form a half-moon shape. Seal the edges either with your fingertips or with the prongs of a fork.
Lightly grease some large baking trays and arrange the borek on them, making sure they do not touch each other. Brush with a little milk or beaten egg and cook in an oven preheated to 225°C, 425°F, gas mark 7 for about 15 minutes or until risen and golden. Serve hot or cold.
NB These lenten borek are sometimes made larger by using 15 cm/6 in squares or circles and filling with 2 tablespoons of the filling.
Savoury fillings–chicken or turkey
Chicken and turkey meat are interchangeable in the recipes below.
450 g/1 lb cooked chicken meat, cut into small pieces
2 tablespoons ghee or butter
2 tablespoons flour
300 ml/½ pint milk
1 teaspoon salt
½ teaspoon white pepper
1 egg, beaten
Melt the ghee or butter in a small pan, add the flour and stir to a smooth paste. Gradually add the milk, stirring constantly, and cook over a low heat until the sauce thickens. Season with the salt and pepper and cook very gently for a few minutes. Remove from the heat, allow to cool a little and then stir in the beaten egg and pieces of chicken. Set aside to cool.
tartar boregi
This is a speciality of the Tartars who were once the scourge of Asia until destroyed by the Russians in the eighteenth and nineteenth centuries.
This recipe from Turkey is made with a milk-based dough—tartar hamuru—and it is served with a tasty tomato sauce.
Dough
450 g/1 lb plain flour, sifted
2 eggs, beaten
100 ml/4 fl oz milk
Filling
25 g/1 oz butter
1 tablespoon flour
300 ml/½ pint chicken stock
350 g/12 oz cooked breast of chicken, chopped
1 teaspoon salt
¼ teaspoon black pepper
¼ teaspoon marjoram
50 g/2 oz white cheese, grated, e.g. feta, Stilton, Lancashire, etc.
50 g/2 oz butter, melted
Sauce
25 g/1 oz butter
6 tablespoons water
1 tablespoon tomato purée
50 g/2 oz grated cheese
Place the flour in a bowl and make a well in the centre. Add the eggs and mix in. Slowly add the milk and knead until you have a soft, smooth dough which is pliable. Cover with a cloth and leave to rest while you prepare the filling.
Melt the butter in a saucepan then remove from the heat and stir in the flour. Stir in the stock, return to the heat and cook gently until the sauce thickens, stirring constantly. Remove from the heat and stir in the chicken, salt, pepper, marjoram and cheese.
Lightly flour a work top and roll out the pastry as thinly as possible. Cut the pastry into 10 cm/4 in squares. Brush each square with some of the melted butter. Place about 2 teaspoons of the filling in the centre of each square and then roll up and seal the ends securely with a fork. Place on lightly-greased baking sheets and brush with any remaining butter. Bake in an oven preheated to 180°C, 350°F, gas mark 4 for 20–30 minutes or until golden.
Meanwhile, place all the sauce ingredients in a saucepan and cook over a moderate heat for about 2 minutes. Remove the boreg from the oven, place on a serving dish, pour the sauce over them and serve.
Other savoury fillings
kabak boregi–courgette
450 g/1 lb courgettes, peeled and grated
2 eggs, beaten
150 g/5 oz grated feta cheese (or Cheddar, Gruyère, etc.)
½ teaspoon salt–taste before adding more as feta cheese is quite salty
¼ teaspoon black pepper
1 teaspoon dried mint
Place the courgettes in a fine sieve and squeeze out as much of the water content as possible. Place the pulp in a bowl, add the remaining ingredients and mix well.
sokhi borek– onion
900 g/2 lb onions
2 tablespoons salt
3 tablespoons walnuts, chopped
1 teaspoon chilli pepper
1 teaspoon dried mint
100 ml/4 fl oz tahina paste
Slice the onions, sprinkle with the salt, squeeze tightly and set aside in a bowl for about 8–10 hours.
Drain the onions, which should now have lost much of their bitterness. Add the remaining ingredients and mix thoroughly.
oughi dobrag brains borek
One of the popular dishes of our kingdom [Vaspoorakan—the region of Lake Van—Armenia] is one called oughi dobrag —a bag of brains... You take ten mutton's brains, wash and soak them in water with vinegar, then boil in salt water. After which, they are cut to small pieces and to which all kinds of greens are added. We use tarragon, onions, garlic, parsley, dill, sumac, purslane and many others.
This then is wrapped—like a bag—in khumor [dough] and fried in temag [sheep's tail fat]. The brains of calves are superior to those of mutton, for the latter, when eaten in large quantities, will make a person short-sighted and slow-witted like the animal itself.
(From eleventh century Armenian illuminated manuscript.)
A modern version of this recipe is the following one.
450 g/1 lb sheep's brains, soaked in cold water for 5 hours with 1 tablespoon vinegar. Change the water at least twice
40 g/1½ oz butter
2 tablespoons onion, finely chopped
1 tablespoon cornflour
about 175–225 ml/6–8 fl oz water
10 tablespoons grated cheese
1½ teaspoons salt
¾ teaspoon black pepper
1 hard-boiled egg, chopped
Remove the brains from the water, rinse and cut away and discard the loose outer membranes. Place the brains in a saucepan, add enough water to cover, with a tablespoon of vinegar and bring to the boil. Simmer for 5 minutes then drain the brains and, when cool enough to handle, cut into small pieces.
Melt the fat in a small pan, add the onion and sauté until soft. Stir in the flour and then gradually add the water, stirring constantly and cook over a low heat until the sauce thickens. Add the cheese, salt and pepper and cook for 1–2 minutes, stirring constantly. Remove from the heat, stir in the brains and egg and set aside to cool. Use as a filling for a pastry of your choice.
koteh dolmeh lentil borek
An Iranian recipe, also popular in the Gulf region, this makes a light, crispy savoury meal. Serve with yoghurt and pickles.
Dough
15 g/½ oz yeast
225 ml/8 fl oz warm water
1 tablespoon rosewater
450 g/1 lb plain flour
1 teaspoon salt
½ teaspoon ground cardamom
2 tablespoons melted ghee
Filling
175 g/6 oz brown lentils, washed thoroughly
1.2 litres/about 2 pints water
3 tablespoons ghee
oil for frying
3 medium onions, finely chopped
1½ teaspoons salt
1 tablespoon brown sugar
Dissolve the yeast in 50 ml/2 fl oz of the warm water in a bowl, put in a warm place. When it begins to froth add the rest of the water and the rosewater.
Sift the flour and salt into a large mixing bowl and add the cardamom. Make a well in the centre, add the yeast mixture and knead to a smooth dough. Add the melted ghee and knead for about 10 minutes or until it is soft and elastic. Cover with a cloth and leave in a warm place for about 1 hour or until the dough has doubled in size.
Meanwhile, place the lentils and water in a saucepan, bring to the boil, cover the pan, lower the heat and cook for about 1 hour or until the lentils are soft and the water absorbed. The exact time will depend on the age of the lentils, so add a little more water if necessary. With a fork, or a potato masher mash the lentils to a coarse purée and reserve.
Melt the ghee in a frying pan, add the onions and fry until soft. Add the lentils and fry for a further 5–6 minutes. Season with the salt and sugar, mix well and set aside to cool.
Punch down the dough and knead for a few minutes. Divide the dough into 2 equal portions and form each into a ball. Roll each ball out, on a lightly floured work surface, until about 0.3 cm/⅛ in thick. Cut into 7.5 cm/3 in circles. Put 1–1½ teaspoons of the filling in the lower half of each circle. Dampen the edges with cold water then fold over to make a half-moon shape. Seal the edges with your fingertips or with the prongs of a fork.
Heat some oil in a deep saucepan, add 4–5 dolmehs at a time and deep fry the dolmehs for about 3 minutes, turning once or twice, until golden and puffed. Remove with a slotted spoon, drain on kitchen paper and keep warm while you cook the remaining dolmehs in the same way. Serve warm with salads, yoghurt and pickles.
lahma-bi-ageen meat pizza
If you are poor, you rub garlic on bread and thank Allah!
If you are rich, you spread meat and nuts on bread and thank Allah.
If, like us, you are very poor
You sit opposite the bakery and dream of mouth-watering sfiha.
I have never much cared for things chauvinistic, but even I, who have been brought up amongst people who, next to death, regard patriotism as a taboo subject, am obliged to scream 'Stop, enough is enough! Sfiha is Arabic, but miss-hatz (lahma-bi-ageen or lahmajoon) isn't and that is final!'
When you compare the two recipes below perhaps you may ask what all the fuss is about because they look so much alike—but they aren't. Lahma-bi-ageen, which literally translated means meat and bread, is pastry topped with meat and vegetables similar in concept to the Italian pizza. We know pizzas were popularized in Naples (I did not say commercialized—that honour belongs to the Americans), and we also know that Neapolitans are of Greek extraction and that the Greeks of Naples came from Greece and Byzantium when that Empire disintegrated. These Byzantines were largely Greek-speaking Anatolians, and the Anatolians were the indigenous inhabitants of modern Turkey, Syria and Armenia. Finally, since even today lahma-bi-ageen is completely unknown in Greece and western Turkey, its natural habitat must be northern Syria and Cilician Armenia (southern Turkey). Therefore, lahma-bi-ageen (the original pizza) originated somewhere in the region of Aleppo, Syria. The word pizza, on the other hand, is most probably derived from the Greek and Turkish words piaz (onion) and pita (bread) since the original pizza did not contain tomatoes, but instead great use was made of onions, e.g. pissaladiere—a Provençal pizza of onions, anchovy fillets and olives on yeast pastry.
So you can see that to call this dish Arab because it has an Arabic name is ridiculous. The reason for its name was commercial—by that I mean that when a minority lives amongst a majority the fomer invariably uses the latter's language for commercial purposes. Recently, on a visit to war-torn Beirut (Lebanon) when I was seeking lahma-bi-ageen recipes from family and friends, I was directed to lahma-bi-ageen shops and all, as it turned out, were owned and operated by Armenians; for lahma-bi-ageen, or miss-hatz, is of Armenian origin. I should know because my grandfather, the baker, was the uncrowned king of this great Middle Eastern classic.
Sfiha are small (miss-hatz type) meat tarts popular in Syria, Lebanon and parts of Iraq where, over the centuries, my people have migrated for safety and business reasons.
Sfiha
Dough
600 ml/1 pint lukewarm water
1 tablespoon dried yeast or 12 g/½ oz fresh yeast
900 g/2 lb plain flour
1½ tablespoons salt
3 tablespoons olive oil
Filling
3 tablespoons olive oil or butter
75 g/3 oz pine kernels
1 onion, finely chopped
900 g/2 lb lean minced meat
1 small green pepper, finely chopped
4 tablespoons parsley, finely chopped
1 tablespoon pomegranate syrup or 1 tablespoon sumac powder or 2–3 tablespoons lemon juice
½ teaspoon allspice
½ teaspoon cayenne pepper
1 teaspoon salt
¼ teaspoon black pepper
First prepare the dough. Measure 4 tablespoons of the water into a bowl, sprinkle or crumble the yeast over the top and leave to rest for 3 minutes. Stir to dissolve and set aside in a warm place until it begins to froth.
Sift the flour and salt into a large mixing bowl, make a well in the centre and pour in the yeast mixture. Slowly stir the flour into the liquid until they are well mixed and the dough can be gathered into a ball. If the dough is a little too stiff then add a little more water.
Sprinkle a table top with some flour. Place the dough on the table top and begin kneading. Do this by pressing the dough down, pushing it under with the heels of your hands and folding it back on itself. Knead 2 tablespoons of the olive oil into the dough. Continue until the dough is smooth and elastic. Sprinkle with a little flour now and again to prevent it sticking to the table. Shape into a ball and rub the remaining tablespoon of oil over the surface of the ball, place in a clean bowl, cover with a cloth and leave in a warm place for about 1 hour. When the dough has about doubled in size punch it down with your fists a few times and then divide into about 15 pieces. Roll each piece into a ball about 4 cm/1/½ in in diameter and leave to rest for 30 minutes.
Meanwhile, prepare the filling. Heat 1 tablespoon of the oil in a saucepan and fry the pine kernels for 1–2 minutes. Add the onion, meat, green pepper, parsley, pomegranate syrup (or sumac powder or lemon juice), allspice, cayenne pepper, salt and black pepper. Mix thoroughly, taste and adjust seasoning if necessary. Set aside.
Now roll each ball of dough into a circle about 10 cm/4 in in diameter and 0.6 cm/¼ in thick. When completed take a small piece of filling about the size of an egg and place in the centre of one of the circles and either flatten it and spread it evenly over the dough to within 1.2 cm/½ in of the edge, bringing the edges up slightly all round, or leave the meat as a ball and bring 3 sides of the pastry up over the meat to enclose it and form a triangular pie. Pinch the top securely.
Continue until you have used up all the ingredients. Use the remaining oil or butter to grease some baking sheets and arrange the tarts on them leaving a space between each one. Place in an oven preheated to 180°C, 350°F, gas mark 4 and bake for 20–30 minutes or until the pastry is golden. Do not overcook, especially the open tarts or the meat will dry out.
NB The Sephardic Jews of Syria make a greater use of pomegranate syrup and include 4–5 tablespoons which gives the pizzas a dark brown appearance.
Today chopped tomatoes are often added to the meat mixture, but they were not a traditional ingredient.
Serve as an appetizer or as a savoury with lemon wedges, salad and yoghurt.
lahma-bi-ageen or miss-hatz
Dough
6 g/¼ oz fresh yeast or 12 g/½ oz dried yeast
1 teaspoon sugar
350 g/12 oz plain flour
1 teaspoon salt
½ teaspoon allspice
180–225 ml/7–8 fl oz water, lukewarm
Filling
450 g/1 lb minced lamb
1 onion, finely chopped
1 small bunch parsley, chopped, about 4–6 tablespoons
1 green pepper, chopped
1 clove garlic, chopped
450 g/1 lb ripe tomatoes, blanched, skinned and chopped
1 tablespoon tomato purée
1 teaspoon salt
½ teaspoon cayenne pepper
¼ teaspoon black pepper
Place the yeast and sugar in a small bowl and add a few tablespoons of the warm water. Stir to dissolve and leave in a warm place until it begins to froth.
Sift the flour into a large bowl, add the salt, allspice and yeast mixture together with enough of the water to make a soft dough. Place on a floured working surface and knead vigorously for about 10 minutes, by which time it should be smooth and elastic and come cleanly away from your hands. Gather the dough into a ball, place in a clean bowl, cover with a cloth and leave in a warm place for about 1 hour or until it has doubled in size.
Meanwhile, prepare the topping by placing all the ingredients in a large bowl and kneading until well blended and smooth.
When the dough is ready divide it into golf ball-sized pieces and allow to rest for 10 minutes. Roll out on a floured surface into circles 12–15 cm/5–6 in in diameter. Spread a generous layer of topping over each surface right to the edges. Place on greased baking trays as you make them, leaving a little space between them.
Bake in an oven preheated to 230°C, 450°F, gas mark 8 for 10–15 minutes. The dough should be lightly golden, but still soft enough to fold.
To eat squeeze a little lemon juice over the surface, roll up like a pancake and eat with your fingers. Accompany with a salad.
NB The classic way to serve miss-hatz, however, is with grilled aubergine seorme. Make 2 or 3 slits in each aubergine and cook over charcoal or in a hot oven until the skins are black and the flesh soft when poked with a finger. When cool enough to handle peel off the skins and slice the flesh lengthways into thin strips. Lay 1 or 2 strips down the miss-hatz, roll up and eat with jajig, pickles and fresh salads.
## kibbehs and kuftas
Kibbeh is the pride and joy of the Syrians and Lebanese. It is also one of the oldest surviving legacies of the ancient Assyrian cuisine that has come down to us almost unchanged.
Kibbeh dishes prevail throughout the Mediterranean coastline and they also appear in Armenia and northern Iraq. But it is in Syria and Lebanon, particularly amongst the Christian elements, that kibbeh is regarded as a national dish entwined in semi-religious mystiques. Some women are supposed to have 'kibbeh fingers'. Often women are praised more for their kibbeh-making abilities than, say, for their looks or intellect. To possess 'kibbeh fingers' is a God-given gift akin to having pianist's fingers.
There are many variations of kibbeh. Some are established classics, e.g. kibbeh-naye, kibbeh-tarablousiyeh, kibbeh-bi-sanieh and kharperti-kufta. Others are of a more regional fame, but they all (with just a few exceptions) have the following ingredients in common: burghul (cracked wheat), minced meat, nuts, herbs and spices. Kibbeh can be eaten raw, fried, boiled, grilled or baked. Their preparation, sizes, shapes and methods of consumption are a binding link between the present and the distant past.
I have included as many recipes as possible as I regard kibbeh-type dishes as one of the most original methods of food preparation known to us. Two of the recipes—kibbeh-naye and houm-miss occur in the mezzeh section beginning.
basic kibbeh mixture
The recipe below is a standard one and I suggest you use it wherever possible. Variations are given in full.
Meat for kibbeh must be lean and it must be minced twice. Use lamb if possible as it is the ideal meat. However, if you want to use beef then add 2 tablespoons of cornflour and 1 small beaten egg—this will help the minced beef to act as a binder as beef does not possess the same elasticity as lamb.
175 g/6 oz fine burghul
225 g/8 oz lean lamb, minced twice
1 tablespoon onion, very finely chopped
½ teaspoon black pepper
½ teaspoon chilli pepper
1 teaspoon allspice
1½ teaspoons salt
Place the burghul in a bowl, add cold water and then pour it away thus getting rid of any dirt, chaff, etc. Repeat 1 or 2 times. Spread the burghul out on a baking sheet and leave while you prepare the meat.
Place the meat in a bowl, add the onion, seasonings and 2 tablespoons cold water and knead until well blended and smooth. Add the meat mixture to the burghul and knead until all the burghul is gathered up with the meat into a ball. Keeping your hands damp, knead the mixture for at least 10 minutes or until the kibbeh has the texture of a soft dough. Do not skimp on the kneading time or the texture of the kibbeh will be coarse when cooked. When you have prepared your kibbeh and made the balls for some of the soups, or tarablousieh for deep frying you might decide that you have much more than you need for the present serving. In this case the kibbeh, stuffed or 'blind', will freeze very well until you wish to use them.
amram grilled kibbeh
Prepare the basic mixture. Divide it into 12 equal portions. Roll into balls between damp palms and then flatten them into rounds 7–10 cm/3–4 in in diameter and 1.2–1.8 cm/½–¾ in thick. Cook over charcoal or under the grill for about 10 minutes, turning occasionally. Serve on a bed of lettuce leaves with a yoghurt dressing of your choice spooned over them.
dabgvadz kufta fried kibbeh
ghee or oil for frying
1 recipe of basic kibbeh mixture—to which mix in the following:
½ teaspoon curry powder
¼ teaspoon cinnamon
½ teaspoon paprika
1 egg
Garnish Lettuce leaves, 1 tablespoon parsley, finely chopped, lemon wedges
Shape the mixture as with amram.
Heat the ghee or oil in a large saucepan, drop in the kibbeh, a few at a time, and fry for about 8–10 minutes until golden brown all over. Place on a serving dish garnished with lettuce leaves and sprinkle with the parsley.
To eat, squeeze lemon juice from the wedges over the kibbeh. Serve with yoghurt or jajig and a fresh salad.
NB These can also be made into sausage shapes 10 × 2.5 cm/4 × 1 in or finger shapes 5 cm × 1.2 cm/2 × ½ in.
khorovadz kufta skewered kibbeh
Prepare the basic mixture. Divide the mixture into 12–16 portions and roll into balls between lightly dampened palms. Pass a skewer through one of the balls and then squeeze out the kibbeh to form a thin sausage shape. Continue with the remaining mixture. Cook over charcoal for about 10 minutes, turning occasionally. Serve immediately with yoghurt spooned over them and a bowl of fresh salad and some pickles.
basic kibbeh and kufta fillings
There are several fillings, but the following two are the most traditional.
Meat and onion filling
2 tablespoons oil or ghee
225 g/8 oz minced meat
2 onions, finely chopped
1 teaspoon salt
25 g/1 oz pine kernels or chopped walnuts
½ teaspoon black pepper
½ teaspoon allspice
½ teaspoon cinnamon
1 tablespoon dried rose petals (optional)
Heat the oil or ghee in a saucepan, add the meat and fry for 5 minutes, stirring frequently. Add the onions and salt and fry for about 30 minutes until the meat is cooked. Add the remaining ingredients, mix well and set aside to cool.
You can prepare this filling well in advance, cover and keep in the refrigerator and you can use it for any stuffed kibbeh including kibbeh-bil-sanieh.
Fatty filling
The traditional fat used for this filling is sheep's tail (alya), but as this is not available in the West the most suitable substitute is suet. However, you can even use butter or block margarine.
Suet
You can use lamb or beef. Blend in a mixer or knead together:
225 g/8 oz suet
25 g/1 oz chopped pine kernels or walnuts
½ teaspoon black pepper
½ teaspoon allspice
1 teaspoon salt
Chill for several hours before handling.
Butter
Blend 225 g/8 oz of butter with the above spices and chill for several hours before handling.
kibbeh-bil-sanieh kibbeh baked in a tray
A classic which makes a fine first course or main dish. It can be baked in advance and warmed when needed.
It is popular throughout Syria, Lebanon, Palestine and Israel. It is also known as sineh kufta amongst Armenians, Turks and the Kurds of southern Turkey.
Serve with a bowl of fresh salad, yoghurt and pickles.
Filling
2 tablespoons oil
25 g/1 oz pine kernels or 25 g/1 oz walnuts, chopped
225 g/8 oz minced lamb
1 onion, chopped
1 teaspoon salt
1 teaspoon black pepper
½ teaspoon allspice
¼ teaspoon cinnamon
1 tablespoon parsley, finely chopped
Kibbeh
175 g/6 oz fine burghul
225 g/8 oz lean lamb, minced twice
1 tablespoon onion, very finely chopped
pinch allspice
50 g/2 oz butter
2 tablespoons oil mixed with 4 tablespoons water
1½ teaspoons salt
1 teaspoon black pepper
To make the filling heat the oil, fry the nuts for about 2 minutes and then remove with a slotted spoon and drain. Add the meat to the oil and cook for about 15 minutes, stirring frequently. Add the onion and seasonings and cook for a further 15–20 minutes, stirring frequently. Stir in the nuts and parsley and set aside.
Wash the burghul in a bowl and pour away excess water. Spread the burghul out on a baking sheet and leave for 5–10 minutes. Add the minced lamb, onion and seasonings and knead for at least 10–15 minutes, keeping your hands damp with cold water. Divide the mixture into 2 equal parts.
Butter a shallow circular baking dish 20–22 cm/8–9 in in diameter and sprinkle it with a pinch of allspice. With your fingers spread one half of the burghul mixture evenly over the bottom of the dish. Spread the filling evenly over this.
Arrange the remaining burghul mixture evenly over the top. The easiest way to do this is to break off large lumps of the kibbeh, press it flat between your palms and place on the filling. Fit the pieces together something like a jigsaw and then draw the edges together and smooth over. Wet your hands and press the mixture well down over the filling. Wet a sharp knife and run it around the edge of the dish to loosen the kibbeh. Cut the kibbeh into diamond shapes. Place a small dab of butter on each diamond and pour the oil and water over the top. Bake in an oven preheated to 190°C, 375°F, gas mark 5 until golden brown and crisp around the edges.
NB A Caucasian variation similar to this uses pork (minced twice) in the kibbeh mixture while lamb or beef is used in the filling.
kibbeh tarablousieh syrian 'torpedo' kibbeh
Probably the most famed dish of all—the preparation of which is an art which takes several years to perfect. I understand that in the USA they are called 'Armenian bombs', but they should more accurately be called Tripoli kibbeh for they are the speciality of the second largest city of Lebanon (Tarablous).
Although difficult at first I strongly recommend you persevere as they are magnificent to look at and taste equally excellent.
Basically they are oval shells filled with a meat mixture and then deep fried. Often they are made only 4 cm/1½ in long, fried and then served cold as an appetizer. Traditionally though, they are served warm, often as a main meal, with salads and yoghurt.
The smaller versions are sometimes added to stews of meat and courgettes, pumpkins or aubergines.
Filling
2 tablespoons oil
25 g/1 oz pine kernels or chopped walnuts
225 g/8 oz minced meat
1 onion, finely chopped
1 teaspoon salt
1 teaspoon black pepper
1 teaspoon allspice
1 tablespoon parsley, finely chopped
Kibbeh
225 g/8 oz fine burghul
450 g/1 lb lean lamb, minced twice
2 tablespoons onion, very finely chopped
oil for deep frying
2 teaspoons salt
1 teaspoon black pepper
To Serve lemon wedges
First prepare the filling by heating the oil in a pan, add the nuts and fry for a minute or two. Remove with a slotted spoon and drain.
Add the meat to the oil and cook for about 15 minutes, stirring frequently. Add the onion and seasonings and cook for a further 15–20 minutes. Stir in the nuts and parsley and set aside.
Wash the burghul in a bowl and pour away the excess water. Spread the burghul out on a baking sheet and knead for a few minutes. Add the meat, onion and seasonings and knead for at least 10–15 minutes, keeping your hands damp with cold water.
To stuff the kibbeh, wet your hands and break off a piece of kibbeh about the size of an egg. Hold the ball of kibbeh in the palm of one hand and, with the index finger of the other hand, make a hole in the kibbeh. Press the index finger down into the palm of the other hand squeezing out the kibbeh and making the shell a little thinner. Slowly rotate the ball of kibbeh so that the finger is pressing down on a new part of the kibbeh shell and making it thinner. Continue turning the shell round and round and pressing it up the finger until you have a long oval shape with a slightly wider mouth. The art is to get the shell as thin as possible without cracking it. It will be a little easier to do this if you keep your hands damp. Place a tablespoon of the filling into the shell and then close the opening by drawing the edges together and sealing. Wet your hands again and roll the kibbeh between your palms to smooth off and ensure that it is a real oval shape. Continue in this way until you have used up all the kibbeh mixture and meat filling.
To cook add sufficient oil to a pan to deep fry and heat until hot. Add a few kibbeh at a time and fry until golden brown all over. Remove and drain. Serve hot with lemon wedges. To eat cut the kibbeh in half and squeeze lemon juice over the filling.
anteb yogurtli kufta kibbehs in a yoghurt sauce
A classic beloved of Syrians, Armenians—who call it madzounov kufte and the Lebanese—who call it kibbeh-bi-laban.
There are several variations of these stuffed kibbehs in a yoghurt sauce, but by far the best is from the city of Anteb (gazi Antab) in southern Turkey. It is a fascinating city. It was, for centuries, part of Armenia and then was conquered by the Ottomans. The inhabitants were almost equally divided between Turks, Armenians and Arabs with a scattering of Jews, Kurds, Alouites, Nestorians, Assyrians, Maronites—the list is endless. The result was an intriguing city of rich cross currents and a justifiably famed local cuisine, of which this dish is part.
Use either of the basic fillings suggested. I very much like the suet one, but I am aware that it is rather heavy for the uninitiated.
This is a really substantial meal and should serve at least 6 people. Serve in soup bowls with bread.
Filling
Prepare either of the basic fillings (see recipes) or you can cook these kibbehs 'blind' i.e. without a filling.
Kufta
Prepare 1½ times the basic kibbeh mixture (see recipe), i.e. 350 g/12 oz meat, 250 g/9 oz burghul etc.
Sauce
1–1.5 kg/2–3 lb chicken, cut into joints
2 teaspoons salt
100 g/4 oz chickpeas, soaked overnight in cold water
1½ litres/2–2½ pints yoghurt
2 eggs
50 g/2 oz butter
1 tablespoon dried mint
Half fill a large saucepan with water, add the chicken, salt and chickpeas and bring to the boil. Remove any scum which appears on the surface and cook until the chicken and chickpeas are tender.
If using a suet filling remove it from the refrigerator and make from it small balls about the size of a pea.
If making 'blind' kufta simply break off pieces of the kibbeh mixture and roll into marble-sized balls. Keep your hand damp—it will make the work easier!
If stuffing the kufta break off a piece of kibbeh about the size of a small walnut and roll into a ball. Push your forefinger into the centre and then, holding it so that your forefinger is parallel to the palm of the other hand, press your finger down into the kibbeh thus making it thinner. Slowly rotate the ball in the palm of your hand all the while pressing down with your forefinger making the shell of the ball uniformly thinner. Put a ball of suet or a teaspoon of the meat and onion filling into the hole. Bring the edges of the opening together and seal. Dampen your hands and roll the ball between your palms to give it a round shape and smooth surface. Repeat until you have used all the ingredients.
Mix the yoghurt and eggs together in a small bowl, add a little of the hot chicken stock and stir well. Pour the yoghurt sauce into the large pan of stock with the chicken and chickpeas. Simmer very gently. Add the kufta balls and simmer, very gently, for 10–15 minutes until cooked.
Meanwhile melt the butter in a small pan, add the mint, stir and pour into the soup. Serve the dish hot by placing several kufta in a soup bowl with some of the chicken and chickpeas and plenty of the sauce.
Serves 6–8 people.
yougov kibbeh fried onion kibbeh
Let onions grow in his navel. A curse.
This recipe, and all its variations, is Armenian from southern Turkey. They make excellent appetizers and snacks and are also often served as meals with salads, yoghurt and pickles.
6 tablespoons oil
1 onion, finely chopped
3–4 tomatoes, blanched, peeled and chopped
2 tablespoons tomato purée
1 tablespoon salt
½ teaspoon chilli pepper
225 g/8 oz fine or medium burghul
1 small green pepper, finely chopped
2 spring onions, finely chopped
2–3 tablespoons parsley, finely chopped
Heat the oil in a large saucepan, add the onion and fry until soft and turning golden, stirring frequently. Add the tomatoes, tomato purée, salt and chilli pepper and mix well. Simmer this mixture for a few minutes and then transfer to a large bowl.
Place the burghul in a small bowl, add cold water, stir and pour away the water together with any husks, etc. Add the burghul to the tomato mixture, mix well, cover and leave to cool for at least 20 minutes so that the burghul becomes softened.
In a small bowl mix together the green pepper, onions and parsley.
Knead the kufta for about 5 minutes until well blended. Mix in the chopped vegetables and knead a little longer. Arrange the kufta in a dish and serve with a spoon.
NB A variation from Kilis—southern Turkey—includes sumac which gives a dark red colour and a strong tangy flavour.
Place 3 tablespoons sumac powder in a small pan with 50 ml/2 fl oz water and bring to the boil. Strain the juice through a fine sieve, add it to the kufta mixture and knead as described above.
kibbeh hali alouite kibbeh
A speciality from northern Syria where people of the Alouite sect are found. A mysterious and secretive people, the Alouites have incorporated much from ancient cults in the tenets of their many sects—Nosairis, Haidaris, Charbis, Chamalis and Kalazes. All of which are variants of a pantheistic worship of the sun, the sky, the air and the Syrian lunar deity. They are not Muslims, but followers of Ali to whom God was manifested, and from whom Muhammad was issued. 'An Alouite never speaks' is an apt Syrian expression and to have secured this recipe of theirs was a great achievement!
225 g/8 oz fine burghul
100 g/4 oz plain flour or fine matzo meal or oatmeal
1 egg
Sauce
4 tablespoons oil
1 onion, finely chopped
2 cloves garlic, finely chopped
1 tablespoon ground coriander
3 tablespoons tomato purée diluted in 300 ml/½ pint water
1½ teaspoons salt
2 bay leaves
½ teaspoon allspice
1 teaspoon cayenne pepper
½ teaspoon black pepper
900 ml/1½ pints water
juice 1 lemon
Garnish 2 tablespoons parsley, finely chopped
Wash the burghul in a bowl and pour away excess water. Spread the burghul out on a baking tray and knead for a few minutes. Add the flour, matzo meal or oatmeal and the egg. Mix everything together and then knead until you have a paste thick enough to mould. If you find the mixture a little too sticky then leave it for 15–30 minutes by which time the burghul will have absorbed much of the excess moisture. Keeping your palms damp, shape teaspoons of the mixture into small balls about the size of marbles. Set these balls aside while you prepare the sauce.
Heat the oil in a saucepan, add onion and fry until soft and lightly browned. Add the garlic and coriander and fry for a further 2 minutes, stirring frequently. Add the diluted tomato purée, salt, bay leaves, allspice, cayenne and black peppers and the water and stir well. Bring to the boil, add kibbeh balls and simmer for about 30 minutes or until the sauce thickens then stir in the lemon juice. Transfer to a serving dish and sprinkle with the parsley.
kharperti kufta stuffed kibbeh in tomato sauce
Kharpert (Harput) is a small town in Turkey famed for nothing else perhaps, save this kibbeh dish, which is excellent on a cold winter's day.
Kibbeh see basic kibbeh mixture (recipe)
Filling
2 tablespoons ghee
1 large onion, finely chopped
225 g/8 oz minced lamb
2 tablespoons green pepper, chopped
2 tablespoons parsley, chopped
½ teaspoon dried basil
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon cinnamon
3 tablespoons walnuts, chopped
Sauce
1.8 litres/3 pints stock
3 tablespoons tomato purée
1½ teaspoons salt
1 teaspoon dried mint
Heat the ghee in a saucepan, add the onion and fry until soft. Add the meat and green pepper and fry for about 30 minutes, stirring occasionally, until the meat is cooked. Add the remaining ingredients and set aside to cool. When you have prepared the kibbeh mixture break off lumps and form into balls about 4 cm/1½ in in diameter.
To stuff the kibbeh take one of the balls and, keeping your hands damp, hold it in the palm of one hand and push the forefinger of the other hand into the centre. Press all around the wall while rotating the ball slowly and evenly. Continue until the walls are as thin as possible. Place a tablespoon of the filling in the opening, lightly moisten your hands, draw the edges together and seal tightly. Roll the ball between your palms to give it a round and smooth shape and then press gently between your palms to flatten slightly. Reserve and proceed until you have used up all the ingredients.
Bring the stock to the boil in a large saucepan and stir in the tomato purée, salt and dried mint. Add the kibbehs, a few at a time, and cook for 10–15 minutes or until they rise to the surface. Serve a few kibbehs in each soup bowl with some of the sauce.
vospov kufta lentil kibbeh
Kibbehs and kuftas—the latter is derived from old Aramaic meaning 'minced', shredded—are also made with other ingredients such as lentils, split peas and rice. The latter two are popular in areas where either burghul is unknown or difficult to find such as Iran, Caucasus, Egypt and southern Iraq.
One of the best known of these 'burghul-less' kibbehs is the recipe below which is popular with Armenians, Turks and the Kurds of southern Turkey. It is a traditional Lenten dish for Christian Armenians, but often it is the staple food for thousands of peasants. Serve with yoghurt salads of your choice, fresh salads, pickles, etc.
175 g/6 oz brown lentils, washed and drained
approximately 900 ml/1½ pints water
1½ teaspoons salt
100 g/4 oz fine burghul
200 ml/⅓ pint olive oil
1 onion, finely chopped
2 spring onions, including green tops, finely chopped
1 green pepper, seeded and finely chopped
2 tablespoons parsley, finely chopped
2 tablespoons fresh mint, finely chopped or 1 tablespoon dried mint
Garnish paprika
Place the lentils in a large saucepan with the water and salt. Bring to the boil, lower heat and simmer for about 30 minutes or until the lentils are tender. Add a little more water if necessary. Stir in the burghul and half of the oil. Simmer for a few minutes then turn off the heat, cover and set aside for 15 minutes—at the end of which time the water should have been absorbed.
Meanwhile, heat the remaining oil in a small pan, add the chopped onion and fry until soft and golden.
Empty the lentil and burghul mixture into a large mixing bowl and add the cooked onion with the oil. Knead the mixture well, keeping your hands damp with warm water, until smooth. Mix in half the spring onions, green pepper, parsley and mint. Taste and adjust seasoning if necessary. Keeping your hands moist shape the mixture into small patties about 2.5–4 cm/1–1½ in long and 2 cm/¾ in wide. Arrange on a serving dish and sprinkle with the paprika and remaining spring onion, green pepper, parsley and mint. Serve warm.
topig stuffed chickpea and wheat balls
A classic Armenian Lenten dish traditionally served by the churches and charitable organizations.
Serve cold as an appetizer or as a main dish with olive oil sprinkled over them and garnished with lemon wedges.
There are several variations. One of the traditional ingredients is skinless wheat, but if this is not available mashed potato is a suitable substitute.
Filling
3 tablespoons olive oil
1 large onion, finely chopped
2 tablespoons flour
3 tablespoons pine kernels
3 tablespoons raisins or sultanas
2 tablespoons parsley, finely chopped
¼ teaspoon allspice
1½ teaspoons salt
½ teaspoon black pepper
pinch cinnamon
3 tablespoons tahina paste
Kufta shell
100 g/4 oz whole-grain wheat, skinless—if this is not available peel 100 g/4 oz potatoes, boil until tender and mash
175 g/6 oz chickpeas, soaked overnight in cold water
paprika
cumin
100 g/4 oz fine burghul
3 tablespoons onion, finely chopped
1 egg
½ teaspoon paprika
1 teaspoon salt
¼ teaspoon cayenne pepper
olive oil
Garnish lemon wedges
Prepare the filling first—preferably the day before—by heating the oil in a large saucepan. Add the onion and fry until soft. Add the flour and fry for 2–3 minutes, stirring frequently. Remove from the heat and stir in the remaining filling ingredients. Mix thoroughly, transfer to a bowl and chill in the refrigerator—preferably overnight.
To prepare the kufta first half fill a saucepan with water, bring to the boil, add the wheat, turn off the heat and leave the wheat to soak for 2 hours.
Drain and rinse the chickpeas and place in a large saucepan half-filled with water. Bring to the boil then lower heat and simmer until tender. Remove any scum that appears on the surface. Add more water if necessary. Drain chickpeas and, when cool enough to handle, remove and discard the skins by pressing each chickpea gently between thumb and forefinger. Pass the chickpeas (and wheat if using it) through a meat mincer or chop finely.
Place burghul in a bowl and wash with cold water until water poured off is clean. Place the burghul, chickpeas and wheat or mashed potato in a large bowl. Add the onion, egg, paprika, salt and cayenne pepper and knead for about 5 minutes or until the mixture is smooth. Add a little water if necessary, but do not make it too damp or it will be difficult to shape. To make the stuffed kufta break off a piece of the kufta and roll into a ball about 4 cm/1½ in in diameter. Keeping your palms damp hold the ball in one hand and make a hole in it with the index finger of the other hand. Press the index finger down into the palm of the other hand squeezing out the kufta and making the shell thinner. Fill the opening with 2.5–4 cm/1–1½ teaspoons of the filling and then close the opening by drawing the edges together and sealing. Roll the kufta between your palms to make a smooth round shape. Slowly rotate the ball of kufta so that the finger is pressing down on a new part of the kufta shell and making it thinner. Continue until all the filling and kufta have been used up.
Two thirds fill a large deep saucepan with water and bring to the boil. Drop in a few kuftas at a time and simmer for 10 minutes. Remove with a slotted spoon to a large plate. Repeat until all the kuftas are cooked. Sprinkle the kuftas generously with the cumin and paprika and refrigerate for a few hours. Sprinkle with olive oil and serve with lemon wedges.
Serves 6 people.
ttoo kufta a sour meat and kibbeh stew
This stew of meat, chickpeas, onions, mint and marble-sized kibbehs has a slightly sharp flavour from the added lemon juice. It is a regional recipe from Cilician Armenia and is usually eaten as it comes with bread. My mother often added pumpkin or aubergine to make it a very substantial meal. I strongly recommend the pumpkin version.
Sauce
3 tablespoons oil
2 medium onions, sliced
225 g/8 oz lamb, cut into 1.2 cm/½ in pieces
100 g/4 oz chickpeas, soaked overnight
2 tablespoons tomato purée
2 teaspoons salt
1 teaspoon black pepper
2 cloves garlic, crushed
1 level tablespoon mint
2 tablespoons lemon juice
Kufta see recipe for basic kibbeh
Heat the oil in a large saucepan, add the onions and meat and fry for about 5 minutes, turning frequently. Drain the chickpeas and add to the pan with plenty of water. Bring to the boil and then lower the heat and simmer until the meat and chickpeas are tender. Remove any scum which appears on the surface. Stir in all the other sauce ingredients and simmer for a further 20–30 minutes. Taste and adjust seasoning if necessary.
Prepare the kufta as described in the basic kibbeh recipe.
These kuftas are 'blind', i.e. they have no filling. To make them keep your hands damp with cold water, break off small pieces of kibbeh and roll between your palms to form small marble-sized balls. Add these to the sauce and simmer gently for about 30 minutes Serve in soup bowls, accompanied by bread.
Variations
450 g/1 lb pumpkin, peeled and cut into 2.5 cm/1 in cubes or 450 g/1 lb aubergine, cut into 2.5 cm/1 in cubes
Add these vegetables to the sauce when you add the kuftas.
kabat al batatis min burkul
potato kibbeh with apricot filling
This recipe from Iraq, also popular in the Gulf States, has a fascinating and unusual meat and apricot filling associated more with Iranian-Caucasian dishes than with the more simple food of the nomad.
The kibbehs are fried in oil and served with fresh vegetables.
Filling
2 tablespoons ghee or oil
1 onion, finely chopped
350 g/12 oz minced lamb
2 tablespoons chopped almonds or hazlenuts
100 g/4 oz dried apricots, chopped
¼ teaspoon ground cloves
¼ teaspoon cumin
¼ teaspoon nutmeg
½ teaspoon paprika
1 teaspoon salt
½ teaspoon black pepper
3 tablespoons water
Kibbeh
450 g/1 lb potatoes, boiled and mashed
100 g/4 oz fine burghul
25 g/1 oz plain flour
oil for frying
1 egg
1½ teaspoons salt
½ teaspoon black pepper
2 tablespoons water
Garnish fresh vegetables, e.g. radishes, cucumber, tomatoes, lettuce, etc.
First prepare the filling by heating the ghee or oil, adding the onion and frying until soft. Add the meat and fry for 5–10 minutes until browned and the lumps have been broken down. Add the remaining filling ingredients, mix well and cook for 15–20 minutes, stirring frequently. Set aside to cool.
Place all the kibbeh ingredients in a large bowl and knead for about 5–10 minutes. Shape and stuff the kibbeh balls following the instructions for kharperti kufta, keeping your hands damp with cold water.
Heat enough oil in a saucepan to deep fry, add the kibbeh, a few at a time and fry gently for 8–10 minutes or until golden. Remove with a slotted spoon, drain, arrange on a serving dish and keep warm while you cook the remaining kibbeh. Serve hot with fresh vegetables.
## yoghurt dishes
One of the most important ingredients in Middle Eastern cooking, yoghurt (see Glossary for methods of preparation, labna and stabilized yoghurt) is not only used as an accompaniment to dishes, in salads or as a refreshing drink; it also appears in soups and stews—often as a main ingredient—as well as in cakes, breads and desserts.
I have included in this chapter a fragment of the vast repertoire of yoghurt dishes that I like—the choice is very personal indeed and the dishes included are not necessarily the most well known.
Having explained traditional methods of preparation in the Glossary, there are a few recipes following which make use of labna, drained yoghurt.
mast-e-kisei labna with herbs
In restaurants and at home the usual breakfast will be a plate of labna topped with a little olive oil and served with bread. The Iranians like to add chopped fresh dillweed, parsley, mint, chives, tarragon or other herbs that are freshly available.
Mix the labna and chopped herbs together, spread on a plate, garnish with a few chopped spring onions and dribble a little olive oil over the surface. Serve with hot bread for breakfast.
The choice of herbs depends on availability and the mixture is strictly personal and so there is plenty of scope to experiment.
madzna banir yoghurt cheese
A variation of the above from Armenia.
Prepare labna from yoghurt (see Glossary). Shape the labna into walnut-sized balls—a 'melon scoop' makes this an easy job. Arrange the balls in a serving dish and pour a little olive oil over them. Mix together 1 tablespoon fresh mint, 1 tablespoon fresh dillweed and 2 tablespoons chives—all finely chopped—and sprinkle over the balls.
shomin spinach and yoghurt salad
Spinach and yoghurt are like horses and carriages and love and marriage. Although I cannot vouch that all horses and carriages work harmoniously I know for sure that spinach and yoghurt always succeed. There are many such recipes throughout the Middle East. I have selected two classics, one from Armenia and the second from Iran.
450 g/1 lb fresh or 225 g/8 oz frozen spinach
50 g/2 oz butter
1 small onion, finely chopped
salt and black pepper to taste
300 ml/½ pint yoghurt
2 cloves garlic, crushed
Garnish
2 tablespoons finely chopped toasted walnuts (optional)
paprika
Strip the leaves from the stalks of the fresh spinach and wash very thoroughly to remove all grit and sand. Thaw out frozen spinach. Half fill a large pan with water, bring to the boil and add the spinach. Simmer for about 8–10 minutes or until the spinach is just cooked. Strain into a colander and leave until cool enough to handle. Squeeze out as much of the moisture as possible and then chop the spinach.
Melt the butter in a large frying pan, add the onion and fry until soft and beginning to brown. Add the spinach and fry for a further 5 minutes, stirring frequently.
Season to taste with salt and black pepper.
Keep on a low heat while you mix the yoghurt and crushed garlic together in a small bowl. Divide the spinach into 4 portions and arrange each on a small plate in a circular shape. Spoon some of the yoghurt into the centre of each circle and then sprinkle the yoghurt with a little paprika. Garnish with a few chopped walnuts if you wish.
borani-ye-esfenjag
This is usually served with meat dishes but, as the shomin, it can be eaten as an appetizer or a savoury dish with bread.
225 g/8 oz fresh spinach
2 tablespoons lemon juice
1 tablespoon onion, finely chopped
½ teaspoon salt
pinch black pepper
300 ml/½ pint yoghurt
Garnish 1 tablespoon fresh mint, finely chopped, or 1 teaspoon dried mint
Wash the spinach several times in cold water until all the sand and grit has been removed. Strip the leaves from the stalks and discard the stalks.
Bring water to the boil in a large saucepan. Add the spinach, lower the heat and simmer for about 8–10 minutes. Drain the spinach into a colander and leave until cool enough to handle. Squeeze as much moisture as possible out of the spinach, chop it finely and place in a salad bowl. Add the lemon juice, onion, salt and pepper and mix. Add the yoghurt and mix thoroughly. Refrigerate for at least 1 hour. Serve garnished with the mint.
yogurtlu çop kebab braised beef with yoghurt
This recipe is from Trakya (the European part of Turkey) and, more precisely, from the region of Edirne (Greek Adrianople). It is also popular in Greece and Bulgaria. It is of course not a true kebab as the meat is neither skewered nor grilled, but rather a stew of sliced beef cooked in yoghurt.
It is an extremely tasty dish from the days of the Ottoman rule.
3 tomatoes, blanched, peeled and sliced
1 small onion, finely chopped
salt and black pepper
675 g/1½ lb lean braising beef, cut into 1.2 cm/½ in pieces
50 g/2 oz butter, melted
1 teaspoon paprika
1½ teaspoons marjoram
3 whole chillies
hot water
300 ml/½ pint yoghurt
2 eggs
2 teaspoons flour
2 teaspoons malt vinegar
Garnish some parsley, finely chopped
Butter a heavy saucepan or casserole dish—the latter preferably so that you can take it straight to the table and serve from it. Arrange the tomato slices over the base and then cover with the chopped onion. Sprinkle with a little salt and black pepper. Arrange the meat over the onions, pour the melted butter over the top and then sprinkle with a little more salt and pepper, the paprika and the marjoram. Add the chillies and sufficient hot water to half fill the casserole. Bring to the boil and then simmer gently until the meat is tender and very little liquid remains. Add a little more water while cooking if necessary.
Put the yoghurt into a mixing bowl and beat in the eggs, flour and vinegar. Stir the yoghurt mixture into the casserole. Bring just to the boil, stirring gently until the sauce thickens. Remove from the heat, sprinkle with the parsley and serve immediately.
borani-ye goosht a lentil and yoghurt stew
This is a typical Middle Eastern stew of lentils, aubergines, lamb and yoghurt. Popular throughout the many lands, this particular recipe is from Iran—hence the local touch of saffron.
Borani is the Persian word for cold dishes made of yoghurt mixed with various vegetables and herbs, named after Poorandok—the daughter of King Khossow Parviz who, it is said, had a special fondness for yoghurt and yoghurt dishes. The many Indian biriani dishes have undoubtedly been influenced by their Iranian counterparts. Serve it cold with bread or hot with a rice pilav.
1 large aubergine
25 g/1 oz butter
1 onion, finely chopped
450 g/1 lb shoulder of lamb, cut into 1.2 cm/½ in pieces
600 ml/1 pint stock
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon saffron diluted in 1 tablespoon hot water
½ teaspoon oregano
75 g/3 oz whole brown lentils
600 ml/1 pint yoghurt
1 clove garlic, finely chopped
Peel the aubergine, cut in half lengthways and then cut crossways into 0.6 cm/¼ in slices. Arrange the slices over a large plate, sprinkle with salt and set aside for 30 minutes. Then drain, rinse and dry with kitchen paper.
Melt the butter in a large saucepan and sauté the onion until soft and golden. Add the meat and cook for a few minutes, turning frequently. Add the stock, salt, pepper, saffron and oregano and bring to the boil. Cover and simmer for 1 hour. Rinse the lentils and add to the pan, cover and simmer for a further 30 minutes or until the lentils are tender. Add more water if necessary. Add the aubergine slices, cover again and simmer for a further 20–30 minutes.
At this stage it is possible to serve the stew with the yoghurt stirred through it. However, the whole stew is usually cooled slightly and then liquidized to a pulp or pounded to a paste in a large mortar. The yoghurt is then stirred into the pulp and the garlic is sprinkled over the top.
badami goosht lamb with saffron and almonds
A dish that is equally at home in North-West India, Pakistan and Afghanistan as in its original habitat of Iran. The Indian touch can be discerned in the use of a cinnamon stick, ginger and cardamom pods.
An exoitic and colourful dish. Serve with a rice pilav.
½ teaspoon saffron strands soaked in 2 teaspoons hot water
300 ml/½ pint yoghurt, stabilized with 1 egg or 1 tablespoon flour
2 teaspoons salt
900 g/2 lb leg of lamb, boned, excess fat removed, meat cut into 2.5 cm/1 in cubes
50 g/2 oz ghee or butter
1 small stick cinnamon
3 whole cloves
1 onion, finely chopped
2 cloves garlic, finely chopped
1 teaspoon grated fresh ginger
1 teaspoon ground cumin
1½ tablespoons ground almonds
3 cardamom pods, optional
300 ml/½ pint water
1 tablespoon fresh mint, chopped, or 1 teaspoon dried mint
Squeeze the saffron strands to remove as much of the colour and fragrance as possible. Put the stabilized yoghurt into a large bowl and stir in the saffron water and salt. Add the meat cubes, turn until coated and then set aside.
In a large saucepan or casserole melt the ghee or butter, add the cinnamon stick and cloves and fry for a few minutes. Add the onion, garlic and ginger and fry gently for a few minutes until the onion is golden. Now add the cumin and fry for 2 more minutes. Drain the pieces of meat from the marinade, but reserve the marinade, add the meat to the pan and toss in the spices until well coated.
Stir in the yoghurt marinade, almonds, cardamom pods and 300 ml/½ pint water, lower the heat and simmer for about 1 hour or until the lamb is tender and the sauce is thick. Stir frequently to prevent the meat sticking to the pan. Stir in the mint and serve immediately on a bed of pilav.
immos assyrian meat and yoghurt stew
This very old dish, of Assyrian origin, is still popular in Syria and Lebanon. The name, literally translated, means 'cooked in its mother's image'. Before the appearance of yoghurt many such dishes were cooked in milk, a method still popular in parts of Russia and also in Central and Southern India. A rich and creamy meal which is best served with a rice or burghul pilav.
300 ml/½ pint water
1 tablespoon oil
2 onions, sliced
900 g/2 lb leg of lamb, cut into 2.5 cm/1 in pieces
1 teaspoon salt
½ teaspoon black pepper
2 cloves garlic, crushed
1 teaspoon fresh parsley stalks, chopped
600 ml/1 pint yoghurt
1 tablespoon cornflour mixed to a paste with 1–2 tablespoons water
grated rind 1 lemon
Garnish 1 tablespoon fresh coriander or parsley, chopped
Bring the water and oil to the boil in a large saucepan. Add the onions, meat, half the salt, the pepper, garlic and parsley stalks. Cover the pan, reduce heat and simmer gently until the lamb is tender and the liquid has been reduced by two thirds.
Place the yoghurt and cornflour mixture in a saucepan with the remaining salt and bring gently to the boil, stirring constantly. Reduce heat to very low and cook for 8–10 minutes. Add the yoghurt mixture and lemon rind to the lamb mixture and simmer, uncovered, for a further 15 minutes. Pour into a large serving dish and sprinkle with the coriander or parsley.
mansaaf lamb with yoghurt
A famed Arab dish very popular in Jordan, Palestine and Saudi Arabia. Seated in their colourful tents the Bedouins prepare this festive dish (often a whole lamb) which is boiled, piled on top of a bed of pilav and served with thin Arab bread—khubz-el-saj.
With this dish the Bedouins honour their guests. The food is eaten in the right hand—a lump of meat is pulled off, rice is rolled around it and the mixture popped into the mouth.
Below is my slightly simplified version of how to prepare, serve and eat this traditional dish.
900 g/2 lb stewing meat, cut into 5 cm/2 in pieces
1 onion, quartered
1½ teaspoons salt
900 ml/1½ pints water
450 ml/¾ pint yoghurt
1 egg
2 tablespoons vegetable oil
2 tablespoons pine kernels
2 tablespoons almonds
Place the meat, onion, salt and water in a large saucepan and bring to the boil. Simmer for about 45–60 minutes or until the meat is tender. Remove any scum which appears on the surface and add more water if necessary. Remove the meat from the pan and retain the stock.
Put the yoghurt in a large saucepan, add the egg and beat. Slowly bring to the boil, stirring constantly in one direction. Add about 300–450 ml/½–¾ pint of the meat stock, stir well and bring to the boil. Add the pieces of meat and heat through. Remove the pan from the heat.
Heat the oil in a small pan, add the nuts and fry gently until golden. Spread a plain rice pilav on a large serving dish, pile the meat in the middle and cover with half the yoghurt sauce. Sprinkle with the nuts and serve immediately with the remaining sauce in a separate jug.
dami ghalebi ba morgh
rice with chicken and dried fruit
A delicious and decorative dish from Iran.
It is usually cooked in a mould and inverted on to a serving dish showing a golden brown crust. However, it can be layered into a casserole and baked in the oven or steamed over a low heat.
Serve with salads of your choice.
4 large prunes, stoned
8 dates, stoned
8 dried apricots
8 dried peaches (optional)
2 teaspoons salt
350 g/12 oz rice, washed thoroughly under cold water and drained
900 ml/1½ pints water
4 chicken breasts, washed and dried
1 onion, thinly sliced
150 ml/¼ pint chicken stock or water
1 teaspoon salt
½ teaspoon black pepper
100 g/4 oz butter, melted
½ teaspoon saffron
150 ml/¼ pint yoghurt
50 g/2 oz walnuts, chopped
50 g/2 oz raisins
Cut the dried fruits into small pieces, place in a bowl of cold water and set aside.
Place the salt and water in a large saucepan and bring to the boil. Add the rice and simmer for about 20 minutes or until the water has been absorbed.
Meanwhile, place chicken breasts in a large saucepan with the onion, stock, salt and pepper, cover and simmer for about 30 minutes or until tender. Turn occasionally. Cool the chicken and remove and discard any bones. Reserve the stock.
In a small bowl mix together half the melted butter, the saffron, the yoghurt and 1 teacup of the cooked rice.
Drain the dried fruit. If using a mould coat its entire surface with this mixture. Over this mixture arrange first a layer of plain rice, then some dried fruit, some chicken pieces, some of the chopped walnuts and raisins and 1–2 tablespoons of the stock. Continue alternating layers until the mould is full ending with a layer of rice. Bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 1 hour.
To unmould dip the mould up to the rim, in cold water for 2 minutes and invert on to a serving dish.
If using a saucepan or casserole first spread the butter-yoghurt-rice mixture over the base and then alternate the ingredients as described for the mould method. If using an ovenproof casserole bake at 190°C, 375°F or gas mark 5. Otherwise place the saucepan over low heat, wrap the lid in a tea towel and fit firmly on the pan. Steam for between 30–45 minutes.
dajaj souryani assyrian chicken with yoghurt
The Assyrians, once a mightly nation who 'came down like hungry wolves', are still found scattered throughout Turkey, Iraq, Syria and Iran. Their glorious days are over, but they continue to practice their age-old customs.
The recipe below is from Baghdad and was given to me by an Assyrian student I befriended in Britain. He asked me, in one of my books to 'include our chicken and emphasize that it is Assyrian—that is very important'. I have kept my promise and I hope you like it.
1.35 kg/3 lb roasting chicken, cut into 8 serving pieces
4 tablespoons butter or ghee
1 onion, finely chopped
1 green pepper, thinly sliced
600 ml/1 pint chicken stock
1 teaspoon salt
½ teaspoon black pepper
2 tablespoons sumac powder
2 tablespoons ground almonds
300 ml/½ pint yoghurt
Garnish
1 teaspoon cayenne pepper
1 teaspoon cumin
Melt the butter or ghee in a large saucepan and cook the chicken pieces until golden brown all over. Remove the chicken pieces from the pan to a large plate and keep warm. Add the onion and green pepper to the pan and sauté for a few minutes until the vegetables are soft. Add the stock, salt, pepper and sumac and bring to the boil. Return the chicken pieces to the pan, cover, lower the heat and simmer for 40–60 minutes or until the chicken is tender. Transfer the chicken pieces to a serving dish and keep warm.
Add a few tablespoons of water to the ground almonds and stir to a smooth paste. Stir into the juices in the pan and bring to the boil stirring constantly. Remove from the heat and stir in the yoghurt. Pour the sauce over the chicken and sprinkle with the cayenne pepper and cumin. Serve with a rice or burghul pilav.
nourov jud chicken with pomegranates
What's in your pocket?
that's what I want.
In your shirt pocket?
An apple you've found?
It's a pomegranate.
It's yours if you give
as many kisses as
the pomegranate has seeds.
Pomegranates can have
a thousand red seeds!
A thousand kisses?
How could that be?
When two love
become one love
It's easily done.
A thousand come quickly,
A thousand and one!
(Armenian folk song)
A family favourite which is simple and very attractive to look at when decorated with the sparkling seeds of the pomegranate.
1.35 kg/3 lb chicken
50 g/2 oz butter
1 onion, thinly sliced
300 ml/½ pint chicken stock or water
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon cumin
600 ml/1 pint yoghurt, stabilized with 1 egg or 1 tablespoon plain flour
Garnish
2 pomegranates, skins removed and red seeds separated and retained in a bowl
Wash and dry the chicken and cut into 8 serving pieces. Melt the butter in a large saucepan and sauté the onion until golden. Add the chicken pieces, stock, salt, pepper and cumin and bring to the boil. Cover and simmer for about 45 minutes or until tender, turning occasionally. When ready to serve remove from the heat and spoon a few tablespoons of the hot sauce into the yoghurt. Gently stir the yoghurt into the saucepan and heat through. Arrange the chicken joints in a serving dish, pour the sauce over the top and then sprinkle the pomegranate seeds all over.
saray ordek duck in yoghurt
One of the many creations of the great nineteenth-century chef-hotelier Tokatlian of Istanbul.
There are some Middle Eastern superstitions about duck featuring in your dreams: to see one means to leave home for distant lands; to hunt one means to wish to return to a distant loved one; to eat one means to live in poverty abroad—how wrong can you be! To eat duck nowadays means to wallow in unashamed luxury—just look at the prices in the shops!
1.35–1.8 kg/3–4 lb oven ready duck
25 g/1 oz butter
5 tablespoons dry white wine
450 ml/¾ pint yoghurt stabilized with 1 egg or 1 tablespoon plain flour
450 ml/¾ pint chicken stock or water
1 teaspoon salt
½ teaspoon black pepper
10 button onions
50 g/2 oz green olives, stuffed
50 g/2 oz black olives, stoned
Garnish
2 spring onions, finely chopped
2 tablespoons parsley, finely chopped
Wash and clean the duck, prick all over with a fork to let the fat run out during cooking. Melt the butter in a large pan, add the duck and brown it all over. Remove the duck to a plate, sprinkle with the wine and keep warm.
In a large, deep saucepan mix the yoghurt and chicken stock together, season with the salt and pepper, add the duck, cover and cook over a low heat until the duck is tender, turning it a few times during cooking.
Meanwhile, cook the onions in the butter remaining in the first pan and, when soft and golden, add the olives, stir and remove from the heat.
Arrange the duck on a large plate; either whole or cut into quarters; and surround with the onions and olives. Garnish with the spring onions and parsley. Serve the sauce separately.
## ganachi–cooked vegetables
A table without vegetables is like an old man devoid of wisdom.—Arab saying.
'Never serve boiled vegetables' was one of my mother's first pieces of advice to her future daughter-in-law. 'Fry, stew, braise, pour sauces over, but never boil in water. It isn't our custom and remember, a man's heart is reached through his stomach.' How right she was!
In the Middle East vegetables are only ever boiled, drained and served when feeding the infirm and the sick.
The area is famed for its wealth of excellent, high quality vegetables, all of which can now be found in Europe and America. Some, such as courgettes, okra, aubergines, green peppers and pumpkins have been, until recently, little known and appreciated, but through the influx of people from the Mediterranean regions and the Far East they are becoming daily more available. Visit your local Greek or Indian grocer and see how the aubergines, okra, silverbeet, artichokes, flat-leaved parsley, avocados, yams, etc., are displayed there.
When served cold, vegetables should be cooked in olive oil, but when served hot they can be cooked in butter, ghee or a vegetable oil. They are eaten raw with a little seasoning and lemon juice, or with a yoghurt dressing; they are baked in the oven, grilled, stuffed with fruits, nuts, rice or meat mixtures. They are stewed with dried legumes, high in food values, such as lentils, chickpeas and beans, and with meats.
Whenever possible use fresh vegetables and only when not available resort to frozen or tinned versions—however good these may be they can never equal their fresh counterparts.
fried vegetables
Deep fried vegetables are very popular in the Middle East. They are either cooked plain or dipped in egg or in egg and flour.
Traditionally olive oil is used, but since this item is increasingly becoming almost a luxury other oils such as corn oil or, a Middle Eastern favourite, sunflower seed oil are good substitutes.
fried aubergines
450 g/1 lb aubergines, cut crossways into 0.6 cm/¼ in slices
2 tablespoons salt
150–300 ml/¼–½ pint oil
2 eggs, beaten
1 clove garlic, crushed (optional)
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon dillweed
1 teaspoon dried mint
Arrange the aubergine slices on a plate, sprinkle with the salt and set aside for 30 minutes. Rinse under cold water and dry with kitchen paper.
Heat the oil in a large frying pan.
Mix the eggs with the garlic, salt, pepper, dill and mint.
Dip the aubergine slices, a few at a time, into the egg mixture and then fry, turning once, until golden on both sides. Remove with a slotted spoon, drain and keep warm while you fry the remaining slices in the same way. Serve warm as an accompaniment to all kinds of meat dishes.
It is also sometimes served cold as an appetizer.
fried courgettes
Courgettes are thinly sliced—to help them lose their capacity for absorbing too much oil—and then either simply fried in oil, or dipped in egg and fried or dipped into a batter made from 100 g/4 oz plain flour, 150 ml/¼ pint of water and 1 egg and then fried.
Serve with lemon wedges and garnished with chopped parsley, tarragon or mint fried in the same oil.
fried broad beans
450 g/1 lb young broad beans
900 ml/2 pints lightly salted water
flour
oil for frying
Remove the tops and tails of the broad beans and pull off the strings, but do not cut or shell. Bring the lightly salted water to the boil in a large saucepan, add the beans and cook for 4–5 minutes. Remove, drain and leave to dry on kitchen paper.
Heat enough oil in a pan to cover the bottom by about 5 cm/2 in. Roll the beans, a few at a time, in the flour and fry in the oil, turning once, until golden all over. Remove, drain and keep warm while you cook the remaining beans in the same way. Serve warm, with lemon wedges, with all kinds of meat dishes.
fried cauliflowers
1 cauliflower, washed and separated into florets
1.2 litres/2 pints lightly salted water
2 eggs, beaten
1½ teaspoons salt
1 teaspoon black pepper
oil for frying
fine breadcrumbs
Bring the lightly salted water to the boil in a large saucepan, add the florets and cook for about 10 minutes or until just tender. Drain in a colander and dry with kitchen paper.
Mix the eggs with the salt and pepper in a shallow dish.
Heat enough oil in a large saucepan to cover the base by about 5 cm/2 in. Dip the florets, a few at a time, into the eggs and then roll in the breadcrumbs to coat completely and fry in the oil until evenly browned. Remove with a slotted spoon, drain and keep hot while you cook the remaining florets in the same way. Serve as an accompaniment to all kinds of meat dishes or with natural yoghurt or with tarator salsasi (see recipe).
fried carrots
450 g/1 lb carrots, peeled and cut diagonally into 0.6 cm/¼ in slices
1.2 litres/2 pints lightly salted water
Beer batter
100 g/4 oz plain flour, sifted
1½ teaspoons salt
175–200 ml/6–7 fl oz beer
oil for frying
Boil the carrots in the lightly salted water for 5 minutes. Drain and pat dry with kitchen paper.
In a bowl mix together the flour, salt and beer until you have a smooth batter—add a little more beer if necessary.
Heat enough oil in a frying pan to cover the bottom by about 5 cm/2 in. Dip the slices, a few at a time, into the batter and fry gently until tender and golden on both sides. Remove with a slotted spoon, drain and keep warm while you cook the remaining slices in the same way. Serve with all types of meat, poultry or fish dishes.
This beer-batter is popular in Turkey, the Balkans and Southern Russia where it originated.
You can fry other vegetables, e.g. sliced potatoes, pumpkin, turnips, etc. with these methods, i.e. plain, dipped in egg, or egg and flour in a batter.
They can all be served as appetizers with plain yoghurt, a garlic-yoghurt dressing or tarator salsasi (see recipe).
badenjam min tamar-el-hind
aubergines with tamarind
This recipe from the Gulf States is undoubtedly of Indian origin and related to bhagar baigan of North India. Tamarind juice can be purchased from Indian grocers, but if you cannot find it then buy tamarind pods and make the juice as directed below.
900 g/2 lb aubergines, hulled, trimmed and cut into quarters
100 g/4 oz ghee
2 large onions, sliced
1 teaspoon mustard seeds
2 teaspoons ground coriander
1 green chilli, seeds removed, finely chopped
2 cloves garlic, crushed
½ teaspoon turmeric
½ teaspoon cumin
3 tablespoons ground almonds
½ teaspoon salt
½ teaspoon black pepper
100 ml/4 fl oz tamarind juice
If tamarind juice is not available buy tamarind pods and put into about 125 ml/5 fl oz of hot water and leave to soak for 20 minutes.
Melt the ghee in a large frying pan, add the aubergine slices and fry for a few minutes until the skins become brown. Add a little more fat if necessary. Remove with a slotted spoon and reserve. Now add the onions and mustard seeds to the pan and fry for about 5 minutes or until the onions are golden, stirring frequently. Add the coriander and chilli and fry for a further minute or two. Add the garlic, turmeric, cumin, almonds, salt and pepper and mix well. Add the tamarind juice (or if it is not available squeeze out the pods and add the required amount of liquid) to the pan together with the aubergine slices. Turn carefully to mix, cover the pan and simmer for about 15 minutes or until aubergines are tender, stirring occasionally. Serve hot with meats of your choice.
nourov sumpoog
fried aubergines with pomegranate sauce
One of the many features of the Iranian and Caucasian cuisines is their clever use of fruit and nuts in food. Pomegranates, quinces, plums, apricots, cherries, etc. are mixed with meat, poultry, rice and other dishes with unusual and fascinating results.
The simple fried aubergines take on a new dimension when prepared with pomegranates and walnuts.
Ideal with kebabs of all kinds, particularly of pork and veal.
450 g/1 lb aubergines, cut crossways into 0.6 cm/¼ in slices
2 tablespoons salt
150–300 ml/¼–½ pint oil
1 green pepper, seeded and finely chopped
3 tablespoons pomegranate sauce (see below)
2 tablespoons cold water
1 clove garlic, crushed
2 tablespoons walnuts, coarsely chopped
¼ teaspoon salt
½ teaspoon oregano
Garnish
1–2 tablespoons pomegranate seeds
First prepare the pomegranate sauce. The quantity given below should make about 100 ml/4 fl oz. Use what you need and then reserve the rest in a glass jar for future use. Squeeze the juice from the seeds of 6 large, ripe pomegranates and place in a saucepan with 75 g/3 oz sugar. Heat slowly, stirring until the sugar dissolves and then simmer until the mixture thickens to a syrupy consistency. Leave until cold then pour into a glass jar and seal.
Arrange the aubergine slices on a large plate, sprinkle with the salt and set aside for 30 minutes. Rinse under cold running water and pat dry with kitchen paper. Heat the oil in a large frying pan, add a few of the slices at a time and fry, turning once, until golden on each side. Remove, drain and reserve while you fry the remaining slices in the same way. Arrange the slices over a serving dish and sprinkle with the chopped green pepper.
Dilute the 3 tablespoons pomegranate syrup in the water, add the garlic, walnuts, salt and oregano and mix thoroughly. Pour this mixture over the aubergines and chill before serving. Garnish with the pomegranate seeds.
fasulyeh-bi-banadora beans with tomatoes
This is a typical vegetable dish popular throughout the Arab lands, as well as Turkey and the Caucasus.
You can prepare other vegetables such as cauliflower, carrots, peas, etc. in the same way. It can be served hot with pasta dishes, kebabs and roasts or cold with sughtorov-madzoon or other yoghurt dressing and bread.
450 g/1 lb fresh beans, whole
about 75 ml/3 fl oz olive or vegetable oil
1 large onion, finely chopped
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon allspice
2 large tomatoes, coarsely chopped
450 ml/¾ pint water
Wash the beans, snip off the ends and cut into 5 cm/2 in pieces.
Heat the oil in a large saucepan and sauté the onion until soft but not brown. Add the beans, salt, pepper and allspice. Cover and cook for 5 minutes.
Uncover, add the tomatoes and water and cook over a medium heat for about 30 minutes, stirring occasionally, until the beans are tender and much of the water evaporated. Serve hot or cold.
bamia basrani okra with tomatoes
A recipe from Iraq which is also popular throughout the Gulf States. There are several such dishes in the Middle East, all making use of okra—still a relatively unknown vegetable in the West. This recipe from the city of Basra—famed for its delicious dates—makes use of coriander, turmeric and tomatoes. Use fresh okra if possible. However, if using the tinned variety the cooking time will be considerably less.
Serve hot with meat and poultry dishes or cold with bread and fresh vegetables.
700 g/1½ lb fresh okra
5 tablespoons olive oil
2 onions, thinly sliced
2 cloves garlic, quartered
450 g/1 lb tomatoes, blanched, peeled and sliced
½ teaspoon turmeric
½ teaspoon coriander
1 teaspoon salt
½ teaspoon black pepper
juice 1 lemon
Wash the okra thoroughly and cut off the stems taking care not to cut into the vegetable.
Heat the oil in a large saucepan and sauté the onions and garlic until soft and lightly browned. Add the okra and cook for a few minutes, stirring gently, but frequently. Now add the tomatoes and continue frying for a few more minutes. Add the turmeric, coriander, salt and pepper. Cover with water, bring quickly to the boil then lower the heat and simmer for 30–45 minutes or until the okra is tender. Stir in the lemon juice and cook for a further 10–15 minutes. Serve hot or cold.
nokhod rawandiz chickpeas, lentils and spinach
The Kurds and Assyrians of Rawandiz in Northern Iraq prepare this dish as often as possible. Although each nation claims it as their own it is, in fact, just as popular with the Iranians, Turks and Arabs of the region as it is with them. A simple, old dish; serve it with yoghurt or as an accompaniment to meat dishes.
2 tablespoons butter or ghee
2 large onions, thinly sliced
225 g/8 oz fresh spinach, washed thoroughly, drained, stemmed and coarsely chopped or 100 g/4 oz frozen spinach, thawed
1 tablespoon lime or lemon juice
About 100 ml/4 fl oz water
1 teaspoon dillweed
¼ teaspoon nutmeg
1 teaspoon salt
½ teaspoon black pepper
175 g/6 oz brown lentils, washed
75 g/3 oz pre-cooked or canned chickpeas
Melt the butter or ghee in a large saucepan. Add the onion and fry until soft and golden. Add the spinach, lime or lemon juice, water, dillweed, nutmeg, salt and pepper and bring to the boil. Now add the lentils, sufficient water to cover by about 1.2 cm/½ in, cover the pan and simmer for 20 minutes.
Add the pre-cooked chickpeas, cover and cook for a further 10–15 minutes until the lentils are cooked. Stir occasionally and add a little more water if there is any danger of the mixture burning. Turn into a serving dish and serve hot.
shesh havij carrots with nuts
This recipe is from Iran. It is tasty and attractive and is traditionally served with a plate of saffron rice pilav and a bowl of fresh salad. It also goes particularly well with poultry and game. Use pomegranate juice if it is available as it gives extra zest to the dish.
3 tablespoons butter or ghee
1 large onion, finely chopped
350 g/12 oz carrots, peeled and thinly sliced crossways
5 dates, stoned and thinly sliced
1 tablespoon raisins or sultanas
1 tablespoon white wine vinegar
1 tablespoon pomegranate juice or 2 tablespoons lemon juice
4 eggs
1 teaspoon salt
½ teaspoon black pepper
Garnish
1 tablespoon almonds, slivered
1 tablespoon pistachios, slivered
Melt the butter or ghee in a saucepan. Add the onion and fry until soft and turning golden. Add the carrots and fry for a few minutes, stirring frequently to coat the carrots with the fat. Add the dates, raisins, vinegar and pomegranate or lemon juice and mix well. Cover the pan, lower the heat and simmer for 30 minutes. Transfer this mixture to a large frying pan or shallow casserole dish.
Break the eggs into a bowl, add the salt and pepper and beat well. Pour the eggs over the carrot mixture and cook over a very low heat until set. Sprinkle with the nuts and serve immediately.
afelia mushrooms with coriander
Afelia is a Cypriot speciality. There are many such named dishes and the one item they have in common is the abundant use of coriander. It will go well with any meat, poultry or fish dish.
50 ml/2 fl oz oil
450 g/1 lb small mushrooms, wiped clean and thickly sliced
100 ml/4 fl oz dry red wine
1½ teaspoons salt
½ teaspoon black pepper
1–2 (depending on taste) teaspoons coriander seeds, crushed
Heat the oil in a large saucepan, add the mushrooms and fry, turning frequently, until the juices evaporate. Lower the heat, add the wine, salt and pepper, cover and cook for 10 minutes. Finally add the coriander seeds and cook for 3–4 more minutes, stirring frequently. Transfer to a serving dish.
Variations
Afelias of potatoes, carrots, cauliflower, beans, etc. can be prepared in this same way.
Shakarov tutum glazed pumpkin
A Caucasian speciality, excellent with kibbeh and pilav dishes and especially with poultry and pork dishes.
225 ml/8 fl oz water
175 g/6 oz sugar
2.5 cm/1 in piece root ginger, peeled and halved
1.35 kg/3 lb pumpkin, peeled and cut into 2.5 cm/1 in cubes
pinch salt
juice 1 lemon
Garnish
2 tablespoons almonds, blanched
1 tablespoon pistachios, slivered
Bring the water to the boil in a large saucepan. Stir in the sugar and root ginger. Add the pumpkin and salt and cook until tender, occasionally stirring carefully. Remove and discard the ginger. Stir in the lemon juice and transfer the pumpkin to a serving dish.
Toast the almonds and pistachios under a hot grill for a few minutes, turning frequently to prevent burning. Sprinkle the nuts over the pumpkin and serve.
Serves 6 people.
baghala ba kadoo broad beans with lettuce
A magnificent vegetable dish from Iran that goes well with rice pilavs, roasts and kebabs.
If by chance you are entertaining a Yazidi (a tribe of Kurds who live in eastern Turkey and northern Iran) never, under any circumstances, offer them lettuce either as a salad or, as in this recipe, cooked. They will be extremely offended for the Yazidies believe that the Devil (Shaytan) lurks in the lettuce leaves, and since they are Devil worshippers you would have insulted them!
450 g/1 lb fresh or dried broad beans
1 lettuce, preferably Cos
12 small button onions, peeled or ½ onion, finely chopped
½ teaspoon dried thyme
6 tablespoons parsley, finely chopped
2 teaspoons salt
½ teaspoon black pepper
5 tablespoons water
3 tablespoons butter, melted
If using dried broad beans then soak overnight. If using fresh beans then shell and remove the transparent skins. Wash the lettuce, shake off excess moisture and chop. Place the broad beans, lettuce, onions, thyme, parsley, salt, pepper and water into a saucepan and mix well. Cover and cook over a low heat for about 30 minutes, stirring occasionally. Remove from the heat, stir in the melted butter and serve.
levivot fried potato pancakes
Before the arrival of potatoes these pancakes were made with cream cheese, but the Jews of Russia substituted and took this, as well as many other European-inspired dishes, with them to Israel.
Levivots are still called latkes by the Ashkanazim Jews of America and Europe, and naturally there are many such recipes. The one below is of Russian-Jewish origin and makes an excellent change for sautéed potatoes.
Traditionally served on the Feast of Chamcah—the Feast of Lights.
3 large potatoes
1 small onion
2 small eggs, beaten
3 tablespoons self-raising flour
1 teaspoon salt
½ teaspoon white pepper
oil for frying
Peel the potatoes and grate finely so that they are almost reduced to a pulp. Place in a sieve to drain for 10 minutes. Squeeze as much liquid as possible from the potato pulp and then place in a mixing bowl. Grate the onion and add it to the potatoes. Add the eggs, flour, salt and pepper and mix well until you have a smooth batter.
Heat a little oil in a frying pan and when it is hot put in tablespoons of the batter, flattening each with the back of the spoon to make the pancakes about 7.5 cm/3 in in diameter. Cook over a moderate heat until brown on one side. Turn and cook on the other side. About 5–7 minutes on each side will allow time for the pancakes to cook through. Remove and drain on kitchen paper. Serve immediately.
batata musulyeh potatoes with yoghurt
A Kurdish dish from Musul, North Iraq.
The Kurds, of course, are the greatest people in the world! I think not, but so they claim; and I quote a few lines from J. Murray's Sketches of Persia to give an idea of the fiercely proud character of this sadly little known race of man:
The evening before we went to Sennah I read the introductory pages of the history of the Kurds. It is written by a native and, according to this patriotic author, all the virtue and courage this world has ever known was nurtured amid the wilds and mountains of Kurdistan. Its inhabitants, he affirms, attained great glory in former ages and would have subjected the universe, but for the caution of the prophet Muhammad who, struck by the fierce look and gigantic form of a Kurd ambassador, prayed to God that this formidable race might never be united. This prayer was heard, adds my author, and the warriors of Kurdistan have ever since been at variance with each other.
Now we know, but seriously—they have produced one of the greatest leaders of history, namely Saladdin. Also many delicious dishes still comparatively unknown and this is one of them.
6 large potatoes, peeled
25 g/1 oz butter or ghee
3 tablespoons yoghurt
4 egg yolks
½ teaspoon sweet marjoram
½ teaspoon cayenne pepper
1 teaspoon salt
2 tablespoons sesame seeds
4 tablespoons walnuts, very finely chopped
3 tablespoons toasted breadcrumbs
oil for frying
Garnish lettuce leaves, tarragon leaves, spring onions and radishes
Cook the potatoes in boiling, salted water until they are tender. Drain them and mash until smooth. Put the mashed potatoes into a large bowl with the butter and yoghurt and stir until well blended. Add 2 of the egg yolks, the marjoram, cayenne pepper, salt and sesame seeds and mix well.
Take a spoonful of the potato mixture, roll between wet palms to form a walnut-sized ball. Repeat with remaining potato mixture.
Beat remaining egg yolks together in a small bowl and mix the walnuts and breadcrumbs together on a plate. Heat some oil in a large frying pan.
Dip each potato ball into the beaten egg, roll in the walnut mix and fry, a few at a time, turning until they are golden all over. This will only take a few minutes. Drain on kitchen paper, arrange on a bed of lettuce leaves and serve with garnishes.
ailazan armenian vegetable casserole
Ailazan means 'different kinds'—suggesting that any type of vegetable can be used. The recipe below is a family favourite.
Serve it hot with roasts, grills or pilavs and cold with cold meats and poultry.
1 aubergine, cut crossways into 1.2 cm/½ in slices
1 green pepper, seeded and cut into 8 pieces
2 carrots, peeled and cut into 0.6 cm/¼ in rounds
1 onion, thinly sliced
1 courgette, cut into 1.2 cm/½ in rounds
100 g/4 oz French beans, trimmed and halved
100 g/4 oz peas
3 tomatoes, blanched, peeled and coarsely chopped
2 tablespoons parsley, finely chopped
2 tablespoons mint, finely chopped
1 clove garlic, crushed
1 tablespoon sumac powder
2 teaspoons salt
1 teaspoon black pepper
70 ml/2½ fl oz olive oil–if to be eaten cold or 50 g/2 oz butter
150 ml/¼ pint water
Place all the vegetables in a large ovenproof casserole, placing the tomatoes on top of the rest. Sprinkle the parsley, mint, garlic, sumac, salt and pepper over the top. Dot with the butter or pour in the oil. Add the water, bring to the boil, cover and place in an oven preheated to 180°C, 350°F, gas mark 4. Cook for about 1 hour or until the vegetables are tender, stirring gently occasionally. Taste and adjust seasoning if necessary.
chatzilim mevushalim israeli aubergine casserole
This is the Israeli 'ratatouille'. Serve with meat dishes.
2 aubergines, cut into 1.2 cm/½ in slices
3 tablespoons dry mustard
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon cayenne pepper
50 g/2 oz butter, melted
1 large onion, thinly sliced
2 large tomatoes, sliced
1 stalk fennel, diced
2 green peppers, seeded and cut into 0.6 cm/¼ in slices
½ teaspoon allspice
Coat each aubergine lightly with the mustard, sprinkle with the salt, black and cayenne pepper and arrange on a large plate. Grease a large casserole with a little of the butter and arrange half the aubergine slices over the bottom. Now arrange the onion, tomatoes, fennel and green peppers over the aubergines. Top with the remaining aubergine slices and sprinkle with the allspice. Dribble the rest of the butter over the top then cover the casserole and bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 1¼ hours or until the vegetables are tender. Serve warm.
sabahna doual arab vegetable casserole
The literal meaning of this dish is 'Seven States'. A typical vegetable casserole which is popular throughout the Middle East. Serve with any meat, poultry or fish dish.
175 ml/6 fl oz oil
1 large onion, thinly sliced
225 g/8 oz haricot beans, soaked overnight in cold water
1.2 litres/2 pints water
2 large potatoes, peeled and thinly sliced
3 carrots, peeled and cut crossways into 0.6 cm/¼ in slices
1 celeriac, peeled and sliced
6 spring onions, chopped
2 cloves garlic, thinly sliced
1½ teaspoons salt
½ teaspoon black pepper
½ teaspoon allspice
2 bay leaves
Garnish 2–3 tablespoons parsley, finely chopped
Heat the oil in a large saucepan, add the onion and fry until soft.
Drain the beans and add to the pan with the water. Bring to the boil, then lower the heat and simmer for 1–1½ hours or until the beans are tender. Add the potatoes, carrots, celeriac, onions, garlic, salt, pepper, allspice and bay leaves. If necessary add a little more water to just cover the vegetables and then simmer until they are all cooked and most of the liquid has evaporated. Pour into a large serving dish and sprinkle with the parsley.
ayvali sabzisi kurdish vegetable casserole
A dish popular throughout Kurdistan, Northern Iran and the Caucasus, where the use of fruit in casseroles and stews is an old one. A colourful dish which has an interesting combination of flavours and textures. It is traditionally served with a burghul or rice pilav and grilled meats.
900 g/2 lb red cabbage, damaged leaves removed and cut into 4 quarters
100 g/4 oz butter or ghee, melted
1½ teaspoons salt
½ teaspoon black pepper
½ teaspoon ground cinnamon
2 tablespoons lime or lemon juice
2 large quinces or cooking apples
1 tablespoon sugar
2 tablespoons walnuts, coarsely chopped
Garnish
2 tablespoons fresh tarragon, finely chopped or 1 tablespoon dried tarragon
Remove the stalks and thick ribs of cabbage then wash under cold water, drain and chop coarsely. Grease a large casserole dish with a little of the melted fat and arrange the cabbage in the bottom. Sprinkle with the salt, pepper, cinnamon and lemon or lime juice and pour the melted butter or ghee evenly over the top. Cover the dish and place in an oven preheated to 180°C, 350°F, gas mark 4 for 1¼ hours.
Meanwhile, peel the quinces or cooking apples, core and cut into 0.6 cm/¼ in thick slices. Remove the casserole from the oven, arrange the slices over the cabbage and sprinkle with the sugar and walnuts. Cover, return to the oven and cook for a further 30–40 minutes or until the cabbage is well done. Serve sprinkled with the tarragon.
## dolmas–stuffed vegetables
Dolma is a Turkish word meaning to fill, to stuff. The idea is very old and very simple; vegetables, rice, burghul (cracked wheat), meats, fruit, nuts and spices are mixed and then stuffed into or used to fill vegetables or fruits. Vegetable and fruit corers have been found in sites as far apart as Knossos, Crete and Medzamor, Armenia (5–6000 years ago). Mentions are made too of wrapped vine leaves both in Greek and early Persian historical accounts, but it was probably during the late Middle Ages in Constantinople that the true dolmas of today were first perfected.
With the conquest of that city by the Ottomans in 1453, and with the subsequent enlargement of their empire, many aspects of the arts, science and domestic culture not only changed names but also spread through the newly acquired territories. This is why dolma-type dishes appear in North Africa, the Balkans, Saudi Arabia, Iran and as far away as the borders of India.
Dolma dishes were first the prerogative of the 'Porte'—Emperor or Sultan—they were created for the gratification of the rich and the mighty. The Ottomans, whose own cuisine had been extremely poor, avidly took up everything that came their way. It is interesting to note that even today there are no dolma or dolma-type dishes in the lands from where the original Turkic-Mongolian tribes came, i.e. Turkmenistan, Uzbekistan and Khazakstan.
To prepare dolma is an acquired art. It is time consuming, a little elaborate, but the final result is—as it was meant to be—regal. The basic requirements are time, a corer and a generous amount of imagination.
There are several standard fillings which are used throughout the Middle East. There are also some little known regional variations, some of which I have included in this section. It is important to note that stuffed vegetables can be served hot or cold. Usually the cold versions do not include meat and are offered as part of the mezzeh table or as savouries with sughtorov-madzoon—yoghurt-garlic dressing.
Hot dolma are always served as a main meal. In the past—and still today with the peasants and mountain folk—certain vegetables were cored and dried under the sun for winter use, a primitive form of preserving. Today fresh vegetables are available throughout the year and this process has been abandoned by the urbanized housewife who has also abandoned the habit of frying the stuffed vegetables in ghee or oil before cooking. This process enriches the flavour of the meal, but makes it rather heavy.
Both vegetables and fruits can be stuffed. The following are by far the most common ones—artichokes, aubergines, courgettes, peppers, onions, potatoes and tomatoes. The following items are also wrapped around fillings—vine leaves, cabbage leaves and silverbeet leaves. Although these are often called dolmas this is an erroneous description. They should be named sarma (Turkish) or patoug (Armenian) as I have done.
cold dolma fillings
The following fillings are general favourites, although they are often flavoured with different spices and herbs.
The quantities given are enough for 1–1.2 kg/2–2½ lbs of vegetables, but this does of course vary according to the size and particular vegetable used and the amount of pulp scooped out of it.
It is approximately equivalent to 6 medium aubergines, 8 medium-large tomatoes or 8 medium green peppers or courgettes.
You can also use 350 g/12 oz vine leaves—see derevi blor.
filling 1
Rice, raisins and onion—Istanbul style.
225 g/8 oz long grain rice, washed thoroughly under cold water and drained
1 large onion, finely chopped
3 tablespoons parsley, finely chopped
2 tablespoons raisins
2 teaspoons salt
½ teaspoon black pepper
filling 2
Typical Middle Eastern type.
175 g/6 oz long grain rice, washed thoroughly under cold water and drained
3 tomatoes, blanched, peeled and chopped
1 large onion, finely chopped
3 tablespoons parsley, finely chopped
2 tablespoons pine kernels
1 tablespoon raisins
1½ teaspoons salt
1 teaspoon dillweed
½ teaspoon black pepper
½ teaspoon ground cinnamon
filling 3
An Iranian favourite.
50 g/2 oz chickpeas, soaked overnight in cold water, then cooked in simmering water and drained
100 g/4 oz long grain rice, washed thoroughly under cold water and drained
3 tomatoes, blanched, peeled and chopped
1 large onion, finely chopped
1 teaspoon dried mint
1 teaspoon dried tarragon
1 teaspoon dillweed
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon cinnamon
To prepare the fillings put all the ingredients into a large bowl and knead until well blended. Then use to stuff the vegetables of your choice.
Fill each one until ¾ full—this will leave room for the rice to swell while cooking.
filling 4
Rice and mushroom—a Caucasian favourite.
4 tablespoons ghee
1 large onion, finely chopped
450 g/1 lb mushrooms, wiped clean and coarsely chopped
1½ teaspoons salt
½ teaspoon black pepper
½ teaspoon allspice
1 tablespoon chopped fresh dill or 1 teaspoon dried dillweed
Heat the ghee in a large frying pan, add the onion and fry until soft. Add the mushrooms and sauté for 2–3 minutes, stirring frequently. Add remaining ingredients, stir and proceed to stuff vegetables as directed.
filling 5
An all vegetable filling.
3–4 tablespoons oil
2 large onions, finely chopped
2 sticks celery, finely chopped
6 medium carrots, peeled and grated
2 large tomatoes, blanched, peeled and chopped
3 tablespoons parsley, finely chopped
1 teaspoon salt
½ teaspoon black pepper
2 tablespoons tarragon or dill, finely chopped, or 2 teaspoons dried tarragon or dill
Heat the oil in a frying pan, add the onions and fry until soft. Add the celery, carrots and tomatoes and fry for a further 3–4 minutes, stirring frequently. Stir in the remaining ingredients and remove from the heat.
Fill the vegetables of your choice with any of the above fillings. Those using filling 5 can be baked in the oven (190°C, 375°F, gas mark 5) with a sauce of 1.2–1.8 litres/2–3 pints water, juice ½ lemon and ½ teaspoon salt.
Any vegetable with a rice or burghul filling should be placed in a saucepan, held down with a plate and weight, covered with a sauce and cooked over a medium heat. For a typical sauce see derevi blor.
derevi blor vine leaves filled with rice and nuts
350 g/12 oz vine leaves—fresh ones are perfect, but if they are not available you can buy packets from Continental or Middle Eastern stores.
Filling
150 ml/¼ pint oil
2 onions, thinly sliced
1 green pepper, seeded and thinly sliced
175 g/6 oz long grain rice, washed thoroughly under cold water and drained
½ teaspoon chilli pepper
1 teaspoon allspice
1 teaspoon salt
1½ tablespoons tomato purée
25 g/1 oz chopped almonds
1 tablespoon parsley, chopped
Sauce
1 tablespoon tomato purée
600–900 ml/2–3 pints water
3–4 cloves garlic, crushed
1 teaspoon salt
½ teaspoon chilli pepper
3 tablespoons lemon juice
Wash the leaves in cold water, place in a saucepan and add enough water to cover the leaves. Bring to the boil and simmer for about 15 minutes and then drain into a colander.
Heat the oil in a large saucepan, add the onions and green pepper and cook for 5–10 minutes, stirring occasionally, until the onions are soft, but not brown. Stir in the rice, chilli pepper, allspice, salt and tomato purée and cook gently for a further 10 minutes, stirring occasionally to prevent the mixture sticking. Remove from the heat, stir in the chopped almonds and parsley, turn into a large bowl and leave to cool.
To make each blor, spread a leaf out flat, veins uppermost. Cut off the stem. With the cut end towards you place I tablespoon of filling near the cut end. Fold the cut end over the filling and then fold the two sides over the filling towards the centre. Roll the leaf up towards the tip and you will be left with a small cigar-shaped parcel. When you have used up all the filling use any remaining leaves to cover the bottom of a medium saucepan—this helps to prevent burning. Pack the parcels carefully and closely into the saucepan in layers and then place a plate on top to cover as many as possible and hold it down with a small weight—this prevents them from moving while cooking and becoming undone.
Mix the ingredients for the sauce together in a bowl and pour into the saucepan. The sauce should completely cover the blor. If it doesn't then add a little more water. Bring to the boil, lower the heat and simmer for 1½–2 hours. Add more water if necessary. Remove from the heat, take off the weight and plate and remove one blor to test if the leaf is tender. Allow to cool, remove from the saucepan and arrange on a serving dish. Garnish with some lemon wedges.
lahana dolmasi
cabbage leaves filled with rice and nuts
A popular recipe from Turkey. You can also use any of the other fillings suggested. Serve cold or warm with lemon wedges.
1–1.35 kg/2–3 lb head of white cabbage, thick core removed
Filling
3 tablespoons oil
2 onions, finely chopped
100 g/4 oz long grain rice, washed thoroughly under cold water and drained
1 tablespoon pine kernels
1 tablespoon blanched almonds, coarsely chopped
1 tablespoon raisins
½ teaspoon allspice
pinch paprika
1½ teaspoons salt
½ teaspoon black pepper
225 ml/8 fl oz water
juice 1 lemon
Garnish 2 tablespoons parsley, chopped and lemon wedges
Bring a large saucepan two-thirds full of lightly salted water to the boil. Place the cabbage in the water and boil for 7–8 minutes. Remove the cabbage and, when cool enough to handle, carefully peel away the outer leaves taking care not to tear them. Place the leaves in a colander to cool.
When it becomes difficult to remove the leaves return the cabbage to the water and boil for a few more minutes. Continue removing the leaves until you have all that you need. Reserve the small inner leaves. Meanwhile, heat the oil in a saucepan, add the onions and fry until soft. Add the rice, pine kernels and almonds and fry, stirring frequently, until the nuts begin to turn golden. Add the raisins, spices and water and bring to the boil. Reduce the heat, cover and simmer until the liquid is absorbed. Remove from the heat, stir in half the lemon juice and set aside.
To fill a leaf place one on a board, veins uppermost, and cut out the hard stem. With the cut end towards you place 1 tablespoon—exact amount depends on the size of the leaf—near the cut end. Fold the cut end over the filling and then fold the two sides over the filling towards the centre. Roll the leaf up towards the tip and the result will be a cigar-shaped parcel. Continue in this way until you have used up all the leaves and filling. Use any remaining leaves to cover the base of a medium saucepan. Pack the parcels carefully and closely into the saucepan in layers and sprinkle with the remaining lemon juice. Place a plate over the leaves to cover as many as possible and hold down with a small weight—this prevents leaves from unwrapping while cooking. Pour in enough water to cover and bring to the boil. Lower the heat and simmer for about 1 hour or until the leaves are tender. Add a little more water if necessary. Remove from the heat and take off weight and plate. When cool enough to handle arrange on a serving plate. Serve garnished with the parsley and lemon wedges.
vospov litsk
vine leaves filled with lentils, burghul and prunes
A regional dish from the Caucasus. It is usually served during the forty days of Lent. In the Caucasus they usually use fresh plums and plum syrup. Unless you can find these I suggest you use prunes and pomegranate juice or lemon juice.
225 g/8 oz vine leaves
Filling
75 g/3 oz brown lentils, rinsed
50 g/2 oz coarse burghul
4 tablespoons oil
1 small onion, finely chopped
6 stoned prunes, soaked overnight and then sliced
1 tablespoon raisins
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon dried mint
½ teaspoon nutmeg
1 tablespoon pomegranate juice or lemon juice
225 ml/8 fl oz water
Sauce
1 tablespoon oil
1 teaspoon salt
water
Wash the vine leaves in cold water, place in a saucepan half filled with water and bring to the boil. Simmer for 15 minutes and then drain in a colander.
Meanwhile, boil the lentils in lightly salted water for 15 minutes and then strain and set aside.
Wash the burghul in a bowl until the water you pour off is clear.
Heat the oil in a saucepan, add the onion and fry until soft. Add the lentils, burghul, prunes, raisins, salt, pepper, mint and nutmeg and stir. Add the pomegranate or lemon juice and the water and simmer over a low heat until the liquid has been absorbed. Remove from the heat and leave to cool. To fill the leaves see derevi blor. Use any remaining leaves to line the base of a saucepan. Pack the filled vine leaves carefully into the pan in layers.
Place an inverted plate over the top to cover as many as possible and place a weight on the plate. This will prevent the vine leaves moving about and becoming undone during the cooking. Add the oil, salt and sufficient water to cover. Bring to the boil, then lower the heat and simmer for 1–1½ hours. Add more water if necessary. Remove from the heat and take off the weight and plate. When cool arrange on a serving plate.
hot dolmas
Below are a few of the more popular fillings.
filling 1
A rice filling popular with Armenians and Turks.
450 g/1 lb lamb, minced twice
175 g/6 oz long grain rice, washed thoroughly under cold water and drained
1 small onion, finely chopped
1 large tomato, blanched, peeled and chopped
2 tablespoons parsley, finely chopped
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon ground cinnamon
filling 2
A rice filling Iranian-style.
450 g/1 lb minced lamb
100 g/4 oz long grain rice, washed thoroughly under cold water and drained
50 g/2 oz yellow split peas, soaked overnight and then cooked until just tender
1 small onion, finely chopped and sautéed in 2 tablespoons butter
2 tablespoons parsley, finely chopped
1 teaspoon salt
½ teaspoon black pepper
¼ teaspoon nutmeg
½ teaspoon ground cinnamon
filling 3
Another Iranian favourite used in dolmeh-ye bademjan—aubergine dolma.
1 large onion, finely chopped and sautéed in 50 g/2 oz butter
450 g/1 lb minced lamb or beef
2 tablespoons tomato purée
6 tablespoons lemon juice
1 teaspoon salt
½ teaspoon black pepper
3 spring onions, finely chopped
4 tablespoons parsley, finely chopped
1 tablespoon fresh mint, chopped or 1½ teaspoons dried mint
1 tablespoon fresh tarragon, chopped or 1½ teaspoons dried tarragon
100 g/4 oz long grain rice, washed thoroughly under cold water and drained
3 hard-boiled eggs, shelled and chopped (optional)
1 tablespoon sugar
filling 4
An Assyrian favourite used in batinjan mishi—aubergine dolma.
450 g/1 lb lamb, minced twice
1 small onion, finely chopped and sautéed in 2 tablespoons butter
100 g/4 oz long grain rice, washed thoroughly under cold water
2 tablespoons raisins
2 tablespoons pine kernels
1½ teaspoons salt
½ teaspoon black pepper
3 large tomatoes, blanched, peeled and chopped
½ teaspoon dried marjoram or basil
½ teaspoon ground cinnamon
filling 5
A burghul and meat filling which is an Armenian recipe—sumpoogi litsk—aubergine dolma. Burghul is used in much the same way as rice and can be used instead of rice in any of the above recipes.
450 g/1 lb minced lamb
1 small onion, finely chopped
175 g/6 oz coarse grain burghul, washed under cold water and drained
1 green pepper, seeded and chopped
1 tablespoon parsley, finely chopped
1–2 cloves garlic, crushed
1 tablespoon tomato purée
1 tablespoon allspice
1½ teaspoons salt
½ teaspoon chilli pepper
½ teaspoon black pepper
Below is described a method for preparing a typical dolma meal. It applies to any of the above fillings.
The vegetable prepared here is the aubergine, but you can prepare in a similar fashion courgettes, tomatoes or green peppers—the latter two do not need to be soaked in salted water. Vine leaves can also be filled with any of the above fillings.
patliçan dolmasi aubergine dolma served warm
Preparation of the aubergines—Cut the stalks off each aubergine. With an apple corer remove as much of the flesh as possible from each vegetable taking care not to split or make a hole in the shells. Leave to soak in cold salted water for 30 minutes. Rinse under cold running water. They are now ready to be stuffed.
Preparation of the filling—Place all the ingredients for the filling of your choice in a large bowl and knead until well blended. Place some of the filling in each vegetable until it is ¾ full. This ensures that there is room for the rice or burghul to swell during cooking. Arrange the vegetables in a large saucepan, cover with an inverted plate and hold it in place with a weight. This prevents the vegetables from moving while cooking and keeps the fillings in place.
Preparation of the sauce—The simplest sauce consists of 1.8–2.4 litres/3–4 pints of water seasoned with 1 teaspoon salt, ½ teaspoon black pepper and the juice of 1 lemon. To this basic sauce you can, if you wish, add one or more of the following ingredients—1 tablespoon tomato purée, 2–3 bay leaves, 1 teaspoon of either mint, thyme, dillweed, basil or marjoram.
One of the tastiest sauces is the one usually used to cook sumpoogi litsk. Mix together the following:
1.8–2.4 litres/3–4 pints water
3 tablespoons tomato purée
2 teaspoons salt
½ teaspoon chilli pepper
1 tablespoon dried mint
4 cloves garlic, halved
juice 1 lemon
Another interesting sauce is the sweet and sour one used for dolmeh-ye bademjan.
1.8–2.4 litres/3–4 pints water
100 ml/4 fl oz vinegar preferably tarragon vinegar
100 g/4 oz sugar
1–2 teaspoons saffron
Mix whatever ingredients you are going to use for the sauce in a large bowl and pour into the saucepan. The sauce must cover the vegetables by at least 2.5 cm/1 in and so add more water if necessary. Bring to the boil and then lower the heat and simmer for 1 hour. The cooking time will vary a little depending on the thickness of the vegetable shells. To taste if they are tender use a fork. If the prongs enter easily then the dolma is cooked. Add a little more water if necessary while cooking. Serve the dolma with a little of their sauce and with yoghurt and/or salads.
sheik-el-mahshi
courgettes filled with meat and nuts
Literally translated this famed Arab dish means 'King of stuffed vegetables', for it contains just meat and nuts and does away with rice—hence a regal meal. Although sometimes aubergines are substituted for courgettes, I think the latter are much more successful. This recipe is from Northern Syria. Serve with a plain rice pilav and fresh salad.
12 small or 8 medium courgettes
oil
Filling
450 g/1 lb minced lamb or beef
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon allspice
2 tablespoons parsley, finely chopped
1 small onion, finely chopped
1 tablespoon tomato purée
100 g/4 oz pine kernels or 100 g/4 oz chopped walnuts
Sauce
25 g/1 oz butter
1 clove garlic, crushed
1 tablespoon tomato purée
600 ml/1 pint water
1 teaspoon salt
½ teaspoon black pepper
Cover the base of a large frying pan with about 1.2 cm/½ in oil.
Slice the stalk ends off the courgettes. Remove as much flesh as possible from each courgette using an apple corer. Ideally the shell should be about 0.6 cm/¼ in thick. Take care not to split or make holes in the shells. Wash the courgettes under cold running water and dry with kitchen paper. Heat the oil in the frying pan, add the courgettes a few at a time and fry gently for about 10 minutes, turning from time to time. When cooked set aside on kitchen paper to drain and cool.
To make the filling put the meat and onion in a saucepan and cook over a moderate heat for 20–30 minutes, stirring frequently to prevent sticking. Stir in the remaining filling ingredients and cook for about a further 10 minutes. Spoon the meat mixture into the courgettes. Lay the vegetables side by side in an ovenproof dish.
Prepare the sauce by heating butter in a saucepan, adding the garlic and tomato purée and cooking for 3–4 minutes, stirring occasionally.
Add the water, salt and pepper, bring to the boil and pour over the courgettes. Place the dish in an oven preheated to 190°C, 375°F, gas mark 5 and bake for about 30 minutes or until the courgettes are tender. Test with a fork.
badenjan mahshi min tamar
aubergines stuffed with dates and nuts
A fascinating recipe from Babylon—a small town in Iraq near the ancient capital with its hanging gardens.
Dates and hazelnuts are the two outstanding products of the land (not counting the black oil) and this recipe makes a clever use of both.
Serve with a rice pilav of your choice or as an accompaniment to poultry dishes.
4 medium aubergines, about 175 g/6 oz each, hulls removed
5–6 tablespoons butter or ghee
1 onion, finely chopped
1 clove garlic, crushed
1 large green pepper, seeded and thinly sliced
3–4 mushrooms, thinly sliced
2 large tomatoes, blanched, peeled and coarsely chopped
8 stoneless dates, thinly sliced
3 tablespoons hazelnuts, coarsely chopped or slivered almonds
½ teaspoon salt
pinch turmeric
3–4 tablespoons orange or apple juice
Cut each aubergine in half lengthways, arrange the halves in a baking dish and bake in an oven preheated to 200°C, 400°F, gas mark 6 until the flesh is soft. Remove and leave until cool enough to handle. Scoop the flesh out with a spoon leaving a shell about 0.6 cm/¼ in thick. Reserve the flesh.
Melt the fat in a saucepan, add the onion and fry until soft. Add the garlic, green pepper, mushrooms and tomatoes, stir well and fry for about 5 minutes. Chop the aubergine flesh and add to the pan. Add the remaining ingredients, mix thoroughly and cook for a further 3–4 minutes, stirring frequently. Remove from the heat.
Grease a large, shallow baking dish and arrange the halved aubergines in it, side by side. Spoon the date and nut mixture into the aubergines and bake in an oven preheated to 190°C, 375°F, gas mark 5 for 12–15 minutes. Remove from the oven and serve.
batatat mahshi
potatoes stuffed with meat and nuts
Popular with all, particularly in Saudi Arabia.
12 medium-large potatoes, peeled
3 tablespoons ghee or butter
Filling
450 g/1 lb minced lamb or beef
1 small onion, finely chopped
1 tablespoon tomato purée
100 g/4 oz pine kernels or chopped walnuts
2 tablespoons seedless raisins
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon allspice
Sauce
3 tablespoons tomato purée
450 ml/¾ pint water
1½ teaspoons salt
½ teaspoon black pepper
6–7 tablespoons oil
Hollow out the potatoes with an apple corer or small teaspoon. Melt the ghee or butter in a pan, add the potatoes and fry for 2–3 minutes, turning frequently.
To make the filling put the meat and onion in a saucepan and cook over a moderate heat for 20–30 minutes, stirring frequently to prevent sticking. Stir in the remaining filling ingredients and cook for a further 10 minutes. Spoon this filling into the potatoes.
Lightly oil the inside of a large casserole dish and pack the potatoes in tightly in a single layer with the openings uppermost.
Dilute the tomato purée in the water and add the remaining sauce ingredients. Pour the sauce over the potatoes. If the potatoes are not half covered then add a little more water. Bring to the boil then place in an oven preheated to 190°C, 375°F, gas mark 5 and cook uncovered for 10 minutes. Cover and continue to cook for 45–50 minutes or until the potatoes are tender, but not soft enough to fall apart. Serve with pilav and salads.
tapoukhai adama im tered
potatoes stuffed with spinach
This is an attractive Israeli dish which is served as a starter or with roasts and kebabs and is often topped with sughtorov-madzoon—yoghurt-garlic dressing.
4 large potatoes, peeled
Filling
25 g/1 oz butter
1 onion, finely chopped
225 g/8 oz spinach, washed very thoroughly
4 tablespoons stock
¼ teaspoon allspice
1 hard-boiled egg, shelled and chopped
1 teaspoon salt
½ teaspoon black pepper
2 tablespoons matzo meal (optional)
oil for frying
Garnish lettuce, finely chopped
pinch paprika
To serve 150 ml/¾ pint sughtorov-madzoon, see recipe
Place the potatoes in a pan of water, bring to the boil and simmer for about 10 minutes. Drain and set aside until cool enough to handle.
Meanwhile, heat the butter in a saucepan and sauté the onion until soft.
Squeeze excess water from the spinach, chop it and add to the onion together with the stock. Cook for 5–7 minutes or until the spinach is limp. Stir in the allspice and drain off any liquid that has not evaporated. Put this mixture into a bowl and add the chopped egg, salt, pepper and matzo meal and mix well.
Cut the potatoes in half lengthways and carefully scoop out the centres leaving a shell about 1.2 cm/½ in thick. Fill the centres with the spinach mixture. Heat some oil in a large frying pan, add the halved potatoes and cook until the potatoes are golden and crusty. This is easiest if you add enough oil to just come to the rim of each halved potato. Garnish a serving dish with the chopped lettuce, arrange the potatoes on top and spoon a little of the yoghurt sauce over each. Sprinkle with the paprika and serve.
foudja-al-jaddah
apples stuffed with chicken and nuts
From Jeddah, Arabia this is a most sophisticated dish of undoubted Persian origin. Serve with a rice pilav and pickles.
8 medium cooking apples
Filling
225 g/oz/2 lb cooked chicken flesh, minced or finely chopped
1 teaspoon salt
½ teaspoon black pepper
50 g/2 oz breadcrumbs
6 cloves
50 g/2 oz chopped nuts
¼ teaspoon turmeric
75 g/3 oz raisins
½ teaspoon cinnamon
¼ teaspoon ginger
water
sugar
50 g/2 oz butter
Wash the apples, core them and scoop out enough flesh to make a hole about 2.5–4 cm/1–1½ in in diameter.
In a small bowl mix the chicken with the salt, pepper and breadcrumbs. Add the cloves, nuts, turmeric, raisins, cinnamon and ginger and mix thoroughly. Now stuff the apples with the mixture, but do not press too hard. Arrange the apples in a large ovenproof dish and add sufficient hot water to come about half-way up the apples. Sprinkle some sugar on to each apple. Cut the butter into 8 pieces and put a knob on top of each apple. Bake the apples in an oven preheated to 180°C, 350°F, gas mark 4 for 30–40 minutes. Take care not to overcook or the apples will split. Serve immediately.
sehki Itzoog stuffed melon
Only an Armenian would stuff a melon, cry over a large chunk of rock [meaning Mount Ararat—the symbol of Armenia] and claim all defeats are really victories.
(Turkish saying.)
As the Turks say, this is a classic of the Armenian cuisine—an unusually fragrant, delightful and original dish. A speciality of the region of Van (Western Armenia), this dish is also prepared using pumpkins.
Serve with a rice or burghul pilav of your choice.
1 large melon, cantaloupe or honeydew
Filling
2 tablespoons oil
1 small onion, finely chopped
225 g/8 oz minced lamb or beef
75 g/3 oz long grain rice, washed thoroughly under cold water and drained
50 g/2 oz pine kernels
50 g/2 oz raisins
½ teaspoon cinnamon
1 tablespoon sugar
300 ml/½ pint water or stock
salt and pepper to taste
Wash the melon. Slice about 2.5 cm/1 in off the top and reserve if for later use as a lid. Clean out and discard the seeds. Scoop out about a cupful of the flesh and chop it up.
To prepare the filling heat the oil in a saucepan, add the meat and onions and fry, stirring frequently, for a few minutes. Add all the remaining ingredients and the chopped flesh, stir well and simmer until the liquid has been absorbed. Leave to cool and then spoon into the melon. Replace the reserved top and secure it with wooden toothpicks.
Set the melon in a greased baking dish just large enough to hold it comfortably. Bake in an oven preheated to 180°C, 350°F, gas mark 4 for about 1 hour or until tender. Cut and serve in wedges.
ashdaragi dolma
apples and quinces stuffed with nuts
Ashdarag is a city in Armenia famed for its delicious fruits. If quinces are unavailable use more apples or unripe pears. Serve as an appetizer with bread or as a main dish with a pilav.
Filling
450 g/1 lb minced lamb or beef
1 small onion, finely chopped
1 teaspoon basil
1 teaspoon allspice
1½ teaspoons salt
½ teaspoon black pepper
75 g/3 oz cooked long grain rice
75 g/3 oz chopped walnuts
4 cooking apples
4 quinces
50 g/2 oz dried apricots, coarsely chopped
50 g/2 oz prunes, stoned and coarsely chopped
300 ml/½ pint stock
salt and pepper to taste
Garnish 1 tablespoon parsley, finely chopped
First prepare the filling by placing the meat in a saucepan and cooking over a low heat for 15–20 minutes, stirring frequently to prevent sticking. Add the chopped onion, basil, allspice, the salt and pepper, stir and cook for a further 10–15 minutes or until the meat is cooked. Stir in the cooked rice and chopped walnuts, mix well and set aside.
Core the fruit and then scoop out much of the flesh leaving a shell about 1.2 cm/½ in thick. Discard the flesh. Fill the fruit with the meat mixture. Arrange the fruit upright in an ovenproof dish and sprinkle the dried fruits around them.
Bring the stock to the boil, season and carefully pour into the dish. Place in an oven preheated to 180°C, 350°F, gas mark 4 and bake for 20–30 minutes. Keep an eye on the fruit and as soon as they show any sign of splitting remove immediately as they are cooked. Serve with its own sauce and sprinkled with the parsley.
Variation
The Iranians also stuff apples dolmeh sib and quinces dolmeh beh. The usual filling is 450 g/1 lb minced meat with 2–3 tablespoons cooked rice, ½ teaspoon cinnamon, 1 teaspoon salt and ¼ teaspoon black pepper.
The sauce is usually 300 ml/½ pint water with juice of 1 lemon, 2 tablespoons brown sugar or 1 tablespoon honey. Cook as above.
karni yarik anatolian baked, stuffed aubergines
This dish literally means 'the stomach cut up'! It is a favourite aubergine dolma of Turks, Armenians and Kurds—in fact of every Anatolian, present and past!—including, I am sure, St George, St Nicholas (Santa Claus) and St Paul of Tarsus.
This is a family recipe. Serve with rice pilav, salads and pickles.
4 medium aubergines
6 tablespoons oil
Filling
2 tablespoons butter
1 small onion, finely chopped
350 g/12 oz minced lamb or beef
2 tomatoes, blanched, peeled and chopped
3 tablespoons parsley, finely chopped
1 teaspoon salt
½ teaspoon black pepper
2 cloves garlic, finely chopped
Sauce
150 ml/¼ pint stock or water
1 teaspoon salt
1 teaspoon dillweed
1–2 tablespoons lemon juice
Cut the stems and hulls off the aubergines and discard.
Peel each aubergine lengthways in 1.2 cm/½ in strips to create an attractive stripey appearance. Heat the oil in a large saucepan or frying pan, add the aubergines and fry gently, turning frequently, until they are soft on all sides. Remove with a slotted spoon and drain on kitchen paper. Arrange the aubergines side by side in a shallow ovenproof dish.
Melt the butter in the pan with any remaining oil, add the onion and fry until soft, stirring frequently. Add the meat and fry, turning frequently until it is lightly browned. Add half the chopped tomatoes and cook for a further 5 minutes, stirring from time to time to prevent sticking. Add the parsley, salt, pepper and garlic, mix well, cook for 10 minutes and remove from the heat. One at a time slit the aubergines lengthways to within 1.8–2.5 cm/¾–1 in of either end and only down into the middle of each vegetable.
Gently ease each slit open a little to form a pocket. Fill each pocket with some of the minced meat and place the remaining tomato over the top. Add the stock or water, sprinkle with the salt and dillweed and cover. Bake in an oven preheated to 180°C, 350°F, gas mark 4 for 30 minutes. Uncover, add the lemon juice, recover and cook for a further 10–15 minutes. Remove and serve.
## pilavs
For a handful of pilav he sold his soul.—Armenian saying.
Rice and rice-based dishes are the basic diet of most Middle Easterners. Rice first appeared in the region some 3000 years ago via the Far East, probably with the advent of the Indo-Iranian tribes (modern Persians, Kurds, Armenians and countless others whose names have now disappeared).
With the Arab conquest of Persia rice appeared in the bazaars of Baghdad, Damascus and Alexandria, whence it travelled to Moorish Spain and, in time, to most of Europe.
However, while in the West rice is still often regarded as a side accompaniment to main dishes, in the East it has acquired a vast cuisine of its own, rich and imaginative. The masters of this cuisine are undoubtedly the Iranians whose poets have called it 'the staff of life' and the 'soul of Allah', and whose housewives have, over the ages, created some of the most delightful and exquisite dishes known to us.
This chapter deals only with pilavs, i.e. meatless rice dishes, which are served as accompaniments to meat, poultry and fish dishes. Other rice dishes appear under sections on meat, poultry and fish.
Rice is never served plain boiled. It is cooked with salt, water and some fat— ghee or butter or, as in centuries past, imag or alya—sheep's tail fat. There are several methods of cooking rice and the following three are probably the most popular.
It is easier to cook rice by volume, i.e. for the first cup of rice (175 g/6 oz) use two cups (450 ml/¾ pint) water and for the next cup of rice use 1½ cups (350 ml/12 fl oz) water. Always use long grain rice, e.g. Basmati. Italian or Spanish short grain rice is normally used in risottos and puddings. If you can find Basmati use it as it is the nearest to that used in the Middle East.
method 1
Roz—plain rice pilav. General favourite throughout the region, except Iran, Iraq and the Gulf States.
50 g/2 oz butter or ghee
250 g/9 oz long grain rice, washed thoroughly under cold water and drained
1 teaspoon salt
600 ml/1 pint water, boiling
Melt the butter or ghee in a saucepan. Add the rice and fry, stirring frequently, for 2–3 minutes. Add the salt and boiling water and boil vigorously for 3 minutes. Cover the pan, lower the heat and simmer for about 20 minutes or until all the liquid has been absorbed. The grains should be tender and separate and there should be small holes in the surface of the rice. Turn off the heat, remove the lid, cover with a teatowel, replace the lid and leave to 'rest' for 10–15 minutes. Fluff up the rice with a fork and serve.
method 2
Popular throughout Syria, Lebanon and Turkey.
600 ml/1 pint water
½ teaspoon salt
250 g/9 oz long grain rice, washed thoroughly under cold water and drained
50 g/2 oz butter or ghee
Bring the water to the boil. Add the salt and rice and boil vigorously for 3 minutes. Cover the pan, lower the heat and simmer for 20 minutes or until the water has been absorbed. Remove from the heat, remove lid, cover with a teatowel, replace lid and leave to 'rest' for 10–15 minutes. Meanwhile melt the butter or ghee in a small pan and pour evenly over the rice after it has rested. Cover the pan again and leave for a further 5 minutes before serving.
method 3
Chelo—Iranian plain rice.
Another Westerner—W. H. Forbis in his book Fall of the Peacock Throne—following in the footsteps of earlier travellers, e.g. J. Fryer, Sir J. Chardin, Lord Curzon, etc., is filled with profound admiration and describes the step by step preparation of this 'jewel' of the Iranian cuisine. I have quoted his words more out of interest than as a recommended method.
The Iranian housewife goes through 14 steps to make a bowl of chelo, crusty steamed rice. Starting with two and a half cups of good long-grain rice, she washes it and rinses it three times in lukewarm water. She soaks it overnight, covered, in heavily salted water. The next day she sets two quarts of water to boiling with two tablespoons of salt, and adds the drained, soaked rice in a stream. She boils the rice for 10–15 minutes, stirring it once or twice, than puts it in a strainer and rinses it with lukewarm water. Next she melts half a cup of butter, and puts a third of it in a cooking pot, to which she adds two tablespoons water. She spoons the boiled rice into the pot so as to make a cone, and pours the rest of the butter evenly over it. She covers the pot with a folded teatowel, to make the rice cook evenly, and then puts on the lid. She cooks it for 10–15 minutes over a medium heat, and for 45 more minutes over a low heat. She places the pot in cold water, to make the rice come free from the bottom of the pan. She turns it out so that the golden crust on the bottom, which is the specific asset that makes Iranian rice the world's best, flecks and accents the whole fluffy mound of distinctly separate grains. She puts 2 or 3 tablespoons of rice into a dish and mixes in a tablespoon of saffron. She pours the coloured rice over the rest, and she is done.
Here is my much simplified version of chelo—and the final result, I assure you, will be equally successful.
250 g/9 oz long grain rice, washed thoroughly under cold water and drained
2 tablespoons salt
about 1.2 litres/2 pints water
50 g/2 oz butter, melted
Place the rice in a deep bowl, add 1 tablespoon of the salt and enough cold water to cover by 2.5 cm/1 in. Leave to soak for 2 hours.
Bring about 2 litres/1¾ pints of water, and the remaining salt, to the boil in a heavy saucepan with a close fitting lid. Drain the rice thoroughly then pour it slowly into the water so that it doesn't go off the boil. Boil for 5 minutes and then drain into a sieve. Pour 150 ml/¼ pint water and half the melted butter into the saucepan, add the rice and pour the remaining butter over the top. Cover the pan with a teatowel, fit on the lid and lift the ends of the cloth on to the lid so that there is no danger of fire. Steam over a very low heat for about 25–30 minutes or until the liquid has been absorbed and the grains are tender and fluffy. Leave to rest for 10 minutes before serving.
chelo ta dig steamed crusty rice
Ta dig, meaning literally 'bottom of the pan', is a crusty, crunchy layer at the bottom of the saucepan and is golden brown in appearance. To achieve this result either (a) mix 75 g/3 oz of the par-boiled rice with 1 egg yolk and spread it over the bottom of the pan before adding the rest of the rice, or (b) mix 75 ml/3 fl oz of yoghurt with ½ teaspoon of saffron and 75 g/3 oz par-boiled rice and spread this mixture over the bottom of the pan before adding the rest of the rice.
Cook as the chelo, but steam for 15–20 minutes longer than stated. The result is a golden layer of crunchy ta dig which is then broken up and cut into small pieces and arranged around the rest of the rice, golden side up.
kazmag pilav rice and dough pilav
This is an Azerbaijanian variation from the Caucasus.
Make a dough by mixing together 1 egg and 75 g/3 oz flour, knead well and roll out to fit the base of the pan.
Brush the base of the pan with a little melted butter, add the dough, brush its surface with a little more butter and then pile in the par-boiled rice and top with remaining melted butter.
Steam as with chelo, but for 15–20 minutes longer.
To serve pile the rice on to a large plate and garnish with wedges of the golden brown kazmag crust. It is sometimes sprinkled with pomegrante seeds for extra colour.
Saffron pilav
Saffron is used to give colour to pilavs and it has an intrinsic delicate flavour. Its appearance on a dinner table gives that extra glow (the Middle Easterners are very fond of the colours gold and crimson) to the already colourful table. The simplest saffron pilav would be to follow method 1 for plain pilav and to add ½ teaspoon saffron to the rice when it is frying.
However, a much more exotic variation, popular in the Gulf States and particularly with the Afghans and Baluchis of south east Iran, usually includes cloves, cinnamon and cardamom.
Often, particularly in Iraq and the Gulf States ¾–1 teaspoon of turmeric—'poor man's saffron'—is substituted for the real thing since saffron is an extremely expensive spice—even in the lands where it is cultivated.
sehriyeli pilavi pilav with vermicelli
In the old days, before commercial versions inundated the shops, women at home prepared their own versions of vermicelli, noodles and pastas. They made many different kinds, e.g. 'elephant's ears', 'shells', 'balls', 'angel's hair', etc. Using their forefingers and thumbs the women shaped the dough into thin threads about 2.5–4 cm/1–1½ in long which were then dried and stored in jars for later use in pilavs and stews. The Muslims prepare this dish on the second day of the New Year 'so that one may have as many children as the sehriyelis in the pilav'. Armenians eat this pilav on Easter Sunday and Christmas Day and call this dish 'Angel's hair pilav' because of some moot superstition about being worthy of an 'Angel's hair'!
50 g/2 oz butter or ghee
25 g/1 oz vermicelli, broken into 2.5 cm/1 in pieces
250 g/9 oz long grain rice, washed thoroughly under cold water and drained
2 tablespoons raisins (optional)
1 teaspoon salt
600 ml/1 pint water, boiling
Melt the butter or ghee in a medium saucepan. Add the vermicelli and sauté until light golden. Add the rice and raisins and proceed as with roz—plain rice pilav, from method 3.
plov azzari ginger and sesame pilav
This recipe is from the Caspian coastline and is a speciality of the Turkish speaking Azerbaijanian people of the former Soviet Republic and Iran.
50 g/2 oz butter or ghee
250 g/9 oz long grain rice, washed thoroughly under cold water and drained
½ teaspoon ground ginger
2 teaspoons sesame seeds
600 ml/1 pint water, boiling
1 teaspoon salt
½ teaspoon black pepper
Garnish 50 g/2 oz slivered, toasted almonds
Melt the butter in an ovenproof casserole, add the rice and fry gently, stirring frequently, for about 10 minutes. Add the ginger and sesame seeds and fry for a further 2–3 minutes. Add the water, salt and pepper, stir and bring to the boil.
Place the casserole, uncovered, in an oven preheated to 180°C, 350°F, gas mark 4 for about 35 minutes, or until the water is absorbed and the rice tender. Turn the oven off and, after 10 minutes, toss the rice with a fork. Leave for a further 15–20 minutes and then toss again. Remove from the oven, sprinkle the almonds over the top and serve.
domatesli pilavi tomato pilav
Popular everywhere, with many variations, tomato pilav is a must with all kebabs. Has a light pink colour and a delicious flavour.
50 g/2 oz butter or ghee
3 large tomatoes, blanched, peeled and chopped
1 small onion, finely chopped
1 clove garlic, finely chopped
½ teaspoon dried basil
2 tablespoons parsley, finely chopped
½ teaspoon black pepper
1 teaspoon salt
250 g/9 oz long grain rice, washed thoroughly under cold running water and drained
600 ml/1 pint water, boiling
Melt the butter or ghee in a medium saucepan. Add the tomatoes, onion, garlic, basil, parsley, pepper and salt and sauté stirring frequently, for a few minutes until the onion is soft. Stir in the rice and fry for 2–3 minutes. Proceed as for roz—plain rice pilav method 3.
sultan resat pilavi aubergine pilav
A Turkish dish named after a Sultan who swore by this pilav—Sultans, of course, swore a great deal in their time! Variations of this pilav appear throughout Iran, Armenia and northern Syria and Iraq.
Often eaten as a savoury, with yoghurt and pickles, by the Anatolian villagers, this pilav is nevertheless excellent with all roasts and kebabs.
2 large aubergines, peeled
50 g/2 oz butter or ghee
1 onion, roughly chopped
2 large tomatoes, blanched, peeled and chopped
50 g/2 oz vermicelli, broken into 2.5 cm/1 in pieces
1 teaspoon salt
½ teaspoon black pepper
175 g/6 oz long grain rice, washed thoroughly under cold water and drained
900 ml/1½ pints water
a few leaves fresh mint
Cut the aubergines into roughly 1.2 cm/½ in cubes. Place the cubes on a plate, sprinkle with salt, place another plate over the top and leave for 30 minutes.
Meanwhile, melt the butter or ghee in a large saucepan, add the onion and fry until soft, but not brown. Stir in the tomatoes, vermicelli, salt and pepper and fry for about 3 minutes.
Rinse the aubergine cubes thoroughly and dry on kitchen paper. Add the aubergines to the saucepan and fry, stirring frequently, for a further 2–3 minutes. Stir in the rice, water and mint leaves and bring to the boil. Lower the heat and simmer for 15–20 minutes until the liquid is absorbed and the rice tender. Leave to 'rest' for 10–15 minutes and serve.
tutumi pilav pumpkin pilav
Marrows and pumpkins are much used in the Armenian cuisines. This is a pilav of pumpkin with sultanas, apricots and rice and it is typical of the Caucasian-Iranian sweet and sour dishes.
Dried apricots are sometimes substituted by thinly sliced apricot paste.
175 g/6 oz long grain rice, washed thoroughly under cold water and drained
225 ml/8 fl oz water
1 teaspoon salt
1 teaspoon dillweed
50 g/2 oz sultanas
3–4 dried apricots, thinly sliced
450 g/1 lb pumpkin, peeled and sliced lengthways into 1.2 cm/½ in slices
50 g/2 oz sugar
75 g/3 oz melted butter
Place the rice in a saucepan with the water, bring to the boil and then simmer for 10 minutes. Stir in the salt, dillweed, sultanas and apricots and simmer for a further 5 minutes. Strain into a sieve and run cold water through the rice.
Lightly butter a baking or casserole dish. Arrange half the pumpkin slices over the bottom and sprinkle with a third of the sugar and a third of the melted butter.
Put the rice into a bowl, add half the remaining sugar and butter and mix thoroughly. Spread the rice over the pumpkin slices.
Pour the remaining butter into a frying pan and sauté the remaining pumpkin slices for a few minutes, turning once.
Arrange the pumpkin slices decoratively over the rice and sprinkle with the remaining sugar and any butter left in the pan. Cover and place in the centre of an oven preheated to 180°C, 350°F, gas mark 4 and bake for 40–45 minutes. Remove from the oven and serve.
ciftlik pilavi farmer's pilav
An Anatolian pilav that makes good use of all the vegetables available. A vegetable risotto, often eaten with bread and a bowl of yoghurt as a main dish.
Allah hu
Pilav su
Allah hay
Kahve chay!
This is a skit on the Arabic sounds which the villagers recite thinking of them as prayers, but in reality which simply mean:
God ho
Pilav water
God hi
Coffee tea!
An allusion to the deep ignorance of the Anatolian peasants. (Quoted from A Village in Anatolia, Mahmut Makal.)
1 carrot, peeled and cut into 0.6 cm/¼ in rings
4 mushrooms, wiped clean and sliced
1 small green pepper, thinly sliced
3 tablespoons garden peas
50 g/2 oz butter for frying
½ teaspoon black pepper
50 g/2 oz butter
250 g/9 oz long grain rice, washed thoroughly and drained
1 teaspoon salt
600 ml/1 pint water, boiling
1 tablespoon sultanas
2 tablespoons parsley, finely chopped
Bring a saucepan half-filled with lightly salted water to the boil, add the carrot rings and simmer until just tender. Remove with a slotted spoon and reserve. Add the mushrooms, green pepper and peas and simmer for 5 minutes. Remove and reserve with the carrots.
Melt 50 g/2 oz of the butter in a pan, add the vegetables and the black pepper and fry for 2–3 minutes, stirring frequently. Set the vegetable mixture aside.
Melt the remaining 50 g/2 oz of butter in a large saucepan, add the rice and fry, stirring frequently, for 2–3 minutes. Add the salt and water. Cover the pan and simmer for 10 minutes then remove the lid, add the fried vegetables, sultanas and parsley and mix thoroughly. Recover and continue cooking until all the liquid had been absorbed. Uncover, turn off the heat and leave to rest for 10–15 minutes. Fluff up with a fork and serve.
turkestan pilavi carrot pilav
In Turkey this dish is often cooked in a mould with stunning effect. It is an attractive golden yellow speckled with black, brown and green. However, it is equally tasty and almost as attractive if cooked and presented in the normal way.
This recipe was given to me by a Turkish friend from Kaysari, Central Anatolia, who assured me that it was authentic since his 'ancestors brought it over from beyond the mountains'. Nevertheless there is more than a little touch of Iran about it, particularly in the use of cinnamon, rosewater and pistachios. Serve with kebabs and roasts.
75 g/3 oz ghee or butter
at least 225 g/8 oz grated carrots
½ teaspoon whole black peppercorns
1 tablespoon seedless raisins
1 teaspoon sugar
½ teaspoon ground cinnamon
350 g/12 oz long grain rice, washed thoroughly under cold water and drained
750 ml/1¼ pints water, boiling
1½ teaspoons salt
1 tablespoon rosewater
Garnish
2 tablespoons parsley or tarragon, finely chopped
1 tablespoon pistachios, chopped
Melt the ghee or butter in medium saucepan, add the grated carrots and peppercorns and fry for 3 minutes, stirring frequently. Sprinkle in the raisins, sugar and cinnamon, stir and fry for another minute. Add the rice and fry for 2–3 minutes, stirring constantly. Add the water and salt, stir and boil vigorously for 3 minutes. Cover pan, reduce heat to very low and simmer for about 15 minutes or until the liquid has been almost absorbed.
Now either lightly oil a round mould and spoon the rice mixture into it. Sprinkle with the rosewater, cover and place in an oven preheated to 180°C, 350°F, gas mark 4 for a further 5–7 minutes; or continue cooking in the saucepan for about another 5 minutes or until all the water has been absorbed.
Remove mould or pan from the heat and leave to rest for 10–15 minutes. Invert a plate over the mould and turn it out or spoon the mixture from the pan into a serving dish. Sprinkle with the parsley or tarragon and the pistachios and serve.
muhamar sweet pilav
A speciality of the Gulf States, this recipe from Bahrain makes clever use of saffron, cardamom, honey and rosewater.
This pilav is said to be the favourite of the pearl divers—a vocation now almost extinct due mainly to newly acquired oil wealth, but which was once the only industry of the islands.
Serve it with kebabs of all kinds particularly those of fish and prawns—another favourite of the region.
3 tablespoons rosewater
½ teaspoon saffron threads
4 cardamom pods, cracked
600 ml/1 pint water, boiling
250 g/9 oz long grain rice, washed thoroughly under cold water
2 teaspoons salt
3 tablespoons honey or 2 tablespoons sugar
50 g/2 oz ghee or butter
Pour the rosewater into a cup, add the saffron and cardamom pods and leave to rest.
Place the boiling water in a saucepan, add the rice and salt and boil for 10 minutes. Drain into a colander. Place the rice in a bowl, add the honey or sugar and mix thoroughly.
Melt the ghee or butter in a medium saucepan, add the rice, cover the pan and cook over a very low heat for 15 minutes. Uncover, add the saffron mixture, mix well, recover and simmer for another 10–15 minutes or until the rice is tender. Leave to rest for 10 minutes before serving.
roz-bil-tamar pilav with almonds and dates
A Bedouin dish. The love of the desert to the Bedouin seems to have been even stronger than that of the sea to the sailor. Even in their days of glory the Califs were more at home in their desert tents than in the ornate palaces of Damascus or Baghdad. My favourite Arab poet Abulla-el-Mahari (973–1058) expresses this Bedouin passion most succinctly with these words:
A-weary am I of living in town and village—
And oh, to be camped alone in a desert region,
Revived by the scent of lavender when I hunger
And scooping into my palm, if I thirst, well-water!
50 g/2 oz butter or ghee
250 g/9 oz long grain rice, washed thoroughly under cold water and drained
1 teaspoon salt
600 ml/1 pint water, boiling
Garnish
50 g/2 oz butter
50 g/2 oz blanched almonds
75 g/3 oz stoned dates
50 g/2 oz seedless raisins or sultanas
1 teaspoon rosewater
Cook the rice following the instructions for roz, method 1—plain rice pilav.
While the rice is 'resting' melt 25 g/1 oz butter or ghee in a large frying pan. Now add the almonds and fry, stirring frequently, until they begin to turn a light golden colour. Add the remaining butter, the dates and the raisins or sultanas. Fry for a few more minutes, stirring frequently. Remove from the heat and stir in the rosewater. To serve spoon the rice on to a serving dish and arrange the fruit and nut mixture over the top.
harsaniki pilavi wedding pilav
One evening Nasrudin quarrelled with his wife and shouted at her so fiercely that she fled for refuge to a neighbouring house, where he followed her. As it happened, a wedding feast was in progress, and the host and guests did all they could to calm him down, and vied with each other to make the couple reconciled, to eat and enjoy themselves. The Mulla said to his wife:
'My dear, remind me to lose my temper more often—then life really would be worth living!' (The Pleasantries of the Incredible Mulla Nasrudin)
An Armenian favourite from Yerevan, this dish is exotic and colourful like Armenian illuminated manuscripts and richly patterned carpets. A must—as the name suggests—at weddings and festive occasions. Serve with all roasts, kebabs and stews.
50 g/2 oz butter or ghee
250 g/9 oz long grain rice, washed
thoroughly under cold water and drained
1 teaspoon salt
600 ml/1 pint water, boiling
Sauce
50 g/2 oz butter
50 g/2 oz apricots, soaked overnight in cold water
50 g/2 oz prunes, soaked overnight in cold water and stoned
50 g/2 oz seedless raisins or sultanas
50 g/2 oz blanched almonds, split
2 tablespoons honey
1 tablespoon hot water
Melt the butter or ghee in a saucepan. Add the rice and fry for 2–3 minutes, stirring frequently. Add the salt and water, stir and boil vigorously for a few minutes. Lower the heat, cover and simmer for a further 20 minutes or until all the water has been absorbed. Turn off the heat, remove the lid, cover the pan with a clean teatowel, replace the lid and leave to 'rest' for 10–15 minutes. Loosen the grains with a fork.
Meanwhile, prepare the sauce which will garnish the pilav. Melt the butter in a saucepan. Add the fruit and nuts and fry, stirring frequently, until the nuts are lightly browned. Mix the honey and water together and pour it into the saucepan. Lower the heat and cook for about 10 minutes, stirring frequently, until the mixture has thickened.
To serve, pile the pilav on to a serving dish and pour the sauce over the top.
tzavarov pilav cracked wheat pilav
Burghul, tzavar, is the staple food of Armenians, most Anatolians and Kurds. Although it is used by Syrians, Lebanese and Cypriots in their kibbeh dishes, burghul is only 'second fiddle' to rice outside the Anatolian Plateau—where, most probably, it originated several millenia ago. A very touching reference is made to it in a thirteenth century Armenian colophone (Jerusalem Museum) where the scribe had added on the margins of a page the following words:
For generations my people [Armenians] hidden in mountain ravines and caves lived on tzavar [burghul] and a little rice, when available, while the barbarian hordes [Seljuks and Mongols] ravaged our biblical land, destroying churches, castles, burning villages, everything, everything! that lay in their way. But we survived thanks to the God given goodness of those tiny grains that possessed the waters of the seas, the rays of the sun and all the goodness of life.
Blessed is the Lord, blessed be the life giving grains of tzavar and of rice.
Burghul-based pilavs are often substituted for those of rice, adding variety and interest and indeed some, including me, regard burghul pilavs as superior to their rice counterparts. Use large-grained burghul which is found in all Middle Eastern, Indian and health food stores.
250 g/9 oz large grained burghul
50 g/2 oz butter or ghee
1 small onion, finely chopped
450 ml/¾ pint water, boiling
1 teaspoon salt
½ teaspoon black pepper
Put the burghul into a bowl or fine sieve and wash several times until the water runs clear. Leave to drain.
Melt the butter or ghee in a saucepan. Add the onion and fry gently until soft and golden. Add the burghul and fry for 2–3 minutes, stirring frequently. Add the boiling water, salt and pepper and stir well. Bring to the boil and boil vigorously for 5 minutes. Lower the heat and simmer for 8–10 minutes or until the water has been absorbed. Turn off the heat, cover the pan with a clean teatowel, clamp on the lid and leave to rest for 10–15 minutes.
Variations
telahaysov tzavari pilav burghul pilav with vermicelli
Follow the recipe for sehriyeli pilavi , but use burghul instead of rice.
Similarly you can prepare most of the rice pilavs with burghul, e.g. tomato pilav, mushroom pilav, spinach pilav, aubergine pilav, etc.
mujaddarah cracked wheat and lentil pilav
A medieval dish sometimes called 'Esau's favourite' or 'the food of the poor'. Mujaddarah appears in several countries under differing names. It is known as kitry in Iraq, adas pollo in Iran, muaddas in the Gulf States and, in far away India as kitcheri.
In the Middle Ages (still today in India) mujaddarah included several vegetables, e.g. carrots, peas, aubergines, etc. as well as meat or fish, but today the basic ingredients are simply rice or burghul, lentils and onion.
It is often eaten as a main dish with fresh salads, yoghurt and pickles. The recipe below uses burghul, but you can substitute an equal amount of rice.
175 g/6 oz brown lentils, washed and drained
900 ml/1½ pints cold water
300 ml/½ pint olive or vegetable oil
2 large onions, thinly sliced
2 teaspoons salt
½ teaspoon black pepper
300 ml/½ pint boiling water
175 g/6 oz large burghul, washed in a bowl with cold water until the water runs clear
Put the lentils into a saucepan, add the 900 ml/1½ pints of water and bring to the boil. Lower the heat, cover and cook for 25–30 minutes or until the lentils are almost cooked and the water mostly absorbed.
Meanwhile, in a frying pan heat the oil, add the sliced onions and fry, stirring frequently, until they are dark golden, but take care not to burn them. Reserve half the onions and the oil.
Stir the other half of the onions into the lentils. Add the salt, pepper and the boiling water and bring to the boil. Stir in the burghul, cover and simmer for a further 15–20 minutes or until the lentils and burghul are tender and the water absorbed. Remove from the heat and leave to rest for 10–15 minutes. Pile the mujaddarah on to a plate and garnish with the remaining onions and oil.
Variations
adas pollo
This is a simpler version without the onion and with 25 g/1 oz each of toasted slivered almonds, and raisins sprinkled over the top of the cooked pilav.
kitry
This uses 2 cloves crushed garlic fried in a little oil instead of the onions and also includes 1 tablespoon tomato purée and ½ teaspoon turmeric.
## kebabs
With the discovery of fire came cooked meat, then barbecues, and kebabs. Ancient civilizations knew all about grilling meats of all kinds on fire. That was nothing new. What was new, however, was, and this came much later, the art of marination, basting and smothering the meats in herbs and spices.
Homer, in his Iliad, describes how Achilles played host to Odysseus outside the walls of Ilium.
Petroclus put down a big bench in the firelight, and laid on it the backs of a sheep and a fat goat and the chine of a great hog rich in lard. Automedon held these for him, while Achilles jointed them, and carved up the joints and spitted the slices. Meanwhile, Petroclus... made the fire blaze up. When it had burned down again and the flames disappeared, he scattered the embers and laid the spits above them, resting them on logs... When he had roasted it and heaped it up on the platters... Achilles divided the meat into portions.
In the Middle East, scenes like this are still an everyday occurrence. To celebrate a birth, wedding, anniversary or religious festival large groups of people drive to the fields or hills, prepare impromptu grills—usually in small depressions in the ground light dry branches, slaughter lambs and skin them. The innards are removed—the heart and livers are set aside as they make excellent kebabs in their own right—and the lambs are rinsed in a nearby brook. Long wooden or metal rods are pushed right through the lamb from the breast to the hindquarters and the legs are trussed. The fire is blazed and then 'when it had burned down again and the flames disappeared', the spits are laid above them.
Next time you have a large party try this whole lamb kebab—it will be the talk of the year!
A fascinating method—still popular in Anatolia, Caucasus, Iran and, I understand, with the Aborigines of Australasia—is to make a shallow (60–90 cm/2–3 ft) pit approximately 90 × 180 cm/3 × 6 ft in the ground and a bed of charcoal laid in the pit. The lamb is laid on the glowing charcoal and the whole is covered with earth. Five to six hours later the lamb will be cooked. It is then removed, brushed to remove the soil clinging to the skin and placed on a large tray. Two hefty people shake the tray until the meat drops from the bones. The meat is sprinkled with salt, black pepper and herbs and eaten.
In the Middle East lamb has always been the most popular meat. Indeed, to an Arab 'meat' simply means lamb or mutton, although in the past kid and gazelle were also eaten as well as camel's meat which is hung from high ceilings in the semi-dark meat markets.
Cattle are seldom bred, except for buffalo—particularly in Egypt and Southern Iraq, but 'Buffaloes are never killed for food; never, indeed, unless they are dying anyway of some disease... their lives are passed in a rich and placid leisure immune alike from fear and frustration. They are maintained in privileged luxury for the sake of their milk and their dung.' (Food in History)
Today beef and veal are gradually becoming more popular. Pork, of course, is only eaten by Christian populations (Greeks, Armenians, Georgians and the Maronites of Lebanon and Syria).
Finally a word of warning. There are too many dishes falsely labelled kebab. The word simply means 'cooked meat'; in the oven or on fire; and is derived from one of the early Indian languages, not Turkish as is often claimed.
In this section I have grouped together kebabs of meat, chicken, game and fish—all either basted or marinated and most skewered and cooked over charcoal. All kebabs are traditionally served on a bed of rice or burghul with fresh salads, pickles, yoghurt and yoghurt drinks.
The choice is wide and I suggest you experiment.
sis kebab
This is by far the most famous Middle Eastern dish and has countless local variations. This kebab was most probably created by shepherds who spent months alone on the hills and, deprived of home cooking and the availability of domestic utensils, produced this simple dish of chunks of meat—lamb or goat—threaded on to wooden sticks and grilled over dried wood.
However, grilled meat does not necessarily make kebabs. For the secret of a good kebab lies in the art of marination, and any kebab that is not marinated or basted is not a kebab, but merely grilled meat or barbecued meat.
Sis kebab is traditionally prepared with lamb (leg) and the following points should be noted when preparing the meat:
(a) Remove tough membranes and ligaments.
(b) Cut the meat across the grain.
(c) Marinate the meat for at least 8 hours.
Below is a standard recipe with several suggestions for marinades.
900 g/2 lb lamb (or beef) cut into 2.5 cm/1 in cubes
A favourite Armenian marinade
150 ml/¼ pint oil
1 teaspoon allspice
150 ml/¼ pint red wine
salt and black pepper to taste
A Greek favourite
150 ml/¼ pint oil
juice 1 lemon
2 onions, chopped and crushed to extract the juice—use garlic press or extractor
2 bay leaves
2 teaspoons oregano
pulp of 2 tomatoes
salt and black pepper
A Turkish favourite
150 ml/¼ pint oil
2 onions, chopped and crushed to extract juice–use garlic press or extractor
1 teaspoon cinnamon
salt and black pepper to taste
A yoghurt marinade
300 ml/½ pint yoghurt
juice 1 onion, chopped and crushed to extract juice–use garlic press or extractor
salt and black pepper to taste
Wine marinade
150 ml/¼ pint red wine
5 tablespoons vinegar
1 clove garlic, crushed
¼ teaspoon black pepper
½ teaspoon crushed dried mint
3 sprigs parsley
Mix all the ingredients for the marinade of your choice in a large bowl. Add the pieces of meat and turn until they are all well coated with the marinade. Cover and leave in the refrigerator for at least 6–8 hours or preferably overnight.
When ready to cook thread the pieces of meat on to skewers and cook over charcoal or under a grill—the latter will not, of course, produce the same end results but it is more convenient. Turn the skewers occasionally so that the meat is cooked evenly. Cook for 12–15 minutes or until the meat is brown and cooked on the outside, but still a little juicy in the centre.
Serve on a bed of pilav with salads of your choice.
Variation
Often the pieces of meat are alternated with pieces of onion, halved tomatoes, bay leaves, pieces of green pepper, mushrooms, etc.
kebab-e-barg fillet of lamb kebab
This is the national dish of Iran when served with chelo rice. It is usually made from lamb fillet. The fillet is cut from the bone, laid in a strip and sliced open lengthways, flattened out and cut crossways to the grain into 5 cm/2 in pieces.
Fillet of lamb is rather expensive since in the West butchers cut the fillet (with the bone) into lamb chops. I suggest you use a 1–1.5 kg/2–3 lb shoulder of lamb instead. Ask your butcher to cut it into 0.6 mm/¼ in thick and 12.5 cm/5 in long slices. Then pound each piece and place in a shallow dish with the marinade.
1–1.5 kg/2–3 lb shoulder of lamb
Garnishes
chelo rice–see recipe
4 egg yolks
sumac
4 raw onion rings (optional)
Marinade 1
3 tablespoons oil
1 large onion, finely chopped or grated
3 tablespoons lemon juice
salt and black pepper to taste
Marinade 2
300 ml/½ pint yoghurt
salt and black pepper to taste
1 large onion, finely chopped or grated
Prepare the meat as described above and lay the strips in a shallow dish.
Mix the ingredients of the marinade of your choice together and pour over the meat. Turn the strips of meat until they are well-coated and then cover and refrigerate for at least 24 hours.
Lay the strips of meat on a kebab grill, and cook fairly quickly, turning frequently so that the meat cooks without drying.
Lay each individual portion on a plate and cover with chelo rice. Sprinkle sumac over the rice and place a pat of butter and an egg yolk on top of each portion of rice. To eat mix the rice, sumac, butter and egg yolk up and garnish with an onion ring if you wish.
Serve with yoghurt, fresh salads and a drink of tan ayran .
mtswadi Caucasian kebab
A recipe from Tiblisi, Georgia—the mythical land of Colchis where Jason and the Argonauts journeyed in quest of the Golden Fleece. Tiblisi (Tiflis) has been the capital for 1500 years. It is a great modern city spread on both sides of the river Kura and is one of the great centres of art and science of the former Soviet Union.
Marinade
3 cloves garlic, crushed
300 ml/½ pint vinegar
2 tablespoons parsley, chopped
1 onion, chopped
½ teaspoon black pepper
1 teaspoon salt
900 g/2 lb lean lamb, cut into 2.5 cm/1 in cubes
75–100 g/3–4 oz streaky bacon, rind and bone removed
25 g/1 oz butter, melted
1 onion
100 g/4 oz mushrooms, wiped clean
50 g/2 oz spring onion, chopped
Garnish
2 tomatoes, sliced
1 lemon, cut into wedges
To serve pomegranate juice—optional (It is possible to buy bottles of pomegranate juice in some delicatessens. Alternatively squeeze the juice from 2 pomegranates, add 1 teaspoon of sugar, bring to the boil, simmer for 2 minutes and leave to cool.)
Mix the marinade ingredients together in a large bowl. Add the cubed meat and stir.
Cover and leave in the refrigerator overnight.
Cut the bacon into 2.5 cm/1 in pieces.
Quarter the onion and separate the various layers.
Thread the lamb on to skewers alternating with pieces of bacon and onion and the occasional mushroom. Cook for 10–15 minutes, turning frequently. Slide the kebabs from the skewers on to a serving dish and sprinkle with the melted butter and spring onions. Garnish with the sliced tomatoes and lemon wedges.
If using pomegranate juice either sprinkle some over the meat or serve in very small individual dishes and dip the chunks of meat in it before eating.
karski shashlig lamb and kidney kebab
Eastern Anatolia—land of mountains, sheep and ruined Christian churches and castles—was once a hive of industry and commerce with centres such as Erzerum, Trabzond, Kars, and Van. All are still there, but depleted of their wealth and populace.
During the first Russo-Turkish wars of the last century, (Alexander Pushkin gives a vivid account of the times in his Journey to Erzerum), Tzarist soldiers were first introduced to this and similar dishes which they took back with them to 'Mother Russia'.
This kebab is usually served on small flaming swords which look very attractive.
900 g/2 lb leg of lamb, cut into 8 large pieces
6 lamb's kidneys, halved
Marinade
1 onion, chopped
2 tablespoons parsley, chopped
2 tablespoons vinegar or the juice of ½ lemon
salt and pepper to taste
Garnish lemon wedges
Mix the marinade ingredients in a large bowl. Add the lamb cubes and halved kidneys, turn to coat with the mixture and leave for 6–8 hours or overnight if possible. Thread 2 pieces of meat on each skewer, each piece sandwiched between kidney halves, i.e. 2 pieces of meat and 3 kidney halves on each skewer. Cook slowly, turning and basting frequently. Serve with lemon wedges and a plain rice pilav.
Variation
Another kebab from Kars makes use of lamb chops and is also usually served on small flaming swords.
Marinade
150 ml/¼ pint oil
150 ml/¼ pint red wine
1 clove garlic, crushed
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon allspice
8 lamb chops
4 tomatoes
8 mushrooms, wiped clean
2 green peppers, each cut into 8 pieces
16 pieces onion
Mix the marinade ingredients together in a large bowl. Add the chops and turn until they are well coated. Cover and leave in the refrigerator overnight. If you have long skewers thread 2 chops on to each skewer alternating them with 1 tomato, 2 mushrooms, 4 pieces of green pepper and 4 pieces of onion. You can also cook the chops on a rack and just thread the vegetables on to skewers and then apportion them after they are cooked.
hirino souvlaki pork kebab
The Middle Eastern pork repertoire is very limited. The Muslim religion forbids its use and only Christians officially eat it; although I know many a good Arab and Jew who secretly relish it—may Allah forgive them!
The Greeks make a lovely kebab of pork served with salads, olives and cheese. You will find this kebab, with many local variations, served in most Cypriot and Greek restaurants.
900 g/2 lb belly pork
Marinade
juice 1 lemon
1 onion, finely chopped
1 clove garlic, finely chopped
3–4 bay leaves
150 ml/¼ pint white wine
4–6 tablespoons oil
2 teaspoons curry powder (optional)
½ teaspoon turmeric (optional)
Garnish 2 lemons, cut into wedges
Cut the meat into 1.8 cm/¾ in cubes without removing the fat. Mix the marinade ingredients in a large bowl, add the meat, mix well, cover and leave for 5–6 hours or, preferably, overnight in the refrigerator. Thread the meat on to skewers and cook over charcoal for 15–20 minutes, turning frequently so that the fat doesn't burn. Serve with the lemon wedges which you squeeze over the meat.
kharapak khorovadze
pork kebab with pomegranate juice
A recipe from the mountains of Kharapak high up in the Caucasus. The meat is marinated in pomegranate juice giving it a most unusual flavour and sharpness. If you wish to prepare your own see the Glossary. Serve with pilav of your choice and salads.
900 g/2 lb boned leg of pork cut into 4 cm/1½ in cubes
Marinade
150 ml/¼ pint oil
1 large onion, finely chopped
1½ teaspoons fresh tarragon, chopped or 1 tablespoon dried tarragon
1½ teaspoons salt
1 teaspoon black pepper
To serve pomegranate juice
Mix the marinade ingredients together in a large bowl. Stir in 3 tablespoons pomegranate juice, add the meat, mix well, cover and leave in the refrigerator overnight. Thread the pieces of meat on to skewers and grill for 20–30 minutes, turning frequently until the meat is cooked through. Serve the kebabs accompanied by a small bowl of pomegranate juice into which you dip each piece of meat before eating.
kasbi mishwi liver kebab with garlic
Liver, either lamb's or calf's makes excellent and succulent kebabs. The recipe below is from Jordan.
900 g/2 lb liver, lamb or calf, soaked in cold water for 20 minutes then drained
Marinade
5–6 cloves garlic, crushed
1½ teaspoons dried mint
100 ml/4 fl oz oil
1½ teaspoons salt
½ teaspoon black pepper
Garnish lemon wedges
Remove any skin and sinew from the liver and then cut into 5 cm/2 in cubes. Mix the marinade ingredients together in a large bowl then add the pieces of liver and turn until well coated. Cover and set aside at room temperature for about 30 minutes. Thread on to skewers and cook over charcoal, turning frequently, for 8–10 minutes. Do not overcook or the liver will dry out. Serve with a little of the remaining marinade spooned over the top and garnished with lemon wedges.
Variation
liver with tomato
This is an Armenian favourite.
900 g/2 lb liver
Marinade
2 tablespoons tomato purée
1 clove garlic, crushed
juice 2 lemons
1 teaspoon chilli powder
1 teaspoon salt
1 teaspoon cumin
Mix all the marinade ingredients together in a large bowl, add the pieces of liver and turn until well coated. Leave at room temperature for 30 minutes. Thread on to skewers and cook as above.
kafta kebab minced meat kebab
Kafta kebab (kafah or kofteh in old Aramaic and Persian) is the general name given to all kinds of minced meat kebabs—lamb, beef or a mixture of the two. It is by far the most popular form of kebab. There are many variations throughout the region and some are regional specialities. I have noted a few to show the possibilities, but you can also experiment at your leisure.
The meat should be minced twice, although in Iran it is often minced three or four times. The secret of a good kafta kebab lies in its seasoning. Apart from salt and pepper, chopped onion and/or parsley, cumin, coriander, mint, chilli pepper and cinnamon are often used in differing combinations.
basic recipe
900 g/2 lb lamb (or beef or a mixture of the two), minced twice
2 onions, very finely chopped
1–2 eggs
salt and pepper to taste
Put all the ingredients into a large bowl and knead until very smooth. Take a lump of meat about the size of an egg and, with damp hands, pass a skewer through it and then squeeze the meat out gently until the kebab is thin and sausage-shaped. Continue until you have used up all the meat. Cook on a well oiled grid, turning frequently, for about 10 minutes.
soong kebab mushroom kebab
Ingredients as for the basic recipe
32 button mushrooms
Prepare the meat as described above. Divide the mixture into 24 small balls. Thread the balls and mushrooms alternately on to skewers, starting and ending with a mushroom. Allow 3 meatballs and 4 mushrooms per skewer. If the mushrooms show a tendency to split soak them for a little while in warm water. Cook as with the basic recipe.
urfa kebab
A favourite from Urfa, a city in Turkey.
Ingredients as for the basic recipe
3–4 aubergines
Prepare the meat as described in the basic recipe. Divide the mixture into 24 small balls.
Cut the hulls off the aubergines and then slice crossways into 1.2 cm/½ in thick rings.
Thread the meatballs and aubergine slices alternately on to skewers. Allow 3 or 4 of each on each skewer depending on length of skewers and size of fire. Cook as in the basic recipe, turning frequently. The aubergines usually take a little longer to cook than the meat and so continue cooking, turning frequently to prevent the meat burning, until the aubergines are tender. Serve with hot bread and a garnish of onion rings and tomato slices.
lulu kebab
This is an Armenian favourite.
900 g/2 lb minced lamb
2 tablespoons parsley, finely chopped
2 onions, very finely chopped
salt and black pepper to taste
Garnish
8 medium tomatoes
3–4 tablespoons parsley, finely chopped
1 medium onion, finely chopped
Put the meat, parsley, onions, salt and pepper into a large bowl and, with damp hands, knead for several minutes until well blended, smooth and malleable. Take a lump of meat about the size of an egg, pass a skewer through it and, with damp hands, gently squeeze the meat out along the skewer until it is thin and sausage-shaped. Continue until you have used up all the meat. Cook on a well-oiled grid, turning frequently, for about 10 minutes.
At the same time thread the tomatoes on to skewers and cook them over the fire. When the kebabs are cooked and ready to serve remove the tomatoes to a plate and skin them. Chop the tomato flesh, put in a bowl and mix in the chopped parsley and onion. Use the tomato mixture as a base and serve the kebabs on top.
kebab-e-koubideh
An Iranian favourite.
700 g/1½ lb lamb or beef, minced twice
1 large onion, finely chopped
½ teaspoon salt
Garnish sumac, 4 egg yolks and butter
Mix the meat, onion and salt together with damp hands until smooth. Take a lump of meat about the size of an egg and pass a skewer (preferably a flat one) through it. Squeeze the meat out firmly until it is thin and more rectangular than round in shape. Continue until you have used up all the meat. Cook over charcoal, turning frequently, for 8–10 minutes.
To serve slide 2 kebabs from the skewers on to a plate and then cover with a mound of chelo rice and sprinkle with some sumac. Top with an egg yolk and pat of butter. Stir the rice so that the egg, melted butter and sumac are all mixed together.
This dish is usually eaten with a spoon rather than a knife. Some Iranians also like to eat this dish with a slice of raw onion and a grilled tomato.
sheftalia Cypriot sausage kebab
A classic kebab of Greek origin, but equally popular with Armenians. Caul fat (panna) from the pig, which is the outer covering of the paunch, can be purchased from most butchers. In the Middle East it is sold by street vendors who follow their mules from street to street offering—amongst other things—sausage casings, lamb's heads, sweetbreads, livers, kidneys, hearts, etc. When panna is opened flat it has a patterned appearance of fat on very fine tissue. Usually served as a main dish with a rice pilav, salads and pickles these kebabs also make marvellous appetizers.
450 g/1 lb lamb or beef, minced 3 times
450 g/1 lb fatty pork, minced twice
1 large onion, peeled and minced
4 tablespoons parsley, finely chopped
2 teaspoons salt
½–¾ teaspoon black pepper
225 g/8 oz panna (caul fat from pig)
warm water
In a large bowl mix the meats, onion, parsley, salt and pepper together and knead for about 3 minutes or until smooth.
Dip the panna into a bowl of water for about 2 minutes. Remove and carefully open, one at a time, on a clean working top. Cut the panna into 10 cm/4 in squares using scissors or a very sharp knife.
Scoop out a spoonful of the meat mixture and shape it into a sausage about 5 cm/2 in long. Place this sausage near one edge of a piece of panna. Fold the edge and sides over and roll up firmly. When all the sausages are made thread them on to flat skewers, 2–3 per skewer and grill over charcoal, turning frequently. The panna will slowly melt away keeping the sausages moist. Serve immediately.
poultry and game
Chickens are excellent for grilling over a charcoal fire. Use only young ones—900–1350 g/2–3 lb in weight or poussins (baby chicken—4–6 weeks old)—which have proportionately more meat on their bodies in relation to their weight.
All kinds of poultry and game make excellent kebabs, e.g. duck, goose, turkey, guinea fowl, capon, quail, woodcock, partridge, pheasant and pigeon. Many of these are, unfortunately, not so easily available or are seasonal and therefore expensive. Rabbit also makes an excellent kebab and is available all the year round.
The basic preparation for grilling whole chicken is as follows:
Tie the wing tips over the breasts and fasten the neck skin to the back with a skewer. Push the spit through the bird from the tail end towards the front—the spit should emerge between the branches of the wishbone. Tie drumsticks and tail together.
Alternatively you can cut each chicken in half lengthways. If using a long spit thread the chicken halves crossways on to the spit piercing each one through the thigh meat. If using individual skewers then push 2 lengthways through each half. This makes it easier to turn without it slipping around the skewer.
At this stage you can either place the chicken in a marinade for a few hours or brush the bird with oil or a baste and cook it immediately. Cook over charcoal turning and brushing with any remaining marinade or with the oil or baste frequently. To test if the chicken is cooked pierce a thigh where the meat is thickest. If the juice runs clear and not pink then the chicken is ready.
Below are a few marinades which are ideal for whole, halved or jointed chicken:
A typical marinade from Turkey
300 ml/½ pint oil
3 cloves garlic, finely chopped
juice 1 lemon
salt and black pepper to taste
Blend the ingredients in a large bowl, add the whole or jointed chicken and mix well. Leave at room temperature for at least 2 hours. Use any remaining marinade as a baste while cooking.
A Kurdish baste
75 g/3 oz butter, melted
½ teaspoon whole savory
½ teaspoon rosemary
½ teaspoon thyme
½ teaspoon marjoram
½ teaspoon sweet basil
Mix all the herbs together in a small bowl and stir in the melted butter. Thread the chicken on to skewers and brush all over with the baste. Cook brushing frequently with the baste.
An Arab favourite from Libya and Egypt
1 tablespoon paprika
1 teaspoon cumin
½ teaspoon chilli pepper
salt to taste
75 g/3 oz melted butter
Mix all the ingredients together and use to baste the chicken frequently while cooking.
judi kebab chicken kebab
A classic from Armenia where chicken is marinated in oil, spices and garlic. This recipe uses poussins, but you can use one larger chicken and cut it into serving pieces. Serve with pilavs or bread and salads of your choice.
4 poussins
Marinade
150 ml/¼ pint oil
2 cloves garlic, crushed
1 lemon, thinly sliced
1 tablespoon sumac (optional)
2 teaspoons salt
1 teaspoon black pepper
Sauce
1 clove garlic, crushed
juice ½ lemon
2 tablespoons olive oil
¼ teaspoon cumin
Wash and dry the poussins. Cut each one into 8 pieces, i.e. 2 breasts, 2 wings, 2 drumsticks and 2 thighs.
Mix the marinade ingredients together in a large bowl. Add the chicken pieces, stir to coat, cover and refrigerate for at least 8 hours or overnight. Remove the pieces from the marinade and pat with kitchen paper, but do not dry.
Thread the pieces on to wide, flat skewers so that each skewer holds the 8 pieces of one poussin. Cook over charcoal, turning and basting regularly with the remaining marinade. Cook for 15–20 minutes then remove from the fire and slide the kebabs off the skewers on to a large serving dish. Mix the sauce ingredients together and sprinkle it over the chicken immediately before eating.
Variation
An equally famed Iranian chicken kebab kabab-e-joojeh which is always accompanied by chelo pilav, a pat of butter and sumac.
4×450–700 g/1–1½ lb poussins, cut as above or 2×900–1200 g/2–2½ lb chicken, each cut into 8 pieces
Marinade
225 g/8 oz finely chopped onion
6 tablespoons fresh lemon juice
1½ teaspoons salt
¼ teaspoon nutmeg
¼ teaspoon cinnamon
Baste
50 g/2 oz butter
pinch ground saffron dissolved in 2½ teaspoons warm water
Mix the marinade ingredients in a large bowl. Add the chicken pieces and turn until well coated. Cover and leave in the refrigerator overnight.
If using poussins thread all the pieces of one poussin on to each skewer, but if using 2 larger birds thread one wing, breast, thigh and drumstick on to each skewer. Melt the butter in a small pan and then stir it and the saffron into any remaining marinade. Brush the kebabs with this mixture and grill for 15–20 minutes, turning and basting occasionally. Serve the kebabs with chelo rice. Place a pat of butter on top of the rice and sprinkle liberally with sumac. As the butter melts mix the rice up so that the butter and sumac are distributed evenly throughout the rice.
kebab me auff israeli chicken kebab
Mrs Stein entered a kosher poultry store and asked the price of stewing chickens.
'Forty cents a pound,' said the butcher.
'Forty cents!' cried Mrs Stein. 'Why, just around the corner
Ellenberger sells for thirty-six cents a pound.'
'If Ellenberger sells stewing chickens for thirty-six cents a pound, why don't you buy there?' asked the butcher impatiently.
'Because he happens to be out of them today.'
'Look, lady,' said the butcher, 'as soon as I run out of stewers I'll sell them to you foronly twelve cents a pound—and you can't beat that price anywhere!'
(Encyclopedia of Jewish Humour)
A colourful and tasty recipe from Israel. The meat is marinated in wine, oil and garlic and garnished with fresh vegetables.
4 large chicken breasts
Marinade
juice 4 lemons
150 ml/¼ pint oil
150 ml/¼ pint dry red wine
2 cloves garlic, finely chopped
2 bay leaves
1 teaspoon ground coriander
1 teaspoon salt
½ teaspoon black pepper
Garnish
2 green peppers, seeded and quartered
2 onions, quartered
2 sticks celery, each cut into 4
4 large mushrooms, wiped clean and trimmed
2 avocados
50 g/2 oz sugar
1 tablespoon lemon juice
150 ml/¼ pint red wine
fresh mint leaves
Skin and bone the chicken breasts and cut into 2.5–4 cm/1–1½ in pieces.
Mix the marinade ingredients together in a large bowl. Add the pieces of chicken, toss to coat well then cover and refrigerate overnight.
Thread the chicken pieces on to skewers alternating with pieces of green pepper, onion, celery and a mushroom. Brush the meat and vegetables with any remaining marinade and grill for 10–15 minutes, turning and basting frequently.
While the kebabs are cooking cut the avocados in half and remove the stones. Mix the sugar, lemon juice and wine together and pour into the halved avocados. Garnish with the mint leaves and serve with the kebabs.
pasianni khorovadze pheasant kebab
This 'classic' from the Caucasus is usually served with tkemali sauce made of prunes, garlic and coriander.
2 pheasants
50 g/2 oz melted butter
salt and pepper to taste
Tkemali sauce —see recipe
Split each pheasant in half and wash thoroughly and dry. Brush each piece on both sides with some of the butter and sprinkle them with salt and pepper. Thread on to skewers and cook for about 30 minutes, turning and basting occasionally with the remaining butter. Take care not to overcook as pheasant flesh tends to dry out quickly. Serve on a bed of rice pilav with the tkemali sauce spooned over the top.
Variation
Iraqis and Iranians often use a marinade of:
½ teaspoon ground saffron diluted in 3–4 tablespoons hot water
6 tablespoons olive oil
juice 1 large lime or lemon
1 teaspoon salt
½ teaspoon paprika
The halved pheasants are marinated in this mixture for 2–3 hours before cooking. Serve on a bed of saffron pilav with yoghurt and salads.
fish kebab
Fish is healthful for the eyes.—Nedashin 54b.
Of the hundreds of different fish available to the Middle Eastern housewife about thirty are commonly used and easily available at the numerous fish shops scattered the length of the Mediterranean coastline. They are; auberjack, anchovy, bass, bonito, block, carp, comber, crab, eel, grey mullet, gurnard, lobster, mackerel, mussels, octopus, prawns, perch, red mullet, salmon, sardines, sea-bream, prawns, sole, squid, sturgeon, swordfish, trout, turbot and whiting. Of these the most popular are red and grey mullet, sea-bream, bass, swordfish, carp and trout. Certain recipes suggest a particular fish, but wherever possible I have also recommended fish that are locally available in the West.
There are two main methods of cooking fish in the Middle East: (a) frying in oil; (b) grilling over charcoal. Both methods are used in the numerous seaside restaurants of Egypt, Lebanon, Turkey and Cyprus.
Before grilling, the fish slices are marinated in herbs, spices and wine mixtures and then they are either threaded on to skewers or contained in a lightly oiled double grill. Sometimes aromatic herbs are thrown on to the coals to give extra aroma to the fish. Always make sure the grilled fish is soft and juicy inside with a crisp brown skin. To achieve this baste regularly with the marinade or with melted butter flavoured with a little lemon juice.
Large as well as small fish are suitable for charcoal cooking, but I suggest you make several diagonal slits on the sides and then tie with string to retain the shape. Some people like to fill these slits with garlic, bay leaves, cloves, thyme, etc.
A favourite Armenian marinade
150 ml/¼ pint oil
1 clove garlic, crushed
1 teaspoon salt
3 bay leaves
grated rind 2 oranges
A Caucasian favourite
4 tablespoons pomegranate juice
1 teaspoon lemon juice
1 teaspoon salt
½ teaspoon black pepper
150 ml/¼ pint yoghurt or soured cream
A Syrian favourite
4 tablespoons oil
4 tablespoons lemon juice
1 clove garlic, crushed
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon ground coriander
3 bay leaves
A Turkish favourite
300 ml/½ pint beer
1 tablespoon chives, chopped
2 tablespoons parsley, finely chopped
1 tablespoon dry mustard
2 cloves garlic, crushed
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon oregano
A Gulf speciality
In the Gulf region the Arabs like to spread puréed dates over the fish and set aside for 30 minutes.
To make the purée put some dried and stoned dates into cold water and leave to soak for 45 minutes. Rub the dates through a sieve or blend in a liquidizer with a little water to make a soft paste. With your hands spread the mixture over both sides of each fish and leave for about 30 minutes. Grill in the usual way.
kiliç şişte swordfish kebab
The fish of the Turks, swordfish is found in abundance along the Mediterranean coastline as well as the Black Sea. It is marinated and then grilled over charcoal. If swordfish is not available I suggest you use cod or halibut.
Serve with a pilav of your choice and tarator—see recipe.
900 g/2 lb swordfish
Marinade
1 onion, cut into 0.6 cm/¼ in slices and then separated into rings
3 tablespoons lemon juice
2 tablespoons oil
1½ teaspoons salt
1 teaspoon paprika
½ teaspoon black pepper
about 20 large bay leaves
Garnish 1–2 tablespoons parsley, finely chopped
Skin and bone the fish and cut into 2.5 cm/1 in cubes.
In a large bowl put the onion rings, 1½ tablespoons lemon juice, 1 tablespoon oil and the salt and paprika and black pepper. Add the pieces of fish and toss to coat well with the marinade. Marinate for 2–3 hours at room temperature.
Meanwhile, soak the bay leaves in boiling water for ½–1 hour and then drain.
Remove the fish from the marinade and thread on to skewers, alternating the pieces with bay leaves. Press firmly together so that the flavour passes from the leaves to the fish. Mix the remaining lemon juice and oil together and brush the kebabs with this baste. Cook over charcoal for about 10 minutes or until golden, turning and basting frequently. Serve with rice and tarator sauce.
NB If you do not wish to use the tarator sauce I suggest you make a simple dressing by mixing together 2 tablespoons lemon juice, 3 tablespoons olive oil, 2 tablespoons finely chopped parsley and ½ teaspoon paprika. Spoon this over the kebabs.
trabzon yilan baligi turkish eel kebab
From Trabzon on the Black Sea coast this recipe is a favourite of the Laz people—Georgian Muslims who emigrated from the Caucasus in the seventeenth century to the Ottoman Empire. Lazes are famed for their beautiful women, wild dances (lezginka) and great acumen for business.
There are several types of eel, any of which will do well.
900 g/2 lb conger eel, skinned and filleted, washed, dried and cut into 2.5 cm/1 in pieces
1 teaspoon salt
1 teaspoon black pepper
4 slices white bread, crusts removed and cut into 2.5 cm/1 in pieces
8 small tomatoes, halved
4 bay leaves
3 tablespoons oil
Garnish lemon wedges, tarragon leaves, shallots or spring onions
Rub the eel cubes all over with the salt and pepper. Thread the pieces of eel, bread cubes and tomato halves alternately on to 4 skewers, placing one bay leaf in the centre of each skewer. Brush the kebabs all over with the oil. Cook for 8–10 minutes, turning frequently until the bread is toasted and the eel cubes are tender when pierced with a sharp knife. Serve immediately on a bed of rice pilav with lemon wedges and a side plate of tarragon leaves, shallots or spring onions.
taparagan mackerel in a spicy tomato sauce
A family favourite. My mother always used mackerel, but trout, red or grey mullet or any fish steaks will do equally well. The fish is marinated, grilled and served with a salad–taparagan (literally meaning 'the wanderer') of onions, tomato purée and spices.
Serve with fried vegetables and a pilav of your choice and/or fresh salads.
4 medium-sized mackerel
Marinade
1 tablespoon tomato purée
4 tablespoons oil
4 tablespoons lemon juice
1 teaspoon salt
1 teaspoon allspice
½ teaspoon chilli pepper
Salad
2 large onions, thinly sliced
1 tablespoon parsley, chopped
1 level tablespoon tomato purée
2 tablespoons olive oil
2 tablespoons lemon juice
1 level teaspoon salt
1 level teaspoon allspice
¼ teaspoon chilli pepper
Cut the heads and tails off the fish. Slit down the whole length of the stomach and remove the insides, backbone and as many of the other bones as possible. Wash thoroughly.
Mix the marinade ingredients together in a shallow bowl and rub the marinade over both sides of each fish. Fold each fish over to take its original shape, cover and leave in the refrigerator overnight.
When ready to cook lay the fish flat out on an oiled double grill and cook for 10–15 minutes, turning once or twice. Take care not to overcook or they will become dry. If the fish are large and your grill not very big then you may find it necessary to cook 2 at a time, keeping the first 2 warm.
While the fish are cooking prepare the salad by mixing together in a bowl the tomato purée, olive oil, lemon juice, salt, allspice and pepper. Stir in the sliced onions and chopped parsley.
When the fish are cooked place them on a large platter, arrange the salad around the edges and garnish with lemon wedges.
## fish dishes
fried fish
Next to cooking fish over charcoal or under the grill the most popular method is frying in sizzling hot oil. Arabs squeeze a little lemon juice over the fish and eat it with fresh vegetables.
Below is a simple method for frying fish in oil.
Wash, clean and scale the fish if necessary. Leave small fish whole, but cut larger ones into steaks or fillets. Pat dry with kitchen paper. For each 500 g/1 lb of fish use 150 ml/¼ pint oil. Heat the oil to sizzling in a large pan, add the fish and fry for 5–10 minutes. Turn at least once and shake the pan once or twice to prevent the fish sticking. Remove the fish, drain and transfer to a large serving dish. Sprinkle with a little salt and chopped parsley and squeeze a little lemon juice over the top.
Sometimes the fish is dredged in flour or flour seasoned with salt, pepper and spices, e.g. cumin, paprika, ground coriander, etc. before deep frying.
Arabs often fry bread in the fish oil and serve it as a garnish.
a typical flour batter
100 g/4 oz flour
200 ml/7 fl oz warm water or milk
1 tablespoon olive oil
1 teaspoon salt
1 teaspoon black pepper
1 egg, beaten
Sift the flour into a bowl and make a well in the centre. Gradually stir in the water or milk until you have a smooth batter. Fold in the remaining ingredients and mix thoroughly. Dip the fish into the batter and deep fry until cooked through and golden.
an istanbul-style batter
100 g/4 oz flour
1 teaspoon baking soda
1 teaspoon salt
¼ teaspoon black pepper
¼ teaspoon cayenne pepper
1 bay leaf, finely chopped
½ teaspoon oregano
1 egg, beaten
2 tablespoons raki (ouzo)
150 ml/¼ pint water
Sift the flour and soda into a large bowl and mix in the salt, black pepper, cayenne pepper, bay leaf and oregano. Add the beaten egg and mix in with a wooden spoon. Mix in the raki. Gradually stir in the water and beat until the mixture is smooth. Dip the fish into the batter until completely coated and then deep fry in hot oil until cooked through and golden.
NB The batter in the 2 recipes above is sufficient for 900 g/2 lb of small fish, e.g. sardines, sprats, anchovies, etc. which have been cleaned and washed, but which have their heads and tails left on.
fish with sauces
Another attractive and tasty way of presenting fish is to first fry it and then to serve with a sauce. There are many sauces, a few are given below.
samak magli kousbariyeh
fried fish egyptian-style
900 g/2 lb small whole fish or large one, thickly sliced
300 ml/½ pint oil
Sauce
2 tablespoons oil
1 large onion, thinly sliced
3 large tomatoes, blanched, peeled and coarsely chopped
100 g/4 oz hazelnuts or walnuts, chopped
50 g/2 oz pine kernels
a little water or dry white wine
3 tablespoons parsley, finely chopped
1½ teaspoons salt
½ teaspoon black pepper
½ teaspoon allspice
Fry the fish or fish slices as described at the beginning of this section.
Heat the oil in a large saucepan, add the sliced onion and fry until soft. Add the tomatoes and continue to simmer until soft. Add the nuts and fry for 2 minutes. Add enough water or wine to cover and season with the parsley, salt, pepper and allspice. Simmer for a few minutes then add the fried fish, turn to coat with the sauce and simmer for 12–15 minutes. Serve with a rice pilav, salads and bread.
samak-bi-tahina Sesame cream sauce
An Arab favourite, particularly in Lebanon, Syria and Palestine. Traditionally this dish is colourfully and artistically decorated with olives, pomegranate seeds, parsley, radishes, etc.
You can either fry or bake the fish before adding the sauce. If you are frying then follow the directions. This recipe, however, is for baked fish.
Serve cold either as part of a buffet spread or a main dish with a pilav and salads.
900–1350 g/2–3 lb whole fish, e.g. snapper, sea-bass, John Dorey, etc.
2 tablespoons lemon juice
1½ teaspoons salt
½ teaspoon black pepper
2–3 tablespoons olive oil
Sauce
2 cloves garlic, crushed
1 teaspoon salt
10 tablespoons tahina cream
juice 1 lemon
6 tablespoons water
3 tablespoons parsley, finely chopped
Garnishes An attractive combination of the following: pomegranate seeds, olives, chopped parsley or tarragon, roasted pine kernels, thinly sliced radishes, cucumbers, etc.
Clean and scale the fish if necessary. Leave the head on, but remove the eyes. Rinse and dry and make 2–3 slits on each side of the body.
Mix the lemon juice, salt, pepper and oil together, rub all over the fish, place in a shallow dish, cover and refrigerate for 1 hour.
Oil a shallow baking dish, put the fish in it and pour any remaining oil and lemon juice mixture over the top. Bake in the centre of an oven preheated to 180°C, 350°F, gas mark 4 for 30–40 minutes or until the fish is cooked. Test with a fork—if the fish flakes easily then it is done. Baste with the pan juices occasionally and do not overcook. Lift the fish carefully on to a large serving dish, cover and chill.
Meanwhile, prepare the sauce by placing the crushed garlic and salt in a bowl and mixing in the tahina. Stir in the lemon juice and water and mix until smooth and creamy. Add the parsley. Spread half the sauce smoothly over the fish leaving the head and tail uncovered, and decorate attractively with the garnishes. Chill and serve with the remaining sauce.
Variation
Often, especially in Lebanon, the fish is boned after baking. Then the flesh is seasoned with 1 teaspoon salt, ½ teaspoon black pepper and then re-shaped to its original form.
The head and tail are replaced and the body is covered with the sauce and then garnished.
barbouni red mullet in wine
Red mullet is, perhaps, the most popular fish throughout the Mediterranean coastline. It is the famed Sultan Ibrahim of the Arabs, barbounya of Turkey, and barbouni to the Greeks. It is a fish that fries and bakes well and has a peculiarly 'Mediterranean' taste to it. This recipe from Cyprus—hence the inclusion of wine—is particularly good.
Serve with pilav, salads and pickles.
900 g/2 lb red mullet, entrails removed
olive oil
Wine sauce
300 ml/½ pint dry white wine
3 cloves garlic, crushed
1 tablespoon tomato purée
2 tablespoons parsley, finely chopped
1 teaspoon dried tarragon
25 g/1 oz dry breadcrumbs
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon fennel
Cut the fins off the fish. Wash thoroughly inside and out.
Line a large baking dish with silver foil, leaving enough to fold over and cover the fish. Brush the foil with olive oil and place the fish in the dish.
In a bowl mix together the wine, garlic and tomato purée and then add the parsley, tarragon and breadcrumbs. Sprinkle the fish with the salt, pepper and fennel and then pour in the wine sauce. Fold the foil over to seal in the fish. Bake in an oven preheated to 190°C, 375°F, gas mark 5 for 25–30 minutes or until the fish is tender and flakes easily. Remove the dish from the oven and carefully lift the fish on to a serving dish. Pour the sauce over the top and garnish with a little parsley.
Tomato is extensively used with fish and there are several fine tomato-based sauces. Here are two which are ideal with fried, baked or grilled fish. The quantities given are adequate for 900 g/2 lb of fish.
loligi salsa
An Armenian favourite.
50 g/2 oz butter
1 small onion, finely chopped
5–6 medium tomatoes, blanched, peeled and finely chopped
2 cloves garlic, crushed
2 tablespoons parsley, finely chopped
1 bay leaf
1 teaspoon salt
½ teaspoon black pepper
Melt the butter in a saucepan, add the onion and fry until soft. Add the remaining ingredients and cook over a low heat for 15 minutes or until the sauce thickens. Stir frequently. Pass through a sieve and serve warm.
tomates salsasi
A recipe from Izmir, Turkey which is popular with the Greeks as well as the Turks.
700 g/1½ lb ripe tomatoes, coarsely chopped
2 teaspoons salt
1 tablespoon sugar
¼ teaspoon black pepper
25 g/1 oz butter, melted
1 tablespoon flour
¼ teaspoon basil
½ teaspoon oregano
Place the tomatoes in a saucepan and cook over a low heat for about 10 minutes, stirring frequently. Add the salt, sugar and pepper and cook for a further 10 minutes. Pass the tomatoes through a sieve into another pan and discard the skin, seeds, etc. Cook the strained tomatoes over a low heat for a further 15 minutes.
In a small bowl mix the melted butter and flour together and then stir in 3–4 tablespoons water until you have a smooth paste. Add this mixture to the tomatoes and stir until blended. Add the basil and oregano and cook over a lower heat for a few more minutes and then serve.
hamsi tavasi anchovy with rice
From the Black Sea coastline of Turkey, this is a speciality of Trabzon's Laz people. Serve with fresh salads and pickles.
700 g/1½ lb fresh anchovies, cleaned and deboned
4 tablespoons salt
75 g/3 oz butter
1 onion, finely chopped
250 g/9 oz long grain rice, washed thoroughly under cold water and drained
600 ml/1 pint boiling water
2 tablespoons hazelnuts, halved or pine kernels or blanched slivered almonds
1½ teaspoons sugar
1 tablespoon sultanas
½ teaspoon allspice
1 teaspoon salt
½ teaspoon cinnamon
½ teaspoon black pepper
Garnish
¼ teaspoon dillweed
½ teaspoon paprika
1 tablespoon sumac powder, optional
Place the anchovies in a large pan, sprinkle with the salt and set aside.
Melt 50 g/2 oz of the butter in a large saucepan. Add the onion and fry for a few minutes until soft. Add the rice and fry for 5 minutes, stirring frequently. Add the water, nuts, sugar, sultanas, allspice, salt, cinnamon and black pepper and stir well. Bring to the boil and cook vigorously for 3 minutes. Reduce the heat, cover the pan and continue cooking for 15–20 minutes until all the liquid has been absorbed.
Grease a large casserole dish. Rinse the anchovies under cold water and arrange half of them in a single layer in the dish. Tip the rice mixture into the casserole and level it out then arrange the remaining anchovies over the top. Melt the remaining butter and pour it over the top. Cover the dish and cook in an oven preheated to 190°C, 375°F, gas mark 5 for about 15–20 minutes or until the fish are cooked. Remove from the oven and sprinkle with the dillweed, paprika and sumac. Serve hot with salad.
sayyadiyah fish with rice
A popular Arab speciality with numerous variations.
Sometimes the fish and rice are prepared and served separately, but in the recipe below they are cooked together. Serve with a bowl of fresh salad.
50 g/2 oz butter
900 g/2 lb halibut steaks, each cut in half
2 tablespoons lemon juice
½ teaspoon salt
¼ teaspoon black pepper
2 tablespoons parsley, finely chopped
Stew
6 tablespoons oil
1 onion, finely chopped
2 tablespoons pine kernels
1 tablespoon sultanas or raisins
½ teaspoon allspice
saffron rice—prepare plain rice pilav with 250 g/9 oz rice, etc., but stir ½ teaspoon powdered saffron into the rice while it is frying
2 tablespoons lemon juice
2 tablespoons parsley, finely chopped
1 teaspoon salt
½ teaspoon black pepper
Sauce
4 tablespoons oil
1 tablespoon pine kernels
1 tablespoon dried mint
1 tablespoon lemon juice
½ teaspoon cumin
Melt half the butter in a large shallow baking dish and add the pieces of fish. Sprinkle with the lemon juice, salt, pepper and chopped parsley. Dot the remaining butter over the fish. Bake in an oven preheated to 160°C, 325°F, gas mark 3 and cook until the fish flakes easily. Remove from the oven, leave to cool and then flake the fish and remove and discard the bones. Set the flaked fish aside.
To prepare the stew heat the oil in a large saucepan, add the onion and sauté until soft. Add the nuts, raisins, allspice and saffron rice as well as the lemon juice, parsley, salt and pepper. Mix all the ingredients together carefully. Spread half this mixture over the base of a large shallow baking dish. Spread half of the reserved fish over the top. Cover with the remaining rice and top with the rest of the fish.
Now prepare the sauce by heating the oil in a small frying pan. Add the nuts, mint, lemon juice and cumin and sauté for a few minutes, until the nuts are golden, stirring frequently. Pour the sauce evenly over the surface of the casserole. Bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 15 minutes or until the ingredients are heated through. Remove and serve with a salad.
Variation
A favourite Egyptian method is to stew the sayyadiyah.
Cut one onion in half lengthways and then cut into thin half-moon shaped slices. Fry in 6 tablespoons oil until golden. Add the fish steaks and season with 1½ teaspoons salt, ½ teaspoon black pepper and ½ teaspoon powdered saffron. Add sufficient water to cover and simmer over a low heat for about 10 minutes.
Break up the fish into smaller pieces, discarding the bones, add 350 g/12 oz washed, long grain rice plus enough water to cover the mixture. Cover and simmer for 15–20 minutes or until the rice is tender and fluffy.
barghoon-el-bahar bil roz
prawn fried rice
One of the numerous prawn recipes from the eastern coastline of Saudi Arabia. Use saffron rice as it gives both extra fragrance and a charming appearance to the dish.
saffron rice (prepare as with plain rice pilav using 250 g/9 oz long grain rice, etc. but stir ½ teaspoon powdered saffron into the rice while it is frying)
3 tablespoons oil
1 stick celery, cut into 1.2 cm/½ in pieces
100 g/4 oz French beans cut into 5 cm/2 in pieces
4 spring onions, thinly sliced
2 tablespoons soy sauce
450 g/1 lb prawns, frozen will do as well as fresh
2 eggs
2 tablespoons blanched almonds
Prepare the rice according to the recipe and keep warm.
Heat the oil in a large saucepan and fry the celery, French beans and half the spring onions for 3–4 minutes, stirring all the time. Add the soy sauce and the prawns and continue to cook, stirring gently, for a few minutes. Break the eggs into a small bowl, lightly beat and add to the pan. Lower the heat and continue stirring the mixture until the eggs are cooked.
Meanwhile, under a grill, cook the almonds for a few minutes until golden brown.
Transfer the rice to a large serving plate and top with the prawns and vegetables. Garnish with the remaining spring onions and sprinkle with the toasted almonds.
Variations
A Kuwaiti favourite—nachbous—makes use of turmeric and curry powder and has a garnish of prawns, tomatoes and bananas.
plain rice pilav, see recipe, and use 250 g/9 oz rice, etc.
40 g/1½ oz butter
2 eggs, beaten
450 g/1 lb prawns, shelled and deveined or frozen prawns thawed
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon turmeric
1 teaspoon curry powder, optional
½ teaspoon paprika
1 tablespoon soy sauce
2 pineapple rings, thinly sliced
Garnish
1 tablespoon parsley, finely chopped
8 whole prawns, cooked
2 tomatoes, thinly sliced
thinly sliced cucumber
8–10 radishes halved
1 banana, peeled and sliced into thin rings
Prepare the rice.
Melt the butter in a large pan, add the cooked rice and fry for 2 minutes, stirring constantly. Add the beaten eggs and cook until well mixed, stirring constantly. Chop the prawns finely and add to the pan together with the salt, black pepper, turmeric, curry powder, paprika, soy sauce and sliced pineapple. Mix well and cook for 8–10 minutes, stirring occasionally. Spoon the mixture into a glass bowl and press down hard to form a mould. Turn out on to a serving plate and garnish the top of the mould with the parsley and prawns. Decorate around the mould with the tomatoes, cucumber, radishes and banana slices. Serve with fresh salad.
stuffed fish
Most fish taste excellent when stuffed with fruit, vegetables, nuts, grains, etc. In the Middle Eastern cuisine it is the Greeks, North Africans, Armenians and Israelis who have perfected this method of cooking fish.
Any large fish is suitable, e.g. trout, sea-bass, mackerel, red mullet, sturgeon or carp. The fish should be thoroughly cleaned, washed and dried. If there are any roes leave them in as they are considered a great delicacy.
letzvadz tzook
A popular Armenian stuffed fish recipe.
Serve with a rice or burghul pilav or roast potatoes and salads.
4 medium-sized whole fish, thoroughly washed, cleaned and dried
Stuffing
6 tablespoons oil
450 g/1 lb carrots, peeled and coarsely grated
2 tablespoons parsley, finely chopped
2 cloves garlic, crushed
1 teaspoon salt
½ teaspoon black pepper
2 medium onions, cut into rings
Sauce
1 tablespoon tomato purée
2 tablespoons oil
150 ml/¼ pint boiling water
½ teaspoon salt
¼ teaspoon black pepper
Garnish 2 tablespoons olive oil mixed with juice of 1 lemon
To prepare the stuffing heat the oil in a saucepan, add the carrots and fry over a low heat for about 10 minutes, stirring occasionally. Stir in the parsley, garlic, salt and pepper. Divide the mixture into 4 and fill each fish. Oil a large baking dish and arrange the fish in it, but separated from each other by the onion rings.
Mix the tomato purée, oil and water together in a small bowl and season with the salt and pepper. Pour the sauce over the fish and bake in an oven preheated to 190°C, 375°F, gas mark 5 for 30–40 minutes or until the fish are tender and turning brown. Place on a serving dish and arrange the onion rings around them. Pour the tomato sauce over the top. Serve the oil and lemon juice in a small jug to pour over the fish.
Variation
A Lebanese version uses water chestnuts, pine kernels and rice in the filling.
The quantities given below are for a 1.8–2.24 kg/4–5 lb whole fish and would therefore be ideal for a buffet or dinner party.
1×1.8–2.5 kg/4–5 lb whole fish, cleaned, washed and dried
Stuffing
250 g/9 oz cooked plain rice pilav, see recipe
150 ml/¼ pint dressing made of oil, lemon juice, garlic, salt and pepper
1 tin (approx 150 g/5 oz) water chestnuts (or use fresh if available), halved
50 g/2 oz chopped spring onions
1 green pepper, thinly sliced
25 g/1 oz pine kernels
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon oregano
50 g/2 oz butter, melted
Garnish lemon wedges, thinly sliced radishes
Mix all the stuffing ingredients, except the melted butter, together in a large bowl. Stuff the fish with this rice mixture. Reserve any filling left over. Sew up the opening with a fine thread. Place the fish in a large greased baking dish and pour the melted butter over the top. Cover and bake in an oven preheated to 180°C, 350°F, gas mark 4 for 30–40 minutes, basting frequently with the butter, or until the fish is tender but not overcooked. Remove from the oven and carefully transfer the fish to a serving plate. Remove the thread.
If there was any stuffing left over, heat it through in a small pan and arrange around the fish.
Garnish with the lemon wedges and thinly sliced radishes.
dag memula im chatzilim
An Israeli version.
The usual fish is carp, but trout, cod or red mullet will do well. Serve with salad, roast or fried potatoes and pickles. The stuffing is made of aubergines, eggs and cheese.
4 medium-sized whole fish, cleaned, washed and dried
Marinade
2 teaspoons salt
½ teaspoon black pepper
juice 1 lemon
100 ml/4 fl oz oil
2 bay leaves
1 onion, thinly sliced
Filling
2 medium aubergines
3 eggs, hard-boiled, shelled and mashed
50 g/2 oz feta cheese, grated
2 cloves garlic, crushed
¼ teaspoon mustard
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon cumin
1 tablespoon mayonnaise
1 teaspoon lemon juice
1 teaspoon olive oil
Vegetables
2 tablespoons oil
2 spring onions, sliced
2 cloves garlic, crushed
2 tomatoes, sliced
2 green peppers, thinly sliced
Mix all the marinade ingredients together in a large shallow dish, add the fish, turn to coat with the marinade and leave for at least 4–5 hours.
Meanwhile, prepare the filling by cooking the aubergines, either in an oven or under the grill, until the skins are black and the flesh soft when poked. When cool enough to handle peel off the skin, scraping off and reserving any flesh which may come away with it. Chop the flesh and put into a bowl. Add the remaining filling ingredients to the aubergine pulp and mix thoroughly. Remove the fish from the marinade, reserving the marinade, and fill each fish with some of the filling.
To prepare the vegetables heat the oil in a pan and sauté the onions until golden. Add the garlic, tomatoes and green peppers and cook for about 5 minutes. Place this mixture in the bottom of a large, greased casserole dish. Arrange the fish on top of the vegetable mixture. Cover with the reserved marinade, place in an oven preheated to 180°C, 350°F, gas mark 4, and bake for about 30–45 minutes or until well cooked. Serve immediately.
sourbour trout stuffed with herbs
A speciality of Khuzestan, Iran popular with the Arabic speaking people of the region. Serve with a pilav or bread and salad and cooked vegetables.
4 medium-sized trout or mackerel, cleaned, washed and dried
2 tablespoons oil
Stuffing
40 g/1½ oz butter
2 tablespoons parsley, finely chopped
2 tablespoons tarragon, finely chopped
4 spring onions, finely chopped, including heads
1 tablespoon sweet basil
1 tablespoon coriander leaves, finely chopped
2 tablespoons mint, chopped
2 tablespoons radish leaves, finely chopped
2 tablespoons tamarind juice—see Glossary
½ teaspoon salt
½ teaspoon turmeric
Garnish lemon wedges, radishes
First prepare the stuffing by melting the butter in a pan. Add all the prepared herbs and onion and fry for 2–3 minutes, stirring constantly. Add the tamarind juice, salt and turmeric and fry for 2 more minutes before removing from the heat. Divide the stuffing into 4 and fill each fish. Secure the openings with toothpicks or thin thread. Heat the oil in a large baking dish, add the fish and turn in the oil. Bake in an oven preheated to 200°C, 400°F, gas mark 6 for 20–25 minutes or until the flesh flakes easily. Baste occasionally. Remove from the oven and carefully transfer the fish to a large serving plate. Garnish with the lemon wedges and radishes.
ishkan noushov stuffed fish with almond sauce
A classic, ishkan is the salmon trout found only in Lake Sevan, over 1370 m/4500 ft above sea level in the Caucasus.
The stuffing is made of almonds and spices and the sauce of almonds, wine and cream. This is a brilliant dish of rare sophistication. Use trout and serve with a burghul pilav or potatoes and salad.
4 medium-sized trout, cleaned, washed and dried. Do not cut off heads
2 tablespoons oil or 25 g/1 oz butter
Stuffing
100 g/4 oz ground almonds
juice 2 lemons
1 teaspoon salt
1 teaspoon black pepper
1 teaspoon cumin
a little water
Sauce
100 g/4 oz ground almonds
1 glass white wine
150 ml/¼ pint single cream
salt and black pepper to taste
a little water or milk to mix
Garnish a few thin strips green pepper, lemon wedges
Mix the ingredients for the stuffing together in a bowl and add just enough water to produce a thick paste. Divide the mixture into 4 and fill each trout. Heat the oil or butter in a large baking dish, add the fish and turn to coat in the oil or butter. Bake in an oven preheated to 200°C, 400°F, gas mark 6 for 20–30 minutes or until tender.
Meanwhile, in a small saucepan mix the almonds for the sauce with the wine and cream and season to taste with the salt and pepper. Add sufficient water to produce the consistency of single cream. Bring to the boil and simmer very gently for about 10 minutes. Stir frequently or the mixture will stick to the pan. If the sauce becomes too thick for your liking then add a little water or milk.
Arrange the fish on a large dish and garnish with the lemon wedges. Stir the slices of green pepper into the sauce, pour a little over each trout and serve the rest separately.
levrek sultan murat
sea-bass with potatoes and artichokes
A recipe from Izmir, Turkey, a culinary centre of great repute famed for its kufta, sweets and fish dishes—of which this is one. Named after an Ottoman Sultan who—so we are told—was rather fond of this particular fish.
Serve with pilav and salads.
700 g/1½ lb sea-bass fillets—or you can use cod or halibut, etc.
2 large potatoes, peeled and cut into 1.2 cm/½ in cubes
5 tablespoons vegetable oil
½ teaspoon salt
2 artichokes
½ teaspoon salt
1 tablespoon lemon juice
juice 1 large lemon
3 tablespoons flour seasoned with ½ teaspoon salt
10 tablespoons melted butter or oil
6–8 mushrooms, quartered
2 tablespoons parsley, chopped
Garnish lemon wedges
Cut each fish fillet into 6–8 pieces.
Put the potato cubes in a pan, cover with water, bring to the boil and simmer for 2 minutes then drain. Heat the oil in a saucepan, add the potato cubes, sprinkle with ½ teaspoon salt, and fry gently for about 15 minutes, turning occasionally until golden all over.
Meanwhile, prepare the artichokes by first peeling the tough outer skin from the stem and cut 0.3 cm/⅛ in off the stem end. Remove any discoloured outer leaves and cut about 1.2 cm/½ in off the top of the remaining leaves. Quarter the artichokes lengthways, remove the pinkish leaves from the centre and scrape out and discard the hairy choke. Drop the artichoke quarters into a saucepan of water with the salt and tablespoon of lemon juice. Bring to the boil and simmer for 15 minutes.
Place the juice of 1 lemon in a small bowl, toss the pieces of fish in it and then roll them in the seasoned flour. Heat the butter or oil in a large saucepan, add the fish pieces and sauté for 4–5 minutes, turning occasionally. Transfer to a serving dish and keep warm.
Add the drained artichokes, potatoes and mushrooms to the pan and sauté for 3–4 minutes, stirring frequently. Stir in the parsley. Pile the vegetables over the fish and garnish with the lemon wedges. Serve immediately.
gormeh sabzi ba mahi iranian fish stew
An Iranian speciality which is colourful and spicy with spinach, fenugreek, limes and turmeric. There are very few interesting fish dishes of Iranian origin as most fish is either fried or grilled. This, however, is a rich stew and the traditional fish would be sturgeon or perch, but any white fish will do.
If you cannot obtain dried or fresh limes use lemon juice instead. Serve with a pilav of your choice.
3 tablespoons red kidney beans, washed under cold water
60 g/2½ oz butter
3 tablespoons parsley, finely chopped
2 leeks, thinly sliced and thoroughly washed to remove all sand
450 g/1 lb spinach thoroughly washed, shaken dry and coarsely chopped
2 sticks celery, thinly sliced
1 tablespoon powdered fenugreek or 2 tablespoons chopped fresh fenugreek
300 ml/½ pint fish stock or water
5 dried limes or 2 fresh limes or juice 2 lemons
900 g/2 lb fish fillets, e.g. perch, sturgeon, haddock, halibut, etc.
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon turmeric
Place the kidney beans in a saucepan, cover with lightly salted water and bring to the boil. Simmer for about 1 hour or until the beans are tender.
Melt the butter in a large saucepan, add the parsley, leeks, spinach, celery and fenugreek and sauté for a few minutes, stirring all the time. Add the stock, stir well, cover and cook over a low heat for 10 minutes. Cut each dried lime at one end and drop into the vegetable mixture or halve the fresh limes and add, or add the lemon juice. Add the fish, cover the pan and continue cooking for 30 minutes or until the fillets are flaky and tender.
Drain the kidney beans and add to the stew together with the salt, pepper and turmeric. Cook for 5 more minutes and then serve immediately with a pilav.
kagitta barbunya red mullet in foil
A most attractive way of cooking fish. In the past it was wrapped in parchment, but foil is usually used today. The flavour of the fish is retained and a succulent, tender flesh produced. Fish smells are also minimized.
The fish is wrapped whole with a few sprigs of parsley or mint and lemon wedges, or with tomato slices and herbs.
Although I have used a recipe popular in Istanbul, this fish—Sultan Ibrahim to the Arabs—is equally popular with Lebanese and Syrians.
Serve with cooked vegetables and/or pilavs.
4 medium red mullet
3 tablespoons olive oil
4 tablespoons parsley, finely chopped
juice 1 lemon
½ teaspoon salt
¼ teaspoon black pepper
¼ teaspoon paprika
½ teaspoon thyme
Leave the heads and tails on the fish but make a small slit in one side of each and clean out the insides thoroughly. Wash under cold running water and pat dry. Rub the fish with the olive oil. Now place each fish on a piece of foil and wrap each one up so that it is completely enclosed. Place on a baking sheet and cook in the centre of an oven preheated to 190°C, 375°F, gas mark 5 for 30–45 minutes.
Meanwhile, in a small bowl mix together the remaining ingredients. When the fish are cooked remove from the oven, unwrap and transfer the fish to a serving platter. Sprinkle with the parsley mixture and serve.
## meat dishes
lamb
Tell me, love of my soul, where you graze your sheep
and where you rest them at noon?
Why should I sit here, like a nomad
among your companion's flocks, and the men saying
If you don't know where, my beauty, try
the sheep tracks—take your goats off to graze
by the shepherd's tent.
(The Song of Songs)
In the Middle East lamb is the most widely used meat—indeed one can say it is the meat, since beef, pork, etc. are unavailable to most due to geographical and religious limits.
I have, therefore, arranged the section on meat dishes in the following manner: it begins with whole lamb, followed by leg of lamb, chops, and ends with offal. In short, every part of the animal is shown as being used—barring the sheep's eyes, but including the head, feet and sweetbreads!
This is followed by a few recipes of beef and pork of merit and the section is completed with minced meat kebabs—an art form in which the Middle Eastern housewife is supreme.
In the chapter on kebabs whole lamb kebab has been described, suffice therefore to say that the same treatment applies to the preparation for lamb roasted in the oven.
Whole roast lamb, sheep or goat is a festive dish and it is absolutely magnificent when stuffed with rice, meat, nuts and fruits.
Baby lamb or kid is the ideal choice; the meat is tender and tastes excellent. Use a 9–11 kg/20–25 lb lamb and order in advance from your butcher. They are not difficult to obtain, but do make sure that your oven is large and deep enough to take the whole lamb before you contemplate tackling this fabulous dish of 'kings and nomads'. I have included two recipes for stuffed whole lamb since I suggest that if you are going to go to the trouble of cooking a whole lamb then you might as well stuff it as well—this way it will feed more people!
Stuffings vary from region to region are two traditional ones.
butzun kuzu dolmasi stuffed whole lamb
This is a favourite throughout the region.
First prepare the lamb, ask the butcher to remove and discard the entrails, but to keep the kidney, liver and heart and the lumps of fat at the back of the kidneys—these can all be chopped and used in the stuffing.
If the lamb is frozen allow to thaw for about 24 hours. Wash thoroughly in a large sink and then dry.
Rub inside and out with salt and pepper and sprinkle with the juice of 1 large onion. You can also rub in a mixture of spices, e.g. 1 teaspoon each of ground coriander, cumin, ginger and turmeric.
Stuffing
About 1.35 kg/3 lb long grain rice, washed thoroughly under cold water and drained
1 teaspoon powdered saffron (optional)
6 tablespoons oil
3 large onions, finely chopped
225 g/8 oz almonds, coarsely chopped
225 g/8 oz walnuts, coarsely chopped
100 g/4 oz raisins
3 tablespoons salt
3 teaspoons black pepper
1 tablespoon allspice
Garnish sprigs parsley
Bring a very large saucepan, half filled with lightly salted water, to the boil. Add the rice, and saffron if using, and simmer until just tender—15–20 minutes—and then drain.
Heat the oil in a large pan, add the onions and fry until soft. Remove the onions with a slotted spoon and add to the rice. Add the almonds, walnuts and raisins to the pan and fry until the almonds are lightly golden, stirring frequently. Add to the rice together with the salt, pepper and allspice and mix well with a large fork. Stuff the lamb tightly with the mixture and sew up the opening with a long, strong needle and thread.
Put the lamb in a large baking dish, brush with oil and cook in an oven preheated to 160°C, 325°F, gas mark 3 for about 4–5 hours, turning at least once. The exact cooking time will depend on the size of the lamb, but it should be very tender—so that the meat is almost falling away from the bones. Baste regularly with the pan juices. Transfer to a large serving tray, open the stomach and allow some of the stuffing to spill out and then garnish with sprigs of parsley.
Serve with fresh salads of your choice.
kharoof mahshi
lamb stuffed with chicken stuffed with eggs
One of the great dishes of the world, the 'Babushka' of all stuffed recipes—unless, of course, one were to stuff a camel with the lamb!
9–11 kg/20–25 lb lamb, prepared as described in the previous recipe
900 g/2 lb chicken, cleaned
1 teaspoon salt
½ teaspoon turmeric
¼ teaspoon black pepper
3 hard-boiled eggs, shelled
Stuffing
900 g/2 lb long grain rice, washed thoroughly under cold water and drained
1 teaspoon powdered saffron (optional)
4–5 tablespoons rosewater
100 g/4 oz fat from the kidneys, minced or 6–8 tablespoons oil
3 large onions, finely chopped
the lamb's liver, washed and finely chopped
the 2 kidneys, washed and finely chopped
100 g/4 oz blanched almonds, chopped
100 g/4 oz pine kernels
100 g/4 oz pistachio nuts
100 g/4 oz raisins
3 tablespoons salt
3 teaspoons black pepper
1 teaspoon cinnamon
1 teaspoon cumin
Wash and dry the chicken, rub the cavity with a mixture of the salt, turmeric and pepper and set aside.
Bring a very large saucepan half filled with lightly salted water to the boil. Add the rice, saffron and 3 tablespoons rosewater and simmer for 15–20 minutes or until just tender, then drain.
Heat the fat or oil in a pan, add the onions and fry until soft. Add the chopped liver and kidneys and fry, stirring frequently for 5–7 minutes. Remove with a slotted spoon and add to the rice.
Add to the frying pan the nuts, raisins, salt, pepper, cinnamon and cumin and fry for 2–3 minutes, stirring frequently. Add to the rice and mix well together with a large fork.
Insert the hard-boiled eggs into the chicken and then spoon in a little of the rice mixture. Secure the opening with thread or a small skewer. Partly stuff the lamb with the rice, then put the stuffed chicken in and continue to fill the cavity with the rice. Secure the opening with a long, strong needle and thread.
Brush with oil and bake in an oven preheated to 160°C, 325°F, gas mark 3 for 4–5 hours or until the meat is very tender. Exact time will depend on the size of the lamb. Turn the lamb at least once and baste occasionally with the pan juices. Place on a large serving tray, open the stomach and allow some of the stuffing and the chicken to spill out. Garnish with parsley and sliced fresh vegetables such as cucumber, tomatoes, radishes, spring onions, lemons, etc.
roast leg of lamb
A popular way of cooking is to bone the meat, season it with herbs and spices and then bake in the oven with vegetables. Below are a few examples, all for a 1.8–2.25 kg/4–5 lb leg of lamb which should be sufficient to feed 8–10 people.
kari vodk
An Armenian favourite.
1.8–2.25 kg/4–5 lb leg of lamb, boned and trimmed of excess fat
2 cloves garlic, crushed
½ teaspoon thyme
½ teaspoon rosemary
1 tablespoon fresh mint, finely chopped
1 teaspoon salt
½ teaspoon black pepper
4 cloves garlic, halved lengthways
50 g/2 oz melted ghee or butter
Mix the crushed garlic, thyme, rosemary, mint, salt and pepper together. Spread this mixture over the inside of the leg of lamb and then tie the leg up with string to resemble its former shape. Make small, 2.5 cm/1 in deep incisions with the point of a sharp knife over the outside of the leg and put a half clove of garlic into each slit.
Put the leg into a large baking dish, fat side up, pour the melted fat over the top and bake in an oven preheated to 180°C, 350°F, gas mark 4 for 2½–3 hours or until cooked to your liking. Baste regularly with the pan juices.
Variations
A Yemeni favourite fakhed kharouf Yemani has the bone removed in such a way that a pocket is left which is then filled with dates.
First the inside and outside of the meat is rubbed with a mixture of 2 tablespoons oil, 1 tablespoon salt, 1 teaspoon black pepper, ½ teaspoon each of thyme and rosemary and 2 tablespoons chopped fresh mint or 2 teaspoons dried mint. Put 2 bay leaves and 100 g/4 oz chopped dates into the pocket and secure the opening with a small skewer or strong thread. Bake as with the recipe above.
The Syrians and Lebanese stuff the leg pocket with 50 g/2 oz chopped, dried figs, 50 g/2 oz stoned, chopped prunes and 50 g/2 oz raisins mixed with ½ teaspoon thyme, ½ teaspoon sage and 25 g/1 oz melted butter. Cook as above.
The Greek arni lemonato is similar to kari vodk except that the meat is generously rubbed with lemon juice.
Use 2 large lemons, halved and after rubbing pour the excess juice into the baking dish and baste often.
Note More often than not vegetables are cooked with the meat. Any of the following will do well:
8–10 small globe artichoke hearts—add to the meat for the last hour of cooking and sprinkle with additional herbs, salt, pepper and lemon juice, baste occasionally
450 g/1 lb or more potatoes, peeled, quartered or thickly sliced
2 or 3 large onions, quartered
450 g/1 lb tomatoes, thickly sliced
450 g/1 lb celery sticks, cut into 7.5–10 cm/3–4 in pieces and blanched in boiling water for 5 minutes before adding to the meat pan
sarapli kuzu pirzolasi lamb chops in wine
This recipe is from Istanbul.
about 8 lamb chops, trimmed of excess fat
40 g/1½ oz butter
225 g/8 oz mushrooms, thinly sliced
1 clove garlic, crushed
1 teaspoon salt
¼ teaspoon black pepper
2 large tomatoes, blanched, peeled and chopped
¼ teaspoon basil
¼ teaspoon oregano
100 ml/4 fl oz dry white wine
In a large saucepan cook the lamb chops in their own fat juices for 5 minutes, turning several times, until browned on both sides. Remove and keep warm. Add the butter to the pan, then add the mushrooms and garlic and fry for 5 minutes, stirring frequently. Return the chops to the pan, add the remaining ingredients and mix well. Cover the pan and simmer for about 30–45 minutes or until the chops are tender. Serve with a rice pilav and salads of choice.
budugov miss lamb chops in a fruit sauce
An Armenian version from the Caucasus. Use the same amount of chops as for sarapli kuzu pirzolasi.
about 8 chops
50 g/2 oz butter
300 ml/½ pint water
1 teaspoon salt
¼ teaspoon black pepper
½ teaspoon cinnamon
½ teaspoon garam masala
75 g/3 oz raisins or sultanas
175 g/6 oz dried apricots, soaked overnight in cold water and then drained
Melt the butter in a large saucepan, add the chops and fry for a few minutes until browned on both sides. Pour off most of the fat and then add all the remaining ingredients. Bring to the boil, reduce heat to low, cover and simmer for 30–45 minutes, or until the chops are tender. Stir occasionally to prevent sticking. Arrange a pilav of your choice around a dish and spoon the chops and fruit sauce into the centre and serve immediately.
maghleh-bil-khel fried liver with vinegar
This particular recipe is a speciality of Tripoli, Lebanon. Although it recommends lamb's liver there is no reason why calf's liver cannot be used with this and the following recipe.
Liver with vinegar is very tasty. Use wine vinegar and serve with a pilav of your choice.
25 g/1 oz ghee
700 g/1½ lb lamb's liver, gristle removed, sliced
1½ teaspoons salt
½ teaspoon black pepper
1 large onion, finely chopped
2 cloves garlic, crushed
1½ tablespoons dried mint
1 teaspoon flour
150 ml/¼ pint wine vinegar
about 3 tablespoons water
½ teaspoon paprika
Melt the fat in a large frying pan, add the liver, sprinkle with the salt and pepper and fry until browned on both sides. Remove the liver with a slotted spoon, drain and reserve. Add the onion and fry until soft, stirring frequently. Add the garlic, mint and flour and mix well. Stir in the vinegar, a little water and the paprika and cook for 5–6 minutes, stirring constantly. Add the liver, a little bit more water to cover if necessary, lower the heat and simmer for 10–15 minutes. Serve with a rice pilav and fresh salads.
khorak-e jegar va gholveh
liver and kidney stew
'Khorban! Jegar!'—My love! My darling! (Persian love expression.)
A recipe from Iran. Serve with pilavs and roast or fried vegetables.
40 g/1½ oz butter
1 large onion, finely chopped
1 lamb's liver (or calf's), gristle removed, chopped
2 hearts, tough valves, etc. removed, chopped
4 kidneys, skin and cores removed, chopped
450 g/1 lb tomatoes, blanched, peeled and chopped
225 ml/8 fl oz water
juice 1 lemon
1 teaspoon salt
¼ teaspoon black pepper
Melt the butter in a large saucepan, add the onion and fry until soft. Add the liver, hearts and kidneys and fry for a further 5 minutes, stirring frequently. Add the remaining ingredients and bring to the boil. Lower the heat and simmer for about 30 minutes or until the meat is cooked.
Variation
Kirshuh is a dish from Yemen similar to the one, except that it has a lot more spices—which give extra zest.
To the ingredients above add 1 teaspoon turmeric, 1 teaspoon coriander, ½ teaspoon cumin and 2 crushed cardamom pods.
Reduce the quantity of tomatoes from 450 g/1 lb to 225 g/8 oz.
Garnish with chopped parsley or coriander leaves.
larshon im rotef
lamb's tongue with a wine and raisin sauce
An Israeli recipe with a tasty sauce.
10–12 lamb tongues, washed, scraped and soaked in water for 3 hours
2 bay leaves
2 cloves
½ teaspoon allspice
1 tablespoon salt
Sauce
40 g/1½ oz butter
1 small onion, finely chopped
25 g/1 oz flour
350 ml/12 fl oz stock from the tongue
25 g/1 oz sugar
50 ml/2 fl oz dry white wine
1 tablespoon lemon juice
3 tablespoons seedless raisins
½ teaspoon ginger
Drain the soaked tongues, place in a large saucepan, cover with water, add the bay leaves, cloves, allspice and salt and bring to the boil. Remove any scum that appears on the surface then lower the heat, cover the pan and simmer for 2 hours.
Fifteen minutes before the end of the cooking time prepare the sauce by melting the fat in a saucepan, adding the onion and frying until soft. Stir in the flour and cook for another minute. Strain about 350 g/12 oz of stock from the tongue pan and stir into the onion mixture. Cook over a low heat, stirring constantly until the sauce thickens and begins to bubble. Stir in the remaining ingredients and cook for about 5–7 minutes before removing from the heat.
When the tongues are cooked drain and leave until cool enough to handle. Skin them and remove the gristle and the bone from the root ends. Slice each tongue lengthways in half. Add the halved tongues to the sauce and then return the pan to the heat and simmer for a further 10 minutes. Serve warm with bread, roast vegetables and salads.
Variation
An Iranian version khorak-e-zaban prepares the tongues as with the recipe above, but makes use of a milk and lemon sauce.
Sauce
Melt 40 g/1½ oz butter in a saucepan and stir in 50 g/2 oz flour. Stir in 300 ml/½ pint tongue stock and 300 ml/½ pint water and cook until sauce thickens, stirring constantly. Season with 1 teaspoon salt, ½ teaspoon black pepper, 1 tablespoon lemon juice and simmer for 10 minutes, stirring regularly.
Prepare the tongues as in previous recipe and add to the sauce. Spoon the mixture into a serving dish and sprinkle with chopped parsley or tarragon.
khash or paça hooves, tongue and tripe stew
One of the oldest and most beloved of all Middle Eastern dishes is paça, often called kele paça which, translated from Turkish means 'head and feet'. There are several variations of this dish and most are regional specialities: e.g. in Cyprus the lamb's head is sometimes substituted with boned meat; in Iraq chickpeas, chopped tomatoes and sliced, toasted bread are added to the stew; in Turkey another version includes 350 g/12 oz soaked haricot beans cooked with the hooves; while in Egypt a very ancient dish kawareh-bi-hummus is still prepared where lamb's or calf's feet are cooked with chickpeas, turmeric, salt and pepper for at least 3 hours or until the meat is falling off the bones. It is really a rich soup and is served with hard-boiled eggs (traditionally cooked in their shells in the stew), bread and salad.
The most famed version, however, is the one below from Turkey and Armenia.
4 calf's hooves (lamb or goat is also suitable). Buy ready-cleaned if possible
6 lamb's tongues
900 g/2 lb calf's tripe
3 cloves garlic, crushed
salt and pepper to taste
25 g/1 oz butter
½ teaspoon paprika
1 tablespoon chopped fresh mint or 1 teaspoon dried mint
1 tablespoon parsley, chopped
juice 1 lemon
Soak the feet in a large, deep saucepan of boiling water to loosen the shoes. Remove the shoe from each foot. Singe off the hair and then soak the feet and tripe in water overnight. Place the feet in a large saucepan, three-quarters full with water, add half the garlic and bring to the boil. Remove any scum that appears on the surface and then lower the heat and simmer.
Meanwhile, cut the tripe into 2.5 cm/1 in squares, place in another saucepan, half fill with water then bring to the boil and simmer for 15 minutes. Drain off the water, then add fresh water and cook for a further 15 minutes. Drain the tripe into a sieve and then add to the feet and cook together for about 2 hours. During cooking scum will constantly appear and so remove with a slotted spoon. When the meat separates easily from the foot bones, remove all the bones.
While the feet and tripe are cooking place the tongues in water in another pan and soak until the skin peels off easily. Remove the tongues, skin and then cut into smaller pieces.
When the feet and tripe have been cooking together for about 1 hour add the pieces of tongue to the pan. While the meat is cooking you will find it necessary to top up with water from time to time. Season to taste with salt and pepper.
When all the meat is tender, melt the butter in a small pan, add the remaining garlic and the paprika and sauté for a few minutes, stirring frequently. Stir in the mint and parsley and pour the mixture into the khash. Serve the meat with its own broth in individual soup bowls. Squeeze a little lemon juice over the khash when eating.
pork
Pork, prohibited by both the Jewish and Muslim religions, is only found in Christian lands where its use is still limited due mainly to centuries of Muslim domination. After all, Greece and Cyprus only threw off the Ottoman yoke some one hundred and twenty years ago; while the Caucasians only received their freedom less than a hundred years ago. For generations therefore, the Christians of the Ottoman Empire could not eat pork—or at least could not be seen to be doing so!
Nevertheless, there are some interesting pork dishes to be found in the Middle East. The finest, perhaps, is roast suckling pig—gourounaki pito in Greek, khozi khorovou in Armenian. The recipe below, which is typical, is from Georgia.
roast suckling pig
1 suckling pig, 4.5–5.5 kg/10–12 lb—order in advance from your butcher and ask him to clean and prepare it for you
2 lemons, halved
2 tablespoons salt mixed with 1 teaspoon white pepper
about 150 ml/¼ pint oil
Wash the pig inside and out and then dry. Rub the abdominal cavity and the skin with the cut lemons and then with the mixture of salt and pepper. Place a piece of wood or a ball of aluminium foil in the mouth—this will help keep the mouth open during cooking. Cover the ears and tail with foil to prevent them burning.
Place the pig in a kneeling position on a rack in a large roasting pan. Brush all over with the oil and pour over any juice remaining in the halved lemons.
Roast in an oven preheated to 160°C, 325°F, gas mark 3 for about 4 hours (allow 25 minutes per 500 g/lb). Baste frequently with the pan juices. When the pig is done and is golden and crisp on the outside and tender inside remove from the oven and transfer to a large serving dish. Decorate with parsley or coriander leaves, peaches, apricots, apples, tomatoes, etc. Remove the wood or foil from the mouth and replace it with an apple. Insert a cherry in each eye socket and serve with salads and pilav.
stuffed suckling pig
The pig is often stuffed before cooking. This is particularly popular with the Greeks and the Balkan people.
This makes a very special meal for that very special occasion.
Stuffing—popular with Cypriots and Greeks
25 g/1 oz butter
1 onion, finely chopped
4 sticks celery, finely chopped
225 g/8 oz minced pork or veal or beef
225 g/8 oz lamb's liver or chicken liver, finely chopped
2 cooking apples or quinces, peeled, cored and cubed
275 g/10 oz long grain rice, washed thoroughly under cold water and drained
750 ml/1¼ pints water
100 g/4 oz raisins
2 tablespoons pine kernels or sliced pistachios or slivered almonds
3 tablespoons parsley, chopped
½ teaspoon dried thyme
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon cinnamon
Melt the butter in a large saucepan, add the onion and celery and fry for a few minutes, stirring frequently, until the onion is soft. Stir in the minced meat and liver and fry for about 10 minutes, stirring frequently. Add the remaining ingredients, mix well and bring to the boil. Cover the pan, lower the heat and simmer until the liquid is absorbed.
Spoon this mixture into the pig's cavity after preparing the pig as in the first paragraph in the recipe. Secure the opening with strong thread or skewers. Continue cooking as described. When cooked place on a large serving dish, open the cavity so that some of the stuffing spills out and garnish.
hirino me kithoria pork with quinces
This is a Greek favourite. Pork and quinces go well together and so try this combination if the quinces are available—otherwise substitute cooking apples.
Serve with pilavs, roast vegetables and fresh salads.
25 g/1 oz butter or ghee
900 g/2 lb pork tenderloin, cubed and then lightly pounded
1 large onion, finely chopped
300 ml/½ pint dry red wine
150 ml/¼ pint water
peel of a lemon or small orange
5 cm/2 in cinnamon stick
1½ teaspoons salt
½ teaspoon black pepper
900 g/2 lb quinces, peeled, cored and thickly sliced
1 tablespoon brown sugar
Garnish 2 tablespoons tarragon or mint or parsley, finely chopped
Melt the butter or ghee in a large saucepan, add the pieces of pork and fry for a few minutes, tossing and turning, until evenly browned. Remove with a slotted spoon and keep warm.
Add the onion and fry for a few minutes, stirring frequently, until soft.
Now return the pork to the pan, add the wine, water, peel, cinnamon, salt and pepper and bring to the boil. Cover the pan, lower the heat and simmer for about 1 hour. Uncover the pan, arrange the quince slices over the meat, sprinkle with sugar and recover. Cook for a further 30–45 minutes or until the pork and quinces are tender. Transfer to a large serving dish and garnish.
Variation
Sergevilov khoz
An Armenian favourite from Karabakh.
Use the same quantities of pork and quinces as above.
Fry the pork cubes in 3 tablespoons of butter in a casserole, until evenly browned, then remove with a slotted spoon and reserve.
Add the sliced quinces to the pan and fry for a few minutes. Now add 4 cloves, ½ teaspoon cinnamon, 1 tablespoon honey or brown sugar and the fried pork. Cover and bake in an oven preheated to 180°C, 350°F, gas mark 4 for about 45–60 minutes or until the meat and fruit are tender.
Serve garnished as above.
beef
basagha im bananave tmarin
beef stuffed with bananas and dates
How beautiful, what a joy, my love!
Like a palm-tree you stand,
Your breasts, its bunches of dates.
I said,
'I will climb up this palm-tree
clasping its branches.'
(The Song of Songs)
One of the most delicious recipes from Israel making use of two of the main local fruits—dates and bananas. The idea is old. There are many such recipes in medieval Arabic, Armenian and Turkish manuscripts, but the use of beef with dates, bananas and figs is new.
Serve with roast vegetables and salads.
900 g/2 lb piece of beef—round or flank steak
1 tablespoon made mustard
1½ teaspoons salt
½ teaspoon nutmeg
½ teaspoon basil
½ teaspoon black pepper
2 tablespoons ground almonds
2 bananas, peeled and cut into 0.5 cm/¼ in slices
2 pickled cucumbers, diced
8 dates, stoned and chopped
8 dried figs, stemmed and chopped
oil
With a wooden mallet pound the meat until about 1.2 cm/½ in thick. Do not over pound or the meat will tear. Coat the surface with the mustard. Mix together the salt, nutmeg, basil, pepper and ground almonds and sprinkle over the meat. Cover evenly with the sliced bananas, pickled cucumbers, dates and figs. Roll the meat up very carefully and fasten with string in about 3 places.
Lightly brush the meat all over with oil. Wrap the meat roll in foil and place in a baking dish. Bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 1½ hours or until the meat is tender. Remove from the oven, discard foil and string, place on a dish and serve.
Kala josh beef with bread and yoghurt
What besides beef can you expect from an ox?
A traditional dish from the region of Van and Erzurum in Eastern Turkey (Western Armenia). In one form or another this dish is found in Turkey, Iran and Khuzestan as well as the Caucasus where the cooks of Karabakh substitute yoghurt with soured cream. Serve with a rice or burghul pilav or roast potatoes and home-made pickled apples and pears.
900 g/2 lb fillet of beef, trimmed of fat and gristle
50 g/2 oz butter
1 small onion, finely chopped
1 clove garlic, crushed
salt and pepper to taste
½ teaspoon ground cloves
6 thick slices bread, cut into 2.5 cm/1 in cubes
300 ml/½ pint yoghurt or soured cream
Pound the fillet until thin and slice the meat into 1.2 cm/½ in pieces. Melt the butter in a large saucepan, add the meat and onion and fry, stirring frequently, until the onion is soft and the meat evenly browned—5–7 minutes. Add the garlic, salt and pepper and cook for a further 2–3 minutes, stirring frequently. Add the bread cubes and cook, turning frequently, until evenly golden. Remove from the heat and sprinkle with the ground cloves.
Pour the yoghurt or soured cream into a small pan and warm through, but do not boil. Pile the meat and bread mixture into a serving dish, spoon the yoghurt or soured cream over the top and serve immediately.
chakhokhbili fillet of beef with wine
Meat and red wine are beneficial after blood-letting.—Shabbatt 129a.
A favourite, so it is said, of the Soviet dictator Stalin, whose personal chef from Georgia often prepared this dish for his gratification.
A simple and tasty meal served with a pilav of your choice. This can also be made with lamb or chicken, but I prefer the beef version.
900 g/2 lb fillet of beef
1 teaspoon salt
¼ teaspoon black pepper
100 g/4 oz butter
1 large onion, finely chopped
2 large tomatoes, blanched, peeled, seeded and chopped
150 ml/¼ pint stock
1 large pickled cucumber, sliced
2 tablespoons capers
1 clove garlic, crushed
300 ml/½ pint red wine
Garnish 2 tablespoons coriander, tarragon or parsley, finely chopped
Trim the beef fillet of any fat or gristle, cut into 2.5 cm/1 in pieces and sprinkle with the salt and pepper.
Melt the butter in a large saucepan, add the onion and fry until soft. Push the onion to one side of the pan, add the seasoned meat and fry, turning regularly, until evenly browned. Add the tomatoes and stock and bring quickly to the boil. Add the cucumber, capers and garlic, cover the pan then simmer for about 45 minutes. Stir in the wine and simmer for a further 15–20 minutes or until the meat is tender. Transfer to a serving dish, garnish and serve with a pilav of your choice.
Variation
Prepare the dish as above but eliminate the wine and, 5–10 minutes before the end of the cooking time, stir in 300 ml/½ pint soured cream or yoghurt mixed with 2 tablespoons plain flour.
kofta and gololig minced meat dishes
Meatballs—the range is inexhaustible; starting with the simple kofteler and extending to the regal ashtaraki gololig—a large meatball enclosing a small boiled chicken or poussin which, in turn, is stuffed with a hard-boiled egg. The whole is wrapped in a cheesecloth and cooked in a wine sauce. In between these two dishes are found countless others. I have tried, in this chapter, to include as many of these as possible for they are worthy of greater recognition. Most are known throughout the region, but others are specialities of a particular city or even a village. One thing they all have in common is a smooth texture—the meat is usually minced twice and then kneaded until it becomes extremely soft. The meat, normally lamb but it can also be beef, veal or pork, is then mixed with some combination of rice, burghul, onion, garlic, aubergine, cinnamon, allspice, cumin, coriander, etc., shaped into small marble-shaped balls, or larger walnut-sized ones, flattened to a 'hamburger'-shape, rolled into finger or sausage-shapes; sometimes stuffed with fruit, nuts, eggs, etc. and then either fried, grilled, stewed or braised.
fried kofta meatballs
3 slices white bread, crusts removed
900 g/2 lb lean lamb (or lamb and beef), minced twice
2 eggs
1 clove garlic, crushed (optional)
1 teaspoon cinnamon or allspice
½ teaspoon oregano (optional)
1½–2 teaspoons salt
½–1 teaspoon black pepper
oil for frying
Soak the bread in water and then squeeze dry and crumble into a large mixing bowl. Add the remaining ingredients and knead until the mixture becomes a smooth paste. Keeping your hands damp roll the mixture into small, marble-sized balls.
Heat some oil in a large pan and fry the balls, a few at a time, until they are cooked through and golden. Remove with a slotted spoon and drain. Serve hot or cold with salads and pickles.
Variation
altya kufta An Azerbaijanian speciality from the Caucasus.
Follow the recipe above, but substitute 1 finely chopped onion for the bread. Keeping your hands damp form the mixture into round patties. Dust with flour, dip into 2 beaten eggs and then coat with breadcrumbs.
Fry in oil or butter until cooked through and golden on all sides. Remove with a slotted spoon, drain, sprinkle with chopped parsley or tarragon and serve with roast potatoes, bread and pickles.
patliçan koftesi aubergine meatballs
A speciality from Izmir, Turkey. Other vegetables, e.g. courgettes, leeks, artichoke hearts and pumpkin can also be chopped, minced and then mixed with meat and fried.
450 g/1 lb aubergines, peeled
2 tablespoons salt
4–5 teaspoons oil
1 large onion, finely chopped
225 g/8 oz minced lamb or beef
25 g/1 oz grated cheese, e.g. haloumi, kashkaval or Cheddar
2 tablespoons parsley, finely chopped
½ teaspoon oregano
½ teaspoon sweet basil
½ teaspoon black pepper
½ teaspoon salt
flour
oil for frying
Cut the aubergines crossways into 0.5 cm/¼ in slices, place in a colander, sprinkle with the salt and set aside for 30 minutes. Wash under cold running water and then pat dry.
Heat the oil in a large frying pan, add the aubergine slices, a few at a time, and fry until soft turning once. Add a little more oil if necessary. Drain the slices on kitchen paper and reserve. Add the onion to the pan and fry for a few minutes until soft and golden.
Chop the aubergine slices finely and place in a large bowl with the fried onion. Add the meat, cheese, parsley, oregano, basil, pepper and salt and knead for a few minutes until the mixture is well blended and smooth. Keeping your hands damp shape the mixture into walnut-sized balls. Roll the balls in flour, place on a large plate and refrigerate for 30 minutes.
Heat some oil in a large frying pan and fry the aubergine balls, a few at a time, for about 20 minutes, turning occasionally until cooked through and evenly browned. Remove with a slotted spoon and serve hot or cold.
armlov potatoes and minced meat baked in the oven
An Armenian speciality.
450 g/1 lb potatoes, boiled and mashed
700 g/1½ lb lamb or beef, minced
25 g/1 oz pine kernels
50 g/2 oz seedless raisins
1½ teaspoons salt
½ teaspoon black pepper
4 teaspoons parsley, chopped
½ teaspoon sumac
½ teaspoon cumin
½ teaspoon paprika
½ teaspoon basil
Topping
1 egg, beaten
breadcrumbs
6–8 tablespoons melted butter
Place the mashed potatoes and minced meat in a large bowl and knead until well blended. Add all the remaining ingredients and continue to knead until the mixture is smooth. Lightly grease a baking dish and press the mixture into it, spreading it out evenly. Brush the top with the beaten egg. Sprinkle breadcrumbs generously over the top. Cut into squares or diamond shapes with an oiled knife.
Pour the melted butter evenly over the surface and bake in an oven preheated to 180°C, 350°F, gas mark 4, for about 45 minutes or until cooked through and golden. The length of time will depend on the thickness of the mixture. Serve hot with a bowl of yoghurt and some salads.
kofta-bil-saniya minced meat loaf in a tray
An Arab dish which is also very popular in Israel. The Israelis, who have a penchant for tahina, have created a new dish from the old, and I think it is an improvement on the Arab one. I have included both versions below.
900 g/2 lb lamb or beef, minced twice
1 large onion, finely chopped
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon cumin
1 teaspoon allspice
4 tomatoes, halved
2 tablespoons tomato purée diluted in 150 ml/¼ pint water
25 g/1 oz butter
2 tablespoons parsley, finely chopped
Place the meat, onion, salt, pepper, cumin and allspice in a large bowl and knead thoroughly. Grease a baking dish about 30 cm/12 in square. Spread the meat mixture evenly over the tray so that it is about 2.5 cm/1 in thick. Arrange the halved tomatoes over the meat. Pour the diluted tomato purée over the meat and top with a few pats of butter and the chopped parsley.
Cook in an oven preheated to 190°C/375°F, gas mark 5 for about 1 hour. The meat will shrink away from the sides of the pan and the meat will be dark brown. Remove to a large serving dish and cut into squares. Serve with fresh salad and/or roast potatoes.
Variation
kufta keftidei cem tahina minced meat with tahina
900 g/2 lb lamb or beef, minced twice
1 teaspoon salt
½ teaspoon black pepper
1 tablespoon Worcestershire sauce
1 tablespoon chives, chopped
Prepared tahina
100 g/4 oz dried sesame seeds
1 clove garlic
juice 1 lemon
150 ml/¼ pint water
2 tablespoons olive oil
½ teaspoon salt
¼ teaspoon paprika
pinch cayenne pepper
3–4 tablespoons parsley, finely chopped
Place all the ingredients for the tahina in a liquidizer and blend thoroughly.
Put the meat, salt, pepper, Worcestershire sauce and chives into a large bowl and knead until thoroughly blended and smooth.
Butter a baking dish about 30 cm/12 in square. Spread the meat mixture evenly over the tray so that it is about 2.5 cm/1 in thick. Spread the tahina mixture evenly over the meat. Bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 1 hour or until the tahina is turning a golden brown. Place on a serving dish and cut into squares. Serve with salad and/or roast potatoes.
tzirani gololig meatballs in apricot sauce
A speciality from the Ashtarag region of Armenia making use of prunus armenicus, the fruit of the land—apricots.
A fascinating recipe that has come down to us virtually unchanged from the distant past. If fresh apricots are not available then dried ones are perfectly suitable.
Serve with rice or burghul pilav.
450 g/1 lb lamb, minced
1 medium onion
1½ teaspoons salt
¼ teaspoon black pepper
100 g/4 oz cooked rice
1 teaspoon basil
1 tablespoon parsley, chopped
3 tablespoons arak (or ouzo)
1.2 litres/2 pints stock
Sauce
100 g/4 oz dried apricots, soaked overnight in 300 ml/½ pint cold water
1 clove garlic
1 tablespoon fresh coriander,
or 1 teaspoon ground coriander
1 onion, finely chopped
1 tablespoon flour
1 teaspoon paprika
To prepare the meatballs first pass the minced meat again through a mincer—this time with the onion. Place the mixture in a large bowl and add the salt, pepper, rice, basil, parsley and arak. Keeping your hands damp with cold water knead the mixture until it is smooth. Shape the mixture into walnut-sized balls.
Place the stock in a saucepan and bring to the boil. Add the meatballs and simmer gently for 15 minutes.
Meanwhile, prepare the apricot sauce. Put the apricots, garlic, coriander and 4–5 tablespoons of the apricot water into a liquidizer and blend.
Remove the meatballs from the stock, and when the stock has cooked a little skim off the fat and put it into a small pan.
Heat the fat, add the onion and fry until soft. Add the flour and paprika and fry for 2–3 minutes and remove from the heat. Add the apricot mixture and gradually thin with some of the stock. Pour the mixture back into the stock, stir well, add the meatballs and simmer gently for a further 30 minutes.
kadin badu lady's thigh kofta
Lady's thigh kofta—it had to be a Turk to visualize such imagery. The kofta should be round, soft and smooth like a lady's thighs—at least, that was the fashion when this dish was created!
Serve with salads and a rice pilav or spaghetti.
450 g/1 lb lamb, minced twice
1 onion, finely chopped
100 g/4 oz cooked rice
2 tablespoons white cheese, grated
1 egg
25 g/1 oz flour
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon cumin
1 egg, beaten
oil for frying
Place the meat, onion, rice, cheese, egg, flour, salt, pepper and cumin in a large bowl and knead for 5–10 minutes until the mixture is well blended and smooth. Keeping your hands damp with cold water take a lump of the mixture about the size of a large walnut and roll it into a ball between your palms. Flatten it gently by pressing one palm against the other. Repeat this with the remaining mixture and arrange all the koftas on a baking tray.
Heat some oil in a large frying pan. Dip a few of the koftas at a time in the beaten egg and fry, turning occasionally, until cooked through and golden. Remove with a slotted spoon, arrange on a dish and keep warm while you fry the remaining kofta in the same way. Serve warm.
terbiyeli kofta
meatballs in an egg and lemon sauce
This is popular throughout the region, but particularly with Greeks and Turks. The sauce has several names, e.g. terbiyeli in Turkish, avgolemono in Greek and beid-el-lemoun in Arabic. In this book the sauce is called the latter.
900 g/2 lb lamb, minced twice
3 tablespoons ground rice
4 tablespoons parsley, finely chopped
1 large onion, finely chopped
1 clove garlic, crushed
1½ teaspoons salt
½ teaspoon black pepper
½ teaspoon cinnamon
½ teaspoon allspice
Terbiyeli sauce Follow instructions for beid-el-lemoun.
Place all the ingredients in a large bowl and knead for 5–10 minutes until well blended and very smooth. If you keep your hands damp it will make this easier. Form themixture into marble-sized balls.
Bring a large saucepan half-filled with water to the boil. Season with 1½ teaspoons salt and then add the meatballs and simmer for about 20 minutes or until soft and tender.
Meanwhile, prepare the sauce. Do not let the sauce boil or it will curdle—hence the Turkish name terbiyeli which means 'to behave'.
When cooked lift the meatballs out of the water with a slotted spoon and add to the sauce. Heat very gently for a few minutes and then serve with a rice pilav of your choice.
ismir koftesi meatballs in tomato sauce
Simply delicious. Serve with a rice pilav or with spaghetti and a bowl of fresh salad.
3 slices white bread
450 g/1 lb lean lamb or beef, minced twice
1 egg, beaten
1 clove garlic, crushed
¼ teaspoon cinnamon
½ teaspoon paprika
1 teaspoon salt
½ teaspoon black pepper
2½ tablespoons plain flour
25 g/1 oz butter
5 tomatoes, blanched, peeled, seeded and chopped
1 green pepper, seeded and chopped
300 ml/½ pint water
Remove the bread crusts, soak the bread in a little water then squeeze dry and crumble into a large mixing bowl. Add the minced meat, egg, garlic, cinnamon, paprika, salt and black pepper and knead thoroughly until smooth. Keeping your hands damp with cold water shape the mixture into walnut-sized balls. Sprinkle the flour over a large plate and roll the balls in it.
When all the meatballs are formed and floured melt the butter in a large saucepan. Add the koftas, a few at a time and sauté until evenly browned all over. Transfer to a plate and keep warm.
Add the tomatoes, green pepper and water to the saucepan and simmer for 15 minutes. Return the meatballs to the pan and simmer for a further 15–20 minutes or until cooked through. Transfer to a serving dish and serve with a rice pilav or spaghetti.
Variation
torshi shami
This recipe is from Rasht on the Caspian coast of Iran.
75 g/3 oz breadcrumbs
150–300 ml/¼–½ pint milk
1 large onion, grated
1½ teaspoons salt
½ teaspoon black pepper
1 tablespoon curry powder (optional)
700 g/1½ lb minced lamb or beef
40 g/1½ oz butter
juice 1 large lemon
3 tablespoons tomato purée
600 ml/1 pint water
Place the breadcrumbs in a large bowl with 150 ml/¼ pint of the milk and mix. Add the onion, salt, black pepper, curry powder and meat and knead until the mixture is well blended and smooth. If the mixture is a little dry knead in some more milk. Keeping your hands damp take small apple-sized lumps of the mixture and form into balls. Press each ball between your palms to flatten and then punch a hole through the middle of each with your forefinger to give the koftas a 'doughnut' shape.
Melt the butter in a large saucepan and sauté the koftas, a few at a time, until evenly browned. Keep them warm while you prepare the sauce.
Add the lemon juice, tomato purée and water to the saucepan, stir well and bring to the boil. Return the koftas to the pan and simmer gently for 20–30 minutes, turning once or twice until they are cooked through and most of the sauce has evaporated. Serve with a rice pilav.
The Iranian cuisine is particularly rich in such dishes. Rice is often mixed in with the meat. Kofta-ye-sabzi is such a dish where rice, split peas, onion and parsley is mixed with the meat, seasoned with cinnamon, nutmeg, salt and pepper, shaped into apple-sized balls and simmered in a tomato and lemon sauce. The finest of all Iranian koftas are reputed to come from the region of Tabriz (Iranian Azerbaijan) in the north-east of the country bordering Armenia and Kurdistan—two very old nations stretching back into antiquity.
In the recipe below the meatballs are stuffed with eggs, dried fruit and walnuts and then cooked in a tomato sauce.
koofteh tabrizi
1 onion, finely chopped
700 g/1½ lb lamb or beef, minced at least twice
2 eggs, lightly beaten
75 g/3 oz rice flour or chickpea flour or split pea flour
2 teaspoons salt
½ teaspoon black pepper
¼ teaspoon nutmeg
½ teaspoon cinnamon
Filling
6 hard-boiled eggs, halved
12 dried apricots or stoned prunes
4 tablespoons raisins
12 walnut halves
Sauce
50 g/2 oz ghee or butter
1 small onion, finely chopped
1½ teaspoons turmeric
1.2 litres/2 pints stock or water
2 tablespoons tomato purée
1½ teaspoons salt
½ teaspoon cayenne pepper
Place all the ingredients for the meatballs in a large bowl and knead for several minutes until well-blended and smooth. Keeping your hands damp divide the mixture into 12 balls. With your forefinger or thumb make a depression in one of the balls and widen the hollow and pinch the meat up to form a pot shape. Place half a boiled egg, an apricot or prune, a few raisins and half a walnut in the hollow. Carefully draw the meat together to completely enclose the filling and roll the kofta around between your dampened palms to resume a ballshape. Repeat with the remaining meatballs and filling ingredients.
In a large saucepan melt the ghee or butter, add the onion and fry until soft and a light golden colour. Stir in the remaining ingredients and bring to the boil. Add the koftas, lower the heat and simmer for about 45 minutes, turning the balls once or twice. Serve immediately with a rice pilav, fresh yoghurt and salads.
mussaka meat and aubergine pie
Although today mussaka is regarded as a Greek dish, it was first developed in the days of the Caliphs during the glorious days of the Baghdad of Sinbad, Shehrezade and Aladdin.
The medieval dish called muhklabah was the original form—asimple and tasty meal of minced meat and fried aubergine topped with cheese. To show my impartiality I have included the recipes for muhklabah (Arab), moussaka (Armenian) and melitzanes moussaka (Greek). muhklabah arabstyle moussaka
muhklabah arab-style moussaka
125 g/5 oz butter or ghee
450 g/1 lb lamb or beef, minced
2 medium aubergines, hulls removed
225 g/8 oz brown or long grain rice, washed thoroughly under cold water and drained
900 ml/1½ pints water
½ teaspoon turmeric
1 teaspoon salt
½ teaspoon allspice
½ teaspoon black pepper
2 tablespoons melted butter
Garnish 100 g/4 oz almonds or pistachios, chopped (optional)
Melt 40 g/1½ oz of the butter in a saucepan, add the meat and fry for 10 minutes, stirring frequently.
Halve the aubergines lengthways and then slice crossways into 0.5 cm/¼ in thick slices. Melt the remaining butter in a large frying pan and fry the slices, a few at a time, turning at least once until golden on both sides. Remove with a fork, drain on kitchen paper and reserve. When all the slices are fried, place half the fried meat in the bottom of a large, greased ovenproof dish. Cover with half the aubergine slices and sprinkle with half the rice. Repeat the layering again, finishing with the rice.
Place the water, turmeric, salt, allspice and black pepper in a small saucepan, bring to the boil and simmer for 5 minutes. Pour the liquid over the casserole.
Place in the centre of an oven preheated to 160°C, 325°F, gas mark 3 and bake for 1 hour or until the rice is tender. Remove from the oven, pour the melted butter over the top of the casserole, sprinkle with the nuts and serve with salad.
moussaka armenian-style
This is a family recipe.
450 g/1 lb aubergines, hulls removed
2–3 tablespoons salt
about 8–10 tablespoons oil
Filling
1 large onion, finely chopped
450 g/1 lb minced lamb
1 clove garlic
1 teaspoon salt
½ teaspoon black pepper
2 tablespoons parsley, finely chopped
2 tablespoons chopped walnuts
½ teaspoon allspice
3 tablespoons tomato purée
4–5 tablespoons dry red wine
Topping
2 egg yolks
6–8 tablespoons grated cheese, e.g. Cheddar, Parmesan or kefalotiri
Cut the aubergines lengthways into 0.5 cm/¼ in thick slices, arrange on a plate, sprinkle with the salt and set aside for 30 minutes.
Meanwhile, heat half the oil in a saucepan, add the onion and fry for a few minutes, stirring frequently, until soft. Add the meat, garlic, salt, pepper and parsley and cook for 5 minutes, stirring frequently. Add the walnuts, allspice, tomato purée and wine, stir well and fry for a further 5–10 minutes. Remove from the heat.
Rinse the aubergine slices under cold water and dry on kitchen paper. Heat the remaining oil in a large frying pan and fry the aubergine slices, a few at a time, until golden on both sides. Add more oil if necessary. Remove and drain on kitchen paper. When all the slices are cooked arrange half of them in the bottom of a large, deep, greased baking dish. Spread the meat mixture over them and cover with the remaining slices. Mix together in a small bowl the egg yolks and cheese and spread over the top. Place in the centre of an oven preheated to 180°C, 350°F, gas mark 4 and bake for 45–60 minutes until the meat is cooked and the top is golden.
melitzanes moussaka greek-style
450 g/1 lb aubergines, hulls removed
2 tablespoons salt
75–100 ml/3–4 fl oz oil for frying
Filling
25 g/1 oz butter
4 small shallots, finely chopped
450 g/1 lb lamb, minced twice
2 tomatoes, blanched, peeled and chopped
1 tablespoon chives, finely chopped
50 ml/2 fl oz white wine
1 teaspoon lemon juice
½ teaspoon dried sage
¼ teaspoon black pepper
2 tablespoons parsley, finely chopped
1 teaspoon sugar
¼ teaspoon cinnamon
50 g/2 oz fresh white breadcrumbs
Sauce
175 g/6 oz mizittire, ricotta, kefalotiri or Parmesan cheese
3 egg yolks
350 ml/12 fl oz single cream
¼ teaspoon salt
¼ teaspoon nutmeg
Cut the aubergines lengthways into 0.5 cm/¼ in slices, arrange on a plate, sprinkle with the salt and set aside for 30 minutes.
Meanwhile, melt the butter in a pan, add the shallots and fry until soft. Add the meat and fry, stirring frequently, until browned. Add all the remaining ingredients for the filling, stir well and cook gently, stirring frequently, for about 10 minutes and then remove from the heat.
Rinse the aubergine slices under cold running water and dry with kitchen paper. Heat the oil and fry the slices, a few at a time, until golden on both sides. Remove with a fork and drain on kitchen paper. Arrange a third of the slices over the bottom of a deep, greased casserole dish. Spread half of the filling over them and cover with another third of the aubergine slices. Spread the remaining filling over them and cover with the remaining slices.
To prepare the sauce place the cheese in a bowl and mash with a fork until smooth. Add the egg yolks and whisk until a smooth paste is formed. Gradually stir in the cream, salt and nutmeg and then pour this mixture over the top to cover completely the aubergines. Place in an oven preheated to 180°C, 350°F, gas mark 4 and bake for about 45 minutes or until the top is golden. Remove, cut into wedges and serve with salads. There are many moussaka variations and you can use many vegetables instead of aubergines, e.g. potatoes, pumpkin, tomatoes, etc.
## poultry and game
In our country there are more chickens than even mice! Those two legged imbeciles are everywhere—in the Hamam (Turkish bath), in streets, in the bazaars, fields, cemeteries, churches, bedrooms, over the bed, under the bed, and all they know is cluck, cluck, cluck, cluck. All they're good for is a strong wring of the neck! But they are delicious and so we tolerate them.
Remember, my son, all things that are good and tasty must be tolerated—however much the inconvenience.
Hadji Baba of Isfahan obviously liked his chicken, as do all Middle Easterners and they have, over the centuries, created a large repertoire of chicken recipes. Chicken, as well as other poultry such as duck, turkey and goose and game such as partridge, pheasant, quail, squab, woodcock, etc. are fried, roasted, grilled over charcoal, boiled with vegetables, baked in the oven, stuffed with rice, wheat, meat, nuts and fruits. They are cooked with chickpeas, aubergines and okra as well as yoghurt, pomegranate juice, etc., the list is endless.
The recipes in this section therefore are only a fragment of that rich repertoire. Use roasting chicken for all recipes unless otherwise stated.
tabaka fried chicken with prune sauce
A Caucasian recipe. Usually the chicken is fried and served with salad or, as with this recipe, with tkemali sauce.
4 × 450–700 g/1–1½ lb poussins, washed and dried
2 tablespoons salt
150 ml/¼ pint yoghurt
75 g/3 oz ghee or butter
3 tomatoes, thinly sliced
1 small aubergine, cut in half lengthways and then cut into 0.5 cm/¼ in slices crossways
1 clove garlic, crushed
½ teaspoon ground cinnamon
To serve 150–300 ml/¼–½ pint tkemali sauce—see recipe
Place 1 poussin on a chopping board, back upwards. With a sharp pointed knife start at the neck and cut along one side of the backbone. Turn the poussin around and cut along the other side of the backbone thus freeing it. Break it away from the spoonshaped bone connecting the breasts and remove both bones and the white cartilage. Loosen the skin around the leg and thigh and push it back exposing the thigh joint. Cut it half across and pull the skin back. Repeat with the other leg. Make a slit in each breast below the ribs.
Turn the poussin flesh side down, cover with greaseproof paper and then flatten with a meat mallet. Twist the legs inwards and push them through the holes in the breasts. Repeat with each poussin. Rub them with the salt and spread the flesh sides evenly with half the yoghurt.
Melt 50 g/2 oz of the butter in a large frying pan, add 2 poussins, skin sides down, place a heavy weight on top and cook over a moderate heat for 8–10 minutes. Turn the poussins over, spread with half the remaining yoghurt, weigh down and fry for a further 10 minutes until golden brown, but do not burn. Repeat with the 2 remaining poussin.
Meanwhile, melt the remaining 25 g/1 oz of butter in a saucepan and sauté the tomatoes, aubergine, garlic and cinnamon until soft.
Serve 1 poussin per person accompanied by some of the cooked vegetables and the tkemali sauce.
Variation
auff sum-sum sesame fried chicken
An Israeli recipe where the chicken pieces are coated in sesame seeds to give an attractive dark golden appearance and interesting texture. Serve with roast potatoes and fried or, as in Israel, boiled vegetables.
1.35 kg/3 lb chicken, cut into 8 serving pieces
75 g/3 oz sesame seeds
50 g/2 oz plain flour
1½ teaspoons salt
freshly ground pepper
1 teaspoon paprika
1 large egg, beaten
75 ml/3 fl oz water or chicken stock
oil
Dry the chicken pieces.
Mix the sesame seeds, flour, salt, pepper and paprika on a large plate.
Mix the egg and water or stock together in a shallow dish. Coat the chicken pieces lightly in the sesame seed mixture. Dip each piece into the egg mixture to coat and then coat with the sesame seed mixture again.
Heat the oil in a large frying pan, add the chicken pieces and fry until golden, turning occasionally. Take care not to burn. Transfer to a casserole or baking dish and bake in an oven preheated to 180°C, 350°F, gas mark 4, for about 30–45 minutes or until tender. Serve immediately.
tashreeb dijaj whole chicken in spicy sauce
A simple dish from Iraq. Serve it with a pilav of your choice or with chelo—as the Iraqi housewives would. Incidentally, they do not call the rice chelo, but timman; however the method and result are the same.
Limes are usually used instead of lemons, but if they are not available substitute the latter.
1.35–1.8 kg/3–4 lb chicken, washed and dried inside and out
1 lemon or lime, quartered
2 teaspoons salt
½ teaspoon black pepper
½ teaspoon turmeric
4 tablespoons ghee
2 cloves garlic, halved
300 ml/½ pint water
1 teaspoon salt
2 bay leaves
2–3 cardamom pods, split
Rub the chicken inside and out with the lemon or lime quarters. Save the quarters. Mix the salt, black pepper and turmeric together and rub into the chicken.
Melt the ghee in a large deep saucepan, add the chicken and brown all over, basting and turning regularly. Squeeze the remaining lemon or lime juice over the chicken. Add the garlic, water, salt, bay leaves and cardamom. Bring to the boil, cover the pan and cook over a low heat for 1½–2 hours, turning and basting occasionally.
Remove the chicken from the pan and keep warm. Reduce the pan juices to half its former quantity over a high heat. Discard bay leaves and cardamom pods.
Carve the chicken and serve on a bed of rice with the pan juices poured over the top.
hav ganachov chicken with vegetables
An Armenian recipe, but typical of Turkey, Kurdistan and Iran.
Serve with a pilav of your choice.
2 aubergines, peeled
150 ml/¼ pint oil
1.35–1.8 kg/3–4 lb chicken, cut into 8 serving pieces
50 g/2 oz flour seasoned with 1 teaspoon salt 1 teaspoon black pepper ½ teaspoon chilli pepper
2 medium onions, sliced
2 courgettes, topped, tailed and cut into 1.2 cm/½ in rings
2 green peppers, seeded and cut into 2.5 cm/2 in squares
450 g/1 lb green beans, trimmed and sliced
225 g/8 oz okra, trimmed and sliced (optional)
4 tomatoes, blanched, peeled and sliced or small tin tomatoes, coarsely chopped
300 ml/½ pint water
2 teaspoons salt
1 teaspoon black pepper
½ teaspoon chilli pepper
3–4 bay leaves
Garnish 1 tablespoon parsley, finely chopped
Cut aubergines crossways into 1.2 cm/½ in rings, arrange on a dish, sprinkle with salt and set aside for 30 minutes. Then rinse under cold water and pat dry.
Meanwhile, heat half the oil in a large pan. Coat the chicken pieces in the seasoned flour and fry until golden on all sides. Remove with a slotted spoon and reserve then heat the remaining oil in the pan. Add the aubergines and remaining vegetables, except tomatoes, and fry for a few minutes, turning carefully. Place all the vegetables in the bottom of a casserole, arrange the chicken joints over the top and place the tomatoes over them.
Bring the water, salt and black and chilli peppers to the boil in a small pan and pour into the casserole. Cover and cook in an oven preheated to 190°C, 375°F, gas mark 5 for about 1 hour or until chicken is tender. Sprinkle with the parsley and serve.
One day... things were going wrong for Boloz Mugush. His wife was threatening to leave him (that was not so bad), his sons had gone to war and business was poor; did I say poor? I meant non-existent. Must do something, he thought, while passing by the large estate of the Melikovs.
Had he been more observant he would have noticed a Russian militia standing nearby when he rushed through the gates, over the low timber fence and grabbed two fat chickens—but before the chicken could cluck, cluck, Boloz Mugush found himself in the clutches of the law.
Next day in court he stood before the chief justice, surrounded by Russian soldiers.
'Are you Bo- Bo- Boloz Mu-Mu- Mugush, the defendant?'
'Yes he is,' interjected the officers in harmony.
The prisoner, who was most inexperienced in matters legal hesitated, then whispered, 'No, your worshipful honour, your magnificent, your majesty. I am not the other one. I am Mugush who stole the chicken.'
dajaj-bil-mishmishiyeh
chicken with pasta and apricot sauce
A very old dish from Iraq described in Al-Baghdadi's medieval cookery manual. I understand home-made pasta—rishta—is still used with this dish and a recipe for it is included in this book. However, you can also use either spaghetti or vermicelli.
Either apricot or prune sauce is used with this dish. The latter is more popular with the Arabs of Khuzestan, Iran (see recipe).
A similar Caucasian version makes use of tkemali sauce and is known as chicken mussaka.
Serve with fresh salads.
2.25 kg/5 lb chicken cut into joints
40 g/1½ oz ghee or butter
juice 1 small lemon
about 150 ml/¼ pint water
1 teaspoon turmeric
1 teaspoon salt
½ teaspoon black pepper
¼ teaspoon chilli pepper
2 cardamom pods, cracked
450 g/1 lb rishta, spaghetti or vermicelli
3 tablespoons oil
1 teaspoon cinnamon
1 large tomato, blanched, peeled and sliced
1 green pepper, seeded and cut into rings
Sauce
225/g/8 oz dried apricots or 225 g/8 oz amarind bastegh, soaked overnight
1 teaspoon sugar or honey
1 tablespoon lemon juice
1 tablespoon rosewater
Remove the skin from the chicken joints.
Melt the ghee or butter in a large saucepan, add the lemon juice, water, turmeric, salt and pepper and stir well. Bring to the boil, add the chicken pieces and cardamom pods. Cover the pan, lower the heat and simmer for 45–60 minutes or until the meat is tender. Remove from the heat and leave to cool a little. Remove the chicken pieces and cut the flesh from the bones. Reserve the stock. Either cut the meat into smaller pieces or chop coarsely.
Meanwhile, bring a large pan half filled with lightly salted water to the boil, add the pasta and boil for 7–8 minutes or until almost tender. Strain into a colander.
Heat the oil in a large casserole dish, add the pasta and fry for 2–3 minutes, stirring and turning with a fork. Remove and reserve half the pasta. Spread the remaining pasta over the bottom and arrange the chicken pieces over the top. Sprinkle with the cinnamon and cover with the remaining pasta. Top with the tomato and green pepper slices and pour in the reserved chicken stock. Bake in an oven preheated to 180°C, 350°F, gas mark 4 for 30–40 minutes.
Meanwhile, prepare the apricot sauce. Place the soaked apricots or thinly sliced amarind in a saucepan, add a scant 150 ml/¼ pint water and simmer until tender. Add the sugar or honey and the lemon juice. Reduce the mixture to a purée in a blender and just before serving stir in the rosewater. Pour a little of the sauce over each serving of the chicken and pasta.
Serves 4–6 people.
budughov judig chicken with fruits
A regal dish from the Caucasus, the chicken is cooked with apricots, prunes, sultanas and wine.
There are many chicken and fruit dishes from the Caucasus and Iran and I have also noted below a simple Turkish-Kurdish version.
50 ml/2 fl oz oil
1 large chicken, cut into 8 serving pieces
50 g/2 oz seasoned flour
1½ tablespoons tomato purée
300 ml/½ pint water or stock
50 g/2 oz prunes, stoned
50 g/2 oz apricots, quartered
2 tablespoons sultanas
1 teaspoon salt
½ teaspoon black pepper
1 teaspoon sumac
300 ml/½ pint red wine
Garnish a little extra sumac
Heat the oil in a large frying pan. Coat the chicken pieces in the seasoned flour and fry in the oil, a few at a time, until golden on all sides. Transfer the pieces to a casserole.
Add the tomato purée to the frying pan and stir in the water or stock. Stir in the dried fruits, salt, black pepper, sumac and wine and bring to the boil. Pour the sauce into the casserole and place in an oven preheated to 180°C, 350°F, gas mark 4. Bake for about 1 hour or until the chicken is tender. Transfer to a serving dish and sprinkle with a little sumac. Serve with a burghul or rice pilav.
Variation
ayvali tavugi chicken with quinces or apples
Fry 1 chopped onion in a little oil, add the chicken pieces and fry, turning frequently, for about 45 minutes, or until tender. Peel, core and thickly slice 4 quinces or apples and add to the pan with 1 teaspoon salt, ½ teaspoon allspice and 300 ml/½ pint stock or water.
Cook for a further 20–30 minutes or until the fruit is soft.
quwarmah ala dajaj curried chicken
A spicy exotic chicken dish from the Gulf region of Arabia, related to the many similar Indian dishes and undoubtedly of Indian origin. Although the Gulf States are part of the Arab world it must be remembered that not only are a great number of people of Indian extraction, but centuries of trade with the Indian sub-continent have had a very strong social and cultural influence on the ethnic Arabs. This is particularly so with the island of Bahrain where, after Arabic, Hindu is the most widely used language.
Serve with muhamar (see recipe) or any pilav dish.
1.35 kg/3 lb chicken, cut into 8 serving pieces
2 teaspoons salt
½ teaspoon nutmeg
1 teaspoon cumin
½ teaspoon paprika
½ teaspoon ground cardamom
½ teaspoon black pepper
1 teaspoon turmeric
40 g/1½ oz ghee
2 large onions, finely chopped
2 cloves garlic, crushed
1 teaspoon grated fresh ginger
1 teaspoon chilli powder
5 cm/2 in cinnamon stick
2 large tomatoes, blanched, peeled and chopped
2 loomi (dried limes) pierced with a fork or the thinly peeled rind of 1 lemon
1 teaspoon salt
300 ml/½ pint water
In a small bowl mix together the salt, nutmeg, cumin, paprika, cardamom, black pepper and turmeric. Rub the chicken pieces all over with half this mixture and reserve the rest. Melt the ghee in a large pan, add the chicken pieces and fry, turning regularly, until they are browned all over. Remove with a slotted spoon and reserve.
Add the onions to the pan and fry for a few minutes until soft. Add the garlic, ginger, reserved spices, chilli and cinnamon stick and continue to fry for a further 5 minutes, stirring frequently. Add the tomatoes, loomi or lemon rind, salt and water and bring to the boil.
Return the chicken pieces to the pan, lower the heat, cover the pan and cook for about 1 hour or until the chicken is tender.
taiyika auff chicken with kumquats and honey
Honey in the mouth won't help bitterness in the heart.
An Israeli recipe.
If you cannot find kumquats—easily available in the USA and Australia, but not in Europe—then use mandarin oranges or ordinary oranges—Jaffa of course!
Serve with salad and potatoes or a pilav of your choice.
1.35–1.8 kg/3–4 lb chicken, halved and with backbone removed
½ teaspoon salt
½ teaspoon black pepper
150 ml/¼ pint fresh orange juice
75 ml/3 fl oz clear honey
1 green chilli, seeded and finely chopped
8 kumquats or mandarins or 4 oranges, peeled, white pith removed and segmented
2 teaspoons arrowroot mixed to a paste with 2 tablespoons water
Garnish
1 lemon, thinly sliced
1 tablespoon parsley, finely chopped
Sprinkle the chicken halves with the salt and pepper. Place in a casserole and pour the orange juice and honey over them. Add the chilli and kumquats or oranges. Place over a moderate heat and bring the liquid to the boil.
Cover the casserole and place in an oven preheated to 180°C, 350°F, gas mark 4. Cook for about 1 hour or until the chicken is tender. Remove from the oven, transfer the chicken halves to a board and cut each one in half. Place the chicken pieces in a serving dish and keep warm.
Stir the arrowroot mixture into the cooking juices and bring to the boil, stirring constantly. Cook the sauce over a low heat for about 3 minutes. Pour the sauce into a sauceboat.
Garnish the chicken with the lemon slices and sprinkle with the parsley. Serve immediately with the sauce.
bursa tavugu chicken with cream and herbs
Bursa is a large prosperous city in north-west Turkey and was once the capital of the Ottomans—before their conquest of Constantinople. The finest cooks of Turkey are reputed to come from there.
This is a very tasty dish usually served with pilavs and salads.
1.35–1.8 kg/3–4 lb chicken, cut into 8 serving pieces
75 ml/3 fl oz oil
2 onions, thinly sliced
450 ml/¾ pint chicken stock
½ teaspoon marjoram
½ teaspoon basil
½ teaspoon salt
½ teaspoon white pepper
1 tablespoon plain flour
3 tablespoons single cream
about 15 stuffed green olives
Garnish 1 tablespoon parsley, finely chopped
Heat the oil in a large casserole, add the chicken pieces and brown, turning occasionally. Remove the chicken and reserve.
Add the onions and fry, stirring frequently, until golden. Return the chicken to the pan and add the stock, marjoram, basil, salt and pepper. Bring to the boil lower the heat, cover and simmer until the chicken is tender. Remove the chicken pieces to a serving dish and keep warm.
In a small bowl blend the flour and cream until smooth. Add the flour and cream mixture and the olives to the sauce and simmer for a few minutes, stirring constantly. Pour the sauce over the chicken, sprinkle with the parsley and serve immediately.
stuffed chicken
'Soud khent elat, vankin haver oudel'—to act the fool and eat the monastery's chicken, in other words to try and get away by acting the fool.
On festive occasions or in honour of a guest, chicken is often served stuffed with rice, nuts, fruits, spices and vegetables. The bird is served on large silver or gilt trays and garnished with fresh vegetables, fruits and mounds of rice pilavs. The stomach is opened and the filling partly spooned out. Then the chicken is carved.
Of the many possible stuffing recipes I have chosen a few typical ones to give just an idea of the wealth and scope of the Middle Eastern repertoire.
dijaj al timman
This dish is from Iraq. The stuffing of rice, almonds, walnuts and raisins is typical of the whole region.
1.35–1.8 kg/3–4 lb chicken, washed and dried inside and out
Filling
50 g/2 oz ghee
1 small onion, finely choped
75 g/3 oz short grain rice, washed thoroughly under cold water and drained
½ teaspoon allspice
225 ml/8 fl oz water
2 tablespoons almonds, blanched, slivered
2 tablespoons walnuts or hazelnuts, chopped
2 tablespoons seedless raisins
1½ teaspoons salt
½ teaspoon black pepper
Basting
50 g/2 oz ghee, melted
1 teaspoon salt mixed with ½ teaspoon black pepper
100 ml/4 fl oz water
Melt the ghee in a saucepan, add the onion and fry until soft. Add the rice and nuts and fry for 3–4 minutes, stirring frequently. Add the raisins, allspice, water, salt and pepper and bring to the boil. Cover and cook over a low heat until the liquid is absorbed. Remove from the heat and leave to cool.
Fill the chicken with this stuffing and close the opening with a small skewer or needle and thread. Put the chicken in a roasting tin, brush with the melted ghee and sprinkle with the salt and pepper. Pour the water and the rest of the melted fat into the tin and bake in an oven preheated to 180°C, 350°F, gas mark 4 for about 2 hours or until the chicken is tender. Baste frequently.
Variation
Syrians substitute 2 tablespoons pine kernels for the almonds and also include in the stuffing 1 tomato, blanched, peeled and chopped, 1 tablespoon finely chopped parsley and ½ teaspoon cinnamon.
Dajaj mahshi—a Lebanese favourite—as well as the ingredients above also includes about 350 g/12 oz fried minced meat.
tzavarov letzvadz variag
An Armenian speciality—the chicken is stuffed with burghul, liver and chickpeas.
1.35–1.8 kg/3–4 lb chicken, washed and dried inside and out
Filling
75-90 g/3-3½ oz butter the liver and heart of the chicken, coarsely chopped
1½ teaspoons salt
½ teaspoon black pepper
1 small onion, finely chopped
1 large tomato, blanched, peeled and chopped
1 small green pepper, seeded and thinly sliced
175 g/6 oz large grain burghul
2 tablespoons parsley, finely chopped
3 tablespoons chickpeas that have been soaked overnight and cooked until tender
½ teaspoon cinnamon
300–350 ml/10–12 fl oz water
Melt the butter in a saucepan, add the liver, heart, salt and pepper and fry for about 3 minutes, stirring frequently. Remove the liver and heart with a slotted spoon and reserve.
Add the onion to the pan and fry until soft. Stir in the tomato and green pepper and cook for a further 2–3 minutes, stirring frequently. Add the burghul and parsley and cook for another 2–3 minutes, stirring constantly. Now add the chickpeas, cinnamon, water and reserved liver and heart. Bring to the boil, lower the heat, cover and simmer for 15–20 minutes or until the liquid has been absorbed. Leave to cool and then fill the chicken cavity. Reserve any leftover stuffing to serve later with the cooked chicken. Close the opening, place in a roasting tin, brush with a little oil and cook in an oven preheated to 180°C, 350°F, gas mark 4 for about 2 hours or until tender. Transfer to a large serving dish, fluff out the filling and serve.
morgh shekumpour
Chicken stuffed with prunes, apples, raisins and apricots
This is a favourite of Iranians. It is beautiful to look at and extremely tasty. Serve with chelo rice or chelo zaffran (saffron pilav).
1.35–1.8 kg/3–4 lb chicken, washed and dried inside and out
Filling
40 g/1½ oz ghee or butter
1 small onion, finely chopped
150 g/5 oz prunes, soaked overnight, stoned and chopped
150 g/5 oz dried apricots, soaked and chopped
50 g/2 oz seedless raisins
2 apples, peeled, cored and chopped
1 teaspoon ground cinnamon
2 teaspoons salt
½ teaspoon black pepper
1 teaspoon brown sugar
Basting
1 teaspoon salt
½ teaspoon black pepper
melted butter
Melt the ghee or butter in a saucepan, add the onion and fry until soft. Add the chopped prunes, apricots, raisins and apples and fry for about 2 minutes, stirring constantly. Season with the cinnamon, salt, pepper and sugar and cook for 2–3 more minutes. Spoon the mixture into the chicken cavity and secure the opening. Rub the chicken with the salt and pepper and then place in a roasting tin. Brush generously with melted butter and then cook in an oven preheated to 180°C, 350°F, gas mark 4 for about 2 hours or until tender. Baste regularly with the pan juices. Serve on a bed of rice.
turkey
Although not as popular as chicken, turkey has recently acquired a degree of importance due to western influence and is fast becoming the 'guest' meal in such countries as Cyprus, Lebanon and Israel.
The locally grown bird tends to be small and tough, but those reared on the Israeli kibbutzes are big, fat and magnificent.
stuffed turkey
The fillings below are for an approximately 3.5 kg/8 lb turkey.
filling 1—a typical Arab stuffing also popular in Israel and Turkey
2 tablespoons ghee or oil
450 g/1 lb minced lamb
3 tablespoons almonds, chopped, blanched
3 tablespoons walnuts, chopped
3 tablespoons pistachio nuts, chopped
3 tablespoons pine kernels, chopped
225 g/8 oz long grain rice, washed thoroughly under cold water and drained
50 g/2 oz raisins
1 tablespoon salt
1 teaspoon black pepper
1½ teaspoons cinnamon
½ teaspoon allspice
600 ml/1 pint water or stock
Heat the ghee or oil in a pan, add the meat and fry for 10–15 minutes, turning and breaking up with a fork. Add the nuts and fry for a further 5–10 minutes, stirring frequently. Add all the remaining ingredients, stir well and bring to the boil. Lower the heat and simmer until all the liquid has been absorbed. Remove from the heat, allow to cool then spoon into the cavity and secure the opening.
filling 2—from Libya and Egypt and also Lebanon
225 g/8 oz cous-cous
175 g/6 oz seedless raisins, soaked in water for 15 minutes
3 tablespoons almonds, chopped, blanched
6–7 stoned, dried dates, thinly sliced
1 teaspoon cinnamon
1 teaspoon nutmeg
1 teaspoon salt
2 tablespoons sugar
25 g/1 oz melted butter
Steam the cous-cous in a steamer or colander over boiling water for 20 minutes until the grains are soft, but still firm. Transfer to a bowl, add all the remaining ingredients and mix well. Pack into the cavity and secure the opening.
filling 3—a Cypriot favourite
25 g/1 oz butter
1 onion, finely chopped
1 celery stick, thinly sliced
225 g/8 oz minced lamb or veal
225 g/8 oz long grain rice, washed thoroughly under cold water
turkey liver, chopped
600 ml/1 pint dry white wine or stock or a mixture of the two
75 g/3 oz raisins
75 g/3 oz almonds, slivered, blanched
1 teaspoon cinnamon
2 teaspoons salt
½ teaspoon black pepper
Melt the butter in a saucepan, add the onion and fry until soft. Add the celery and fry for 2 minutes, then add the meat and fry for a further 5–10 minutes, turning and mashing with a fork. Add the rice and wine or stock, bring to the boil, cover and cook for 10 minutes. Stir in the remaining ingredients and continue to cook until all the liquid has been absorbed. Spoon into the cavity and secure the opening.
To cook the turkey
Rub the skin with half a lemon and a mixture of 1½ tablespoons salt and half a teaspoon black pepper. Place in a roasting tin and pour in either 75–100 g/3–4 oz melted butter or oil and 100 ml/4 fl oz water. Cook in an oven preheated to 180°C, 350°F, gas mark 4 allowing approximately 30 minutes to every 500 g/lb. Baste regularly.
When cooked remove to a serving dish, skim the pan juices and serve in a sauceboat. Serve with salads and pilavs.
fesenjan-e-ordek
duck in a pomegranate and walnut sauce
One of the great dishes of Iran. In an Iranian home, if a guest is to be honoured, he is often served this meal of duck coated in a thick sauce of ground walnuts flavoured with pomegranate syrup.
You can use a large chicken instead of duck. Serve on a bed of chelo or other pilav of your choice.
1 large duck, cut into quarters or 1 large chicken, cut into 8
8 tablespoons oil
1 onion, thickly sliced
½ teaspoon turmeric
450 g/1 lb shelled walnuts, ground—reserve a few whole ones for garnish
900 ml/1½ pints chicken stock
1 teaspoon salt
½ teaspoon black pepper
150 ml/¼ pint fresh pomegranate juice or 3 tablespoons concentrated juice
juice 2 lemons
50 g/2 oz sugar
Heat half the oil in a frying pan, add the onion and turmeric and fry until the onion is soft and golden. Remove the onion with a slotted spoon and transfer to a large casserole. Stir in the ground walnuts and chicken stock and season with the salt and pepper. Bring to the boil and simmer, stirring occasionally, for 20 minutes.
Heat the remaining oil in the frying pan, add the duck or chicken pieces and fry, turning occasionally, until browned all over. Transfer to the casserole, mix to coat with the sauce and then cover and cook over a very low heat for about 1 hour or until the meat is tender. Stir occasionally to prevent the sauce sticking. Skim excess fat off the surface of the casserole
In a bowl mix together the fresh or concentrated pomegranate juice with the lemon juice and sugar. Stir this sweetened juice into the casserole and simmer for a further 15–20 minutes, stirring occasionally. Taste and adjust seasoning if necessary.
Arrange the duck or chicken pieces on a bed of rice, spoon some of the sauce over the top and garnish with the reserved walnuts—either whole or coarsely chopped. Serve the remaining sauce separately.
Khorovadze sag roast goose with apricots
A magnificent dish from Armenia.
Serve with a rice or burghul pilav accompanied by apricot sauce and freshly sliced vegetables and pickles.
3.6 kg/8 lb goose
1 tablespoon salt
Filling
2 large cooking apples, peeled, cored and thickly sliced
1 large onion, thinly sliced
goose liver, chopped
50 g/2 oz walnuts, coarsely chopped
1 teaspoon cinnamon
juice 1 large lemon for basting
100 ml/4 fl oz dry sherry or brandy
Tzirani salsa—apricot sauce
225 g/8 oz dried apricots, thinly sliced
25 g/1 oz brown sugar
2 tablespoons brandy
1 teaspoon rosewater
½ teaspoon nutmeg
50 g/2 oz pistachio nuts, halved or almonds, slivered, blanched, or pine kernels
Wash and dry the goose inside and out. Rub inside and out with the salt.
In a large bowl mix the apples, onion, goose liver, walnuts and cinnamon. Fill the cavity with this mixture and secure the opening.
Place in a large roasting dish and cook in an oven preheated to 180°C, 350°F, gas mark 4 for about 3 hours or until the juices run clear when a thigh is pierced. Do not overcook and baste regularly with the lemon juice.
While the goose is cooking prepare the sauce. Place the apricots in a pan with enough water to cover and bring to the boil. Reduce the heat and simmer for 10–12 minutes. Add the sugar and cook for a further 5 minutes, stirring regularly. Remove from the heat and stir in the brandy, rosewater and nutmeg.
Meanwhile, roast the nuts under a hot grill, turning at least once, until golden. When cooked remove the goose to a serving dish and remove the scum from the juices in the pan. Add the dry sherry or brandy to the pan juices and cook, stirring constantly until the mixture is reduced to a smooth gravy.
Serve the goose on a bed of pilav garnished with sliced tomatoes, cucumbers, radishes, etc. Pour the sherry/brandy pan juices over the goose. Stir the nuts into the apricot sauce and serve in a separate sauceboat. Serves 4–6 people.
Variation
hina yemistii me kastana
A Greek-Turkish favourite with a chestnut stuffing. Follow the recipe, but prepare the stuffing with:
225 g/8 oz chestnuts, peeled and halved
50 g/2 oz butter
1 small onion, finely chopped
2 large cooking apples, peeled and chopped
Melt the butter in a pan, add the onion and fry until soft. Add the halved chestnuts and fry for a further 3 minutes. Remove from the heat and stir in the apples. Fill the cavity with this stuffing.
avaz memula im matzot
goose stuffed with matzo meal and nuts
Geese are extensively reared in Israel and large quantities are exported to France. The next time you taste the famed pâté de foie gras the meat will most probably have come from Israel.
This is a rich and tasty meal. Garnish it with orange and grapefruit slices. Matzo meal can be bought from most large stores and continental shops.
3.5 kg/8 lb goose
25 g/1 oz lard, melted
1 tablespoon grated orange rind
juice 1 lemon
½ teaspoon white pepper
1 tablespoon salt
Stuffing
40 g/1½ oz lard
2 onions, finely chopped
1 stick celery, finely chopped
goose liver, chopped
3 tablespoons parsley, finely chopped
1 apple, grated
2 teaspoons paprika
50 g/2 oz walnuts, coarsely chopped
2 tablespoons pine kernels
100 g/4 oz dried prunes, soaked overnight
175 g/6 oz medium matzo meal
225 ml/8 fl oz chicken stock
3 tablespoons fresh orange juice
2 eggs, beaten
Baste
3 tablespoons brandy
1 tablespoon honey
1 tablespoon orange rind, grated
Remove the giblets and set aside. Wash and dry the goose inside and out.
Mix the melted fat with the orange rind, lemon juice, pepper and salt and brush this mixture all over the inside and outside of the goose.
To make the stuffing melt the lard in a large saucepan and sauté the chopped onion until soft. Add the celery, liver, parsley, apple, paprika, walnuts, pine kernels and chopped prunes and fry for 4–5 minutes. Add the remaining stuffing ingredients and mix thoroughly. Stuff this mixture into the goose cavity and close the cavity with a small skewer or needle and thread.
Place the goose in the centre of a large baking dish and cook in an oven preheated to 180°C, 350°F, gas mark 4 for about 3 hours or until the juices run clear when a thigh is pierced.
In a small cup mix the baste ingredients together and use to baste the goose regularly, especially during the last 30 minutes.
Remove the goose to a serving dish. Pour the pan juices and any remaining baste into a saucepan, spooning off as much of the fat as possible. Remove the skewer or thread from the goose cavity and scoop out the stuffing which you can serve in a separate dish. Arrange the orange and grapefruit slices decoratively around the goose and serve. Heat the pan juices and serve in a sauceboat. Serves 6 people.
kereghani paisan baked pheasant
This dish is traditionally cooked and served in a kereghan, large clay pot, and is particularly popular in the mountain villages of Armenia where pheasants, partridges and other wild game are found in abundance. Other particularly famed local dishes included 'stuffed bear paws' and 'boars head', but unfortunately I have had to exclude these recipes as some of the raw materials are not easy to find!
Serve with pilavs or roast potatoes and salads.
900–1125 g/2–2½ lb pheasant, washed and dried inside and out
2 tablespoons salt
40 g/1½ oz ghee
100 g/4 oz mushrooms, thinly sliced
1 small onion, thinly sliced
1 tablespoon walnuts, coarsely chopped
100 ml/4 fl oz dry white wine
100 ml/4 fl oz stock
Garnish
1 tablespoon parsley, finely chopped
1 tablespoon sumac
Rub the pheasant inside and out with the salt. Melt the ghee in a large casserole, add the pheasant and fry, turning occasionally, until evenly browned all over. Remove the pheasant and reserve.
Add the mushrooms and onion and fry for 3 minutes, stirring regularly. Add the walnuts and fry for 1 minute.
Return the pheasant to the casserole, add the wine and stock and bring to the boil. Cover and cook in an oven preheated to 160°C, 325°F, gas mark 3 for about 1 hour or until tender. Add a little more wine or stock if necessary. Transfer the pheasant to a serving dish, cut into serving pieces and spoon the pan juices over them. Garnish with the parsley and sumac and serve.
yogurtlu tavsan rabbit with yoghurt
This Anatolian recipe works equally well with hare. In Turkey yoghurt is preferred to soured cream which is more popular in the Caucasus.
Wine is used by non-Muslims, while Muslims substitute water—at least in theory. The fact is that wine is both made and drunk in Turkey without much religious polemic. It must be well understood that Turks adopted the Muslim faith more out of political expedience than religious fervour.
Serve with a pilav of your choice; burghul pilav is particularly good with this dish.
1 rabbit or hare, cut into serving pieces
300 ml/½ pint wine vinegar
300 ml/½ pint water
2 tablespoons plain flour
2 tablespoons salt
½ teaspoon black pepper
½ teaspoon oregano
½ teaspoon dillweed
3 tablespoons oil
100 ml/4 fl oz dry white wine or water
2 tablespoons fresh tarragon or mint or parsley, finely chopped
about 150 ml/¼ pint stabilized yoghurt—see Glossary
Garnish
1 teaspoon paprika
1 teaspoon cumin
Place the pieces of rabbit in a bowl, cover with the wine vinegar and water and leave to soak for 1 hour. Drain and dry with kitchen paper.
In a small bowl mix together the flour, salt, pepper, oregano and dillweed. Coat the rabbit joints in the seasoned flour.
Heat the oil in a large frying pan and fry the joints for about 5 minutes or until nicely browned all over. Remove and transfer to a large casserole dish or saucepan. Add the wine or water and the chopped tarragon or mint or parsley, cover and simmer for about 1 hour. Add the stabilized yoghurt, stir well and simmer for a further 30 minutes. Spoon into a serving dish and garnish with the paprika and cumin.
## firin kebabs and khoreshts
Stews and casseroles—the mainstay of any cuisine—are richly represented in the Middle East.
Lamb, beef, poultry, game, fish and sometimes pork (in the Christian regions) are combined with vegetables, pulses, fruits, nuts and herbs to be cooked slowly in the oven (firin in Turkish) or on top of the stove.
Firin kebabs—the title is misleading since none of these dishes are true kebabs, meat on skewers cooked over charcoal—are particularly popular in Anatolia where the peasantry have, over the centuries, developed a method of cooking which stretched the meal as far as possible. Since meat has always been the prerogative of the rich, a cut of meat was combined with other ingredients to make a substantial meal to satisfy many.
Khoreshts—stewed meat in a sauce—are always served with rice and are the mainstay of the Iranian cuisine. Most housewives will prepare a khoresht and rice as the main meal no matter how many other delicious side dishes are prepared. There may often be 3–4 different khoreshts served at one meal.
The repertoire is vast, elegant, tasty and colourful. Variations abound and it has been extremely difficult for me to decide which particular recipes to include. Do not hesitate to vary the choice of meat or fruits or nuts suggested, for once you have mastered the basic principle of making a khoresht you should be able to experiment at leisure.
gatzai kebab pot kebab
A decorative dish from Anatolia and beloved of both Armenians and Turks. Serve it with rice pilav, fresh salad and yoghurt.
450 g/1 lb minced lamb
½ green pepper, chopped
1 small onion, chopped
4 tomatoes, blanched, peeled and chopped
2 tablespoons parsley, chopped
1 tablespoon tomato purée
2 teaspoons salt
1 teaspoon black pepper
1 teaspoon chilli pepper
4 aubergines
4 large potatoes
4 large tomatoes
In a large bowl, mix together the meat, chopped green pepper, onion, tomatoes, parsley, tomato purée, salt, black and chilli pepper. When well blended set aside and prepare the vegetables.
Peel strips of skin lengthwise from the aubergines to give them a striped effect.
Peel the potatoes and wash the tomatoes.
Slice all the vegetables, at 1 cm/½ in intervals, crosswise about ¾ of the way down so that they remain attached at the bottom.
Take each vegetable and fill the gaps with a little of the meat mixture. Arrange the stuffed vegetables tightly in an ovenproof dish so that they keep their shape and hold the meat in place. Half cover them with water seasoned with a little salt. Bake in an oven preheated to 200°C, 400°F, gas mark 6 for 1–1½ hours or until the vegetables are tender.
malatya kebabi
stuffed aubergines baked in the oven
Another aubergine-based firin kebab is this one from Malatya (ancient Melitane) for a while part of the Crusading Kingdom of Eddessa and for centuries under Armenian sovereignty.
This dish of aubergines stuffed with meat and onions and topped with tomatoes and green peppers is a classic of the Ottoman period and equally popular with Armenians, Kurds and Turks. Serve it with rice or burghul pilav.
6 medium aubergines
2 teaspoons salt
8 tablespoons oil
Filling
1 large onion, finely chopped
450 g/1 lb minced lamb or beef
1½ teaspoons salt
½ teaspoon black pepper
½ teaspoon paprika
2 tomatoes, blanched, peeled and chopped
300 ml/½ pint water
Topping
3 tomatoes, quartered
3 green peppers, seeded and quartered
Cut the heads off the aubergines and then cut each one in half lengthways. Remove some of the pulp from each half leaving shells about 0.5 cm/¼ in thick. Arrange the aubergines on a plate, sprinkle with 2 teaspoons of salt and set aside for 30 minutes. Rinse the aubergines under cold running water and pat dry with kitchen paper.
Heat 6 tablespoons of the oil in a frying pan, add the aubergine halves, a few at a time, and sauté for about 3 minutes, turning regularly. Using a slotted spoon transfer the halves to a large, shallow casserole dish and arrange side by side.
Heat the remaining oil in a saucepan, add the onion and sauté until soft, stirring occasionally. Add the meat, salt, pepper, paprika and chopped tomatoes, stir well and sauté for 5 more minutes. Add the water and cook for 30 minutes, stirring regularly. Remove the pan from the heat and, with a tablespoon, fill each halved aubergine with the meat mixture. Place a quartered tomato and green pepper on each half and pour any pan juices into the dish. Bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 30 minutes. Remove from the oven and serve immediately with a pilav.
One day Boloz Mugush asked his nephew 'What is an aubergine?'
Without a moment's hesitation the nephew replied 'A newly born ox whose eyes have still not opened.'
Boloz Mugush, amazed at this brilliant explanation, beams and announces aloud to one and all in the room 'See how bright our nephew is—this neither I nor his father taught him. This he found out for himself!'
papaz kebabi priest's kebab
A very ancient dish mentioned both by Apicius and the Arab historians of the Middle Ages.
Meat is cooked with milk—an aberration in Semitic and hence Islamic customs where milk and meat are not permitted to be eaten together.
As the name suggests this is a dish of Christian origins (Armenian or Greek). Serve with rice pilav or roast potatoes and salads.
50 g/2 oz butter
2 onions, thinly sliced
700 g/1½ lb lean lamb or beef, cut into 6 pieces
600 ml/1 pint water
1 teaspoon salt
1 tablespoon plain flour
450 ml/¾ pint milk
1 teaspoon salt
Melt half of the butter in a large saucepan. Add the onions and fry, stirring frequently, until soft. Add the pieces of meat, cover the pan and cook for 10 minutes. After 5 minutes uncover the pan, stir the meat mixture and cover again. After 10 minutes uncover the pan, add the water and salt, stir and continue simmering until the water has evaporated. Remove from the heat and keep warm.
Heat the remaining butter in a small saucepan, remove from the heat and stir in the flour. Gradually add the milk, stirring constantly, until the mixture is smooth. Season with the salt, return to a low heat and cook, stirring constantly, until the mixture thickens. Transfer the meat and onion mixture to an ovenproof casserole and pour the white sauce over the top. Place in an oven preheated to 180°C, 350°F, gas mark 4 and cook for 20 minutes. Remove from the oven and serve immediately.
kuzu ankara tavasi
ankara-style lamb casserole
A rich, creamy casserole from Turkey which is topped with a crust of lightly cooked eggs. Serve with salads and pickles.
25 g/1 oz butter
900 g/2 lb lean lamb, cut into 6 equal portions
1.35 litres/2 pints water
1 teaspoon salt
2 onions, thinly sliced
1 large carrot, peeled and cut crossways into 0.5 cm/¼ in rounds
75 g/3 oz green beans
450 ml/¾ pint yoghurt
50 g/2 oz plain flour
2 eggs, lightly beaten
Garnish
1 tablespoon paprika
1 tablespoon dried dill
Melt the butter in a large saucepan. Add the pieces of meat and fry for a few minutes, turning occasionally, until they are browned all over. Add the water and salt and bring quickly to the boil. Cover the pan, lower the heat and simmer for 45 minutes. Add the onions and carrot and cook, uncovered, for 30 minutes. Add the green beans and cook for a further 10–15 minutes or until the beans are just tender.
With a slotted spoon transfer the pieces of meat to a large ovenproof casserole. Arrange the cooked vegetables over the meat.
Pour the yoghurt into a bowl and stir in the flour until it is well blended and smooth. Now stir in about 450 ml/1 pint of the warm meat liquid from the saucepan and pour the sauce into the casserole. Bring gently to the boil and simmer over a low heat for 10 minutes.
Pour the beaten eggs over the surface of the casserole and bake in an oven preheated to 190°C, 375°F, gas mark 5 for 10–15 minutes or until the egg is set and golden. Remove from the oven, sprinkle with the paprika and dill and serve with rice and/or potatoes and a salad.
arnaki se filo lamb in filo
A dish of light, crisp pastry filled with a mixture of meat, onion and tomato. Originally the filling was wrapped in parchment, but nowadays aluminium foil is usually used. However, this Greek-Cypriot recipe is for baklava filo. The same dish appears in Turkey as ali pafla kebabi and is usually made with borek hamuru—borek dough. You can use puff pastry instead.
The story goes that in the bad old days the Palikari (originally bandits—later the name was given to partisans fighting for the independence of Greece and Crete) wrapped the meat in parchment so that the aroma would be tightly sealed in and their hideouts not discovered.
I suggest you use baklava filo.
Serve with fresh salad.
25 g/1 oz ghee or butter
2 onions, thinly sliced
700 g/1½ lb lean lamb
1 large tomato, thinly sliced
1 teaspoon salt
1 teaspoon black pepper
225 ml/8 fl oz water
2 tablespoons parsley, finely chopped
Dough
8 sheets baklava filo
60 g/2½ oz unsalted butter, melted
Melt the ghee or butter in a large saucepan, add the onion and fry until soft and turning golden. Cut the meat into 1.2 cm/½ in cubes and add to the saucepan. Cover the pan and simmer for 5 minutes, stirring occasionally. Add the tomato, salt, pepper and water and bring to the boil. Lower the heat and simmer for about 1 hour or until the meat is tender and most of the water has evaporated. Remove from the heat, stir in the parsley and set aside to cool.
Meanwhile, open up one sheet of the filo and brush with the melted butter. Place another sheet of filo over the top. Brush one half of this second sheet with butter and fold the 2 sheets over so that they are half their original size. Place ¼ of the meat mixture in the centre of the filo. Fold the filo over in envelope-style to enclose the meat and brush the edges with butter to hold them down. Continue with remaining meat and pastry until you have 4 'parcels' and brush the surface of each with any remaining butter.
Lightly grease 2 baking trays and place 2 'parcels' on each. Place in an oven preheated to 180°C, 350°F, gas mark 4 and bake for 20–30 minutes or until the pastry is golden. Remove from the oven and serve with a salad of your choice.
kazan kaypapi aubergine with lamb and fruits
An Azerbaijanian dish from the Caucasus that is, in reality, a khoresht rather than a firin kebab.
Makes an excellent use of fruits and spices.
Serve it with a bowl of yoghurt and a rice pilav.
1 large aubergine, peeled and cut crossways into 0.5 cm/¼ in slices
1 tablespoon salt
3 tomatoes, blanched, peeled and thinly sliced
1 onion, thinly sliced
1 green pepper, seeded and thinly sliced
5–6 apricot halves, each halved
5–6 stoned prunes, halved
40 g/1½ oz ghee or butter
450 g/1 lb lean lamb cut into 2.5 cm/1 in cubes
¼ teaspoon cinnamon
¼ teaspoon ground cloves
½ teaspoon allspice
1 teaspoon salt
½ teaspoon black pepper
3–5 tablespoons oil
4 tablespoons lemon juice or, preferably, pomegranate juice
6 tablespoons melted ghee or butter
1 medium quince or cooking apple, peeled, cored and cut into 1.2 cm/½ in pieces
Put the aubergine slices in a colander, sprinkle with the salt and set aside for 30 minutes.
Meanwhile, lightly grease a large casserole dish and spread half the tomato slices in the bottom. Now lay the onion and green pepper slices and half the apricots and prunes over the tomatoes.
Melt the ghee or butter in a saucepan, add the meat cubes and sauté until browned all over, turning frequently. Add the cinnamon, cloves, allspice, salt and pepper and cook for a further 2 minutes, stirring constantly. Transfer the spiced meat to the casserole and spread evenly over the vegetables. Lay the remaining tomato slices, apricots and prunes over the meat.
Heat the oil in a large frying pan. Rinse the aubergine slices under cold water and pat dry with kitchen paper. Fry the aubergine slices, a few at a time, until golden on both sides and then remove and drain on kitchen paper. Add more oil if necessary. Arrange the slices over the top of the casserole. Pour the lemon juice and half the melted ghee over the meat and vegetables.
Heat the remaining melted ghee in a small saucepan, add the pieces of quince or apple and fry for a few minutes, stirring and turning frequently. Arrange them over the aubergines and pour any remaining fat into the casserole. Cover and bake in an oven preheated to 180°C, 350F°, gas mark 4 for about 45 minutes or until the meat and vegetables are tender. Remove from the oven and serve.
hamuth helou lamb stew with dates
This is a rich Iraqi stew of lamb with dried fruits. Traditionally it is prepared with a thick date syrup and, to contrast, dried limes. I have suggested puréed dates (see recipe) instead of date syrup and lemon juice and peel if, as is possible, loomi (dried limes) cannot be obtained. Serve with a pilav of your choice.
50 g/2 oz ghee
900 g/2 lb lean lamb, cut into 2.5 cm/1 in pieces
1 onion, chopped
600 ml/1 pint water
1½ teaspoons salt
5 cm/2 in piece of cinnamon
1 loomi (dried lime) or peel of 1 lemon and its juice
6 stoned dried dates, chopped
8 dried apricots, halved
8–10 prunes, stoned and halved
3 tablespoons raisins
1 tablespoon brown sugar or honey
Melt the ghee in a large saucepan. Add the meat and sauté for 5–7 minutes, stirring frequently. Remove the meat with a slotted spoon and reserve.
Add the chopped onion to the pan and fry until soft. Return the meat to the pan, add half the water, the salt, cinnamon and loomi or lemon juice and peel. Cover the pan, lower the heat and simmer for 45 minutes.
Meanwhile, place the remaining water in a small pan, add the chopped dates and simmer over a moderate heat for 12–15 minutes until the dates soften. Transfer the mixture to a liquidizer and blend to a purée. Add this date purée to the meat mixture together with the apricots, prunes, raisins and sugar or honey. Stir well, recover the pan and cook for a further hour. When the meat is tender transfer the stew to a serving dish and remove the cinnamon stick and loomi or lemon rind.
cholent israeli meat and vegetable casserole
A classic of the Jewish cuisine, developed in the Middle Ages in Central Europe, as a response to religious requirements.
Cholent (also called shalet and shalent) has numerous variations—depending basically on where the particular Jewish family originate. The traditional one, still popular in Israel amongst Russian and Polish Jews, always includes potatoes (the staple food of Eastern Europe), kasha (buck-wheat) and turnips, carrots and meat.
The recipe below is a typical one prepared on a Friday evening and cooked very slowly to be ready for the Sabbath.
A rich and nourishing meal. Serve with a pilav of your choice and bread.
The Oriental Jews, of course, have their own versions, e.g. dfeenah.
50 g/2 oz butter or ghee
1.1–1.3 kg/2½–3 lb brisket of beef
1 onion, chopped
1 clove garlic, finely chopped
1 teaspoon salt
½ teaspoon black pepper
½ teaspoon paprika
¼ teaspoon ginger
¼ teaspoon cinnamon
350 g/12 oz butter beans (lima beans), soaked overnight in cold water
100 g/4 oz pearl barley
10 small potatoes, peeled
2 carrots, peeled and thickly sliced
1 turnip, peeled and cut into 2.5 cm/1 in pieces
2 bay leaves
1 onion, chopped
1 tablespoon flour
½ teaspoon paprika
water for boiling
Garnish 2 tablespoons parsley, finely chopped
Melt the butter or ghee in a large casserole, add the meat, onion and garlic and sauté until brown, turning the meat occasionally. Mix the salt, pepper, paprika, ginger and cinnamon together, sprinkle over the contents of the casserole and cook, stirring frequently, for 3–4 minutes. Add the beans, pearl barley, vegetables and bay leaves. Sprinkle with the flour and paprika and stir well. Add sufficient boiling water to cover the ingredients by about 2.5 cm/1 in. Cover the casserole and place in an oven preheated to 180°C, 350°F, gas mark 4. Cook for 2–3 hours or until the meat is tender. Remove from the oven, slice the meat and arrange in the centre of a large serving dish.
With a slotted spoon remove the beans, barley and vegetables and arrange them around the meat. Sprinkle with the chopped parsley. Pour the pan juices into a sauceboat and serve as an accompaniment.
Serves 6–8 people.
khoresht-e-albaloo meat and cherry stew
Sour black cherries should be used for this dish, but Morello cherries will do. This is a typical Iranian khoresht of meat, fruit and spices.
Always served with chelo rice pilav, but you can serve any pilav of your choice.
The Arab dish lahma-bil-karaz is similar to this dish except that the meat is minced and rolled into marble-sized balls and the dish is served with bread instead of rice.
Other fruits are also used, e.g. green plums—khoresht-e-go jeh, sour grapes—khoresht-e-gooreh and almonds—khoresht-e-ghaghaleh-badmoon, etc.
The cooking method is the same, as are the remaining ingredients and so you can experiment if you like.
50 g/2 oz ghee
1 large onion, finely chopped
700 g/1½ lb stewing lamb or beef, trimmed of excess fat and cut into 2.5 cm/1 in cubes
½ teaspoon cinnamon
½ teaspoon turmeric
100 ml/4 fl oz water
2 teaspoons salt
½ teaspoon black pepper
4 tablespoons lemon juice or 4 tablespoons powdered dried lime
225 g/8 oz sour black cherries, stoned
2 tablespoons sugar—or more to taste
Melt the ghee in a large saucepan, add the onion and fry until soft. Add the meat cubes and fry, turning frequently, until evenly browned. Stir in the cinnamon and turmeric and cook for 2 minutes, stirring frequently. Add the water, salt, pepper and 2 tablespoons of the lemon juice or powdered lime. Mix well, cover the pan, lower the heat and simmer for about 40–45 minutes. Stir in the cherries and sugar, cover and simmer for 10 minutes.
Taste the sauce. The ideal flavour should be sweet-sour. If the sauce is too sweet then add the remaining lemon juice or lime powder, but if it is too sour then add enough sugar to suit your taste. Cover and continue to simmer for a further 20–30 minutes or until the meat is tender. Transfer to a large dish and serve with chelo or other rice pilav of your choice.
Variation
khoresht-e-ghooreh meat and sour grape stew
If you happen to have a vine then try this recipe. Follow the directions above but substitute 450 g/1 lb unripe sour grapes for the cherries and add 1 tablespoon tomato purée to the pan.
Either eliminate the lemon juice or lime powder or add only 1 tablespoon—this will really depend on the sourness of the grapes!
khoresht-e-narengi
chicken and tangerine stew
A tasty stew that typifies everything that is uniquely Iranian. A brilliant arrangement of yellows and golds.
75 g/3 oz ghee or butter
1.3–1.8 kg/3–4 lb oven ready chicken, cut into 8 serving pieces
1 large onion, thinly sliced
½ teaspoon saffron
600 ml/1 pint water
1½ teaspoons salt
½ teaspoon black pepper
juice 1 large lemon
4 tangerines, peeled and with peel reserved
225 g/8 oz carrots, peeled and cut into 1.2 cm/½ in slivers
1–2 tablespoons brown sugar
Garnish
2 tablespoons toasted almonds, slivered
2 tablespoons pistachios, halved
Melt 40 g/1½ oz of the ghee or butter in a large saucepan. Add the chicken pieces and sauté until almost tender, turning until evenly browned. Remove from the pan and reserve.
To prepare the sauce add the remaining ghee or butter to the pan, add the onion and fry until soft. Stir in the saffron and return the chicken pieces to the pan. Add the water, salt, pepper and lemon juice, lower the heat, cover and simmer for 30 minutes.
Meanwhile, with a sharp knife scrape and discard the white pith from the tangerine peel and then cut the peel into thin strips. Add to the stew together with the carrots and brown sugar. Continue to simmer until the peel and carrots are tender.
Separate the tangerines into segments and peel each segment. Add segments to the pan and cook for a further 10–15 minutes. Transfer to a large serving dish, sprinkle with the nuts and serve with chelo or other rice pilav of your choice.
## sauces
There are three main types of sauces—those made with yoghurt, tomatoes or nuts. Wine and cream are little used—except sometimes in Greece, Israel and Northern Caucasia.
Most Middle Eastern dishes are eaten dry, e.g. kebabs, roasts, pilavs, etc. or, as is often the case, are inclusive of their own sauces, e.g. khoreshts, dolmas, stews, etc.
There are, however, a few well known sauces which often accompany vegetable, meat and rice dishes.
Yoghurt is the basis of some of these sauces. The most popular one is sughtorov-madzoon—a garlic-yoghurt sauce. It is versatile, easy to make and is used extensively with stews and vegetables—especially dolmas, pilavs and meat. The recipe is a standard one.
sughtorov-madzoon garlic-yoghurt sauce
300 ml/½ pint yoghurt
1 clove garlic, crushed
½ teaspoon salt
Garnish
½ teaspoon dried mint
1 spring onion, finely chopped (optional)
Pour the yoghurt into a bowl. Add the garlic and salt and mix well. Sprinkle the dried mint, and onion if using it, over the top and serve.
darçinli yoghurt cinnamon yoghurt sauce
This yoghurt sauce includes cinnamon and sugar and is traditionally served as an accompaniment to roast meats as well as salads and vegetables.
300 ml/½ pint yoghurt
2 teaspoons sugar
Garnish 1 teaspoon ground cinnamon
Pour the yoghurt into a serving bowl. Add the sugar and mix well. Sprinkle with the cinnamon and serve.
peynirli yoghurt salsasi
yoghurt-cheese sauce
This is a regional speciality from Turkey which is ideal with cooked eggs, pastas, vegetables, salads and Anatolian pancakes.
50 g/2 oz butter
2 tablespoons flour
450 ml/¾ pint yoghurt
100 g/4 oz grated cheese, e.g. Gouda, Edam or Cheddar
½ teaspoon paprika
½ teaspoon salt
pinch black pepper
Melt the butter in a saucepan. Remove from the heat and stir in the flour. Cook for 1 minute. Beat the yoghurt vigorously and slowly stir it into the flour mixture. Cook slowly over a low heat, stirring constantly, until the mixture thickens. Stir in the cheese, paprika, salt and pepper. Cook slowly until the cheese melts and then serve.
tomato sauces
Although tomatoes are a relatively new arrival to the Middle East they have been incorporated very successfully in countless dishes. They make delicious sauces such as the famed Armenian kebab sauce, the Greek-Turkish saltsa tomata and the spicy dukkous al-tamata from the island of Bahrain. More often than not nowadays tomato purée is used instead of tomatoes.
kebabi salsa kebab sauce
This is our family recipe—one that we also use in our restaurants. There are a few others, e.g. one Lebanese version includes okra, but the basic ingredients are the same.
4 tablespoons cooking oil
1 small onion, finely chopped
1 clove garlic, crushed
1 green pepper, seeded and chopped
2 tomatoes, blanched, seeded and chopped
2 tablespoons tomato purée
600 ml/1 pint water
½ glass red wine (optional)
2 bay leaves
100 g/4 oz peas
1 teaspoon ground coriander
salt and black pepper to taste
Heat the oil in a large saucepan, add the onion, garlic and green pepper and fry until the onion is golden. Add the remaining ingredients, except the peas, and stir well. Now add the peas and bring to the boil. Cook for 15–20 minutes, stirring occasionally. Taste and adjust the seasoning if necessary.
saltsa tomata tomato sauce
This is a versatile sauce used over pilavs, roasts and kebabs.
40 g/1½ oz butter
3 tablespoons onion, chopped
5 large tomatoes, blanched, peeled and finely chopped
1 clove garlic, crushed
2 tablespoons parsley, finely chopped
2 bay leaves
½ teaspoon salt
¼ teaspoon allspice
¼ teaspoon black pepper
Melt the butter in a saucepan, add the onion and fry until soft and transparent. Add the remaining ingredients and cook over a low heat for about 15 minutes, stirring frequently. Serve hot.
dukkous al-tamata spicy tomato sauce
A strong, spicy, garlic-flavoured sauce which is usually served with rice pilavs and roasts.
2 tablespoons oil
8 cloves garlic, crushed
900 g/2 lb ripe tomatoes, blanched peeled and coarsely chopped
1 tablespoon salt
¼ teaspoon paprika
¼ teaspoon cumin
¼ teaspoon ground coriander
¼ teaspoon black pepper
¼ teaspoon ground nutmeg
¼ teaspoon turmeric
¼ teaspoon chilli pepper
Heat the oil in a saucepan, add the garlic and fry for 2 minutes, stirring constantly. Add the tomatoes and salt, lower the heat, cover the pan and simmer for 20–30 minutes. Add all the remaining ingredients, stir thoroughly and cook for a further 5 minutes. Serve with pilavs, vegetables and meat.
beid-el-lemoun egg and lemon sauce
One of the oldest known sauces this was prepared by the ancient Egyptians, Romans and Byzantines. It is extremely popular in Greece as saltsa avgolemono. The Greeks passed it on to the Turks—terbiye and to the Arabs—whose name I have used for this sauce. It is used in soups, with salads and vegetables and particularly with hot and cold fish dishes.
The recipe below is to accompany chicken dishes, but if you wish to use it with fish then use fish stock instead of chicken stock.
450–600 ml/¾–1 pint chicken stock
½ teaspoon salt
¼ teaspoon black pepper
3 egg yolks
juice 2 small lemons
1 tablespoon cornflour
Pour the stock into a saucepan and season to taste with the salt and pepper.
Beat the egg yolks in a small bowl and then add the lemon juice, stirring constantly. Pour the egg mixture slowly into the stock and stir well.
Mix the cornflour with a little water and add to the pan. Heat the sauce gently, stirring constantly using a wooden spoon, for about 10 minutes. Do not bring to the boil or it will curdle. The sauce should by now have a smooth, cream-like consistency. Serve hot or cold.
tahiniyeh garlic and tahina sauce
Used extensively in the Syrian and Lebanese cuisines. It's often eaten as a dip and is excellent with fish dishes and as a salad dressing.
Will keep for several days in a refrigerator.
150 ml/¼ pint tahina paste
juice 2 lemons
300 ml/½ pint milk or water
2 cloves garlic, crushed
1 tablespoon parsley, finely chopped
1 teaspoon salt
½ teaspoon chilli pepper
Pour the tahina into a bowl and stir in the lemon juice. Slowly add the milk, stirring until you have a mixture of a thick, creamy consistency. Add the remaining ingredients and stir well. Serve as required.
tkemali sauce prune sauce
A classic Georgian sauce extensively used in the Caucasus. It is used with chicken and kebabs of meat and fish.
It is easy to prepare and will keep for a long time.
450 ml/¾ pint water
225 g/8 oz prunes
1 clove garlic
1 teaspoon ground coriander
½ teaspoon salt
½ teaspoon paprika
1½ tablespoons lemon juice
Bring the water to the boil in a saucepan, add the prunes, remove from the heat and set aside for 10 minutes. Bring back to the boil and cook briskly for about 15 minutes or until the prunes are tender. Pour the contents of the pan into a sieve placed over a bowl. Reserve the liquid.
Stone the prunes and put the flesh into a liquidizer with the garlic and coriander. Add a little of the reserved liquid and blend well. The sauce needs to have the consistency of thick cream and so adjust the amount of liquid you add accordingly. Transfer this sauce to a saucepan, stir in the salt and paprika and bring to the boil. Remove from the heat and stir in the lemon juice. This is usually served at room temperature.
schoog yemeni sauce
Originally from Yemen this sauce is now, perhaps, more popular in Israel—brought over by the Jews who settled in the Holy Land in the early fifties. Hot and pungent, it is often eaten on its own with bread or as an hors d'oeuvre. Also served with hot vegetable dishes, roasts and kebabs.
3 cloves garlic
1 teaspoon cumin
½ teaspoon salt
3 teaspoons ground coriander
6 fresh chillies
2 tomatoes, blanched, peeled and chopped
juice ½ lemon
½ teaspoon sugar
3 tablespoons water
Put all the ingredients in a liquidizer and blend. Spoon into a saucepan and bring to the boil. Allow to cool and then serve.
tarator garlic and walnut sauce
The origin of this sauce is lost in the mists of time. I would like to think it has something to do with the Tartars and their famed sauce—better known to you and me as sauce tartare. Serve with poultry and fish.
100 g/4 oz walnuts
3 cloves garlic
½ teaspoon salt
2 slices bread, crusts removed
2 tablespoons lemon juice
olive oil
Garnish 1 tablespoon parsley, chopped
Pound the walnuts with the garlic and salt in a mortar or grind in a blender. Soak the bread in a little water and then squeeze out. Spoon the nut mixture into a bowl and add the bread. Mix thoroughly. Little by little add the lemon juice and enough oil for the sauce to become thick and smooth. Serve in small dishes, sprinkled with parsley, beside each plate.
Variations
tarator çamfishtikli pine kernel sauce
Prepare as with the recipe above, but omit the bread and use 100 g/4 oz pine kernels, 2 cloves garlic, 1 teaspoon salt, 2 tablespoons lemon juice and oil.
tarator sade garlic sauce
A Turkish sauce traditionally served with fish dishes, particularly mussels, it also goes well with cold meats and salads.
2 cloves garlic
1 teaspoon salt
4 tablespoons olive oil
2 tablespoons lemon juice
In a small bowl crush the garlic with the salt. Add 2 tablespoons of the oil and set aside for 10 minutes. Stir in the remaining oil and the lemon juice and spoon over fish, meat or vegetables.
## khubz–bread
The staff of life, bread has been the major ingredient in all the Middle Eastern cuisines. In the desert regions the nomads still prepare their bread on a saj—a large, circular, dome-shaped piece of cast iron heated from underneath by dry sticks and camel dung. In the highlands of Lebanon the heat for cooking the bread is generated by burning pine needles, while further north in the mountains of the Caucasus and Iran the bread is traditionally cooked in a tonir—large clay oven—heated by wood or charcoal.
In the Middle Ages, we are informed that there were two basic types of bread in the Middle East—al khubz al huwmara (white flour) and al khubz al khashkar (coarse unhusked flour). Both these breads were glazed with borax (bowrag) which was imported from Armenia. Some of these breads were khushanaj (dry bread) similar to simit and choreg—see recipes; mutbag (envelope) related to pita bread still called khubz Shami or khubz Arabi—see recipes; akras mukallala (crowned loaves); khubz al-abazir (seasoned bread) similar to khubz-el-saluf or mannaeesh. There are literally hundreds of different breads throughout the region, eaten with every meal.
lavash crispy thin bread
The oldest form of bread found in the Middle East, lavash is a thin, crispy bread made from plain flour (wholemeal is also sometimes used).
It comes in many different shapes and sizes from small rounds to oval to large circles often 60 cm/2 ft in diameter. It keeps for a long time without going mouldy. Traditionally enough was often baked at one time to last for 3–4 months and it was then wrapped in linen until required. Normally prepared in a tonir—similar to the Indian tandoor, it can also be prepared, and often is in the Arab lands, on a large, circular, dome-shaped piece of cast iron known as a saj which is heated from underneath by burning wood chippings.
The recipe below is a simplified version ideal for dips, salads and for wrapping around kebabs.
15 g/½ oz fresh yeast or 7 g/¼ oz dried yeast
1 teaspoon sugar
lukewarm water
700 g/1½ lb plain flour
1 teaspoon salt
Place the yeast in a small bowl with the sugar and dissolve in 300 ml/½ pint warm water and set aside for about 10 minutes in a warm place or until the mixture begins to froth.
Sift the flour and salt into a large bowl. Make a well in the centre and slowly work in the yeast mixture and enough warm water to make a stiff dough. Knead on a floured surface for about 10 minutes until the dough is smooth and elastic. Place the ball of dough in a clean bowl, cover with a cloth and leave in a warm place for about 2–3 hours or until it has doubled in size.
Transfer the dough to a floured surface, punch it down and knead again for a few minutes. Return to the bowl, cover and leave for a further 30 minutes. Flour the working surface again. Divide the dough into apple-sized balls. This amount of dough should make about 12–15 balls.
With a long rolling pin roll out each ball into a thin sheet about 20–25 cm/8–10 in in diameter. Sprinkle the working surface with flour now and again to prevent sticking.
Line the bottom of the oven with foil and heat the oven to 200°C, 400°F, gas mark 6. Place one sheet of dough on the foil and cook for about 3 minutes. Remove the cooked lavash and cover with a teatowel while you cook the remaining lavash in the same way. Serve immediately.
If the lavash are not to be used at once then, when completely cold, fold and wrap them in a teatowel and then in plastic or wrap and freeze. When ready to serve sprinkle them lightly with water, wrap in a teatowel and leave for 10 minutes to absorb the moisture and to soften.
khubz arabi arab bread
Eat of the bread made by a woman with a bleeding nose, but do not eat the bread of her who constantly reminds thee of having given it.—Arab wisdom.
Better known as pita bread in the west, khubz Arabi is the ideal bread for eating with most Arab food. Makes about 8 loaves.
15 g/1½ oz fresh yeast or 7 g/¼ oz dried yeast
1 teaspoon sugar
about 300 ml/½ pint tepid water
450 g/1 lb plain flour
½ teaspoon salt
Place the yeast and sugar in a small bowl, dissolve in a few tablespoons of the warm water and set aside in a warm place for about 10 minutes or until it begins to froth.
Sift the flour and salt into a large bowl. Make a well in the centre and pour in the yeast mixture. Add enough of the warm water to make a firm, but not hard, dough.
Knead on a floured working top for 10–15 minutes or until the dough is soft and elastic. If you knead in a tablespoon of oil it will make a softer dough.
Wash and dry the mixing bowl and lightly oil it. Roll the dough around the bowl until its surface is greased all over—this will prevent the dough going crusty and cracking while rising. Cover the bowl with a damp cloth and set aside in a warm place for at least 2 hours until the dough has doubled in size. Transfer the dough to the work top, punch down and knead for a few minutes. Divide the mixture into 8 pieces. Roll them between your palms until they are round and smooth.
Lightly flour the working top and flatten each ball with the palm of your hand, or with a rolling pin, until it is about 0.5 cm/¼ in thick and is as even and circular as possible. Dust the tops with flour and cover with a floured cloth. Leave to rise in a warm place for a further 20–30 minutes.
Preheat the oven to 230–240°C, 450–475°F, gas mark 8–9 putting in 2 large oiled baking sheets half-way through the heating period. When the oven is ready slide the rounds of dough on to hot baking sheets, dampening the tops of the rounds to prevent them browning, and bake for 10 minutes. Do not open the oven door during this time, but after that it is safe to open it to see if the pitas have puffed up. Slide on to wire racks as soon as you remove from the oven. They should be soft and white with a pouch inside.
Variation
pideh
The Armenians like to sprinkle sesame seeds over this bread. The bread is prepared as above, except that wholemeal flour is often used, and before being placed in the oven the tops of the rounds are scored with a knife to form a diamond design and then brushed with milk and sprinkled evenly with sesame seeds.
khubz-el-saluf fenugreek and coriander bread
In the Yemen, as well as the adjacent regions of Muscat and Oman, a spicy bread is prepared which is topped with a paste of fenugreek called hulba which gives the loaves a beautiful aroma.
You can prepare the loaves with plain flour or with half plain and half wholemeal flour—I prefer the latter.
Serve with meat, fish and kebab dishes.
15 g/½ oz fresh yeast or 7 g/¼ oz dried yeast
about 350 ml/12 fl oz warm water
450 g/1 lb plain flour or 225 g/8 oz plain and 225 g/8 oz wholemeal flour
½ teaspoon salt
a little melted ghee
Topping
2 teaspoons fenugreek seeds—soaked in 100 ml/4 fl oz water in a small bowl overnight
1 clove garlic
50 g/2 oz chopped coriander leaves
½ teaspoon salt
1 tablespoon lemon juice
2 tablespoons water
Place the yeast in a small bowl and dissolve it in a few tablespoons of the warm water. Set aside in a warm place until it begins to froth.
Sift the flour and salt into a large mixing bowl and make a well in the centre. Pour the yeast mixture into the bowl and gradually add enough of the water to make a firm but not hard dough. Transfer to a lightly floured working surface and knead for about 10 minutes until the dough is smooth and elastic. Cover and put in a warm place for at least 2 hours or until the dough has doubled in size.
Meanwhile, prepare the topping by draining off the water from the bowl of fenugreek soaked overnight. Transfer the fenugreek to a liquidizer, add the garlic, coriander leaves, salt, lemon juice and water and blend to a paste. Turn into a small bowl, cover and refrigerate until needed.
Preheat the oven to 250°C, 500°F, gas mark 10 and half-way through the heating period slide 2 greased baking sheets into the oven.
Punch the dough down and knead for a few more minutes. Divide the mixture into 8–10 portions and roll between your palms until smooth. Roll or press each ball out to a circle about 18–20 cm/7–8 in in diameter and prick the surface of each in several places with a fork to prevent the dough puffing up while cooking. Brush the top of each loaf with a little melted ghee. Carefully spread a teaspoonful of the topping over each loaf. Slide the loaves carefully on to the hot sheets and bake for 5 minutes. Serve warm.
zeytin ekmegi olive bread
While olives produce forgetfulness of what one has learned, olive oil makes a clean head.—Midrash Tehillim.
A regional speciality from Antakya (historic Antioch, once one of the glories of Roman and later Byzantine empires, today a typical sleepy Turkish town). And although I have given it its Turkish name this bread, made of olives and onions, is a particular speciality of Greeks and Armenians belonging to the Eastern Orthodox church and was traditionally prepared during the forty days of Lent to give extra interest to their limited fare (no meat, eggs or dairy produce were permitted during that time).
Outside Cyprus, where it is well known and popular as elioti, and amongst Armenians, who call it tsit-hats, this bread is also eaten by the members of the Allaoui sect in Northern Syria who call it khubz-el-zeytoun.
A wonderful and extremely tasty bread.
Dough
15 g/½ oz fresh yeast or 7 g/¼ oz dried yeast
1 teaspoon sugar
about 300 ml/½ pint tepid water
450 g/1 lb plain flour
½ teaspoon salt
Olive filling
1 tablespoon olive oil
1 medium onion, finely chopped
15–20 olives, halved and stoned
oil for glazing
Prepare the dough as described in khubz Arabi . Divide the dough into 2 equal parts and roll between palms to form smooth balls. Roll each ball out into a rectangle about 20 × 40 cm/8 × 16 in and 0.5 cm/¼ in thick. Cover with damp cloth while you prepare the filling.
Heat the oil in a small saucepan, add the onion and fry until soft and transparent. Stir in the halved olives and set aside to cool. Then spread half of the olive mixture over one of the rectangles, leaving the edges clear, and then roll up from one of the longer sides to form a loaf shape. Seal the edges and place on a greased baking sheet with the join underneath. Repeat with the remaining rectangle of dough and filling.
With a sharp knife make 3–4 diagonal slashes across the top of each or make the traditional pattern of the cross. Cover with a cloth and set aside in a warm place for 30 minutes. Brush tops with oil and bake in an oven preheated to 190°C, 375°F, gas mark 5 for about 40 minutes or until golden and baked through. Serve warm or cool on a wire rack.
challah Sabbath bread
Three things are good in little measure and evil in large—yeast, salt and hesitation.—Jewish wisdom.
This is the Jewish bread and indeed it is more than a mere bread since it is intertwined with so much of their religion, culture and history.
However, the finest challahs that I have tasted have all been cooked by Arab bakers—there must be a moral there somewhere!
The recipe below makes a large plait about 30 cm/12 in long.
15 g/½ oz fresh yeast or 7 g/¼ oz dried yeast
1 tablespoon sugar
600 ml/1 pint lukewarm water
700 g/1½ lb plain flour
2 teaspoons salt
3 tablespoons oil
1 egg
1 beaten egg yolk
Garnish 2 tablespoons poppy seeds
Place the yeast and sugar in a small bowl, dissolve in a few tablespoons of the warm water and set aside in a warm place for about 10 minutes or until the mixture begins to froth.
Sift the flour and salt into a large bowl and make a well in the centre. Add the oil, egg, yeast mixture and enough of the warm water to make a stiff dough. Knead the dough on a floured surface for about 10 minutes until smooth and elastic. Cover and leave in a warm place for about 2 hours until it has doubled in size. Transfer to the work top, punch down and knead for a few minutes. Divide the dough into 3 parts—1 large, 1 medium and 1 small.
Take the large portion of dough, divide into 3 equal parts and roll each into a sausage about 36 cm/14 in long and plait the 3 together. Repeat with the medium lump of dough and then with the small one. Lay the largest plait on a greased baking sheet, press the medium plait on to it and the small plait on top of that. Brush the whole surface of the loaf with the beaten egg and sprinkle with the poppy seeds.
Bake in the centre of an oven preheated to 180°C, 350°F, gas mark 4 for about 1 hour or until it sounds hollow when knocked on the bottom. Cool on a wire rack.
khubz basali onion bread
Onion bread is a North Syrian speciality from the region of Antakya (Antioch), the Taurus mountains and the plain of Cilicia (Southern Turkey).
The bread of the poor, it was often eaten as a complete meal or, with the city-dwelling merchant classes, as part of their breakfast table with eggs, laban and cheese.
The recipe below is for 1 large loaf, but if you prefer you can make the dough into smaller loaves about the size of khubz Arabi.
450 g/1 lb self-raising flour
1 teaspoon salt
2 teaspoons baking powder
about 300 ml/½ pint water
about 20 black olives, stoned and coarsely chopped
1 medium onion, finely chopped
1 teaspoon cumin
1 teaspoon thyme
½ teaspoon chilli pepper
25–50 g/1–2 oz self-raising flour
8 tablespoons oil
Sift the flour, salt and baking powder into a large bowl and make a well in the centre. Gradually add enough water to make a firm but not hard dough. Add all the remaining ingredients except the oil.
Lightly flour a work surface and knead the dough for about 10 minutes until it is smooth and elastic. You will probably find that the addition of the onions and olives make the dough a little sticky and I suggest you gradually knead in a little extra flour. Knead in the oil. This will not only help to give the dough its particular flavour, but will also help to soften it.
Press the dough out into an oblong about 30 cm/12 in long and 10–13 cm/4–5 in wide and about 1.2–1.8 cm/½–¾ in thick and lay on a greased baking sheet. Alternatively divide the dough into 6–8 portions and press each out into a small round about 1.2–1.8 cm/½–¾ in thick and arrange on greased baking sheets.
Bake in an oven preheated to 180°C, 350°F, gas mark 4 for 45–60 minutes or until the crust is golden. Remove, cool on a wire rack and serve.
## torshi–pickles
Although today there are a few torshi sellers with donkeys found in the streets of the Middle Eastern towns, torshi is still sold in abundance in small, specialist shops often tucked into a corner of the local bazaar. These are fascinating establishments full of colour and flavour where—and here I vividly recall certain instances from my childhood—customers come to taste and compare before making a purchase. Large barrels are filled with different pickled vegetables, e.g. turnips, carrots, aubergines, peppers, small or large cucumbers, etc.
One day my maternal uncle took me with him to the shop of a friend of his in the Dora district of Beirut. After the initial greetings the shop owner and my uncle entered into the second phase—that of bargaining.
'How much are the cucumbers?'
'For you, 50 piasters.'
'Why for me?'
'You are a friend.'
'Cut that out. How much?'
'Alright then 45.'
'25.'
'No way.'
'Come on man, what do you take me for?'
'A good friend.'
'That's what you charge a good friend!'
'Be reasonable. It cost me 40 piasters to produce the stuff. I only use the tenderest, smallest cucumbers which, as you well know I personally select and, as you damn well know, I go all the way up the Bakka valley...
'Give me a handkerchief, I can't control myself.'
'Be reasonable. I have to make a living, what with four growing children and a fifth (blessed be the Lord) on the way, my in-laws, her brother, cousins and me the only one with honourable work, and you...
'We all have our problems, that is no excuse. 35 piasters is too high.'
'I didn't say 35. I said 45, are you deaf or something?'
'45? I thought you said 35, and I was upset. So it's 45 eh!'
'You don't have to buy you know.'
'You are damn right. Your's isn't the only shop.'
My uncle grabbed my hand and we tried to leave.
'Now hold on, what's the matter, we are old friends. Now come back!'
We returned.
'Taste this, feel the flavour, look how delicious it is. Here little one, you try it.'
I did.
'You like it?' he asked me.
I nodded my head.
'You see, even a child knows how delicious my torshi is.'
'What do you expect from children.'
'Be reasonable friend. Life is getting more and more expensive each day. A man has to make a living. Here, I'll throw in a bottle of rosewater.'
'25.'
'40.'
'30.'
'35.'
'Done.'
'Good!'
'Give me 10 kilos and 5 peppers and 5 cauliflowers.'
'Fine, fine. It's always good to do business with a friend who appreciates quality and flavour.'
We left laden with jars of pickles, hailed a horse-drawn cart and went home. My uncle content with himself, his friend content with his lot and I, for my part, happy to be licking at an ice-cream. Overall it had been a successful day. The ritual of buying and selling had been performed to perfection.
I have selected a few pickle recipes that are typical of the region. Most are common throughout the area, but a few are regional favourites. Everything that can be is pickled and what was once a method of preserving perishable goods has, in time, become a way of life.
betingan makbouss aubergines in olive oil
A Syrian-Lebanese speciality, the aubergines are pickled in olive oil and served as a mezzeh, cut lengthways.
Lemons, limes, small cucumbers and small marrows can be pickled in the same way. Keep for 3 weeks before serving.
12 or more small aubergines
5–6 tablespoons walnuts, coarsely crushed
2 teaspoons salt
2 tablespoons paprika
4 lemons, thickly sliced
olive oil or corn oil
Half-fill a large saucepan with water, bring to the boil, add the aubergines and simmer for 10 minutes. Drain and dry with a kitchen towel. With a knife make an incision lengthways into the middle of each aubergine, big enough to take a slice of lemon and a teaspoon of walnuts.
Mix the walnuts, salt and paprika together. Place a slice of lemon in each slit and add a teaspoon of the walnut mixture. Arrange the aubergines in large sterilized jars and completely cover with olive oil. Seal the jars and leave for at least 3 weeks. By this time the aubergines should be soft and fragrant.
Variation
lemoun makbouss lemons in olive oil
Another Syrian-Lebanese speciality. This is for pickled lemons in oil.
Use 20 lemons, scraped and sliced
Sprinkle with 6 teaspoons salt and leave to rest in a colander overnight. Arrange the slices in sterilized jars, sprinkling each layer with a little paprika—use 2–3 teaspoons paprika altogether. Completely cover with oil, seal tightly and leave for 3–4 weeks before serving.
titvash mixed pickles
This is probably the most popular form of pickling throughout the Middle East. There are numerous regional variations, especially in the choice of vegetables, but basically the ones listed below are those most commonly used. I suggest you suit your own taste. It is advisable to make a fairly large quantity of this pickle. The recipe below makes about 4.5 litres/1 gallon, but you can increase or decrease the amounts accordingly.
2 small cauliflowers, separated into florets
8 carrots, peeled, quartered lengthways and cut into 8 cm/3 in pieces
8 small cucumbers, 5–8 cm/2–3 in long
225 g/8 oz green beans, trimmed
6 sweet yellow peppers, quartered and seeded
6 small hot red peppers
6 cloves garlic, peeled and halved
6 fresh dill sprigs
1.8 litres/3 pints water
600 ml/1 pint white wine vinegar
100 g/4 oz salt
Wash the vegetables thoroughly. Pack them into large sterilized jars portioning out the garlic and dill sprigs evenly.
Place the water, vinegar and salt in a large saucepan and bring to the boil. Pour the vinegar mixture over the vegetables until completely covered. Seal tightly and store in a cool place for 6–8 weeks. Serve as a salad with all kinds of meat and vegetable dishes.
mukhalal luft pickled turnips
These are very popular with everyone, particularly Iraqis and Iranians. In this recipe the turnips acquire a beautiful pinky-red colour. They are excellent with all types of kebabs.
1.8 kg/4 lb small white turnips
2 beetroot
leaves from the top of 4–6 celery sticks
2 cloves garlic, or more depending on the number of jars used
1.8 litres/3 pints water
600 ml/1 pint white wine vinegar
8 tablespoons salt
Peel, wash and then quarter the turnips. Peel and slice the beetroot. Pack the quartered turnips into sterilized jars, arranging beetroot slices between the layers. Add some celery leaves and a clove of garlic to each jar. Place the water, vinegar and salt in a large bowl and mix well. Pour this mixture into the jars until the vegetables are completely covered. Seal the jars tightly and store for at least 4 weeks.
torshi-ye hafte-bijar pickled herbs
A unique pickle recipe from Iran where herbs, fresh, dried or pickled, are used extensively. The following list of herbs is merely a suggestion. All kinds of herbs are suitable as long as they are fresh.
Use equal amounts of:
leeks
tarragon
spinach
mint
parsley
celery
basil
beetroot leaves
dillweed
fenugreek
Pickling mixture:
coarse salt
fresh red peppers
peppercorns
dried oregano
dried marjoram
cloves garlic, peeled
black pepper
vinegar
Wash the herbs and leaves and spread out on kitchen paper to dry. When completely dry chop finely. Mix thoroughly in a large bowl and then fill sterilized jars with the herbs. Sprinkle with some salt. To each jar add half a red pepper, a few peppercorns, some oregano and marjoram, 2 or 3 cloves garlic and a little black pepper. Fill the jars to the top with vinegar and seal tightly. Store for at least 2 weeks.
torshi-ye-miveh fruit pickles
The bear has twelve dreams—and they are all about bears.—Armenian saying.
Particularly attractive is the concept of making pickles from fruits such as peaches, cherries, grapes, orange rind, dates, etc. They are kept in wine vinegar and spices, sealed in jars for at least a fortnight and then served with meat and vegetable dishes.
The masters of this kind of pickling are the Iranians and Caucasians. The method is simple and I have included several recipes under this heading as I think that with the excellent fruits now available in the West good pickles can be prepared.
torshi-ye-barg-e holoo pickled peaches
An Iranian recipe.
900 g/2 lb dried peaches
900 ml/1½ pints white wine vinegar
175 g/6 oz sugar
1 teaspoon ground ginger
1 teaspoon ground coriander
2 cloves garlic, finely chopped
1 tablespoon tamarind
1 teaspoon dry mustard
1 teaspoon paprika
½ teaspoon cinnamon
Place the dried peaches in a large bowl, add the vinegar and leave to soak for 2 days. Transfer to a saucepan, add the remaining ingredients and bring to the boil. Lower the heat and simmer for about 1 hour. Pour the mixture into sterilized jars making sure you distribute the peaches evenly. Seal the jars tightly and leave for at least 10 days.
torshi-ye-gilas pickled cherries
Simple and delicious, this is another masterly recipe from Iran.
450 ml/¾ pint white wine vinegar
4 tablespoons salt
900 g/2 lb cherries, discard the stems and over-ripe and discoloured ones
4 sprigs tarragon
10 peppercorns
Place the vinegar and salt in a large saucepan and bring to the boil. Simmer for 3 minutes and then remove from the heat and leave to cool. Wash the cherries and place in sterilized jars. Pour the vinegar mixture into the jars until the cherries are completely covered. Distribute the tarragon leaves and peppercorns between the jars, seal tightly and leave for 3–4 days. Pour the vinegar mixture out of the jars. Make up a new mixture and when it is cool pour it over the cherries. Seal tightly and leave for at least 2 weeks.
khaghoghi titvash pickled grapes
This recipe is from Armenia.
1.8 kg/4 lb firm white grapes, stemmed, washed and with overripe ones discarded
1.2 litres/2 pints white wine vinegar
1 tablespoon salt
Dry the clean grapes on kitchen paper.
Put the vinegar and salt into a saucepan, bring to the boil and simmer for 3–4 minutes. Remove from the heat and allow to cool. Pack the grapes into sterilized jars. Pour the vinegar into the jars until the grapes are completely covered. Seal the jars tightly and leave for at least 10 days.
torshi-ye-khorma pickled dates
The greatest of all pickles, this is absolutely delightful and can be served with all meats. The pride of Iranian and Iraqi picklers.
Both sumac and tamarind can be bought from most continental and Middle Eastern stores.
75 g/3 oz sumac
225 g/8 oz dried tamarind
900 ml/1½ pints water
juice 1 lemon
450 g/1 lb dates, stoned
2 cloves garlic, crushed
½ teaspoon salt
¼ teaspoon black pepper
¼ teaspoon cinnamon
¼ teaspoon nutmeg
Soak the sumac in 450 ml/¾ pint of water in a bowl overnight. Do the same with the tamarind in another bowl. Now strain both the sumac and the tamarind through muslin and reserve the liquid. Place the liquids in a saucepan with the lemon juice and boil for 3 minutes.
Mince or finely chop the dates and add to the boiling liquids. Add the remaining ingredients, stir well and then pour the mixture into sterilized jars and seal tightly. Use after 1 week.
## desserts
To be an Arab and not have a sweet tooth is to be a Muslim and not believe in Paradise.
Most Middle Eastern desserts are very sweet—literally soaked in honey or syrup or a combination of the two. They are made and served in abundance and any occasion (birth, christening, circumcision, wedding—a hot favourite—religious festival, pilgrimage and even a funeral) is a good enough excuse for the housewives to indulge in an orgy of sweet making.
These desserts are exciting and original and most are very different from those found in the West. Their origins are older than most of the races of the Middle East today. The Ancients—Sumerians and Assyrians in particular as well as the Hittites—are known to have had a great penchant for all things sweet.
Although most of these desserts are found—in one form or another—throughout the entire region, there are some which are still local in character. A few of these I have included besides the more famed ones—the latter are the pastry-based sweets, e.g. baklava, kadayifi, etc., which have, in recent years, made their appearance in the West.
Apart from the pastry desserts I have included several halvas—semolina-based sweets, several fritters—the forerunners of the doughnut, fruit-and-nut-based desserts, milk-and-rice-based puddings as well as many dry biscuits and cakes.
Many of these are easy and inexpensive to make while others take more time and require more expensive and sometimes not such readily available ingredients, but you will find that these desserts are well worth the time and trouble.
pastries
baklava flaky pastry with nuts and syrup
This is perhaps the most famous Middle Eastern 'pastry' dessert, whose ownership is claimed by all—Arabs, Turks and Greeks. However, it is not mentioned in any medieval Arab manuscript and it only arrived in the Ottoman court in the late fifteenth to early sixteenth centuries after the conquests of Cilicia and Cappadocia by the Turks.
Baklava, in fact, is an Armenian sweet (Bahk meaning Lent, halva from ancient Akkadian meaning sweet). Traditionally baklava (bahlawah in Arabic) consists of forty layers of flaky pastry—one for each day of fasting—filled with almonds and walnuts and soaked in syrup. It was eaten on Easter Sunday after mass.
There are several variations of this magnificent sweet, a few of which I have included. It is easiest to use ready-made filo, but if you wish to be really authentic you can make your own (see Glossary).
1 packet (450 g/1 lb) filo pastry
225 g/8 oz unsalted butter, melted and with froth removed
225 g/8 oz walnuts, chopped or coarsely ground
Syrup
350 g/12 oz sugar
1 tablespoon lemon juice
350 ml/12 fl oz water
2 tablespoons rosewater
First prepare the syrup by placing the sugar, lemon juice and water in a saucepan and bringing to the boil. Lower the heat and simmer for about 10 minutes or until the syrup leaves a slightly sticky film on a spoon. Add the rosewater and set aside to cool.
Most packets of pastry have sheets 50×28 cm/21×11 in, but it is not easy to find a tin with these dimensions. I use one 30×20 cm/12×8 in and trim the sheets to make them fit. As I am loathe to waste good food I simply slip the trimmings between the sheets in such a way as to maintain an even thickness. The one important thing is that the tin must be at least 2.5 cm/1 in deep.
Grease the baking tin with a little melted butter. Lay 2 sheets of the pastry on top of each other in the tray and then dribble a tablespoon of the melted butter over the second sheet. Repeat in this way until you have 6 or 8 sheets in the tray. While you are layering the sheets try to press on them as little as possible. This ensures that air is trapped between the layers and so enables the sweet to rise. Spread half of the crushed nuts over the last sheet of pastry.
Continue with the layers of pastry and spoonfuls of butter until you have laid down a further 6 or 8 sheets. Spread the remaining nuts over the last sheet.
Continue layering the pastry with spoonfuls of melted butter dribbled over alternate sheets until you have used up all the pastry. Spoon any remaining butter over the last sheet, discarding the milky residue at the bottom of the pan. Lightly brush this butter all over the last sheet so that every bit of pastry is covered.
Cut the baklava into lozenge shapes using a sharp knife and taking care to press as little as possible on the actual baklava. Place the tin in an oven preheated to 180°C, 350°F, gas mark 4 and cook for 30 minutes.
Lower the temperature to 150°C, 300°F, gas mark 2 and cook for a further hour or until the pastry is turning a pale golden.
Set the baklava aside until it is warm and then pour the cold syrup all along the gaps. Set aside until completely cold. To serve first run a sharp knife along the gaps to make sure that all the layers have been completely cut through.
The quantities given here will make 24–30 pieces.
Variations
Below are several suggestions for alternative fillings. In each case prepare the baklava as above and simply substitute the filling of your choice.
almond filling
225 g/8 oz almonds, chopped, blanched
4–5 tablespoons caster sugar
1 teaspoon ground cinnamon
Coconut filling A Turkish favourite.
225 g/8 oz dessicated coconut
6–7 tablespoons caster sugar
2 teaspoons vanilla essence
2–3 tablespoons water
fruit filling Another popular Turkish-Armenian filling. You can use many fruits such as apples, cherries, oranges, pumpkin, etc.
350 g/12 oz apples, peeled and grated
225 g/8 oz caster sugar
1 teaspoon ground cinnamon
Mix the grated apples and sugar together, put them in a muslin bag and squeeze out as much juice as possible.
Empty the mixture into a bowl, stir in the cinnamon and proceed with the recipe.
Pumpkin and walnut filling
225 g/8 oz pumpkin, peeled and grated
225 g/8 oz caster sugar
100 g/4 oz walnuts, chopped
1 tablespoon rosewater
Mix together and proceed as with the baklava recipe.
souarzeh 'bird's nest' pastries
A light, delicate pastry from the Aleppo region of Northern Syria—also popular in Southern Turkey where it is known as Anteb bulbulu—Anteb nightingale.
10 sheets filo pastry
225 g/8 oz unsalted butter, melted
Syrup
450 g/1 lb sugar
450 ml/¾ pint water
1 tablespoon lemon juice
Garnish
6–7 tablespoons pistachio nuts, very finely chopped
Lay the sheets of pastry out flat, on top of each other, on a work top. Each sheet is approximately 50×28 cm/21×11 in. Mark the top one into 6 portions each about 18×13 cm/7 × 5½ in and then cut down through all 10 sheets. Stack the 60 pieces of pastry on top of each other and cover with a damp cloth to prevent them drying out.
Remove 1 piece of pastry and brush the top all over with a little melted butter. Roll up the pastry as you would a cigarette so that you have a roll 13 cm/5½ in long. Carefully bend the roll into a circle and squeeze the two ends of the pastry together. They will stick easily if you dampen your fingers first. Repeat with all the remaining pieces of pastry.
Arrange on lightly greased baking trays about 1.2 cm/½ in apart and brush the outer surfaces of the circles with butter. Place in an oven preheated to 160°C, 325°F, gas mark 3 and bake for 20–25 minutes or until they are just turning a light golden colour. While they are cooking prepare the syrup by placing the sugar, water and lemon juice in a saucepan and bringing to the boil. Boil quite vigorously for about 5 minutes and then remove from the heat. When the souarzeh are cooked place them in a shallow dish, pour the boiling syrup over them and leave for 2 hours to cool. Lift from the syrup and arrange the pastries on a large serving plate. Dust with the finely chopped pistachios.
Makes 60 pastries.
madig almond fingers
Another popular way of using filo pastry is to fill the sheets with nuts and then roll up like cigars. There are countless variations of this sweet. Arabs call it 'Zeinab's fingers' and the Turks vezir parmagi.
These are simple to prepare and will keep for several days. They make an ideal after dinner dessert.
100 g/4 oz unsalted butter
1 packet (450 g/1 lb) filo pastry
Filling
225 g/8 oz almonds, coarsely ground
2 teaspoons cinnamon
2 teaspoons sugar
Syrup
450 g/1 lb sugar
350 ml/12 fl oz water
juice 1 lemon
2 tablespoons rosewater
First prepare the syrup by placing the sugar, water and lemon juice in a saucepan and bringing to the boil. Lower the heat and simmer until the syrup leaves a sticky film on a spoon. Add the rosewater and set aside to cool.
Prepare the filling by mixing together the almonds, cinnamon and sugar.
Melt the butter in a small pan over a low heat. Brush a baking sheet with a little of the melted butter.
Open out the filo pastry and cut along the fold so that each sheet is divided into 2 rectangles. While you are using each piece of pastry keep the others covered as they dry out quickly. Lay a rectangle of pastry on a work top, short side nearest you, and brush the 2 long edges with butter. Arrange a teaspoon of the almond mixture in a ridge across the short edge nearest you. Fold the 2 long edges inwards over the edges of the almond mixture and then roll up to form a cigar shape.
Place each roll on the baking sheet, opening underneath. Brush them all with any remaining melted butter. Cook in an oven preheated to 190°C, 375°F, gas mark 5 for 20–30 minutes or until golden. Dip the hot rolls into the cool syrup and then arrange them on a serving dish.
Makes about 40 pastries.
kunafeh shredded pastry with nuts
Kunafeh swimming in butter,
Bearded with right vermicelli,
God has not given my belly
Half of the words it would utter
Of kunafeh's sweetness
And syrup's sweetness.
Kunafeh lies on the table
Isled in a sweet brown oil,
Would I not wonder and toil
Seventy years to be able
To eat in Paradise
Kunafeh's subtleties?
(The Book of 1001 Nights)
This sweet is known as kadayif to Greeks and Turks, but this is a misnomer as ataif is a pancake and very Persian in origin.
I remember my grandfather preparing these shredded pastries on Christmas and New Year's Eve. He would make a batter from flour and water and pass it through a sieve on to a hot metal sheet heated by charcoal. The dough would set in seconds and we children would sweep it to one side straight into an earthenware bowl.
Kunafeh pastry looks like vermicelli or 'shredded wheat' and it is sold in 450 g/1 lb bags under the name of kadayifi filo. There are many variations of this dessert. The first one below is a family favourite—indeed it is the very same my grandfather used to prepare.
1 packet kunafeh or kadayifi pastry—usually 450 g/1 lb
350 g/12 oz unsalted butter, melted and with froth removed
Filling
175 g/6 oz walnuts, chopped or coarsely ground
2 tablespoons sugar
2 tablespoons cinnamon
Syrup
350 g/12 oz sugar
350 ml/12 fl oz water
juice 1 lemon
2 tablespoons rosewater
First prepare the syrup by placing the sugar, water and lemon juice into a saucepan and bringing to the boil. Lower the heat and simmer until the syrup begins to leave a film on the back of a spoon. Add the rosewater and set aside to cool.
Lightly brush a baking tin, about 30 × 23 cm/12 × 9 in or about 25 cm/10 in in diameter and at least 2.5 cm/1 in deep, with a little of the melted butter. Put the pastry into a large bowl and gently ease apart the strands without breaking them. Remove any hard nodules of pastry which you may find in some brands. Pour three-quarters of the melted butter into the bowl and gently rub all the strands between your fingers until they are all well-coated with the butter. Divide the pastry into 2 equal parts and spread one part evenly over the base of the tin. Mix the filling ingredients together and spread evenly over the pastry, pressing down firmly. Arrange the remaining pastry evenly over the top, tuck in any strands hanging over the sides and press the pastry down firmly. Spoon the remaining melted butter evenly over the top, discarding the white residue in the bottom of the pan. Place in an oven preheated to 180°C, 350°F, gas mark 4 and cook for 30 minutes. Lower the heat to 150°C, 300°F, gas mark 2 and cook for a further 1½ hours or until golden.
Remove from the oven and pour the syrup slowly over the kunafeh, covering as much of the surface as possible. Cover with silver foil, place a large board over the top and add a heavy weight in order to flatten the kunafeh. Leave to cool and then cut into squares or lozenges 3.8–5 cm/1½–2 in in size.
Makes 24–30 pieces.
Variations
kaymakli tel-kadayif
This is a popular Turkish version where the shredded pastry has a 'cream' filling made of milk and semolina. This is a light and delicious sweet.
1 packet (450 g/1 lb) kunafeh pastry
225 g/8 oz unsalted butter, melted and with froth removed
Filling
450 ml/¾ pint milk
25 g/1 oz fine semolina or 25 g/1 oz ground rice
1 tablespoon rosewater
Syrup
350 g/12 oz sugar
350 ml/12 fl oz water
juice 1 lemon
1 tablespoon rosewater
Prepare the syrup as for kunafeh above. Prepare the filling by first bringing the milk to the boil in a saucepan. Add the semolina and rosewater and simmer for 5 minutes, stirring constantly, until the mixture thickens. Remove from the heat and set aside to cool.
Put the pastry into a large bowl and gently ease apart the strands without breaking them. If you squeeze portions between the palms of your hands it will make this easier. Pour the melted butter into the bowl, discarding the milky residue at the bottom of the pan. Gently run all the strands between your fingers until they are all well coated with the butter. Lightly grease a baking tin about 30 × 23 cm/12 × 9 in or about 25 cm/10 in in diameter and at least 2.5 cm/1 in deep. Spread half the pastry over the bottom of the tin.
Pour the filling over the pastry and spread it out evenly with the back of a spoon. Arrange the remaining pastry evenly over the filling. Press down lightly and tuck in any loose strands. Place in an oven preheated to 180°C, 350°F, gas mark 4 and cook for 45 minutes or 1 hour until golden.
Remove and immediately pour the cold syrup evenly over the top. Cover with silver foil, place a board over the sweet and top with a heavy weight in order to compress the sweet. Set aside to cool and then cut into squares or lozenge shapes.
kunafeh-bi-jibn
A speciality of Damascus where the shredded pastry is filled with a soft and unsalted cheese. You can either pour the syrup over the kunafeh when it is removed from the oven or serve it separately.
1 packet (450 g/1 lb) kunafeh pastry
225 g/8 oz unsalted butter, melted and with the froth removed
Filling
350 g/12 oz soft, unsalted cheese, e.g. ricotta, akkawe or mizithra, grated
1 tablespoon sugar
grated rind 1 lemon
1 tablespoon rosewater
Syrup
350 g/12 oz sugar
350 ml/12 fl oz water
juice 1 lemon
2 tablespoons rosewater
Prepare the syrup as with kunafeh , and set aside to cool.
To prepare the filling place the grated cheese in a bowl with the sugar, lemon rind and rosewater and beat with a wooden spoon until it is well blended and soft.
Prepare the pastry and baking tin as described in the second paragraph of instructions in kaymakli tel-kadayif . Spread the cheese mixture evenly over the pastry. Top evenly with the remaining pastry, press down lightly and tuck in any loose strands. Place in an oven preheated to 180°C, 350°F, gas mark 4 and cook for 45 minutes to 1 hour or until golden.
Remove and pour the cold syrup immediately over the kunafeh or set aside to cool without the syrup. Cover with foil, place a board over the top of the sweet and top with a heavy weight in order to compress the sweet. When cold cut into squares or lozenge shapes and serve with the syrup if you haven't already soaked the sweet in it.
balurieh white kunafeh
Also known as the 'Queen of Kunafehs', baluriehs are a Syrian speciality, milk-white in appearance, tightly packed and stuffed with pistachios.
Keep the oven door slightly ajar to achieve the whitish appearance. Like all baklava or kunafeh pastries these balurieh will keep for several days.
This is my favourite pastry. Serve with cream.
450 g/1 lb kunafeh filo
150 g/5 oz unsalted butter, melted and skimmed
1 tablespoon clear honey
Filling
175 g/6 oz pistachio nuts, coarsely chopped
40 g/1½ oz caster sugar
½ tablespoon cinnamon
Syrup
450 g/1 lb sugar
450 ml/¾ pint water
1 tablespoon lemon juice
Garnish 2 tablespoons pistachio nuts, very finely chopped
Pour the melted butter into a bowl, discarding the milky substance in the bottom of the pan, and place the bowl in the refrigerator until the butter is semi-solid. Remove from the fridge, add the honey and whisk until the mixture begins to foam. Pour the mixture into a baking tray about 30 × 20 cm/12 × 8 in and at least 2.5 cm/1 in deep.
Open up the packet of pastry and lay it out on a clean work top. In order to loosen the strands I suggest that you divide the pastry into 4 portions and squeeze each portion between the palms of your hands—as though you are making a snowball—for about 2 minutes. Take 2 sections of the pastry and, without breaking the strands, gently ease the pastry out spreading it over the bottom of the tray.
Mix the filling ingredients together in a bowl. Spread this filling evenly over the pastry in the tray. Gently ease apart the 2 remaining portions of pastry and arrange them evenly over the filling. Press the pastry down firmly and tuck in any strands of pastry hanging over the edge of the tray. Place in an oven preheated to 150°C, 300°F, gas mark 2 and bake for 20 minutes. Keep the door ajar—this will prevent the balurieh from changing colour.
Meanwhile, prepare the syrup by placing the sugar, water and lemon juice in a saucepan and bringing to the boil quite vigorously for 5 minutes and then remove from the heat.
After the balurieh has been cooking for 20 minutes take it from the oven. Very carefully lift the tray to an angle and pour the butter and honey mixture into a bowl.
Now completely cover the sweet with another flat surface such as a kitchen board or the back of another tray and turn the balurieh over on to this surface. Very gently and carefully slide the sweet—now bottom side up—back into the tin. Return to the oven and, still keeping the door open, cook for a further 20 minutes. Remove from the oven. Pour the boiling syrup evenly over the surface of the sweet. In order to get the tight and compact appearance of the sweet put an empty tray over it and hold it down with some kind of heavy weight while the sweet cools.
When cold cut into 5 cm/1½ in squares. Sprinkle some of the very finely chopped pistachio nuts over each square and serve.
Makes 24–30 pieces.
kunafeh mabrouma
rolled kunafeh with pistachios
These kunafeh (burma in Turkish) are shaped in whirls and then cut into 5–7 cm/2–3 in pieces. A fairly dry, but rich sweet with the green pistachios contrasting colourfully with the golden brown of the pastry.
1 packet (450 g/1 lb) kunafeh or kadayifi pastry
350 g/12 oz unsalted butter, melted and with froth removed
Filling
350 g/12 oz whole pistachio nuts, shelled
75 g/3 oz almonds, finely chopped
3 tablespoons sugar
Syrup
350 g/12 oz sugar
350 ml/12 fl oz water
juice 1 lemon
1 tablespoon orange blossom water
Prepare the syrup by placing the sugar, water and lemon juice in a saucepan and bringing to the boil. Lower the heat and simmer until the syrup begins to leave a sticky film on a spoon. Remove from the heat, stir in the orange blossom water and set aside.
Put the pastry into a large bowl and gently ease apart the strands without breaking them. Divide the pastry into 3 portions. Take one of the portions and lay it out flat on a clean work top. Flatten it as much as possible with your hands until it is about 0.6–1.2 cm/¼–½ in thick and then shape it into an oblong approximately 30 × 15 cm/12 × 6 in. With a pastry brush, brush the surface with some of the melted butter.
Take a flat stick about 45 cm/18 in long and approximately 2.5 cm/1 in wide and lay it diagonally across the flattened pastry. Mix the filling ingredients together and then lay a third of the mixture evenly along the stick. Roll the strands of dough around the stick as tightly as possible. Carefully slide the stick out leaving the filling inside. Brush some melted butter all over the roll of pastry. Prepare the other portions of pastry in the same way.
Lightly butter a baking tray about 30 × 20 cm/12 × 8 in. Arrange the 3 pastry rolls in the tray and pour any remaining butter evenly over the rolls, taking care to discard the milky residue. Cook in an oven preheated to 180°C, 350°F, gas mark 4 for 30 minutes, then lower the temperature to 150°C, 300°F, gas mark 2 and cook for a further 1½ hours until the kunafeh is golden. Remove from the oven and pour the cold syrup evenly over the rolls, turning each one so that it is covered all over with the syrup. Leave to cool and then cut, at a slant, each roll into 5–7.5 cm/2–3 in long pieces.
Makes 12–15 pieces.
halawiyat semolina sweets
This section comprises sweets made with ground rice, semolina and cous-cous—each with its own flavourings, textures and colours.
These cream puddings are popular throughout the region, but more so with the Arabic speaking people of Egypt, Syria, Iraq, Lebanon and Palestine.
One of the most popular halawahs is called basbousa in Egypt and Lebanon. It is a semolina-based sweet to which nuts, spices and essences are added. It is then baked in the oven, cut into squares or lozenge shapes and soaked in syrup.
A typical basbousa is the recipe below made with milk, semolina and coconut.
basbousa-bil-joz el hindi
halva with coconut
50 g/2 oz plain flour
1 teaspoon baking powder
350 g/12 oz semolina
225 g/8 oz caster sugar
50 g/2 oz dessicated coconut
100 g/4 oz unsalted butter, melted
225 ml/8 fl oz milk
1 teaspoon vanilla essence
1½ teaspoons cinnamon
Syrup
225 g/8 oz sugar
150 ml/¼ pint water
1 tablespoon lemon juice
First prepare the syrup by placing the sugar, water and lemon juice in a saucepan and bringing to the boil. Simmer for 6–8 minutes or until the syrup has thickened. Remove from the heat and then cool and refrigerate.
Sieve the flour and baking powder into a large bowl. Add the semolina, sugar and coconut and mix well. Pour in the melted butter, milk and vanilla essence and stir until completely mixed. Stir in 1 teaspoon of the cinnamon. Spoon this mixture into a greased tin about 28 × 18 cm/11 × 7 in so that it is about 1.2 cm/½ in thick. Bake in an oven preheated to 160°C, 325°F, gas mark 3 for about 30–40 minutes until the top is crisp and golden brown. Remove from the oven and sprinkle the remaining cinnamon over the top. Cut into lozenge shapes and quickly pour the cold syrup evenly over the basbousa. Serve hot or cold, with a little double cream if liked.
Makes 20–24 pieces.
imrig khavitz semolina halva
An Armenian halva which is also a personal favourite.
Syrup
200 ml/1/3 pint water
75 g/3 oz sugar
1 tablespoon rosewater
40 g/1½ oz butter
75 g/3 oz fine semolina
1 tablespoon pine kernels
1 tablespoon raisins
1 tablespoon almonds, blanched
1 tablespoon cinnamon
First prepare the syrup by placing the water and sugar in a small saucepan and bringing to the boil. Lower the heat and simmer for about 10 minutes or until the syrup coats the back of a spoon. Remove from the heat, stir in the rosewater and set aside to cool.
Melt the butter in a saucepan, add the semolina and stir well. Cook over a medium heat for several minutes, stirring constantly, until the semolina has slightly browned. Add half the pine kernels, the raisins and almonds and cinnamon, and stir well. Gradually pour in the syrup, stirring constantly. Cook for about 5 more minutes until the syrup is absorbed and the mixture has thickened. Remove from the heat and allow to cool for about 5 minutes. Now scoop a tablespoon of the mixture into the palm of your hand, close your fingers around it, press tightly and then place on a serving plate. Repeat until all the halva is thus arranged.
Sprinkle the remaining cinnamon over the halva and stick one pine kernel into each piece. Serve warm with tea or coffee.
One day Boloz Mugush woke up early, 'Good morning wife,' he said. 'I have this terrible craving for imrig khavitz. Make some for lunch, there's a good wife.'
She made a large quantity filled with raisins, almonds and pine kernels. After work Boloz Mugush returned home, took his shoes off, sat on the verandah and ate—nearly all of it.
That night, when the wife had counted her last sheep, he woke her up.
'What's up?'
'I have just had a thunderously fantastic thought.'
'What is it?'
'First bring me the rest of the khavitz and I will tell you.'
She got up, went to the kitchen and returned with the imrig khavitz. He ate, splurging and licking his fingers and lips.
'Well," said the wife, 'what was your brilliant thought? Come on man, tell me, for I'll never be able to sleep otherwise.'
Boloz Mugush stroked his belly. 'The thought,' he said, 'was this—never go to sleep without having finished all the imrigi khavitz that has been made during that day.'
revani semolina cake
This is a Turkish speciality that is also popular throughout the Balkans. It is made of semolina, eggs and nuts and is soaked in syrup. The blanched almonds can be substituted with equal amounts of chopped walnuts or pistachios or hazelnuts. There are countless variations. Grated orange or lemon rind is often added to give the cake a fruity flavour.
Revani can be served warm, but it is at its best cold with kaymak or whipped or clotted cream.
6 large eggs, separated
225 g/8 oz sugar
225 g/8 oz semolina
1 teaspoon baking powder
2 tablespoons blanched almonds, finely chopped
2 tablespoons brandy
pinch salt
Syrup
350 g/12 oz sugar
600 ml/1 pint water
1 stick cinnamon 7.5–10 cm/3–4 in long
Place the egg yoks in a large bowl, add the sugar and beat until smooth and light in colour. Add the semolina, baking powder and almonds and stir thoroughly. Stir in the brandy.
In another bowl beat the egg whites until stiff. Add the salt and whisk a little longer until the egg whites stand up in peaks. Fold the egg whites gently into the semolina mixture.
Grease a cake tin approx 25 × 25 × 5 cm/10 × 10 × 2 in, add the cake mixture and smooth over. Place in an oven preheated to 180°C, 350°F, gas mark 4 and bake for 30–40 minutes or until golden.
Meanwhile, prepare the syrup by placing the sugar, water and cinnamon stick in a small saucepan and bringing to the boil. Reduce heat, simmer for 15 minutes then remove from heat and set aside. When the cake is cooked remove it from the oven and spoon the syrup over it. Use enough syrup to be easily absorbed, but do not make the cake too soggy. Leave to cool and then refrigerate. Before serving cut into 50 cm/2 in squares. This cake can be eaten by itself or with cream.
Makes 25 pieces.
haytaliah floating scented pudding
'She moves like a balouza and floats in the air like haytaliah.'—Compliment paid to belly dancers whose tummies are likened to the jelly-like movement of the sweet.
One of the most beautiful puddings in the world and a true Arab classic. Haytaliah are small squares of cornflour pudding floating in a scented syrup with raisins, almonds, pistachios and a few rose petals to give an attractive touch of colour.
A must during all Muslim festivals. Huge bowls of this pudding adorn all delicatessens and homes. It is particularly loved by children.
To prepare this sweet you first have to make the pudding which is called balouza. Balouza can be eaten on its own, chilled and garnished with chopped pistachios, but the sweet becomes a classic when served as a haytaliah.
Balouza
100 g/4 oz cornflour
1.8 litres/3 pints water
275 g/10 oz sugar
100 ml/4 fl oz orange blossom water or rosewater
50 g/2 oz pistachios, chopped or almonds, chopped, blanched
Garnish 1 teaspoon cinnamon
Syrup
1.2 litres/2 pints water, cold
100 g/4 oz sugar
6 tablespoons rosewater (or more depending on taste)
100 g/4 oz raisins
2 tablespoons pistachio nuts, halved
seeds 1 small pomegranate
a few rose petals, washed and dried
ice cubes
Place the cornflour in a large saucepan and add about 300 ml/½ pint of the water. Stir in the rest of the water and the sugar and stir over a low heat until the sugar dissolves. Bring to the boil, stirring constantly, then lower the heat to a minimum and simmer for about 5 to 10 minutes or until the mixture thickens and coats the back of a spoon. Stir occasionally to prevent burning. Stir in the orange blossom water and cook for a further 2 minutes. Add the chopped nuts and stir thoroughly. Remove from the heat and leave to cool for 2 minutes. If you are going to serve it by itself then pour into glass bowls and chill for several hours. Before serving sprinkle with a little cinnamon.
If making haytaliah pour the balouza into a square or rectangular baking tray moistened with cold water and even it out with the back of a spoon. The balouza should not be thicker than 2.5 cm/1 in. Cool for 15 minutes and then refrigerate for several hours. Remove from the fridge and then cut into 2.5 cm/1 in squares.
Now prepare the syrup by mixing the water and sugar in a large bowl and stirring until the sugar dissolves. Add the rosewater, raisins, pistachio nuts, pomegranate seeds and stir well. Pour into a large serving dish, add a few rose petals and drop in squares of balouza and some ice cubes. Serves about 10 people.
mamounia aleppan halva
This halva—named after a famed Arab Caliph Al-Mamun (813–833)—is a speciality of Aleppo, Syria.
Mamounia is eaten daily for breakfast, is highly recommended to women in labour and is generally distributed to the poor and the weak. A Syrian old wives tale claimed 'A bowl of mamounia a day will keep malaria, typhoid, consumption, etc. etc. away,'—not forgetting the great bogey Sheytan (the Devil). Not being very superstitious I would like to say that I eat it because I like it!
This is a family recipe. Some people like to pour kaymak or double cream over it, but I prefer it as it comes—warm and plain.
100 g/4 oz unsalted butter
100 g/4 oz semolina
1 teaspoon ground cinnamon
Syrup
600 ml/1 pint water
175 g/6 oz sugar
1 tablespoon lemon juice
First prepare the syrup by placing the water, sugar and lemon juice in a saucepan and bringing to the boil. Lower the heat and simmer for 10 minutes, then remove from the heat and set aside.
Melt the butter in a large saucepan. Add the semolina and fry, stirring constantly, for about 5 minutes or until the mixture becomes crumbly in appearance. Pour in the syrup and mix thoroughly with a wooden spoon. Cook for a further 2–3 minutes. Remove from the heat and set aside for 12–15 minutes. Spoon into a serving bowl, sprinkle with the cinnamon and serve while still warm.
mahlebieh almond cream pudding
The most popular of all Middle Eastern rice puddings, mahlebieh (Arabic meaning 'with milk') should be served chilled. It is often decorated with chopped pistachio nuts, pomegranate seeds and/or a mixture of chopped almonds with honey-based syrup scented with orange blossom water.
There are many variations. A particularly tasty one, which I have included below, is flavoured with mastica (gum mastic) and pistachios. It is called sakiz muhallebisi and is from Turkey. This recipe for mahlebieh is a family one which my mother always prepared for Christmas.
100 g/4 oz ground rice
2 level tablespoons cornflour
1.2 litres/2 pints milk
8 tablespoons sugar
2 tablespoons orange blossom water
or rosewater or a mixture of the two
¼ teaspoon grated nutmeg
100 g/4 oz ground almonds
Garnish 1 small pomegranate, seeded
2 tablespoons pistachio nuts, chopped
In a large bowl mix together the ground rice and cornflour. Add about 10 tablespoons of the cold milk and stir until you have a smooth paste.
Bring the rest of the milk to the boil in a large saucepan. Add the sugar and stir until it is dissolved. Slowly pour the hot milk on to the rice paste, stirring constantly. Pour the mixture back into the saucepan and cook over a low heat, stirring constantly, until the mixture thickens. Stir in the orange blossom water or rosewater and the nutmeg and cook for a further 3–4 minutes, stirring constantly. Remove from the heat and stir in the almonds. Pour into a serving bowl, leave to cool and then place in the refrigerator to chill. Before serving decorate with the pomegranate seeds and pistachio nuts.
Variations
sakiz muhallebisi
Prepare as with mahlebieh and when it is cooked, remove from the heat and stir in 100g/4 oz unsalted, halved pistachio nuts and 1 teaspoon powdered mastica with the ground almonds. When chilled decorate with a few pistachio nuts and sprinkle all over with 1 teaspoon ground ginger.
Ground almonds can be substituted by ground walnuts or hazelnuts or desiccated coconut.
In Turkey 2 tablespoons powdered coffee or cocoa are often added to give colour and variation of flavour.
gatnabour armenian rice pudding
A personal favourite. Traditionally served to visitors and well-wishers on the birth of a son. When I asked my mother what people served on the birth of a daughter, she gesticulated with her hands, shrugged her shoulders and said, 'Oh, a glass of orangeade or something like that.'
This is a magnificent dessert, highly recommended by all who have tasted it.
75 g/3 oz round grained rice
1.2 litres/2 pints milk
peel of 1 lemon
100 g/4 oz sultanas
100 g/4 oz sugar
2 tablespoons vanilla essence
50 g/2 oz split almonds, toasted under a hot grill until golden
white rum or any other favourite flavouring, to taste
Wash the rice under cold running water. Place the rice in a large saucepan with 600 ml/1 pint of the milk and the lemon peel. Bring to the boil, lower the heat and then simmer very gently until the rice is tender and the milk absorbed. You may need to add a little more milk and stir occasionally to prevent sticking. Stir in the sultanas and simmer for a few more minutes. Remove from the heat and stir in the sugar, vanilla essence and almonds. If the mixture is very thick stir in a little more milk. Place in the refrigerator to chill.
When ready to serve remove the lemon peel, add more milk if you like to make the consistency you prefer and then flavour with the rum. Spoon into a serving dish and serve.
Serves 6–8 people.
Variations
There are inumerable rice-based puddings in the Middle East starting with the simplest roz-bil-halib (rice with milk) of the Arab villages. There is kishkel fukhara (poor man's sweet) similar to the Turkish sakiz muhallebisi except that the powdered mastic is tied in a small muslin bag and removed after cooking—for further use no doubt—hence the name of the sweet as mastic is expensive. The Iranian shir berenj (sweet rice) includes 2 teaspoons ground cardamom, 1 teaspoon cinnamon and is served with honey; while shol-e-zard (Saffron pudding), as the name suggests, includes 1 teaspoon powdered saffron to give that golden colour for which it is justly famed.
A very interesting variation meghli, from Syria-Lebanon, is also traditionally served on the birth of a son and has a unique aroma of caraway, fennel and aniseed. I do not know what is served on the birth of a daughter, but I suggest that this delicious pudding should be prepared.
asure anatolian vegetable and rice pudding
This unusual rice pudding is prepared with great ceremony on the tenth day of the month of Muhareem (late October) which is also called the month of Asura. By tradition vast quantities are made and distributed among neighbours, friends, relatives and the poor.
One of the oldest surviving recipes that has come down to us, according to tradition, from our mutual ancestor—Noah. It was his wife, daughters and daughters-in-law who created this pudding by accident, so we are told, on their last day in the Ark when they used up all the remaining ingredients and created—it is written on the wings of time—this classic Anatolian dish. Some people purée the rice after it has been cooked and the dish is then called beyaz asure (white asure). Traditionally it is made with the wheat bugday, but this is not readily available outside Anatolia and I suggest you substitute it with large burghul.
50 g/2 oz haricot beans, soaked overnight in cold water
50 g/2 oz chickpeas, soaked overnight in cold water
50 g/2 oz large burghul
50 g/2 oz long grain rice
900 ml/1½ pints water or milk
150 ml/¼ pint milk
225 g/8 oz sugar
50 g/2 oz sultanas
4 dried figs, chopped
4 dried apricots, chopped
2 tablespoons rosewater
40 g/1½ oz walnuts, coarsely chopped
25 g/1 oz pine kernels
25 g/1 oz pistachio nuts, halved
25 g/1 oz butter
Place the haricot beans and chickpeas in separate pans with their soaking water, bring each to the boil and then simmer until tender. Add more water if necessary and remove any scum that appears on the surface. The chickpeas will take longer than the beans.
Rinse the burghul and rice and place in a saucepan with the water or milk. Bring to the boil, lower the heat and then simmer for 20–30 minutes.
When tender strain the beans and chickpeas and add to the rice pan together with the milk and simmer for a further 10 minutes. Add the sugar, sultanas, figs, apricots and rosewater, stir thoroughly and simmer for another 10 minutes. Stir in the nuts and butter. By now the mixture should resemble thick porridge. If you think it is still a little thin then simmer for a few more minutes. Pour the asure into individual bowls or a decorative serving dish and decorate, making patterns with some combination of the following: sultanas, ground cinnamon, pomegranate seeds, chopped blanched almonds, halved walnuts, etc.
Serves 6–8 people.
tavuk gogsu chicken breast pudding
A most unusual pudding made with chicken, milk and cream and served chilled. A truly magnificent sweet of the Ottoman period.
1 chicken breast
enough water or dry white wine to cover the chicken
900 ml/1½ pints milk
300 ml/½ pint single cream
¼ teaspoon salt
175 g/6 oz sugar
3 tablespoons ground rice
2 tablespoons cornflour
Garnish 1 teaspoon ground cinnamon
Place the chicken breast in a saucepan and add enough water or dry white wine to cover by 1.2 cm/½ in. Bring to the boil and then simmer until almost tender. Drain the chicken and then finely shred it or pass it through a mincer. Return the meat to a saucepan, add sufficient water to cover by 2.5 cm/1 in and bring quickly to the boil. With a slotted spoon remove any fatty residue that builds up on the surface. Drain, add fresh water and bring to the boil again. Again remove any residue and then drain and set aside.
Put the milk, cream, salt and sugar in a saucepan and bring slowly to the boil, stirring constantly until the sugar has dissolved. Place the ground rice and cornflour in a small bowl and mix to a smooth paste with a little cold water. Add this mixture to the boiling milk and continue cooking, stirring constantly, until the mixture thickens. Add the chicken, stir and cook over a very low heat for a few minutes, stirring frequently. Pour into individual bowls, sprinkle with the cinnamon and serve.
doughnuts and pancakes
awamyat epiphany doughnuts
These doughnut balls are popular throughout the Middle East and were known in the region long before they were introduced into Europe.
They are called lokma in Turkey, zalabyeh in Egypt, lokumades in Greece and jalabi in Iran. There are many variations. Some of the better known ones are luqumatal quadi—sweet tongue of the judge; dilberdudagi—sweet lips doughnuts and gobegi—lady's navel fritters. All very descriptive!
Awamyat are prepared traditionally on the morning of Epiphany by the Christians of Syria and Lebanon and the recipe below is a simple family one.
'The zalabyeh is forbidden to the dogs,' say the Egyptians since these doughnuts were so highly regarded that non-muslims 'dogs' and the poorer classes were unworthy of them.
Syrup
450 g/1 lb sugar
450 ml/¾ pint water
1 tablespoon lemon juice
1 tablespoon rosewater
1 tablespoon orange blossom water
Dough
½ teaspoon fresh yeast or 1 teaspoon dried yeast
1 teaspoon sugar
300 ml/½ pint water
oil for frying
450 g/1 lb plain flour
½ teaspoon salt
300 ml/½ pint milk
Garnish 1 tablespoon cinnamon
First make the syrup by placing the sugar, water and lemon juice in a saucepan and bringing to the boil. Lower the heat and simmer for about 10 minutes or until it is slightly sticky and just coats the back of a spoon. Remove from the heat, stir in the rosewater and orange blossom water and set aside.
Meanwhile, place the yeast and sugar in a small bowl and dissolve in a little of the water, warmed, taken from the 150 ml/½ pint. Set aside for about 10 minutes or until the mixture begins to froth.
Sift the flour and salt into a large bowl and make a well in the centre. Pour the rest of the water and the milk into the yeast mixture and beat thoroughly. Gradually add this liquid mixture to the flour and gradually beat the flour in until the dough is soft and smooth, but not quite a liquid. Cover and leave to rise in a warm place for 1 hour. Beat the dough at least once more and leave to rest again. The final dough should be a well fermented, sponge-like mixture.
Pour about 5 cm/2 in oil into a large, deep saucepan and heat through until hot.
When the dough is ready wet a teaspoon, take a teaspoonful of the mixture and drop it into the oil. Another method is to take up some of the dough in one hand and gently squeeze it up between thumb and forefinger to form small walnut-sized balls. Fry a few at a time. The balls will rise to the surface shaped like walnuts. Turn them over and remove when crisp, golden all over and cooked through. Drain on kitchen paper. Dip them immediately into the cold syrup and lift out with a slotted spoon. When they are all cooked arrange them in a pyramid shape on a large plate. Sprinkle with the cinnamon and serve while still warm.
Serves 6–8 people.
mushabbak patterned doughnuts
These doughnuts are sold throughout the bazaars by street vendors. They are traditionally patterned in never ending circles, and are often coloured red, golden or pink.
The dough must be soft enough to enable the doughnut to retain much of the syrup.
Syrup see awamyat above
Dough see awamyat above
Food colouring of your choice
oil for frying
To prepare the dough and syrup follow the instructions for awamyat to the last paragraph.
While the oil is heating decide what colours you are going to use and divide the dough into the requisite number of portions. Add a little different food colouring to each portion and beat well. By now the oil should be hot.
Spoon one of the coloured portions into a piping bag with a 0.6 cm/¼ in nozzle and slowly squeeze a thin stream into the oil. You can make any patterns of your choice. Traditional ones are: (a) intertwining circles; (b) a circle filled with an intertwining design; (c) names etc. Fry them until golden and then remove with a slotted spoon and drain on kitchen paper. Dip immediately into the cold syrup, remove with a slotted spoon and arrange decoratively on a serving plate.
Repeat with the remaining coloured portions of dough, taking care to rinse out the bag when changing colour. Serve when cold as a dessert or with coffee.
kaygana anatolian sweet crêpes
These thin batter pancakes, made of eggs, milk and flour, are fried on both sides and served hot. Anatolian-style crêpes have been made by the peasants for centuries and although they can be savoury as well as sweet, it is the latter which is the most popular.
I have given below the basic dough recipe, as well as the recipe for apricot kaygana, but there is no reason why you cannot experiment with apples, strawberries, pistachios and other fruits and nuts or mixtures of the two.
In the villages of Turkey and Armenia these crêpes are made much thicker than their wafer-thin cousins in Europe, but I suggest you try and make them as thin as possible.
Batter
225 g/8 oz plain flour, sifted
½ teaspoon salt
4 eggs
a little vegetable oil
4 tablespoons melted butter
500 ml/16 fl oz tepid milk
Place the flour and salt in a large bowl. Make a well in the centre and add the eggs, one at a time, stirring them in with a wooden spoon. Add the melted butter and stir in thoroughly. Now gradually add the milk, stirring continuously, until you have a smooth batter with no lumps. Cover the bowl with a cloth and leave in a cool place for 1 hour.
With a pastry brush lightly grease a 15–18 cm/6–7 in heavy-based frying pan with a little of the oil. Place the pan over a moderate heat and warm the oil until it is very hot. Remove the pan from the heat and pour 4 tablespoons of the batter into it. Tilt the pan in all directions to help the batter spread. Return the pan to the heat and cook for 30–40 seconds. Shake the pan gently to loosen the crêpe. With a palette knife gently lift the crêpe and turn it over. Brown the reverse side for 20–30 seconds and then slide the kaygana on to an ovenproof plate and keep it warm. Continue making the kaygana, greasing the pan each time, until you have used all the batter.
kayisili kaygana crêpes with an apricot filling
Batter see recipe above
400 g/1 lb fresh apricots
350 g/12 oz caster sugar
juice 1 lemon
100 ml/4 fl oz water
Garnish 100 g/4 oz icing sugar, sifted
Prepare the kaygana following the recipe above and keep warm.
Drop the apricots into a bowl of boiling water for 1 minute, then remove and peel. Cut them in half, remove the stones and then halve the halves. Place the apricots in a saucepan, add the sugar, lemon juice and water, bring to the boil and then simmer gently for 15 minutes, stirring regularly. Remove from the heat and leave to cool. Strain off the juice and reserve.
Bring the kaygana to a work top, take one, spread it out and place 2 tablespoons of the apricots onto one half of the crêpe. Fold the crêpe in half and then over again into quarters. Continue until you have filled and folded the remaining kaygana.
Preheat the oven to 200°C, 400°F, gas mark 6.
Arrange the kaygana in a shallow ovenproof dish and heat for 5 minutes in the oven. Remove from the oven, sprinkle generously with icing sugar and serve warm. In a small pan reheat the apricot juice and pour a little over each kaygana as it is served.
Variation
(a) apples—peel, core and slice 450 g/1 lb apples. Cook with 3 tablespoons water over a low heat with 350 g/12 oz caster sugar until just soft. Stir in 1 teaspoon cinnamon and set aside to cool.
(b) strawberries—450 g/1 lb. Hull, wash and drain and then mash with 225 g/8 oz caster sugar. Stir in 100 g/4 oz crushed cream crackers or sweet tea-time biscuits, and fill the kaygana.
ataif arab pancakes
Ataif are pancakes dipped in syrup and they were traditionally served on festive occasions. Nowadays they can be purchased from bakeries and delicatessens. The recipe below is a family one and will make 15–16 pancakes.
Syrup
225 g/8 oz sugar
150 ml/¼ pint water
1 tablespoon lemon juice
1 tablespoon rosewater
Batter
1 teaspoon dried yeast or
7 g/¼ oz fresh yeast
½ teaspoon sugar
300 ml/½ pint tepid water
100 g/4 oz plain flour
Place the yeast and sugar in a small bowl, add 3 tablespoons of the tepid water, mix to dissolve and then leave to rest in a warm place for about 10 minutes or until the mixture begins to bubble.
Sift the flour into a large bowl, add the yeast mixture and work it in with your hand. Little by little add the remaining water and continue kneading and, when the mixture begins to become liquid, beat until you have a smooth batter. Cover with a tea towel and place the bowl in a warm place for 1 hour.
Prepare the syrup by placing the sugar, water and lemon juice in a saucepan and bringing to the boil. Lower the heat and simmer until the syrup begins to leave a sticky film on a spoon. Remove from the heat, stir in the rosewater and set aside.
Lightly brush the inside of a large frying pan with oil and heat the pan for a few minutes until it is really hot. Reduce the heat to medium. Pour 1 tablespoon of the batter into the pan and tilt the pan around to allow the batter to spread. After a minute or two the batter will begin to bubble and become pale golden. With a palette knife lift the pancake and turn it over to cook the other side. Remove the pancake to a large plate and continue to make pancakes, piling them on top of each other when cooked.
The traditional method of eating ataif is to pour some syrup over each one and spread with kaymak or clotted cream. Sometimes chopped pistachios or almonds are then sprinkled over the top.
You can omit the cream and just pour over some syrup, or spread with honey, and sprinkle with nuts.
You can also spread them with cream and then sprinkle with a little cinnamon and pour a spoonful of mulberry syrup or bekmez (grape syrup) over the top. And, of course, you can stuff them.
ataif-bil-joze pancakes filled with nuts
Batter and syrup see ataif above
Filling
225 g/8 oz walnuts, coarsely chopped, or chopped pistachios or almonds or any combination
4 tablespoons sugar
2 teaspoons ground cinnamon
1 teaspoon orange blossom water
1 teaspoon rosewater
Prepare the pancakes as described above, but only cook one side of each one.
Mix the filling ingredients together. Take one ataif and place it on a working top uncooked side upwards. Place 1 tablespoon of the filling in one half of the pancake. Fold the other half over to make a half-moon shape and pinch the edges together very firmly.
Heat some oil in a large pan until hot and then deep fry the ataif for 2–3 minutes or until golden. Remove with a slotted spoon, drain on kitchen paper. Dip into cold syrup and serve warm or cold.
ataif-bil-jibn pancakes filled with cheese
Batter and syrup see ataif
Filling 225 g/8 oz cheese, e.g. akkawa, ricotta or feta
Prepare the pancakes as described but only cook one side of each one.
Place the cheese in cold water overnight and then drain and grate. This will remove most of the saltiness. Continue as with ataif-bil-joze (above) using the cheese filling instead of the nuts.
fruit and nut desserts
kompostolar fruits cooked in syrup
Wherever one eats in Turkey, and to a lesser extent in the Balkans, one is inevitably offered a komposto at the end of a meal, be it of quinces, apples, pears, apricots, peaches, strawberries, oranges, etc. Dried as well as fresh fruit can be prepared in this way and 2 or more kinds of fruit can be cooked together.
The kompostolar are sometimes flavoured with cinnamon, cloves, rosewater or orange blossom water. I have included a few recipes below, but you can experiment with any fruit, or combination of fruits, of your choice.
ayva kompostosu quinces in syrup
900 g/2 lb firm quinces, peeled, cored and quartered
450 g/1 lb caster sugar
900 ml/1½ pints water
1 tablespoon lemon juice
6 cloves
Wash the quinces under cold running water and drain.
Place the sugar, water and lemon juice in a large saucepan and bring to the boil, stirring until the sugar has dissolved. Add the quinces and cloves and simmer for about 30 minutes or until the fruit is tender. Do not overcook or the fruit will disintegrate. Remove from the heat and leave to cool. Transfer to a serving bowl or individual dishes and chill. Serve with kaymak, cream or yoghurt.
portakal kompostosu oranges in syrup
4 medium oranges, peeled
175 g/6 oz caster sugar
300 ml/½ pint water
3 tablespoons orange blossom water
To ease the removal of any white pith drop the oranges into boiling water for 30 seconds and then remove and strip off as much of the pith as possible. Separate the segments.
Place the sugar and water in a large saucepan and bring to the boil, stirring until the sugar dissolves. Add the orange segments and simmer for 15–20 minutes. Remove from the heat, stir in the orange blossom water and leave to cool. Place in a large serving dish and chill. Serve with kaymak, cream or yoghurt.
çilek kompostosu strawberries in syrup
450 g/1 lb firm, fresh strawberries, hulled and rinsed under cold water
225 g/8 oz sugar
300 ml/½ pint water
juice ½ lemon
4 tablespoons fresh strawberry juice—if available
Place the sugar, water and lemon juice in a saucepan and bring to the boil. Simmer for 10 minutes. Drop in the strawberries and simmer for a further 5–10 minutes. Don't let the strawberries overcook. Remove from the heat and stir in the strawberry juice if you are using it. Leave to cool then pour into a serving dish and chill. Serve with kaymak, cream or yoghurt.
Since nothing is new under the sun here is how Apicius advised his contemporaries to prepare a fruit compôte:
That they clean the hard-skinned early fruits, remove the seeds and keep cold in a pan. In a separate pan crush pepper (allspice or nutmeg?) with dry mint. Moisten with water, a little honey, some raisins, wine and wine vinegar. Mix well and pour over the fruit. To this a little oil was to be added and the whole cooked over a low heat until tender.
The juice was to be thickened with some roux (rice flour), and finally a little pepper (a highly prized commodity of the day) was sprinkled over the top and served.
I suggest, however, you prepare your kompostosu the Turkish way!
izmir kompostosu figs in syrup
A speciality of Izmir (Greek Smyrna), the third largest city in Turkey. Izmir is renowned for her figs and grapes. You need dried figs for this recipe.
150 g/5 oz sugar
juice 1 lemon
600 ml/1 pint water
450 g/1 lb dried figs
Garnish 75 g/3 oz hazelnuts or pistachio nuts, finely chopped
Place the sugar, lemon juice and water in a saucepan and bring to the boil. Simmer for about 5 minutes. Lower the heat and arrange the figs in the syrup. Simmer very gently until the figs begin to 'plump up' and to take on something like their original shape. Remove the figs with a slotted spoon and arrange in a serving dish. Sprinkle with the chopped nuts and serve.
khoshab dried fruit compôte
Khoshab (Persian word meaning 'sweet water') is the name of a historic region of Armenia famed for her mountains, impregnable fortresses and swift-flowing mountain streams.
It is a dried fruit compôte, improvised during the centuries as a welcome dessert in winter when fresh fruit was not available.
There are many khoshab-type desserts throughout the Middle East, particularly in Iran and Turkey. This recipe is from the Kirovakan region of Armenia and is also known as Kirovakan Mirkatan.
225 g/8 oz dried apricots, soaked overnight in cold water
225 g/8 oz prunes, stoned, soaked overnight in cold water
225 g/8 oz dried peaches or pears, soaked overnight in cold water
100 g/4 oz sultanas, soaked overnight in cold water
75 g/3 oz honey
rind of 1 lemon in one piece
½ teaspoon nutmeg
1 stick cinnamon about 5 cm/2 in long
2 tablespoons pine kernels
2 tablespoons brandy
Drain the dried fruit and reserve about 300 ml/½ pint of the soaking water. Place the fruit and soaking water in a large saucepan and add the honey, lemon rind, nutmeg and cinnamon. Bring to the boil then lower the heat and simmer for 20–30 minutes or until the fruits are soft and tender. Remove from the heat and discard the lemon rind. Add the pine kernels and stir gently. Cool to room temperature and then stir in the brandy. Transfer to a deep serving bowl or individual dessert dishes, cover and refrigerate. Serve well chilled.
maadan hasultan
almond mousse topped with apricot sauce
An Israeli sweet, attractive, delicious and typical of the cuisine of a new country that is successfully marrying oriental flavours to a European concept. Will make a most suitable dessert for a dinner party.
Custard
3 eggs, separated
2 tablespoons sugar
300 ml/½ pint milk
Mousse (Sultan's delight)
50 g/2 oz toasted almonds, ground
½ teaspoon vanilla essence
½ teaspoon almond essence
the custard, cool
3 teaspoons gelatine
3 egg whites, remaining from the custard
2 tablespoons sugar
150 ml/¼ pint cream
Apricot sauce
300 g/11 oz fresh ripe apricots, stoned or 150 g/5 oz dried apricots soaked overnight in water
lemon juice
sugar
Garnish 6–8 cherries or fresh strawberries
To make the custard place the egg yolks in a small bowl, add the sugar and mix well.
Bring the milk to the boil in a small saucepan. Pour a little of the milk into the egg mixture, stir and then pour it back into the rest of the milk in the pan. Stir well and lower the heat.
Fill a large saucepan or bowl with cold water and, at the first sign of the milk boiling, remove the mixture from the heat, dip the pan into the cold water and leave to cool. When the custard is cool prepare the mousse by first putting the ground almonds, the essences and the custard into a large bowl and mixing thoroughly.
Place the gelatine in a small bowl with a few tablespoons of water, place over a pan of simmering water and stir until the gelatine has dissolved. Stir the gelatine into the almond mixture.
Whisk the egg whites until stiff and fold in the sugar. Whisk the cream until thick. Fold the egg whites and cream into the almond mixture and stir carefully with a metal spoon until well blended. Either spoon the mixture into a decorative glass dish or divide between 6–8 individual dessert glasses. Do not fill each glass more than three-quarters full. Leave in the refrigerator to set and meanwhile prepare the apricot sauce.
Blend the apricots with a little water to form a purée about the consistency of double cream. Add lemon juice and sugar to taste. When the mousse has hardened pour the sauce over the top. Decorate with cherries or strawberries and serve chilled.
zardalou toush porshodeh
Stuffed apricot balls
Do you know the one word
that holds the beauty of the world,
a whole world of colour,
a kaleidoscope of summer
and inexpressible magic?
You would if you had seen
that tree blossoming.
Not any tree, but the apricot
budding like Semirami's breasts
and greener than the willows in Babylon...
(Harout Gostantian—Armenian poet)
An Iranian dessert of dried apricots shaped into balls and stuffed with nuts. They are simple, delicious and typical of the highly sophisticated cuisine of one of the oldest surviving civilizations in the world. Make sure the dried apricots are soft and tender.
450 g/1 lb dried apricots
3 tablespoons ground pistachio nuts
4 tablespoons icing sugar
2 tablespoons orange blossom water
½ teaspoon ground cardamom
Filling
2 tablespoons ground almonds
2 tablespoons sugar
½ teaspoon cinnamon
Garnish
sifted icing sugar
Wipe the apricots with a slightly damp cloth. Chop very finely or pass through a mincer. Transfer to a large bowl, add the pistachio nuts, icing sugar, orange blossom water and cardamom. Wet your hands and knead vigorously until you get a smooth paste. Wet your hands again and shape the mixture into marble-sized balls.
Mix the ground almonds, sugar and cinnamon together. Take 1 ball, hollow out and fill with a little of the nut mixture. Close the hole, roll the ball between your palms and then roll generously in the icing sugar. Repeat with the remaining balls. If wrapped in waxed paper and stored in an airtight tin these balls will keep for several weeks.
narinchanoush
oranges and peel in syrup and liqueur
A family favourite and one that we found to be very popular in our restaurants. The orange peel can be prepared in large quantities and stored in sealed jars for weeks, and you can use sliced pistachio nuts or chopped hazelnuts or walnuts instead of the almonds.
900 g/2 lb oranges, peeled, reserve the flesh and the rind
900 g/2 lb sugar
900 ml/1½ pints water
1 tablespoon lemon juice
3 tablespoons orange blossom water
Garnish
3 tablespoons almonds, blanched, slivered, toasted
2 tablespoons orange liqueur, e.g. Filfar or similar
Half fill a large saucepan with water, bring to the boil and drop in the pieces of peel. Simmer for about 30 minutes or until the peel is soft. Remove from the heat and drain. When cool enough to handle take a sharp knife and carefully slice off as much of the white pith as possible. Slice the peel into strips less than 0.3 cm/⅛ in thick, if possible.
Place the sugar, water and lemon juice in a large saucepan, bring to the boil and simmer for 5 minutes. Add the sliced orange peel and simmer until the syrup thickens and coats the back of a spoon. Remove from the heat, stir in the orange blossom water and set aside to cool completely.
To prepare the sweet for serving cut the oranges crossways into 0.6 cm/¼ in thick slices. In a shallow dish arrange the slices, overlapping, around the edge leaving a gap in the centre. Spoon some of the orange peel evenly over the slices and pile a little more in the centre. Pour some of the syrup over the slices and then sprinkle with the toasted almonds and the liqueur. Chill before serving.
serov hatsi kadaif
bread cooked with honey and served with cream
This is a highly sophisticated dessert. The authentic way of serving this sweet is with kaymak which is the thick Middle Eastern cream. If you wish to make this then follow the instructions in the first paragraph below. Otherwise use whipped double cream or, preferably, clotted cream and start the preparation at the second paragraph below.
This sweet is the creation of a great 'known' chef—Tokatlian, of Istanbul, who lived in the nineteenth century.
1 round loaf white bread, e.g. Italian or Greek—here I should mention that when I wanted to prepare this sweet in a hurry and had no white bread I used wholemeal instead and I must admit I found it just as delicious
a little milk—about 5 tablespoons
300 ml/½ pint clear honey
juice 1–2 lemons
To serve kaymak (see Glossary and use half the quantity of double cream suggested there) or 300 ml/½ pint double cream, whipped or 300 ml/½ pint clotted cream
Slice the loaf in half and use only the bottom half. Remove the bottom crust leaving a thick slice of bread about 3 cm/1¼ in thick. Place the slice of bread in an oven preheated to 180°C, 350°F, gas mark 4 and leave until toasted, golden and crisp. Remove the bread, sprinkle on both sides with the milk and wrap in a teatowel until the milk is absorbed.
Mix the honey and lemon juice in a small saucepan and bring to the boil. Place the bread in a round ovenproof dish just large enough to hold it and spoon the honey evenly over it.
Bake in an oven preheated to 180°C, 350°F, gas mark 4 for about 30 minutes or until most of the honey has been absorbed and the bread is golden. Cut into pieces and top with the cream.
Variation
This is a quicker and cheaper method—a 'poor man's' version, but tasty nevertheless.
6 thick slices bread
2 tablespoons sugar
150 ml/¼ pint milk
Syrup
225 g/8 oz sugar
450 ml/¾ pint water
1 tablespoon lemon juice
First prepare the syrup by placing the sugar, water and lemon juice in a small saucepan and bringing to the boil. Lower the heat and simmer for about 10 minutes or until the syrup begins to form a sticky coating on the back of a spoon. Remove from the heat and set aside to cool.
Arrange the slices of bread side by side in a large frying pan. Dissolve the sugar in the milk and pour over the bread. Place over a low heat and gradually add the syrup, pouring it evenly over all the slices. Simmer until most of the syrup has been absorbed. Cool and then serve with cream.
## cakes and biscuits
karabij stuffed pastries with natife
A speciality of Aleppo in Syria. These pastries are dipped in a cream called natife and eaten. Natife is made from pieces of wood called bois de Panama which can be bought from good health food shops and Middle Eastern stores. It is often sold in powdered form which makes its preparation simpler.
This is a unique pastry and it is well worth going to the trouble of making it.
Cream
75 g/3 oz bois de Panama, also known as 'halva wood'
225 g/8 oz sugar
1 tablespoon lemon juice
2 tablespoons orange blossom water
4 egg whites
Filling
225 g/8 oz walnuts, finely chopped
100 g/4 oz sugar
1 tablespoon cinnamon
Dough
450 g/1 lb plain flour
225 g/8 oz unsalted butter, melted
3–4 tablespoons water
First prepare the cream by pulverizing the pieces of wood. Place in a bowl with about 150 ml/¼ pint water and leave to soak for 4–5 hours. Transfer the contents of the bowl to a saucepan and bring to the boil. Lower the heat and simmer until the liquid has thickened. Strain the mixture through fine muslin and set the liquid aside.
Dissolve the sugar in 8 tablespoons water in a small saucepan, add the lemon juice and bring to the boil. Lower the heat and simmer until the syrup has thickened—about 10 minutes. Remove from the heat, stir in the orange blossom water and the hot bois de Panama liquid and beat vigorously. Set aside to cool. When the mixture is cold place the egg whites in a large bowl and whisk until very stiff. Gradually add the cold syrup mixture, beating continuously until the mixture froths and expands. Transfer to a serving dish and set aside.
Mix the filling ingredients together in a bowl.
To prepare the dough sift the flour into a large bowl, add the butter and knead. Add the water and continue kneading until the dough is soft and smooth. Divide the dough into walnut-sized lumps. Take one lump, roll between your palms to form a ball and then hollow it out with your thumb, pinching the sides up until they are thin and form a pot shape. Fill the hollow with a little of the nut mixture and then press the dough back over the filling to form a ball. Gently press between your palms to make an oval shape. Repeat until you have used up all the dough and filling. Place the karabij on baking sheets and cook in an oven preheated to 150°C, 300°F, gas mark 2 for 20–30 minutes. Remove before the pastry changes colour—they should still be white. Set aside to cool.
When serving arrange the karabij on a large plate and offer a bowl of the natife cream. Dip the karabij into the cream and eat. The pastries will keep for a long time in an airtight tin, but refrigerate the cream.
Makes 30–40 pastries.
ma-moul Stuffed easter pastries
A Syrian-Lebanese speciality these pastries are traditionally prepared during Easter week. Today they are sold throughout the year.
The recipe below is a family one using dates, walnuts and almonds. The Lebanese often use semolina instead of flour which gives the pastry a coarser and more earthy taste.
Filling
225 g/8 oz dates, stoned
225 g/8 oz walnuts, roughly chopped
100 g/4 oz almonds, roughly chopped—you can use pistachio nuts instead
150 ml/¼ pint water
100 g/4 oz sugar
1 heaped teaspoon cinnamon
Dough
450 g/1 lb plain flour
225 g/8 oz unsalted butter, melted
2 tablespoons rosewater
4–5 tablespoons milk
Garnish icing sugar
First prepare the filling. Chop the dates and place them in a saucepan with the nuts, water, sugar and cinnamon. Cook over a low heat until the dates are soft and the water has been absorbed.
Sift the flour into a bowl, add the melted butter and mix by hand. Add the rosewater and milk and knead the dough until it is soft and easy to mould. Divide the dough into walnut-sized lumps. Take one lump, roll it into a ball and then hollow it out with your thumb, pinching the sides up until they are thin and form a pot shape. Fill the pot with some of the date mixture and then press the dough back over the filling to enclose completely. Roll into a ball and then gently press with your palm to flatten it slightly or, if you have a wooden spoon with a deep curved bowl, mould each pastry with that. Repeat until you have used up all the pastry and filling.
Arrange the pastries on baking sheets. Make interesting patterns with a fork on each pastry. The traditional one is to mark straight lines down the length of each pastry. Bake in an oven preheated to 150°C, 300°F, gas mark 2 for about 30 minutes. Do not let them change colour or the pastry will become hard. Remove from the oven and allow to cool. When cold roll in sieved icing sugar and store in an airtight tin.
Makes 30–40 pastries.
tahinov hats sesame-cream cakes
A speciality from southern Turkey—the Armenian kingdom of Cilicia that flourished in the Middle Ages.
Most delightful cakes, filled with sesame cream (tahina) and sugar, which are ideal with tea and coffee.
Traditionally eaten during the forty days of Lent. these cakes will keep for several weeks.
I like my hats very sweet, but you can vary the sugar content according to taste.
½ teaspoon dried yeast
½ teaspoon sugar
175 ml/6 fl oz water, warm
225 g/8 oz plain flour
½ teaspoon salt
100 ml/4 fl oz cooking oil
Filling
tahina
sugar
ground cinnamon
In a small bowl dissolve the yeast and the sugar in the warm water. Place in a warm place and leave for about 10 minutes or until the mixture begins to froth. Sift the flour and salt into a large bowl, make a well in the centre and pour in the yeast mixture and the cooking oil. Mix together and then knead well until the mixture is smooth. Roll the dough into a large ball, place in the bowl, cover with a tea towel and leave in a warm place until the dough has doubled in size. Pour some tahina into a bowl and stir until it is smooth. Divide the dough into 6 balls. Lightly flour a work top and roll one ball out into a circle about 0.3 cm/⅛ in thick.
Spread a tablespoon of the tahina over the circle of dough and then sprinkle 1–2 tablespoons sugar over the top. Vary sugar according to taste. Sprinkle a pinch of cinnamon over the sugar. Roll the circle up into a sausage, grasp it in your hands and squeeze gently. This closes the sausage and doubles its length. Cut the sausage in half. With each piece, fold the 2 ends over to the middle—one slightly overlapping the other. Press the cake down gently to secure the ends and flatten slightly. Repeat with the remaining balls of dough.
Place the cakes on lightly greased baking sheets. Cook in an oven preheated to 200°C, 400°F, gas mark 6 for 30 minutes or until golden. Remove and leave to cool.
Makes 12 cakes.
zadgva katah easter cake
Traditionally, during Easter, Armenians make this cake and many others. Basically they are dry, bread-like cakes, sometimes with a filling, but often without. They are delicious when sliced and eaten with tea or coffee.
Dough
15 g/½ oz fresh yeast or 7 g/¼ oz dried yeast
225 ml/8 fl oz tepid milk
50 g/2 oz sugar
100 g/4 oz butter, melted
400 g/14 oz plain flour, sifted
½ teaspoon salt
Khoritz (filling)
2 tablespoons raisins
2 tablespoons walnuts, chopped
1 tablespoon brown sugar
1 tablespoon white sugar
1 teaspoon cinnamon
1 tablespoon sesame seeds
1 oz butter, melted
Garnish beaten egg
Place the yeast in a small bowl and add half the milk.
Pour the remaining milk into a large mixing bowl, add the sugar and stir until dissolved. Add the melted butter to this mixture. When the yeast has softened pour the mixture into the mixing bowl and stir. Gradually stir in the flour and salt. When well-blended transfer the dough to a well-floured work top and knead for at least 10 minutes. Roll the dough into a ball, place in a clean bowl, cover with a cloth and set aside in a warm place until it has doubled in size. Remove the dough to a work top, punch down and knead for a few more minutes.
Now prepare the filling by mixing all the ingredients, except the butter, together in a small bowl. Divide the dough into 3 equal parts. Take one part and roll out on a floured surface until 0.3–0.6 cm/⅛–¼ in thick. Brush the surface all over with melted butter. Fold the edges into the centre to make about a 12.5 cm/5 in square. Place a third of the filling in the centre of the square and bring the opposite corners of the square over to cover the filling completely. Then gently roll the cake out until it is about 15 cm/6 in square, taking care not to let the filling ooze out. Place on a greased baking sheet. Repeat with the remaining dough and filling. Brush the top of each with beaten egg and set aside in a warm place for a further 30 minutes.
Heat the oven to 200°C, 400°F, gas mark 6, add the cakes and bake for about 20 minutes or until risen and golden. Remove and cool on wire racks. Serve sliced with tea or coffee.
choreg festive biscuits with sesame seeds
Choreg are dry bread sticks of Armenian origin (chor-ekmek—dry bread), which have, over the centuries, become equally popular in Turkey and Greece—choureks. Traditionally prepared over Christmas—Zenunti choreg and Easter—Zadgi choreg, they are excellent with tea or coffee at any time of the day, either plain or with cheese, jam or honey.
Usually prepared in large quantities they last for a long time when stored in airtight tins.
Mahaleb (from the kernel of the black cherry stone) gives this biscuit its very unique flavour and I strongly recommend that you try to use it although it is expensive and sometimes difficult to find. Good Middle Eastern stores should have it.
1 teaspoon dried yeast or 7 g/¼ oz fresh yeast
1 teaspoon sugar
1 teacup warm water
450 g/1 lb plain flour
pinch salt
100 g/4 oz butter
1 teaspoon mahaleb, crushed
100–225 ml/4–8 fl oz water
1 teaspoon cooking oil
Topping
1 egg, beaten
sesame seeds
Place the yeast and sugar in a small bowl. Add the teacup of warm water, stir to dissolve and then leave in a warm place until the mixture begins to froth.
Sieve the flour and salt into a large bowl. Add the butter and rub it in until the mixture resembles fine breadcrumbs. Stir in the crushed mahaleb. Make a well in the centre of the flour and pour in the yeast mixture. Blend the flour and yeast mixture together until you have a stiff dough. Now add a little of the water at a time and knead until you have a soft dough. The amount of water you add will vary because of the differing qualities of the various brands of flour. When the dough comes easily away from the sides of the bowl remove to a clean work top and knead for about 10 minutes until it is easier and pliable. Add the oil and knead it in. Roll the dough into a ball, place in a clean bowl, cover with a tea towel and leave it to rest in a warm place until it has doubled in size.
When the dough is ready: (a) heat the oven to 200°C, 400°F, gas mark 6; (b) grease baking sheets with cooking oil; (c) place the beaten egg in a small bowl; (d) pour some sesame seeds on to a plate.
Punch down the dough a few times and then break off a piece about the size of a walnut. Roll it between your palms to form a ball. Place it on your working top and roll it to and fro with your palms until you have a long strip which is pencil thin and about 30 cm/12 in long. These strips are made into many different shapes which often vary from family to family. The two described here were favoured by my mother.
The twisted circle
Bring one end of the strip over to meet the other, thus halving its original length.
Lightly roll your palm over the 2 loose ends 2 or 3 times to obtain a twisted strip. Bring the 2 ends of the strip together and press the uncut end over the 2 loose ends to form a circle.
The plait
Break off a third of the strip and press one end of it half way along the remaining strip. Plait the 3 strips of dough together and then press the 3 loose ends together.
Place each choreg on to a baking sheet leaving a little space between each one. Brush each one with the beaten egg and then sprinkle with sesame seeds. Turn the oven off, place the baking sheets inside and leave the choreg to rise a little. Turn the oven up to 200°C, 400°F, gas mark 6 again and cook until golden. As the choreg become golden remove the trays from the oven and pile all the choreg on to one of the trays. Turn off the oven and return the tray to the oven and leave for several hours to dry out. When cold store in an air-tight tin.
gurabiah lover's shortbread
Gurabiah is prepared in most countries of the Middle East as well as North Africa and the Balkans.
Arab by origin these almond-based sweets can be made with flour or semolina. Although fairly simple to make they do need careful handling and careful cooking. The name gurabiah comes from the Arabic word gharib meaning to miss, to yearn, etc. which suggests longing for one's loved one—hence the sweets are often heart-shaped.
By far the most tasty recipe is this one from Syria which just melts in the mouth.
450 g/1 lb unsalted butter
225 g/8 oz icing sugar, sifted
450 g/1 lb plain flour, sifted
blanched, halved almonds
Melt the butter in a small saucepan over a low heat. Spoon off any froth and pour the yellow liquid into a large mixing bowl, discarding the milky residue in the bottom of the pan. Put the bowl in the refrigerator and leave until the butter has solidified.
Beat or whisk the butter until it is white and creamy. Add the icing sugar, a little at a time, and continue beating. Add the flour, a little at a time, and continue to mix until the mixture is stiff. Collect the dough up and knead it by hand until it forms a ball and becomes smooth and pliable. Leave to rest in the bowl for about 10 minutes.
Preheat the oven to 150°C, 300°F, gas mark 2.
On a clean surface shape the mixture into walnut-sized balls. Roll one into a sausage and join the ends to form a circle. Place on a baking sheet and then place an almond over the join. Continue until you have used up all the mixture, placing the gurabiah about 2.5 cm/1 in apart on the baking sheets. Place in the oven and cook for 20 minutes or until the almonds are a very light golden, but the biscuits are still white.
barazeh Shami damascus sesame biscuits
One of the oldest cities in the world, Damascus—of biblical fame—is little known (outside her city walls) for her excellent cuisine. But excellent it is, for some of the finest meat and vegetable dishes and desserts—particularly those made with the fruits of the Orontes valley, and the mastic-based ice cream popular throughout the Arab world—are all of Damascus origin.
These sesame biscuits are typical of that sophisticated and yet extremely simple culinary art evolved over the ages. I would like to think that St Paul, Muhammad and the infamous Genghis Khan all ate these sesame-coated biscuits with their milk or tea.
¼ teaspoon fresh yeast or ½ teaspoon dried yeast
1 teaspoon sugar
225 ml/8 fl oz tepid water
450 g/1 lb plain flour
225 g/8 oz caster sugar
175 g/6 oz butter
225 g/8 oz sesame seeds
50 g/2 oz butter, melted
Place the yeast and sugar in a small bowl, add a little of the warm water, stir to dissolve and place in a warm place for about 10 minutes until the mixture begins to froth.
Sift the flour into a large bowl and stir in the sugar. Cut the butter into small pieces, add to the flour and rub in until the mixture resembles fine breadcrumbs. Make a well in the centre, add the yeast mixture and gradually add the remaining water, mixing with your hand until a dough is formed. Knead for about 5 minutes until the dough is soft and easy to handle. If necessary add a little more water. Cover the bowl with a cloth and leave in a warm place for about 1 hour or until the dough has doubled in size. Remove the dough and place on a lightly floured working top. Punch the dough down and knead for a further 1–2 minutes. Divide it into 2 equal portions.
Sprinkle the work surface with a little more flour and roll out one portion until about 0.3 cm/⅛ in thick. Using a 10 cm/4 in cake cutter cut out as many rounds as possible. Repeat with the remaining portion of pastry.
Grease 2 or 3 baking sheets. Spread the sesame seeds out on a plate. Lightly brush both sides of each biscuit with butter and then dip into the sesame seeds so that both sides are coated.
Arrange, 2.5 cm/1 in apart, on the baking sheets and bake in an oven preheated to 160°C, 325°F, gas mark 3 for 20–25 minutes or until the biscuits are golden and dry. Remove from the oven, cool completely on wire racks and store in an airtight tin.
## sweets
rahat lokum turkish delight
The most famed sweet of Turkey, known throughout the world, is rahat lokum. It is often better known simply as lokum—'giving rest to the throat'.
Although this sweetmeat is produced in many countries no make can equal the famed produce of the House of Hadji Bekir—a family business reputed to have been established for over 250 years and still going strong in Istanbul.
Traditional lokum consist of the pulp of white grapes or mulberries, semolina, flour, rosewater, honey and apricot kernels. There are many variations, some of which I have indicated below.
The recipe below is one my mother often prepared which I am assured is based on one passed down to her from her grandmother.
Rahat lokum is easy to prepare and has a certain mystique surrounding it—one of romance, adventure, abandonment and leisure!
butter
450 g/1 lb sugar
300 ml/½ pint water
1 teaspoon lemon juice
25 g/1 oz gelatine dissolved in 100 ml/4 fl oz hot water
½ teaspoon vanila essence
1 tablespoon pistachio nuts, halved
3 drops food colouring, e.g. red, yellow or gold
1 tablespoon rosewater
50 g/2 oz icing sugar
25 g/1 oz cornflour
Grease a 15 × 15 cm/6 × 6 in baking tin with the butter and set aside.
Put the sugar, water and lemon juice in a saucepan and bring to the boil. Continue boiling until the temperature reaches 120°C, 250°F on a sugar thermometer. If you do not have one then drop a little of the syrup into a bowl of cold water. If it has reached the required temperature it will form a hard ball. Remove the pan from the heat and leave to stand for 10 minutes. Stir in the dissolved gelatine and the vanilla essence and beat with a wooden spoon until the mixture is well blended.
Pour half the mixture into the baking tin, and sprinkle the halved nuts over it.
Stir the rosewater and colouring into the remaining mixture and mix well. Pour this mixture into the tin and set aside in a cool place overnight.
Sift the icing sugar and cornflour on to a large plate. Turn the lokum out on a clean board and cut into 2.5 cm/1 in cubes. Toss the cubes in the sugar mixture and make sure they are thoroughly coated. Shake off any excess sugar. Either wrap individually in waxed paper or store in an airtight container.
Variations
You can use chopped walnuts or almonds instead of the pistachios or you can prepare the lokum plain, i.e. without any nuts.
Mulberry or strawberry lokum
Cook 225g/8 oz of the fruit in a little water. Drain off any liquid, mash the pulp and incorporate into the mixture above, omitting the other flavourings.
Crème de menthe lokum
Replace the food colourings and the flavourings above with 2½ tablespoons crème de menthe and ½ teaspoon green food colouring.
Gikolat lokumi
Prepare the lokum as with the recipe for rahat lokum and after you have cut it into cubes dip them individually into melted hot chocolate and dry on a greased wire rack.
Covered with dessicated coconut
When the lokum has been cut into cubes roll them individually in dessicated coconut until completely covered.
tamrieh date sweets
Legend goes that when the prophet Muhammad meditated for forty days in the wilderness he lived on dates and goat's milk. Arab historians relate how the early Arab armies marched on date, rice and fruit. While in our time Bedouin (and there are still millions of them regardless of the pace of industrialization and urbanization) in the deserts still make great use of the fruit of the oasis—the date.
In Oman and Yemen fresh dates are dipped in ghee and eaten with relish. Throughout the desert regions most Arab countries cultivate this fruit in a big way. Particularly successful is Iraq whose dates are renowned worldwide.
There are three basic date types: Kahastawari, Khadrawi and Zhehdik—all of which are dried. The best known fresh dates are Baban and Berhi and they usually come from Israel and Lebanon.
The fruit is versatile. When it is being dried a thick dark syrup is exuded which is used in many dishes creating a sweet-sour flavour. This syrup is often mixed with butter marys and eaten with bread and kaymak.
I have selected a few date desserts under the general heading Tamrieh, but dates also appear in many other recipes in this book.
halawah temar date halva
One of the simplest and most popular date sweets found throughout Arabia, Iraq and the Gulf States. Its brilliance lies in its simplicity.
450 g/1 lb dates, stoned and chopped
225 g/8 oz walnuts, coarsely chopped
225 g/8 oz almonds, coarsely chopped
2 tablespoons toasted sesame seeds (optional)
icing sugar
Mix the dates and nuts together in a large bowl and knead until smooth.
If using sesame seeds knead these in too.
Lightly dust a board with icing sugar. Place the ball of dates and nuts on the board and with a rolling pin dusted with icing sugar roll it out into a square about 1 cm/½ in thick. With a sharp knife cut into 2.5 cm/1 in squares. Dust a serving plate with icing sugar, arrange the squares over it and then dust with a little more icing sugar. Serve with kaymak, clotted or double cream.
This will keep if wrapped in wax paper and stored in an airtight tin.
mischlachat ha negev
date rolls stuffed with nuts
Dates are not, of course, the prerogative of the Arabs. All Middle Eastern people love and use them in their dishes.
The following Israeli sweet, the creation of a well known local chef Roger Debasque, is a good example of a variation on a traditional theme. Dedicated to the 'Negev Expedition' it can be made in large quantities and kept in an airtight tin after being wrapped in waxed paper.
4 eggs
5 tablespoons double cream
600 ml/1 pint milk
10 tablespoons sugar
green food colouring
200 g/7 oz butter
150 g/5 oz plain flour
40 dates, stoned
150 g/5 oz shelled hazelnuts, whole
250 g/9 oz roasted almonds, chopped
1 tablespoon vanilla essence
1 tablespoon almond essence
Topping 40 almond or walnut halves
Separate the eggs and reserve the whites for another recipe. Mix the yolks and cream together in a small bowl.
Bring the milk to the boil in a saucepan, add the sugar and 1–2 drops green food colouring to obtain a light green colour.
In another saucepan melt the butter, add the flour and stir constantly over a low heat until the flour begins to fry. Add the boiled milk gradually, stirring constantly to ensure it is smooth and free from lumps. When the mixture turns smooth, hard and even stir in the egg yolk mixture. Place in the refrigerator to cool.
Meanwhile, stuff each date with 3 hazelnuts.
When the dough mixture is cool place on a work top and knead until pliable. Add the chopped almonds and the essences and knead a little more. Flatten the dough and cut 40 strips about 5.5 × 0.6 cm/2½x ½ in. Fold a strip of dough lengthways around each date and then decorate each with half an almond or walnut. Serve as a sweetmeat after a meal and with turkish coffee and tea.
## jams and preserves
The word murababbiyah meaning jams or preserves, is of Arab origin, but this method of fruit preservation is of much older vintage. Our ancestors developed special techniques to help enjoy the fruits of summer throughout the year. Jams and preserves are two of the more important ones.
Traditionally honey was used in the making of these preserves, but in time sugar was substituted—since honey tends to thicken and often crystallize the preserve.
For my money the greatest preserve of all is varti-anoush—rose petal jam. This jam is popular throughout the Middle East, the Balkans and India and I have tried several commercial brands, but nothing can beat the home-made version.
One day a great admirer of the Hodja sends him a large pot of home-made rose petal jam. The Hodja is delighted, but unfortunately had to attend the funeral of a close friend. He places the pot in a safe place and warns his pupils (at this stage Hodja was a schoolmaster) not to touch the contents of the pot for 'You can never trust anyone nowadays. I suspect an adversary of mine has sent it and it may well be poisoned.'
He left. Immediately his nephew retrieved the pot of jam and called to his fellow pupils, 'Let's eat!'
Some objected saying that they were afraid it may be poisonous.
'Stupid,' said the nephew, 'he wanted you to believe so. Look, it's nothing of the sort.' Saying this, he ate a spoonful. 'Mmm–beautiful.'
His classmates followed suit and in a matter of minutes there was no rose petal jam in the pot.
'Leave it to me,' said the nephew, 'I know how to tell my uncle.'
He took Hodja's pen sharpener and broke it.
When Hodja returned he was aghast to see his pen sharpener cracked. 'Who is the guilty party?,' he angrily demands and immediately everyone points to his nephew who bursts into tears.
'Explain,' demands the Hodja.
'Uncle,' says the nephew, 'my pen sharpener broke so I borrowed yours and, clumsy as I am, I broke it. I knew you would be angry with me. I felt so sick I wanted to kill myself. I thought of the school well, I wished to drown, but I remembered my friends drink of that well. And then, just then I remembered the pot of jam and how you thought it was poisoned. So I ate it all and, begging forgiveness from all and everyone sat down on my cushion and waited for the angels to take me to hell—but, nothing happened.' He continued to weep, so loud and with such gusto that some of his classmates began to accompany him.
'Oh God,' whispered the Hodja, 'I know you have been generous with your distribution of wisdom and guile amongst the lot of our clan, but this is too much—I am all for sapience and cunning, but this young dog is teaching me new tricks.'
varti-anoush rose petal jam
I have tried to prepare this jam with roses from my garden, but unfortunately the petals remain a little tough and there is little fragrance. Make sure, therefore, that you boil the petals until really tender and I suggest that you add rosewater to get that wonderful fragrance. Use fresh red petals, cut off their white ends and wash thoroughly—especially if they have been sprayed.
450 g/1 lb fresh rose petals, red and with as strong a fragrance as possible
juice 2 lemons
600 ml/1 pint water
450 g/1 lb sugar
3–4 tablespoons rosewater, depending on strength
Put the washed rose petals into a large glass bowl, squeeze half the lemon juice over and leave for 10 minutes.
You can cook the petals whole, but because they might be tough I suggest you pass them through a mincer. My mother used to knead them by hand, but this took her ages. Put the petals into a large saucepan with any of the lemon juice left in the bowl (this will help to set the jam). Add the water, bring to the boil and then lower the heat and simmer until the petals are tender. This may take anything from 10 minutes to 1 hour depending on the petals. Add the sugar and remaining lemon juice and bring to the boil, stirring continuously until the sugar dissolves. Simmer, stirring frequently, until the syrup thickens—about 10 minutes. Remove from the heat and stir in the rosewater. Leave to stand for a few minutes and then skim off any scum on the surface. After about 15 minutes stir and then pour into warm sterilized jars. Seal when cold.
ayva reçeli quince jam
Once very popular in Western Europe the quince nowadays is little used except for making preserves—excellent ones at that. Try this recipe from Turkey which is also a great favourite with Kurds, Armenians and Caucasians. It has an exquisite flavour emanating from the cloves, cinnamon and rosewater.
6 large quinces, peeled, cored and quartered
450 g/1 lb sugar
900 ml/1½ pints water
2 tablespoons lemon juice
5 cm/2 in stick cinnamon
3 whole cloves
3 tablespoons rosewater
Halve the quartered quinces. In a large saucepan bring the sugar and water to the boil, stirring constantly until the sugar has dissolved. Add the quince slices, lemon juice, cinnamon and cloves and boil for 3 minutes. Lower the heat and simmer for about 2 minutes, stirring frequently, until the syrup thickens and coats the back of a spoon. Remove from the heat and discard the cinnamon stick and cloves and stir in the rosewater. When completely cold seal tightly.
sumpoogi kaghtsr whole aubergine preserve
One of the great classics of the Armenian cuisine. Use small, 5–7.5 cm/2–3 in, long aubergines. Serve individually with a little syrup.
900 g/2 lb small aubergines, peeled
2.3 litres/4 pints water
2 tablespoons lime powder (obtainable from your chemist)
1½ teaspoons cinnamon
1 teaspoon cloves
1 tablespoon rosewater
900 g/2 lb sugar
2 litres/3½ pints water
Mix the water and lime powder together in a large saucepan, add the aubergines and leave them to soak overnight. To keep them submerged invert a large plate over them and weigh it down.
Next day rinse the aubergines in a colander under cold running water. Remove as much moisture as possible from the aubergines by squeezing each one between your palms. Place them in a large saucepan, add water, bring to the boil and simmer for 5 minutes. Drain, rinse once more under cold water and squeeze each one again between your palms to remove all traces of bitterness.
Place the sugar and water in a large saucepan and bring slowly to the boil, stirring constantly. Drop the aubergines into the syrup, add the cinnamon and cloves, lower the heat and simmer for 2 hours or until the syrup is thick. Stir carefully so as not to break up the fruit. Remove from the heat, stir in the rosewater and leave to cool. Spoon into warm sterilized jars and when completely cold seal tightly.
badrijani muraba aubergine jam
A recipe from Batumi on the Black Sea coast. Makes a beautiful jam.
1.8 kg/4 lb small, peeled aubergines
2 tablespoons salt
4 lemons
4 cloves
2.5 cm/1 in piece fresh root ginger, peeled and bruised
1.3 kg/3 lb sugar
100 g/4 oz crystallized ginger, chopped
Cut the aubergines into 0.6 cm/¼ in cubes. Place the cubes in a large colander, sprinkle with the salt and set aside for 30 minutes.
Peel the lemons and cut the rinds into thin strips. Squeeze and reserve the juice.
Rinse the aubergine cubes under cold water and then pat dry. Put the cubes in the top part of a steamer and half fill the bottom part with water.
Or put in a colander over a saucepan and cover with the lid. Bring to the boil and then steam the aubergines until tender—about 10–15 minutes. Remove from the heat and transfer the aubergines to a large pan and add the lemon juice.
Put the lemon rind, cloves and ginger in a muslin bag, tie tightly and add to the pan. Add the sugar, mix thoroughly, cover the pan and leave to stand undisturbed for at least 24 hours.
Next day bring slowly to the boil, stirring continuously until the sugar has dissolved and then simmer for 10 minutes, stirring frequently. Add the crystallized ginger, raise the heat and boil vigorously until the jam thickens. Remove from the heat, discard the muslin bag and leave to cool. Spoon into warm sterilized jars and seal when completely cool.
murababbiyeh-bil-amar date preserve
Delicious preserves from Iraq which are also popular throughout Syria and the Gulf region. For this preserve you need fresh dates, the large yellow or red varieties which sometimes appear in the shops, usually air-flown from Israel or Morocco.
1 kg/2½ lb fresh dates
3 litres/5 pints water
toasted slivered almonds
450 g/1 lb sugar
6 cloves
3 tablespoons orange or tangerine rind, thinly sliced
2 tablespoons lemon juice
Peel the dates and place them in a large saucepan with the water. Bring to the boil and then simmer for about 1 hour or until they are tender. Remove from the heat and, with a slotted spoon, transfer the dates to a large plate. Reserve the liquid. When the dates are cool enough to handle use a small, sharp knife to remove and discard the stones. Push 2 slivers of toasted almonds into each date.
Sprinkle a third of the sugar evenly over the bottom of a medium saucepan. Lay a third of the dates and 2 cloves over the sugar. Repeat these layers twice more.
Make the reserved liquid up to 1 litre/1¾ pints with cold water if necessary and pour into the pan. Cover the pan and leave to rest overnight. The next day bring this mixture to a quick boil and boil vigorously for 5 minutes. Lower the heat and simmer for about 30 minutes. Stir in the orange or tangerine rind and the lemon juice and simmer until the syrup is thick enough to coat the back of a spoon. Remove from the heat and cool. Spoon into warm sterilized jars and when cold seal tightly.
engouyzi-anoush green walnut preserve
A superb classic. Naturally you do need a walnut tree. If one is not available—read on, and wish you had a walnut tree!
But first a few cautionary words. The walnuts must be picked when the green fruit is not yet full size and the inner shell is still soft. To test I suggest that you use a strong needle and pierce the nut in 3–4 places. If there is no resistance then the nuts are suitable to preserve. Walnuts will stain your hands black so use rubber gloves.
This recipe is from the Caucasus where the finest walnut preserves are made. Although it is also known in adjacent lands, e.g. it is called morabaye gerdu in northern Iran, and as far as Cyprus where it is known as glyko karithi, its home is definitely the Caucasus—and in particular, Georgia. Lime powder (slake lime) helps to harden the outer skin of the nuts otherwise they will disintegrate while cooking. You should be able to purchase it from your local chemist. Do explain that it is for cooking purposes.
about 50 walnuts
8 tablespoons lime powder (slake lime)
900 g/2 lb sugar
600 ml/1 pint water
1½ tablespoons lemon juice
5 cloves
10 cm/4 in piece cinnamon bark
Remove the outer green walnut shells. Place the walnuts in a large pan and soak in water for 2 days, changing the water 3 times a day. On the third day drain the nuts, add fresh water, stir in the lime powder and soak the nuts for 24 hours. Next day drain the walnuts and rinse very thoroughly under cold running water. Pierce each nut in several places with a thin skewer and then soak in fresh water for a further 2 days. Drain into a colander.
Place the sugar, water and lemon juice in a saucepan and bring to the boil. Lower the heat and simmer until the syrup thickens and coats the back of a spoon. Add the nuts, cloves and cinnamon and cook for 3 minutes. Remove the pan from the heat and leave to cool. Return the pan to the heat, bring to the boil, simmer for 1 minute and remove. Repeat this process twice more.
Finally remove the pan from the heat and discard the cloves and cinnamon. When cool spoon into warm sterilized jars and when completely cold seal tightly. Serve 2 at a time with a little of its syrup as a finishing touch to a meal or with coffee.
tutumi kaghtsr pumpkin preserve
Pumpkin makes an excellent preserve. You can shred the vegetable instead of cutting it into squares, and some people like to add three tablespoons of slivered almonds or sliced pistachio nuts at the end of the cooking process.
This recipe is from Armenia.
1 small pumpkin, peeled and seeded. You need about 1.1 kg/2½ lbs flesh
2.3 litres/4 pints cold water
2 tablespoons lime powder (slake lime)
900 g/2 lb sugar
1.2 litres/2 pints water
2.5 cm/1 in cinnamon stick
3 cloves
2 tablespoons lemon juice
175 g/6 oz sugar
1 teaspoon vanilla essence
3 tablespoon almonds, slivered, or pistachios, sliced (optional)
Cut the pumpkin into 0.6 cm/¼ in thick slices. Now cut these slices into 4 cm/1½ squares or shred. In a large deep pan mix the water and lime powder together and soak the pumpkin in it overnight. Next day drain the pieces into a colander and then rinse very thoroughly under cold running water to remove all traces of the lime. Drain thoroughly.
In a large pan bring the sugar and water to the boil, stirring constantly until the sugar dissolves. Add the pieces of pumpkin, lower the heat and simmer about 1 hour. Pour some of the syrup into a smaller pan and to it add the cinnamon, cloves, lemon juice and 175 g/6 oz sugar. Bring to the boil, simmer for 2–3 minutes and pour this mixture back into the pumpkin pan. Lower the heat and simmer until the syrup thickens and coats the back of a spoon. Remove from the heat and stir in the vanilla essence and nuts if using them. When cool spoon into warm, sterilized jars and, when completely cold, seal tightly.
mourbaba-al-bousfeir preserved orange rolls
A popular Middle Eastern preserve where the orange peel is rolled, strung on to thread and then cooked in syrup.
You can prepare grapefruit, lemon and watermelon peel in much the same way. The finest such preserve of all is called mouraba-al-kabbad, a Syrian-Lebanese speciality made with the peel of bitter oranges.
Serve as a dessert or with tea or coffee.
6 large Jaffa oranges
500 g/18 oz sugar
600 ml/1 pint water
1 tablespoon lemon juice
Lightly grate the surface of the oranges to remove the shine. Cut the rind, skin deep, into 6 vertical sections and peel them away from the flesh. With a sharp knife carefully remove as much of the white pith as possible from the pieces of peel without cutting them. Tightly roll up each piece of peel and then with a needle and heavy thread string up the rolls. I suggest 18 rolls (from 3 oranges) on each string. Tie the ends of the threads together to form 2 garlands.
Put the threaded rolls into a large saucepan, cover with cold water, bring to the boil and then drain. Repeat this process at least 2 more times—this will remove the bitter taste. Place the rolls in the saucepan, cover with cold water again, bring to the boil and cook for about 30–40 minutes or until the rolls are tender. Drain them and pat dry.
Place the sugar, water and lemon juice in a large saucepan and bring to the boil. Add the strings of rolls and simmer for about 45 minutes or until the syrup is thick and the rolls are beautifully glazed. Remove from the heat and leave to cool. Carefully remove the strings and store the rolls, in their syrup, in sterilized jars. Seal tightly and store in a cool place.
Variation
mouraba-al-griffon preserved grapefruit rolls
Follow the recipe above but:
a) use 4 grapefruit
b) cut the rind of each into 8 vertical sections
c) boil in water 4 times instead of 3 to remove bitterness.
Morabaye hendevaneh
preserved watermelon rind
This recipe is from Iran, but the basic concept is the same throughout the region. The Greeks and Arabs would add a thinly peeled strip of lemon rind, a 5 cm/2 in piece cinnamon bark and 3–4 tablespoons toasted almonds towards the end of the cooking time.
700 g/1½ lb watermelon rind, peel green skin from the rind and remove all pink flesh
900 g/2 lb sugar
1.2 litres/2 pints water
1½ tablespoons lemon juice
1 tablespoon crushed cardamom
5 cm/2 in stick cinnamon (optional)
1 thinly peeled strip lemon rind (optional)
3–4 tablespoons toasted, blanched almonds (optional)
Cut the rind into 2–2.5 cm/¾–1 in cubes. You will probably end up with about 400–450 g/14–16 oz of cubed rind. Put the rind in a large saucepan, cover with water and bring to the boil. Lower the heat and simmer for about 1 hour or until the rind is translucent and tender. Drain into a colander.
Place the sugar, water and lemon juice in a large saucepan and bring to the boil. Add the crushed cardamom seeds and the cinnamon and lemon rind if using them.
Add the drained rind and simmer for 15 minutes, skimming off any froth that appears on the surface. Remove the pan from the heat, cover and set aside for 18–24 hours.
Next day return the pan to a moderate heat and bring to the boil. Lower the heat and simmer for about 20 minutes or until the syrup has thickened and coats the back of a spoon. Remove cinnamon and lemon rind and add the almonds if using them, stir well then remove from the heat and set aside to cool. Spoon into warm sterilized jars and seal when completely cold.
One day Hadji Baba of Isphahan was confronted by a young man who wished to test the reputed wisdom of the master clown.
'Tell me Hadji,' he asked, 'which is the most useful and perfect fruit?'
'Why, it's the watermelon of course,' replied Hadji Baba calmly.
'How come?'
'Because one eats the fruit, munches the seeds and makes a lovely morabaye with the skin.'
'I thought the skin was for one's donkey?'
'No way, young ass. I wouldn't give it to you, let alone to a donkey!'
Then, smiling to himself, he went his way..
orojig grape juice and walnuts
This sweet is also known as sweet soujouk, but in Eastern Turkey and Armenia where it originated it is better known as orojig or roejig meaning 'round balls'. Basically, it is halved walnuts dipped in a grape juice and sugar syrup. There is an equally popular sweet where a halved walnut is alternated with a blanched almond, etc. and then dipped in the syrup—this is known as goshdig—meaning 'crooked one' since it lacks the evenness of the former and is coarser in shape.
Sometimes other fruit juices are used instead of grape juice, but in my opinion nothing beats the original version given below.
Needless to say this sweet should be dried under the sun, but as is often the case one has to improvise and near a radiator or in an airing cupboard will do.
350 g/12 oz plain flour, sifted
450g/1 lb sugar
100 g/4 oz cornflour
3.4 litres/6 pints grape juice (or apple juice or orange juice)
halved walnuts
icing sugar, sifted
Mix the flour, sugar and cornflour together in a large bowl. Add a little of the grape juice and mix to a smooth paste. Little by little add the rest of the juice, stirring all the time.
Take a length of string and a strong needle and thread the first 2 walnut halves back to back into the middle of the piece of string. The nuts on either side of the 2 centre nuts should face the same way as those 2 nuts.
Tie the 2 ends of the string to a stick leaving a gap of at least 7.5 cm/3 in so that the nuts do not touch. The flat surface of the nuts must be facing upwards so that the thickened grape juice does not slip off.
Pour half of the grape juice mixture into a large saucepan and gradually bring to the boil, stirring constantly. Lower the heat and simmer, still stirring, until the mixture thickens. Dip the strung nuts into this juice 4–5 times, or until they have picked up most of the liquid. Hang to dry overnight above something in which to catch the drips.
Repeat the previous paragraph of instructions with the remaining juice, but this time leave to dry for several days. When completely dry cut into 15 cm/6 in lengths, roll in icing sugar and store in an airtight container.
## ice cream
Dondurma or ice cream is the name given to a sweet confection usually made of cream or milk (and in recent years with yoghurt), then flavoured and frozen.
China is claimed to be the original homeland of this invention, but I really can't accept this mythical reasoning since at no time have the Chinese ever mastered or developed this wonderful sweet.
It all probably started in Northern Persia and from there passed to the Greeks and Romans. It arrived in Britain via France in the seventeenth and eighteenth centuries, but it was already well established in the Middle East over one thousand years ago. The Khalif of Baghdad (in the tenth century), while on a pilgrimage, ordered his servants to bring snow, loaded on the back of countless camels, from the mountains of Persia so that he could drink cold sherbet flavoured with orange or mulberry juice.
In my childhood ice cream was made—as it was in Britain up until the early decades of this century—in large barrel-shaped cylinders with hollow centres filled with ice and salt. The whole thing was continually rotated with a thick wooden pole enabling the ice cream to freeze.
The finest ice creams in the Middle East, in my opinion, are still made in Damascus, Syria. Particularly the brilliantly white dondurma kaymakli which is literally hung up like an animal carcass in the ice cream shops and parlours—and in the middle of summer. Mastic (gum arabic) acts as the glueing element and hardens the ice cream. The recipe below is for this marvellous sweet. Try it and you will, I hope, agree with me that it is truly a great work of art.
dondurma kaymakli mastic ice cream
1 teaspoon powdered sahleb or 1½ tablespoons cornflour
900 ml/1½ pints milk
300 ml/1½ pints single cream
225 g/8 oz sugar
½ teaspoon powered mastic
3 teaspoons orange blossom water
Garnish chopped nuts of your choice
Dissolve the sahleb or cornflour in a little of the cold milk.
Pour the remaining milk into a large saucepan and stir in the cream and sugar. Bring slowly to the boil, stirring constantly until the sugar has dissolved. Pour a little of the hot milk into the sahleb or cornflour mixture and mix well. Pour into the large saucepan, add the mastic and simmer, stirring constantly, until the mixture thickens. Remove from the heat, stir in the orange blossom water and beat thoroughly with a wooden spoon. Pour the mixture into freezing trays and freeze.
Three to four hours later remove the trays, spoon the ice cream into a large bowl and beat lightly. Return to the trays and freeze again. Repeat this once more. Serve the ice cream sprinkled with the nuts of your choice. Serves 8 people.
seftali dondurmasi peach ice cream
This is a particularly refreshing ice cream. The recipe is from Southern Turkey.
1 tablespoon powdered gelatine
350 ml/12 fl oz water
175 g/6 oz sugar
5 fresh peaches
3 tablespoons lemon juice
Garnish 1 tablespoon pistachio nuts, finely chopped
Put the gelatine to soften in a small bowl with 4–5 tablespoons of the water.
Meanwhile, boil the rest of the water in a saucepan. Add the sugar and continue to simmer for about 5 minutes. Remove from the heat, stir in the gelatine mixture until it dissolves and set aside to cool.
Blanch, peel and slice the peaches. Using a liquidizer purée the peach flesh. Add the purée to the sugar mixture, together with the lemon juice and mix thoroughly. Transfer to a freezing tray and freeze for 1 hour. Remove from the freezer and beat with a fork for 1–2 minutes. Return to the freezer until the edges have frozen, but the centre is still soft. Remove and beat again. Repeat this process once more and then leave in the freezer until hard.
Half an hour before serving transfer the ice cream to the refrigerator to soften a little. Serve sprinkled with the chopped nuts. Serves 6 people.
glidat egosim pistachio ice cream
This is an Israeli ice cream, the creation of a brilliant chef, Roger Dabasque. A simple and exceptionally attractive sweet that will delight all. Use unsalted pistachio nuts which are available from most Middle Eastern stores.
3 eggs, separated
6 dessertspoons sugar
pinch salt
100 g/4 oz pistachio nuts, unsalted and shelled
300 ml/½ pint double or whipping cream
few drops green food colouring
Place the egg yolks in a Pyrex bowl with half the sugar and beat thoroughly. Add the salt, place the bowl over a pan of simmering water and cook until the sugar dissolves and the mixture reaches a toffee consistency. Place in the refrigerator to cool.
Chop the pistachios finely. Beat the egg whites in a bowl until stiff and then fold in the remaining sugar. Whip the cream in another bowl until stiff.
Place the egg yolk mixture in a large bowl and then fold in the egg whites, cream, three-quarters of the pistachio nuts and just enough food colouring to give a pale green appearance. Sprinkle the remaining nuts over the bottom of 1 large or 2 smaller freezing trays and then pour the mixture over the top. Freeze and then turn out on to a serving dish when required.
paludeh Sorbet
In the Middle Ages there was a popular sweet in Baghdad called faludaj which was made of ground almonds, sugar and rosewater, about which the Arab chronicler Ibn-al-Jawzi remarked that:
'Had Moses come to Pharaoh with Faludaj he would have accepted Moses' mission, but alas, he came to Pharaoh with the stick.' Quoted from Social Life under the Abbasids.
Unfortunately the recipe for the above sweet does not exist, but what is available is a sorbet bearing the same name which is pure white with large crystals sparkling in it which is served with a syrup of choice poured over the top.
The recipe below is one of Iran's favourite and the topping is made of mulberries, although there is no reason why other syrups, e.g. strawberry or raspberry, cannot be used.
paludeh mulberry Sorbet
Sorbet
1 tablespoon powdered gelatine
2 litres/1¾ pints water
275 g/10 oz sugar
2 tablespoons sultanas
1 tablespoon rosewater
2 tablespoons pistachio nuts, finely chopped
Topping 75–100 ml/3–4 fl oz mulberry syrup or 225 g/8 oz mulberries mashed with sugar to taste
Put the gelatine in a small bowl with 4–5 tablespoons of the water to soften for 5 minutes.
Meanwhile, bring the remaining water to boil in a saucepan. Add the sugar and simmer for 5 minutes, stirring constantly until the sugar dissolves. Add the softened gelatine and continue to stir until it dissolves. Remove the pan from the heat and stir in the sultanas, rosewater and pistachio nuts. Pour the mixture into freezing trays and set aside to cool.
When cold place in the freezer for 2–3 hours or until almost frozen. Remove from the freezer and spoon the mixture into a bowl. Stir lightly with a fork—this should distribute the sultanas and nuts evenly. Return to the trays and freeze until solid.
When ready to serve spoon the sorbet into individual dishes and pour a little of the mulberry syrup or the mashed mulberries over the top.
Serves 8 people.
oghi sorbet raki sorbet
Sorbet
450 ml/¾ pint fresh orange juice
150 ml/¼ pint fresh lemon juice
600 ml/1 pint water
350 g/12 oz sugar
2 tablespoons orange blossom water
Topping
12 strawberries (or raspberries or mulberries)
1 small apple, peeled, cored and thinly sliced
1 small pear, peeled, cored and thinly sliced
1 apricot, peeled, stoned and thinly sliced
3 tablespoons raki
Strain the orange and lemon juice.
Place the water and sugar in a saucepan and bring slowly to the boil, stirring constantly until the sugar dissolves. Simmer for 5 minutes. Remove from the heat and stir in the orange and lemon juice and the orange blossom water. Set aside to cool.
When cold pour into freezing trays and place in a freezer for 2–3 hours or until nearly solid. Remove from the freezer, spoon the mixture into a bowl and beat lightly with a fork. Return to the trays and place in the freezer until solid.
Meanwhile, prepare the fruit-raki mixture. Place all the prepared fruit in a bowl, add the raki, toss so that the fruit is coated and then set aside for 45 minutes to marinate. To serve scoop several tablespoons of the orange sorbet into each individual dish and spoon some of the fruit mixture over each.
An ideal after-dinner sweet.
Serves 8 people.
## khumichk–drinks
One day King Djem was sitting in his tent, watching his archers practising, when there appeared in the sky a great bird hardly able to fly because of a snake which had wrapped itself around its neck. This would be an intolerable sight to an Aryan, since birds belonged to Good Creation while reptiles were the most frightful of the Bad. Djem ordered one of his archers to aim at the snake and kill it, and to take care not to harm the bird. The arrow delivered a mortal blow to the snake, which immediately released its prey and fell to the ground, while the bird flew off and disappeared over the horizon.
Not many moments passed before it reappeared and landed on the ground in front of Djem, and as if wishing to show its gratitude dropped some seeds from its beak at his feet.
These seeds, which were of a kind no one had seen before, were picked up and planted, and in a little time put forth a plant which grew and flourished in its season, producing beautiful fruit in great bunches. It was the vine.
The king noticed that the delicate skin of the lovely fruit enclosed a liquid content which would be easy to separate from the pips; so he set his servants to work to do this, and enclosed the resulting juice in a jar. After a few days the king decided to taste it, no doubt assuming it would be something like mead or similar drinks. However, he was repelled by such a strange, bitter taste that he thought it must be poisonous and kept it on one side with the thought—or so the oriental narrator candidly opines—that it might one day be useful in affairs of state.
It so happened however that Djem had a very beautiful and dearly loved girl slave. One day when he was out hunting she was taken ill with violent pains in the head so that she was unable to have a moment's rest. Nothing that others could do could bring her solace. At last, driven mad and in despair, the poor girl decided to kill herself and remembered the poison the king had put aside. She opened the jar and began to drink. She drank so much that she fell asleep, and when she awoke she found that she was perfectly well again. When Djem returned, she told him what had happened. As a result the king changed his opinion of the nature of the beverage whose recipe he had discovered; instead of using it for devious purposes of state he used it as a medicine, with such success in so many cases that wine came to be known to the old Persians as Darou-e-Shah, meaning 'the king's medicine'.
The World of the Persians, J. A. de Gobineau.
Thus according to ancient vedic tradition wine was invented, a drink that, one way or another, originated in the Middle East. Noah, we are told, after descending from the Ark became a husbandsman 'and he planted a vineyard; and he drank of the wine, and was drunken; and he was uncovered within his tent.'
The Greek God Bacchus was reputed to have been born in Asia Minor, and throughout the ages the vineyards of Armenia and Iran were famed. Yet with the arrival of Islam this rich viticulture was almost devastated, so much so that the Pahlevi Persian word for wine sharab (the king's water) came to suggest all types of drinks made of syrups, fruits etc.
There is a great paucity of alcoholic beverages in the Middle East due primarily to religio-social demands. The only areas that have continued the age-old tradition of wine making are—apart from mainland Greece—Cyprus, Lebanon and the Caucasus—all Christian lands where the influences of Islam have remained minimal. Where Islam has taken root, wine has been uprooted.
Today in Israel, Lebanon, Cyprus and the Caucasian Republics there exists a flourishing wine industry. Particularly good are the wines of Cyprus, Georgia and Israel, the araks of Lebanon, the brandies and champagnes of Armenia. When all is said and done the drinking habits of the Middle Easterner are those of a sober, sweet-tongued teatotaller and what he has lost on drinks alcoholic he has made up tenfold with drinks non-alcoholic.
The Egyptians have various kinds of sherbets or sweet drinks. The most common kind is merely sugar and water, but very sweet; lemonade is another; a third kind, the most esteemed, is prepared from a hard conserve of violets, made by pounding violet-flowers and then boiling them with sugar. This violet-sherbet is of a green colour. A fourth kind is prepared from mulberries; a fifth from sorrel. There is also a kind of sherbet sold in the streets which is made with raisins, as its name implies; another kind, is a strong infusion of liquorice-root, and called by the name of that root; a third kind, is prepared from the fruit of the locust tree, and called in like manner by the name of the fruit. Manners and customs of the Modern Egyptians, Lane.
The love of the Middle Easterner for things sweet is well known. However, little is known to date of the methods whereby he achieves this 'sweetness'. Here is a little story about Al-Hallaj who drew straight out of rivers and streams delicious water smelling of attar of roses and camphor. To achieve that effect he took a brand-new water pot or pitcher, dissolved high-quality white sugar in rosewater and reduced it to a stiff paste over a fire. He then put it in the new pot and the clay vessel selected for the purpose, spreading it in an even layer over the inside walls. The vessel absorbed it and so it became like an internal coating. He kept the water pot beside him. When anyone asked him for a drink, he used it to draw the water, waited for a moment to let some of the paste dissolve in the water and then gave it to the man. Those who drank it were unaware of all this. They thought the water had been transformed into rose syrup and they believed whatever he wanted them to. Al-Hussein ibn Mansour al-Hallaj—Muslim theologian and mystic. The Subtle Ruse.
'Sous-sous', chanted the street vendor. 'Aryan-aryan', echoed another trying to out-do the songs of the donkey and horse-drawn carts and the moans of the hamals who often acted as human donkeys. These vendors carry a selection of colourful drinks made of mulberry or sous, yoghurt, lemons, oranges, rosewater, tamarind, etc. The large glass flasks are held by a strap balanced on the shoulders. In recent years modernization has brought a certain amount of change to the lives of the street vendors; some are now in kiosks where, apart from the age-old favourites, they also offer their thirsty customers, with the aid of certain modern contraptions, freshly squeezed juices from almost any fruit or vegetable of their choice.
The recipes below are a personal selection from the extremely rich non-alcoholic repertoire of the Middle East. All are well worth preparing if only to remind oneself how much more delectable the products of nature are when compared with all things commercial and synthetic.
visne serbeti cherry syrup
The most successful drinks are those made of fruits and of those the most famed, especially amongst Turks and Iranians, is a syrup made from big, black cherries which was called 'the Queen of Queens' by an infatuated Persian poet. It has a unique sweet, yet tart flavour.
450 g/1 lb large dark sour cherries washed
900 g/2 lb sugar
600 ml/1 pint water
¼ teaspoon vanilla essence
Place the sugar and water in a large saucepan and bring to the boil, stirring constantly until the sugar dissolves. Lower the heat and simmer for 15 minutes. Add the cherries and simmer for a further 20 minutes, stirring frequently, until the syrup thickens. Remove from the heat and cool for 10 minutes.
Strain the mixture through a muslin bag into a bowl and add the vanilla essence. If possible suspend the bag over the bowl and let the liquid drip through. When cold squeeze the bag tightly to extract all the liquid. Pour the syrup into sterilized bottles and seal. To serve dilute to taste with water and ice.
tout sherbat mulberry syrup
A fabulous drink beloved by all orientals from Morocco to Iran. Use large black, ripe mulberries—the kind that stain your hands dark red. Place them in a muslin bag and squeeze to extract all the juice. Measure the volume of juice and pour it into a saucepan.
To the juice add double the volume of sugar and 1 tablespoon of lemon juice for every 300 ml/½ pint of juice. Bring slowly to the boil, stirring constantly until the sugar has dissolved. Lower the heat and simmer, stirring occasionally, until the syrup thickens—it should coat the back of a spoon. Remove from the heat and leave until cold. Pour into sterilized bottles and seal. To serve dilute to taste with water and ice.
nouri osharag pomegranate syrup
A favourite drink of mine. I remember making my first pomegranate syrup when I was ten-years-old, with fruit which came from our own tree in the garden.
A pleasant, sharp, refreshing drink which is a brilliant red.
15 large, ripe pomegranates, peeled
juice 1 large lemon
½ teaspoon orange blossom water
175 g/6 oz sugar
Using a lemon squeezer or juice extractor squeeze out as much of the juice as possible from the pomegranate seeds. Strain the juice through muslin into a large jug. Add the lemon juice, orange blossom water and sugar and stir constantly until the sugar has dissolved. Chill the juice. Serve in tall glasses with crushed ice. It looks magnificent.
sharbate rivas rhubarb syrup
A particularly good drink beloved of the Iranians and ideal in the summer months.
45 g/1 lb fresh rhubarb
600 ml/1 pint water
550 g/1¼ lb sugar
Wash and trim the rhubarb and cut into 2.5 cm/1 in pieces. Place the water in a saucepan and bring to the boil. Add the rhubarb, lower the heat and simmer for 20 minutes or until soft. Remove from the heat and leave to cool for 10 minutes. Strain the mixture through a muslin bag into a bowl. Squeeze to extract all the liquid.
Measure the juice and make up to 600 ml/1 pint with water if necessary. Return the juice to the saucepan, add the sugar and bring to the boil, stirring constantly, until the sugar dissolves. Boil for 5–6 minutes and then set aside until cold. Pour into sterilized bottles and seal.
To serve pour 5–6 tablespoons of the syrup into a glass, stir in 6–7 tablespoons water and crushed ice.
sekanjabin sweet-sour mint syrup
Vinegar may relieve a toothache, but is injurious to healthy teeth. Shabbatt 11a
'Sekanjabin,' said my sister who is married to an Iranian, 'is well worth including in your book. It is one of those versatile concoctions that the people here love. You can eat it by dipping Cos lettuce leaves in it—as they do for breakfast, brunch or as a dessert. You can drink it and, among the party-giving set, it is made into a punch.' And she gave me these recipes.
about 500 ml/17 fl oz water
350 g/12 oz sugar (or more to taste)
100–150 ml/4–6 fl oz (to taste) wine vinegar
4 large sprigs fresh mint, washed and drained
Bring the water to the boil in a saucepan. Add the sugar and stir constantly until it has dissolved. Add the vinegar and simmer for 20 minutes or until the syrup has thickened and coats the back of a spoon. Add the mint, stir well and set aside to cool. The syrup should now have the consistency of thin honey.
Strain the syrup when cold into a sterilized bottle and seal.
To serve
a) | For breakfast or as a dessert — separate the leaves of a Cos lettuce, wash and pat dry. Pour some syrup into a bowl. Fold the leaves, dip into the syrup and eat—very refreshing.
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b) | As a drink — about one third fill a glass with syrup. Top it up with water, or, as in fashionable circles, with mineral water or soda water. Stir, drop in some ice cubes and serve.
c) | As a punch — grate a small, sweet cucumber into a punch bowl. Add syrup, ice cubes and soda water. Stir well and garnish with mint sprigs and cucumber slices.
tan yoghurt drink
Yoghurt, one of the major ingredients in the Middle Eastern cuisine, also makes a fine, refreshing drink called tan or aryan or dough.
This is undoubtedly the most popular drink of all, served at home, in restaurants, by street vendors and, in recent years, in supermarkets in litre-sized bottles. The proportions below are for one person. Simply increase the ingredients in proportion to the number of people to be served.
2 tablespoons yoghurt
300 ml/½ pint water
¼ teaspoon salt
¼ teaspoon dried mint
some ice cubes
Spoon the yoghurt into a glass and very slowly stir in the water to make a smooth mixture. Stir in the salt and mint. Drop in a few ice cubes and serve.
If preparing this drink for more than one person then mix in a large jug first.
Variation
Iranians often substitute natural water with mineral water to give a lightly gaseous effect to this drink. Try it with one of the bottled varieties available.
Another delightful variation to the simple tan is this Armenian favourite:
salori-tan yoghurt and prune juice
450 ml/¾ pint juice
450 ml/¾ pint natural yoghurt
2 teaspoons lemon juice
¼ teaspoon ground cinnamon
Place all the ingredients in a large jug and blend thoroughly. Pour into glasses and serve with a few cubes of ice.
Ideal on a warm summer's day this drink can also be made with other fruit juices, e.g. apricots, strawberries, etc.
chay tea
Tea has been known in the region for far longer than in the West. Indeed, when it first appeared amongst the conquering Arab armies in the ninth century it was highly disapproved of by the ruling élite, but the Iranian people and the Mongolian Turks regarded it as their national drink, and still do.
Although today in most Middle Eastern lands coffee has replaced tea as the social beverage, in Iran and the Gulf regions as well as Iraq and Kurdistan, tea still predominates. Indeed the Iranians, like the Russians, have created a whole school of tea drinking filled with mystique. Its brewing, its service and the very manner in which it is drunk have an aura of deep-rooted tradition which demands, in the first place, that the water is boiled in a samovar which is often an ornately decorated work of art and which creates a peculiar atmosphere of cosiness at the tea table. I am also assured that the tea itself tastes better when brewed with boiling water drawn from a samovar. The tea is never drunk with milk, and sugar is never added to it. A cube or, as in the past, a lump of concentrated sugar is put in one's mouth and tea is drunk through it. My father recalled, while living in Iran early last century, seeing people seated around a low table passing to each other a large block of crystallized sugar which was gently licked by one and all as they imbibed the aromatic black teas.
The finest tea comes from Iran, although in recent years both Turkey and particularly the Republic of Georgia have developed rich tea plantations. Incidentally, both these ventures were first started by pioneering English men who had received their experience in India.
In the Arab lands, where I spent most of my childhood, tea is only drunk in winter. It is served in small cups and sipped from a teaspoon and never from the cup—that would be regarded as impolite! To the Arabs tea has medicinal qualities and is often made with herbs and spices. Flavoured tea is recommended to the sick, the weak and the hypochondriacs! A popular one is chay-bi-yanasoun—tea mixed with aniseed. Below is the recipe for it. The proportions given are for one person. Simply increase the ingredients in proportion to the number of people to be served.
chay-bi-yanasoun aniseed tea
1 teaspoon tea
½ teaspoon powdered aniseed
sugar to taste
Make the tea in the ordinary way. Add the aniseed powder and sugar, if required, stir and allow to settle. Serve.
Chay-bi-nana mint tea
Although this has become the national brew of North Africa it is also very popular throughout the Middle East. It is considered to be good for upset stomachs, colds and flu. Serves 4.
3 teaspoons green tea
handful fresh, whole mint leaves or 1 tablespoon dried mint
sugar to taste
Warm the pot with a little hot water and pour out. Add the tea leaves, pour a little more hot water into the pot, swirl around again and pour out the water, but not the leaves! Add the mint and amount of sugar required. Add about 900 ml/1½ pints boiling water and steep for 5 minutes. If any mint surfaces remove it. Taste and serve. Do not add any sugar once it has been poured into the cup.
haygagan tey cinnamon and clove tea
The Armenians have their own famed tea—an aromatic infusion of cinnamon, cloves and tea leaves called Haygagan tey. Inevitably it is recommended for anyone feeling one degree under—whatever the ailment! Serves 4.
4 cups fresh cold water
4 cm/1½ in piece cinnamon
2 whole cloves
1 tablespoon tea leaves
sugar to taste
Place the water, cinnamon and cloves in a small pan and bring to the boil. Lower the heat and simmer for 5–7 minutes. Turn off the heat and stir in the tea leaves. Steep for 2–3 minutes and then strain into cups. Serve with sugar.
ainar tea with nuts
Infusions are extremely popular with Middle Easterners. Sweet basil, jasmine, rose petals, coriander, sage, verbana, etc. are all added to boiling water and drunk with gusto. One of the most popular ones, a Lebanese speciality, is ainar which is traditionally served when a child is born. Serves 5.
1 tablespoon caraway powder
1 tablespoon cinnamon powder
1 tablespoon aniseed
⅛ teaspoon ground nutmeg
5–7 tablespoons pine nuts, walnuts, almonds or a mixture
sugar to taste
Place all the spices in a small pan with 5 cups of water and bring to the boil. Simmer for 5 minutes. Strain through muslin into a tea pot. Place 1 tablespoon, or a little more, of nuts in the bottom of each cup. Add the required amount of sugar. Fill the cups with the hot, spiced water and serve.
Drink from a teaspoon, a sip at a time.
asal du kouzbara Coriander honey tea
Another infusion is this recipe which is from Arabia and which is drunk hot or cold. It is highly recommended for colds and flu.
The recipe makes enough for one person. Increase proportions accordingly.
1 cup water
2 teaspoons honey
½ teaspoon ground coriander
Warm the water in a small pan. Dissolve the honey in the water, add the coriander and stir well. Serve hot or cold.
I cannot finish without telling the story of Boloz Mugush and his experience with a cup of tea.
One day the great wit and a friend, Hamo, were thirsty.
'Let's go to the Chaykhana [tea house] for a drink,' said one. They ordered a large glass of black tea.
'You drink your half first,' said Hamo, 'because I have a cube of sugar that's just enough for me.'
'Add it now,' said Boloz Mugush, 'and I shall drink only my half.'
'Impossible. One cube is not enough for both of us. The tea will never be sweet enough.'
Boloz Mugush went to see the proprietor. A few minutes later he returned, all smiles, with a tablespoon of salt.
'Problem solved, Hamojan,' he said. 'I am drinking first as agreed and I want my tea with lots of salt.ߣ
kahwah arab coffee
'Let's have a cup of coffee and talk politics.' —Arab expression for a relaxing time.
Originally a poetic name for wine in old Arabic, the word was transfered to mean a drink made from the berry of the coffee tree (coffea arabica) and was first popularized about the thirteenth century in Arabia amongst dervishes and Muslim pilgrims who took the idea of brewing a strong cup of coffee back with them to all corners of Asia and Africa. Claudia Roden's fine book, Coffee, gives a full history.
In time coffee was introduced to Europe. Coffee houses sprang up everywhere. The pioneers were Armenians (as with yoghurt, backgammon, Turkish-type cigarettes and Turkish delight, etc.). In France there were Pascal and Krikor, in London Pasqua Rosée and in Italy Manuel Armeno.
In the Middle East the drinking of coffee still retains a strong religious-social mystique about it. Perhaps it is the early religious use of coffee that has given it a ceremonial character in the world of Islam. The dervishes of old drank coffee to keep awake during the nights given to religious devotion. The drink was kept warm in a large red earthenware vessel, each dervish receiving some in turn from his superior, who dipped their small bowls into the jar. They sipped the coffee while they chanted 'Allah w' akbar!' (God is great). After the dervishes were served, the jar was passed round to the rest of the congregation. Never was a religious ceremony performed without coffee being drunk.
Today, centuries after it became secularized, coffee drinking is still in the Middle East an activity enmeshed in ritual, practised at all times throughout the day.
As the most important drink of the region coffee is claimed by Greeks, Turks, Armenians and Arabs as their very own. The truth however, is much simpler. It belongs to none, but to all. I am, of course, referring to coffee prepared in the oriental way—often called 'Turkish' or 'Greek' coffee which the French traveller Thevenot described as 'black mud' and said 'one must drink it hot, but in several instalments, otherwise it is no good. One takes it in little swallows for fear of burning oneself–in such fashion that in a café one hears a pleasant little musical sucking sound.'
You can have Arab coffee with sugar, without sugar and with spices. It is all a matter of taste and tradition.
There are several versions of this 'black mud' and some of the more important ones I have included below.
When buying the coffee remember to ask for 'Turkish' coffee beans and make sure they are 100 per cent pulverized and fresh.
Coffee is traditionally prepared in a jaswah (Arabic) or ibrik (Turkish)—a smallish, long-handled metal pot (copper or brass).
The recipe below is for one person so you can increase the proportions accordingly. The amount of sugar depends on personal taste, but the usual quantity is 1 teaspoon per person called orto in Turkish, mazbout in Arabic, michag in Armenian and metrios in Greek.
1 teaspoon sugar, 1 coffee cup water, 1 teaspoon coffee
Mix the sugar and water together in the jaswah and bring to the boil, stirring until the sugar has dissolved. Add the coffee, stir well and bring to the boil. As the coffee froths up remove the jaswah from the heat and allow the froth to subside.
Return the jaswah to the heat until the froth reaches the brim. Remove once again. Repeat this process 2 more times. Remove from the heat and pour into the coffee cup.
Do not add more sugar and do not stir or you will disturb the sediments at the bottom of the cup.
Variations
Armenian: As above, but add I crushed cardamom seed and 2 drops orange blossom water.
Anatolian: As above, but add 2 drops rosewater.
Cypriot: As above, but add a few drops of cold water to the coffee in the cup.
Ottoman: This is the version first popularised in the West—it is thick and very sweet. Allow at least 2 teaspoons sugar per person.
Bedouin: Known as kahwah-al-hilo this coffee is very thick and flavoured with cardamom and saffron.
1 teaspoon coffee
1 coffee cup water
½ cardamom seed, crushed
½ teaspoon saffron, powdered
Combine the above ingredients in a jaswah, stir to dissolve and bring to the boil. Reduce heat to very low and leave for 20 minutes until reduced and thick. Add sugar to taste—normally very little if any at all.
The coffee will have a slightly bitter flavour. Drink a few sips at a time.
After the coffee is drunk one always thanks the Lord—'Shoukran Allah', then the cups are turned upside down on to the saucers and one's fortune is read. Coffee cups are read (like tea leaves) by those in the know. Theirs is an hereditary gift that few mortals possess. My aunt was one such gifted person to whose abode came women from far and wide for a chat and a quick read. She had the gift of the gab and, most certainly, a deeper understanding than most of the human psyche. They sat around hanging on to every word she uttered—a letter, perhaps soon, from a distant relation bringing (most certainly) good news; and a paper (a cheque!) perhaps; one was to travel—perhaps across the seas; a small ailment in the family, but nothing to worry about, not serious (Praised be the Lord); then there was a certain young man waiting, hoping. Where? There, just near that blob of coffee grounds; a stag—a sign of prosperity and good times. Life would be much better soon (after the cup was drunk).
## glossary
This is a selected glossary with emphasis on the rather unusual vegetables, herbs, spices and ingredients.
Most of the ingredients used by Middle Eastern housewives are now easily available in Western shops, although a few are still difficult to trace. There are also certain basic rules that apply to the preparation of some recipes and these have been included—although every individual recipe is fully explained.
Aubergine
Also known as eggplants, aubergines are indigenous to the Indian subcontinent, whence they spread to the Middle East and later, via Muslim Spain, into Europe.
They are popular throughout the region particularly with Turks and Armenians who boast over 200 recipes.
Aubergines are used in soups, make excellent dips and salads, and are stuffed with meat and/or rice or burghul. They also feature in many stews and casseroles and preserves. Recipes give details of preparation in most cases, however, as a general rule when sliced or cubed, sprinkle with salt, cover with a plate and set aside for about 30 minutes. This will ensure that the bitter juices, which are particularly so if the aubergines are from Spain or the Canary Isles, are released. Rinse the slices or cubes, pat dry with kitchen paper and proceed with the specific recipe.
The ideal way to cook the flesh for many of the dips and salads is over charcoal, but a hot oven or grill will do instead. They are ready when the skin has turned black and the flesh is soft. (They can also be cooked in a microwave oven.) Make 2–3 slits with a knife, place in a dish and cook for 4–5 minutes. In this case the skin will not change colour.
Peel off the skin and proceed with the required recipes as described.
Baharat
This is similar to allspice (bahar), but includes cumin, coriander, pepper and paprika.
Black-eyed beans
Native of Africa. Very popular in Ethiopia, Egypt and, in recent years, they have become more and more popular in the Middle East for they are sweeter and much quicker to prepare than some of the other beans.
Make fine soups and salads.
Broad beans, small
Also known as Egyptian brown beans. They are used only in their dried form.
First cultivated in Egypt, these are the national food of the Fellahin who make the classic ful medames.
Soak the beans for 14–18 hours and then cook very slowly.
Burghul (cracked wheat)
This is hulled wheat which is steamed until partly cooked, dried and then ground into 3 grades:
Large—used for pilavs and stuffings
Medium—for fillings
Fine—for kibbehs, kuftas and salads
The national cereal of Armenia, originated most probably with the first civilizations of Assyria and Urartu, but was replaced when rice was introduced via Iran and the Gulf regions.
Burghul is still very popular with Armenians and is extensively used by Syrian and Lebanese cooks in the preparation of the classic kibbehs. Armenians use burghul for pilavs, dolma fillings, salads and several puddings.
Burghul is sold in most good Indian and Middle Eastern shops.
Borekler
The Turkish name for pie-like pastries with meat, cheese, spinach, brain and many other fillings. These are either baked in the oven or deep fried.
Borekler are particularly popular with Armenians and Turks, though their origin can be traced to the Mongolian and Chinese.
Butter, clarified (see ghee)
Cardamom
Originally from the Indian subcontinent, this spice is particularly popular with Iranian, Iraqi and Gulf cooks.
It is available in pods, as seeds or ground.
Arab coffee often includes one or two crushed pods or a little ground cardamom to give it a characteristic pungent flavour. Is also used in pilavs and sauces.
Carob
An evergreen tree of Mediterranean origin. The dried pods are eaten as a snack for their sweet, chocolatey flavour. The juice of the carob is substituted for syrup and when mixed with tahina makes the classic tahini-roub.
Dry carob can be purchased from most health shops and carob syrup from Middle Eastern grocers.
Chickpeas (garbanzo beans)
First cultivated in Egypt, chickpeas are used extensively throughout the region. They appear in dips, salads, stews and kuftas. They are also roasted, salted and eaten like peanuts; or roasted, coated with sugar and served with tea or coffee.
Chickpeas need to soak for 24 hours before cooking. Some recipes require them to be skinned. In which case after they have been soaked take each pea between thumb and forefinger and squeeze firmly; it should pop out of its skin.
You can purchase skinless chickpeas from some Indian and Middle Eastern stores.
The most famous dishes are hummus-bi tahini and nvig.
Chilli
First introduced via the Gulf region where it is still widely used.
Chillies are either red or green. Whole, fresh pods can be used, with or without seeds, although it is usually better to remove the seeds as they are very hot.
More often whole, dried chillies or ground chilli is used as these are available all the year round.
Cress (rock cress)
A member of the pleasantly pungent herbs of the mustard family, rock cress grows wild in the Caucasus and northern Iran and is used in stews, salads and rice dishes in these regions. Ordinary watercress will make a suitable substitute.
Cumin (black)
These are small, black aromatic seeds used in pastries, sweets and also to flavour rolls, cakes and breads. They are used by Christians in their Easter breads and cakes but are not known to most of the Muslims of the region. Cypriots and Lebanese use them in their haloumi cheese and the Armenians in tel-banir (hair cheese).
Dates
A member of the palm family, Phoenix dactylifera is indigenous to Arabia and North Africa. It has been cultivated for over four thousand years. It was known to the Ancient Egyptians who cultivated it commercially, as did the Sumerians.
Dates are a staple source of food for the Bedouins, and one of the main ingredients in true 'Arab' cooking of the region.
Dibs (pekmez)
A syrup which is made from the Carob pod in Syria and Lebanon and from grape juice in Armenia and Turkey.
See carob and pekmez.
Dolma
This is the Middle Eastern method of cooking vegetables, such as aubergines, courgettes, peppers, tomatoes, onions, etc., whereby they are filled with meat and/or rice or burghul and flavoured with a variety of nuts and spices.
Most probably of Armeno-Assyrian origin, dolma, although widespread throughout the region, is still most popular amongst Armenians and Anatolians (Turks).
Dried limes
A must in both Iranian and Gulf cooking. These small but very sharp limes are grown extensively in southern Iran, Iraq and Oman. They are grey-brown, walnut-sized and very bitter.
Used in such dishes as gormeh sabzi and dizzi. Also, as in Iran, they are sprinkled with a little salt (to take away the bitterness) and then sucked for the juice.
Fenugreek
Native to the Mediterranean this most unusual herb with tiny reddish seeds is very popular in Iranian, Armenian and Yemeni cooking. The small oval leaves are used in vegetable stews.
The Armenian aboukhd uses it in its powdered form as a thick paste, while in Yemen it is used in sauces.
It has a very pungent, bitter flavour resembling that of burnt sugar.
Feta
A popular Middle Eastern cheese that has a soft, crumbly texture and is normally made from ewe's or goat's milk. Cow's milk produces a much firmer feta. Usually served with lavash or pita bread, either as a meal with fruit or as an hors d'oeuvre or side dish.
In Turkey and Iran a large portion of the populace live on feta, or similar cheeses, which they eat with bread, fresh tarragon, chives, mint, spring onions, etc.
If you like your cheese salty then eat the feta as it comes from the jar or packet (where it is preserved in brine), otherwise soak it for a few hours in cold water, changing the water from time to time.
Ghee, clarified butter
Ghee is pure butter fat which can be heated to high temperatures without burning. It is superior to ordinary butter and has a fragrance of its own. It is used extensively in Indian and Middle Eastern cooking, and is available in all Indian shops, but if you are unable to buy any then you can easily make your own which you then cover and refrigerate until you need it.
Melt 900 g/2 lb unsalted butter in a large saucepan over a low heat. Skim off the foam with a wooden spoon as it appears on the surface. Remove the pan from the heat and let rest for 10 minutes. Now skim off any foam that remains on the surface. Spoon the butter into a container, discarding the milky residue left in the bottom of the pan. Cover the container and refrigerate. 900 g/2 lb of butter should make about 700 g/112 lb of clarified butter which can then be used in all kinds of pastries, cakes and in cooking in general and especially when frying fish.
Haloumi
A salty, sheep's milk cheese made throughout the region. It is matured in whey, is string-like in texture and is often flavoured with chives, mint or black cumin.
Halva
Halawah or halaweh is an ancient Akkadian word meaning simply—sweet. Halva nowadays means different things to different people, but basically it is a sweetmeat which usually has sesame seeds as one of the main ingredients. The other ingredients do vary but include sugar, naffit, bois de Panama, vanilla and nuts of choice.
There are countless variations (as noted in our introduction), but unfortunately it cannot easily be duplicated all that conveniently at home. Halva made at home usually consists of fine semolina, sugar, milk or cream and perhaps nuts and/or raisins. Halva can be purchased from most Middle Eastern stores. The best quality ones are from Egypt, Syria and Lebanon.
The Spanish turron slightly resembles halva—no doubt a legacy of the Moorish occupation of that peninsula.
Kaymak
Kaymak is the thick cream which can literally be cut with a knife. It is usually prepared with buffalo's milk, but can also be made with cow's milk or even sheep's.
Although often made at home it is also often bought from kaymakjis—small shops specializing in dairy produce. To make your own kaymak I suggest you follow one of the simple recipes below.
With double cream
Pour 1.2 litres/2 pints of double cream into a shallow enamelled saucepan. Use as wide a pan as possible to give the cream the greatest possible surface. Bring to the boil over a low heat. Using a ladle remove some cream and then pour it back into the pan. Do this from as high a point as possible so that bubbles are formed. Continue thus for 45–60 minutes. Turn off the heat and leave to rest for 3 hours. Place the pan in the refrigerator for 15 hours or more.
With milk
Pour 1.2 litres/2 pints of milk into a shallow enamelled saucepan. Add 300 ml/½ pint double cream and stir well. Bring to the boil over a low heat and simmer for 2 hours. Turn off the heat and leave the saucepan to rest for 5 hours. Place the pan in the refrigerator for 15 hours or more.
When you remove the pan from the refrigerator a thick layer of cream will have formed. Using a sharp knife free the edges of the kaymak and then cut into strips. Using a spatula remove the strips of kaymak to a large serving plate and then cut into squares or curl into rolls.
Kaymak is beautiful on its own topped with sugar, jam or honey; or served as a topping for pastries.
The nearest substitute is thick clotted cream.
Kebab
Kebab means meat cooked on or in the fire (i.e. oven). It does not really mean skewered cooking although this is often what is understood by the word 'kebab' these days.
Cooking on fire is, of course, one of the oldest methods of cooking, but 'kebab' cooking implies something more, i.e. being marinated in oil and spices and then cooked over wood or charcoal.
There are many variations on this method of cooking which is easy, quick and convenient.
However, there are several dishes, particularly in Turkey, called kebabs, e.g. tas kebab, cop kebab, which are not true (skewered) kebabs in that they are cooked in the oven in a Cassera dish.
Kibbeh
Of Assyrian origin kibbeh is minced meat with onion, burghul and spices. It is a speciality of Syria, Lebanon, Armenia and Kurdistan.
Kibbeh can be eaten raw, grilled on charcoal or baked in the oven. There are many variations with each region boasting its own as the 'finest, truest' kibbeh.
Kishk
A Syrian-Lebanese speciality of burghul fermented with milk and yoghurt. It is then salted, spread out to dry under the hot sun, then ground and stored for winter use. Then it is made into a soup or porridge with milk or water. Kishk can be purchased from some Middle Eastern stores. It is almost the same as the Armenian tarkana. (See recipe for tarkana.)
Mahaleb
This Cilician spice is gathered from the kernel of the black cherry stone. It has a sweet and spicy flavour, is pale brown in colour and is the size of a peppercorn.
Pound it in a mortar and use to flavour breads, cakes and choreg-type dry biscuits.
Unfortunately mahaleb is rather expensive and not very easily found—try good Armenian or Arab shops.
Melokhia
The young shoots of melokhia are harvested and the leaves (5–7.5 cm/2–3 in) are stripped from the stalks and used to make the classic Egyptian soup of the same name, popular since the days of the Pharoahs.
Fresh melokhia is difficult to find outside Egypt, but it is possible to find the dried version in good Middle Eastern stores.
Mezzeh
An Arab word, of Greek origin (mazo), loosely meaning hors d'oeuvre, but more precisely a selection of starters laid out on a large table, which precede a main meal of roast lamb, kebabs, etc.
The mezzeh table, which can include up to 200 starters, is a particular speciality of Syria, Lebanon and Palestine.
Nuts
There are five basic nuts used in Middle Eastern cooking: almonds, walnuts, pistachios, hazelnuts and pine kernels. Although coconuts and peanuts are known and used they are of much more recent origin.
Almonds
The almonds of Turkey have been famed throughout the ages. They are succulent, plentiful and exported in large quantities. They can be bought in shells or ready shelled. They are blanched by pouring boiling water over the kernels, leaving for 3–4 minutes and then draining and squeezing each one so that the kernel pops out of the skin.
To split an almond just separate the two halves with a thin, sharp knife.
To sliver (by far the most popular kind) soak for 5–6 minutes after blanching, cut each nut into 3–4 slivers and then dry out in a warm oven.
To grind either use the time-honoured mortar and pestle or else a blender.
Almonds are used in soups, sauces, pilavs, stuffings and in some of the most famed Middle Eastern desserts and puddings.
They are also roasted, covered in cocoa or with a candy coating.
Walnuts
The finest walnuts in the Middle East come from the Caucasus, particularly the Black Sea coast of Turkey and Georgia.
Walnuts can be purchased in shells or ready shelled. To slice or grind follow the instructions for almonds.
Walnuts appear in Turkish-Armenian and Caucasian dishes and are used in soups, sauces, vegetable salads, stuffings, desserts, pickles and jams.
Pistachio nuts
The pistachio nuts of Syria and Iran are the best. In the West pistachios are sold ready salted, but it is the unsalted version that is the one required for cooking. These are rather difficult to find, but specialist shops do often stock them. They are usually sold in their shells.
They can be sliced or ground as with almonds.
Pistachio nuts are particularly good in pastries and pilavs, and are prominent in Syrian and Iranian cooking.
Hazelnuts
The finest hazelnuts come from the Black Sea coast of Turkey and from Iraq. They are used extensively in Turkish (Laz) and north Iraqi (Kurdish) cuisines. Hazelnuts are used in sauces, stuffings, pilavs, vegetables and desserts. They can be purchased ready shelled and for cooking they should then be blanched and sliced or ground as with almonds.
Pine kernels
Pine kernels appear in Levantine (Mediterranean coastline) foods, particularly those of Syria, Lebanon and Palestine-Israel. They are also known as pignolia nuts and are the kernels from the cones of the stone or umbrella pine. The finest are from Lebanon, those from Turkey tend to be yellower, thicker and smaller. They are used in pilavs, stuffings, kibbehs and desserts.
Okra
Popularly known as 'Ladies fingers' or 'gumbo', okra is native to Africa and has been widely cultivated throughout the Middle East for millennia.
Okra is relatively unknown in the West, but in the East it is used in stews, vegetable dishes and pickles.
Fresh okra can be bought for only a few weeks of the year, while the rest of the time dried or tinned okra is fairly easily found in Middle Eastern and Indian shops.
If you wish to prepare okra then follow the simple method below:
Buy small, young okra with very little fuzz. Wash carefully and then dry with a kitchen cloth. Trim the stems but do not cut into pods. Sometimes okra is tossed in vinegar for 30 minutes—this prevents it becoming slimy during cooking. Use 1 cup of malt vinegar to 1 kg/2.2 lb okra.
Okra freezes well and so it is advisable to buy large quantities while it is available and then freeze it until required. To do so wash and trim the okra. Then in a large saucepan heat 4 tablespoons oil to 1 kg/2.2 lb of okra, add the okra and fry for 5 minutes. Toss very gently using a wooden spoon so that the pods do not break. Remove from the heat and leave to cool. Put into containers, seal and freeze.
Orange flower water
A concentrated liquid flavouring distilled from the blossoms of the bitter orange tree. It is used to flavour ice creams, syrups and pastries, and is very popular with the Iranians, Turks and North Africans. You can buy it in Middle Eastern shops and certain 'gourmet' shops.
Pastry—Filo
Filo is paper-thin dough used in the making of pastries such as baklava. This pastry can be bought commercially from Middle Eastern or Greek shops and it keeps for a long time when refrigerated.
If using commercial pastry follow the instructions below for its uses:
Remove the filo from its wrapping. Each packet usually weighs 450 g/1 lb and contains about 20–25 sheets. Sprinkle a little flour over a work top then open the sheets out flat. Moisten a tea towel until it is evenly damp and spread it over the top of the filo. Remove each sheet as you need it and always remember to recover with the cloth. After shaping the pastry to the required sizes always cover with a damp cloth because filo dries out quickly, especially at the edges. Any dough not used should be wrapped carefully and returned to the refrigerator.
Home-made filo
Here is a simple recipe for home-made filo. All you need is patience, space and plenty of practice to achieve a thin filo, so until that day I suggest that when making baklava or similar pastries you use half the number of sheets recommended in this recipe. Work fast but carefully.
700 g/1½ lb plain flour
1 teaspoon salt
450 ml/¾ pint tepid water
3 tablespoons olive oil
cornflour
Sift the flour and salt into a large bowl. Make a well in the centre and, little by little, add the water and knead by pressing the dough down and pushing it forward several times with the heel of your hand and folding it back on itself. Knead for 7–10 minutes until the dough is soft. Make a ball of this dough, make a large well in the centre, add 1 tablespoon of the oil and knead into the dough. Repeat this with the remaining 2 tablespoons of oil. By now the dough should be very smooth and elastic. Cover the dough with a cloth and leave to rest for 4 hours.
Divide the dough into 18 equal portions and roll each one into a ball. Use the cornflour to lightly flour a working top. Roll out each ball, one at a time, with a floured rolling pin into a 15–18 cm/6–7 in diameter. Place each round on top of the other, each one separated by greaseproof paper. When completed cover all with a damp cloth and leave to rest for 45 minutes.
To shape the dough lift a round, stretch it over the backs of your hands and pull your hands apart slowly and most carefully until the filo is paper thin. You must work carefully but fast since the dough dries quickly. When you have stretched a sheet to about 30–33 cm/12–13 in by 46–51 cm/18–20 in place it on a work top dusted with cornflour. Trim off the thick edges and you will have a sheet of dough about 28–30 cm/11–12 in by 38–41 cm/15–16 in. Cover with a damp cloth and proceed with the other rounds.
Always keep them under a damp cloth and when all are made proceed with the preparation of the pastry of your choice.
Home-made kunafeh filo
Kunafeh or kataifi filo is the shredded pastry-type filo which is used to make many well known pastries.
Kunafeh filo can be bought from Middle Eastern shops. It comes in 450 g/1 lb packets and will keep for a long time in the refrigerator.
I suggest that when making pastries which require this pastry you buy it, but for interest I have given below a method for making home-made filo. Although it is quite simple to make it does require a very special container—a kunaffahiah—a large, shallow metal dish with small holes where the dough flows on to a large, hot metal griddle in rope-like form and solidifies in a matter of seconds. The kunafeh shreds are quickly lifted and separated in a large bowl. When all the dough has turned to shredded pastry proceed with the recipe of your choice.
700 g/1½ lb plain flour
300 ml/½ pint tepid water
300 ml/½ pint milk
Sift the flour into a large bowl, gradually add the water and knead. Add the milk, little by little, and continue kneading until you have a smooth dough. Now start thinning the dough by gradually adding more water, stirring all the time, until the dough has the consistency of batter. Have a large metal griddle ready on a low heat.
With a large soup ladle pour some of the batter into the kunaffahiah. The batter will drop on to the hot metal griddle in a rope-like form and solidify in a few seconds. Lift the cooked pastry into a large bowl and continue until all the batter is cooked.
Pekmez
This is usually carob or grape juice. Carob juice can be bought from good Middle Eastern or health food shops. However, concentrated grape juice cannot be easily found in the west, but it is possible to make your own. Below is a simple method.
2.7 kg/6 lb grapes, stems removed
225 g/8 oz sugar
Rinse the grapes then place in a large saucepan with 2.3 litres/4 pints of water and bring to the boil. Lower the heat and simmer for 15 minutes. Pour through a sieve into a large bowl. Rub the grapes through with the back of a wooden spoon and then discard the pips and skin remaining in the colander. Pour the juice and pulp into a muslin bag and leave suspended over a bowl overnight. What is collected in the bowl is pure grape juice. Pour this juice into a saucepan, add the sugar and bring slowly to the boil, stirring all the time. Lower the heat and simmer for about 15 minutes or until the mixture thickens and becomes a golden yellow colour. Remove from the heat and when cool pour into a sterilized jar, seal and store.
Pomegranate
A fruit indigenous to the region and particularly popular with Caucasians, Kurds, and Iranians. Pomegranates keep for a long time if picked before they reach full maturity and they are stored in the cool, dry cellars most Middle Eastern homes possess.
They are eaten raw as a dessert and the seeds are often sprinkled over pilavs, fish and meat dishes.
Pomegranates make an excellent sharbat and, when the juice is concentrated, it is used in many Iranian and Armenian stews and kebab marinades.
Pomegranate syrup
This is the concentrated juice of the pomegranate fruit and it is sold in some Middle Eastern shops. I suggest you make your own which is much better than the commercial kind which is, anyway, sometimes difficult to find.
8 large, ripe pomegranates
175 g/6 oz sugar
Remove the skin of the pomegranates with a sharp knife. Remove the seeds from their hives by tightly squeezing the segments of the fruit in your palms. Now, unless you have a fruit-juicing machine, place a handful of seeds at a time in a muslin bag and squeeze the juice out into a bowl. Pour the juice into a small saucepan and heat through. Add the sugar and bring slowly to the boil, stirring all the time until it dissolves. Lower the heat and simmer for 15–20 minutes or until the mixture thickens to a syrup. Remove from the heat, leave to cool then store in a glass jar, and use as required.
Pulses/legumes
Some pulses require soaking before cooking. The problem with soaking first is that water-soluble vitamins like thiamine, niacin and riboflavin will be thrown away if the pulses are rinsed after soaking. Therefore I think it is advisable to rinse the pulses very thoroughly before soaking, picking out any small stones, dried beans etc, and then to soak and to cook the pulses in the soaking water.
Pre-soaked pulses
A. | Black-eyed, haricot, lima, red and cannellini beans. There are 2 ways of soaking these beans:
---|---
| • Wash the beans thoroughly under cold running water. Place in a saucepan and for each cup of beans add 3 cups (about 750 ml/1¼ pints) cold water and bring slowly to the boil. Boil for 2–3 minutes, cover, remove from the heat and leave until beans are fully plumped up.
| • Wash the beans thoroughly under cold water. Place the beans in a bowl and for each cup of beans add 3 cups (about 750 ml/1¼ pints) cold water.
| Leave to soak overnight.
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|
|
B. | Dried broad beans, chickpeas and split peas (some people don't soak the latter, but I find that they usually require it):
| • Wash thoroughly under cold, running water.
| • Place in a bowl and for each cup of beans or peas add 3 cups (about
| 750 ml/1¼ pints) cold water and leave to soak overnight. Most dried broad beans need about 48 hours soaking and if the room temperature is warm leave in the refrigerator or the beans may begin to ferment.
Non-soaked pulses
Lentils—brown, green or red—and mung beans do not need to be soaked.
Rinse thoroughly in a strainer under cold running water and pick out any stones, discoloured seeds, etc.
Skinning pulses
Certain recipes ask for skinned pulses.
Broad beans: soak these for 48 hours then squeeze each bean firmly and it should pop out of its skin. If this doesn't work slit the skin with a small knife or, as most Middle Eastern housewives do, with your fingernails and then squeeze.
Chickpeas and lentils: after soaking take a handful and rub between the palms so that the seeds rub against each other. Drop back into the bowl and the skins should float to the surface. Remove these and discard. Another method, and one I used when helping my mother, is to squeeze each chickpea between thumb and forefinger and let the pea jump into the water. A long and tiring process, but necessary for some recipes.
Quince
A small tree with a fragrant but tart fruit of unusual flavour. A quince has a greenish-yellow-golden colour. It is the size of an apple except that it is pear-shaped near the stem and has minute hairs over its surface. It is cooked in a variety of ways, e.g. stuffed with meat and rice, used in soups and stews.
In the West quince jam is well known, but otherwise this rather fascinating fruit is neglected.
Rice
Indigenous to the Far East (China and India) rice arrived in the Middle East sometime in the early centuries of our era, since it is not mentioned in the Bible. If the mainstay of Armenian and Anatolian cuisines is burghul, then that of Iran and the Arabians in general is rice. In the Middle East rice is grown in south-eastern Turkey, southern Iraq, Egypt and Iran.
In Iran there are 3 main qualities of rice (regardless of varieties):–(a) Berenj domsiah—long slender grains (b) Berenj sadri—grains slightly broken (c) Berenj champa—broken grains.
In the rest of the Middle East long grain rice, particularly the Basmati type, is the most popular choice.
The most imaginative rice dishes (pilaf in Turkish and Armenian, pollo in Persian and timman or roz in Arabic) are undoubtedly those of Iran, followed closely by the Anatolian pilavs. The rice dishes of Egypt, and Arabia in general, betray their simple nomadic background in their simplicity and limited choice of vegetables and herbs. However—and this is an important point—nowhere in the entire region will one be served a dish of plain boiled rice similar to those cooked in south-east Asia—the original home of rice.
Rice is used in soups, salads, as a stuffing for vegetables, for pilavs and in puddings and desserts. A bed of pilav is the ideal accompaniment to all kebabs and stews. Rice is also often served on its own with yoghurt as is the custom in Turkish-Iranian and Kurdish villages.
There are several methods for the preparation of rice which are given in the section on rice dishes. Whichever method you use the end result should be a light and fluffy pilav with each grain firm and separated and usually with a slight golden hue from the ghee/clarified butter. In short your rice dish should never have a sticky or mushy consistency. To achieve the perfect end follow the recipe to the letter for the amount of liquid used will be crucial in determining the final outcome of the pilav.
Rosewater
A characteristic fragrance of the Middle Eastern cuisine, it is distilled from rose petals—the finest being the pink-red damask rose. It can be used to flavour puddings, desserts and savouries and can be found in most Middle Eastern shops. Rosewater essence in a concentrated form is also available from many chemists. Use this sparingly in drops rather than spoon measures.
Saffron
A very expensive spice, for it takes something like the stamens of a quarter of a million blooms to produce 450 g/1 lb saffron. Saffron originated in Asia Minor and spread through the Middle East towards India.
To use pound the threads in a mortar then soak in a little liquid to bring out the colour and fragrance.
Used particularly by Iranians, Caucasians and Iraqis saffron gives colour to rice dishes, meat and chicken stews and to some desserts.
Salep
Salep is a ground powder from the dried tubers of various species of the orchidaceae family. It has a gelatinous quality similar to cornflour and is used in hot and cold drinks. It is rather difficult to find and arrowroot or cornflour is often substituted. The finest salep comes from Azerbaijan in the Caucasus. It is also used in ice cream and puddings.
Sesame seeds
Sesame is a rough, hairy, gummy annual about 60 cm/2 ft in height with egg-shaped leaves. It produces black and white seeds which are oily and highly nutritious.
Popular throughout the region since ancient times sesame grows well, indeed it thrives in poor soils, and its seeds yield half of their weight in the oil which is yellow, limpid and will keep for years.
The oil is called tahina, and the seeds are used on breads, cakes, sweets and in the famous halvas.
Sumac
The dried, crushed berries of a species of the sumach tree. Sumac has a sour, lemony taste.
Crush and steep it in water to extract its essence which can then be used in stews instead of lemon juice. It is extensively used in the Caucasian, north Iranian and Kurdish cuisines in their soups, salads and dolmas.
Please note that most sumach trees are poisonous so never make your own, always buy it from a reliable Middle Eastern shop and ask for 'Armenian sumac'.
Syrups
A traditional feature of Middle Eastern pastries and sweets is the syrup used on and/or in them. It is a sugar-based syrup and it is either thin and liquid or thick and almost treacly. Sometimes the North Africans and Caucasians use honey for syrup, the recipe given below is almost a standard one throughout the region.
450 g/1 lb sugar
300 ml/½ pint water
1 tablespoon lemon juice (prevents crystallization)
1–2 tablespoons rosewater or orange blossom water, or sometimes both
Place the sugar, water and lemon juice in a saucepan and bring slowly to the boil, stirring constantly. Remove any scum which appears on the surface. Lower the heat and simmer for 5–10 minutes. When the required thickness has been achieved remove from the heat. Gently stir in the required fragrance and leave to cool for future use. If covered it will keep for a long time.
Note—A simple method to see if the syrup has the required thickness is to see if it coats a metal spoon or, if you have a candy thermometer, allow the syrup to reach 110°C, 225°F and then remove from the heat.
Gavour flurubu (Infidel's syrup)
This is a charming variation on the standard syrup above.
A piece of cinnamon stick about 5 cm/2 in long is added to the syrup at the beginning of its cooking time and is removed when the syrup has cooled. There is no need for any other flavouring.
Tahina
A nutty-flavoured paste made from toasted and crushed sesame seeds. The finest quality tahini comes from Syria and Turkey. There are many cheaper versions which tend to be lighter and more runny.
It originated in Asia Minor and is very popular with Armenians, Syrians, Lebanese, Palestinians, Israelis and the Copts of Egypt. Is used in dips, lenten kuftahs and cakes.
Tahina separates if it has stood for some time, so always stir before using.
Tamarind
'Date of India' is the large bean pod of the tropical tree indigenous to south-east Asia. It is very popular in Iraq and the Gulf states.
Tamarind sauce can be purchased from Asian stores as can 450 g/1 lb weights of the compressed dried pods. If you cannot find the sauce follow the simple method below for making your own.
Salsat-el-sbar
450 g/1 lb tamarind pods
1.2 litres/2 pints water
450 g/1 lb sugar
Clean and wash the pods. Break the pods into smaller pieces and put into a large saucepan with the water. Leave to soak for 24 hours.
The next day rub the pods through a sieve using the back of a wooden spoon, collecting the liquid and pulp in a saucepan. Place the seeds and fibres in a muslin bag and squeeze any remaining liquid into the pan. Add the sugar to the pan. Add more sugar if you want a sweeter sauce. Slowly bring to the boil stirring constantly until the sugar has dissolved. Lower the heat and simmer gently until the syrup thickens. Remove any scum which appears on the surface. Remove from the heat, leave to cool then pour into jars and seal tightly.
Tarator
A Turkish name given to sauces making use of nuts, tahina, garlic and bread. These tend to be rather thick and are served with fish and with vegetables.
Turmeric
Often called the 'poor man's saffron', turmeric is the root of a plant of the ginger family. It has a slightly bitter, resinous flavour and is yellow in colour. It should be used in very small quantities.
Although known throughout the region it is most used in the Iranian, Gulf and Kurdish cuisines in sauces, soups, pilavs and the many Iranian khoresht-type stews.
Vine leaves/cabbage leaves
Vine leaves
The natural abode of the wild vine was the Caucasus from whence it spread south and to east and west.
A hint of its Caucasian origin is to be found in both the Bible—Noah made wine and got drunk on the slopes of Mount Ararat in Armenia; and in the Avesta of the Aryans—King Djem's slave fell ill but was cured by drinking the juice of the wild vine which she mistook for poison.
The vine was, and still is, the national symbol of the Armenian nation.
It is found on ancient monuments, churches and it has even entered the Armenian Church rituals with the blessing of grapes on Holy Mother's Day every August.
Vines are also portrayed on early Parthian friezes, but disappear with the advent of Islam.
Vine leaves were known and used by the Ancient Greeks and Sassanians who 'wrapped' wheat, and later rice and nuts, and cooked them in broth—similar to the sarma and yalançi dolma of today.
You can buy preserved vine leaves from most continental shops, but if you do have access to a vine then you can use the fresh leaves instead.
To prepare preserved leaves: first rinse under cold water then place in a saucepan half filled with water and bring to the boil.
Simmer for 15 minutes and then drain into a colander.
Leave until cool enough to handle.
To prepare fresh vine leaves: stack the leaves on top of each other, drop into a pan of boiling water and simmer for about 5 minutes. If the leaves are older cook for a little longer. Drain into a colander and leave until cool enough to handle.
Cabbage leaves
These are also popular in the Middle East and fresh ones are always used.
To prepare: take about a 900–1400 g/2–3 lb head of white cabbage and remove as much of the thick core as possible.
Bring a large saucepan two-thirds filled with lightly salted water to the boil. Place the cabbage in the water and boil for 7–8 minutes.
Remove the leaves in a colander to cool them. When it becomes difficult to remove the leaves return the cabbage to the pan and boil for a few more minutes. Continue removing leaves until you have all you need.
Yahni
An Arabic term for a standard Middle Eastern method of cooking where the food is braised with onions in olive oil and after the addition of water, vegetables, spices, etc. is cooked over a low heat until the meat and/or vegetables are tender. Popular throughout the region from Egypt to Afghanistan.
Yoghurt
Until a few score years ago yoghurt was almost unknown in the West. It has now arrived and has already created a myth around itself. Yet in the Middle East it has been known and used for thousands for years.
Yoghurt was introduced by the Aryan tribes as they penetrated this region. It was, and still is, an Indo-Aryan speciality for it is only widely used in such areas of Aryan origins as Asia Minor, Kurdistan, Caucasia, Iran, Afghanistan and Northern India.
Yoghurt arrests intestinal putrefaction, has antiboitic properties and is excellent for people with weak digestion and for the aged.
It is an essential part of the Middle Eastern diet and its preparation is a must if one is to master the many delightful yoghurt-based dishes of the region.
Although yoghurt can be purchased commercially I strongly recommend that you make your own at home following the simple method below. There is no need to go to the expense of purchasing 'yoghurt makers' that manufacturers constantly tempt one with.
1.2 litres/2 pints milk
1 soupspoon yoghurt—the starter (culture of the bacteria Bulgaris)
Bring the milk to the boil in a saucepan and when the froth rises turn off the heat.
Allow the milk to cool to the point where you can dip your finger in, and count up to 15 or, if you have a thermometer, where the temperature registers 45°C or 115°F.
Beat the spoon of yoghurt in a cup, add a tablespoon of the warm milk, beat vigorously and pour into the milk. Empty the milk into an earthenware or glass bowl and stir for a minute. Cover the bowl with a large plate and wrap in a towel or tea towel. Place in a warm place, e.g. near a radiator or in an airing cupboard and do not disturb for about 10 hours. Remove the wraps and place the covered bowl in the refrigerator.
The yoghurt is now ready to use. It can be kept for up to a week in the fridge. If using this yoghurt as a 'starter' for a new batch then use it within three days (after this the balance of the bacteria in the culture alters and the quality of the new yoghurt will be poorer).
To stabilize yoghurt
If you are to use yoghurt in hot dishes such as soups, sauces or stews, which require boiling it is necessary to stabilize it first otherwise it will curdle.
Either Stir a tablespoon of flour into a little water until you have a smooth paste and add to the yoghurt before you heat it.
Or Beat an egg into the yoghurt before cooking.
Note once the yoghurt has been stabilized and boiled it cannot be used as a 'starter' as the bacteria dies at a high temperature.
Drained yoghurt/labna
Certain recipes call for drained yoghurt or labna. To prepare this follow the simple recipe below.
Line a colander with a piece of damp muslin or sew a piece of muslin into a bag with a drawstring top. Spoon the yoghurt into the colander or bag and leave to drain. Suspend the bag over the sink. The whey will drain away leaving a light, soft, creamy cheese called, in Arabic, labna.
## bibliography
C. F. Abbott, Under the Turk in Constantinople, Macmillan and Co. Ltd, 1920.
M. N. Adler, The Itinerary of Benjamin of Tudelo, Oxford, 1970.
M. M. Ahsan, Social Life under the Abbasides, Longman Group, 1979.
A. J. Arberry, Aspects of Islamic Civilization, G. Allen and Unwin Ltd, 1964.
E. Ashtor, A Social and Economic History of the Near East in the Middle Ages, Collins, 1976.
Gertrude Bell, Persian Pictures, Jonathan Cape Ltd, 1973
Gertrude Bell, Syria, The Desert and the Sown, Heinemann 1907.
Issa J. Boullata, Modern Arab Poets, Heinemann 1946.
Richard J. Burton, The Book of 1001 Nights, 1885.
Mahmoud Darwish, The Music of Human Flesh, Heinemann, 1980.
J. A. de Gobineau, The World of the Persians, Minerva, Geneva, 1971.
Firdusi, Shah Nameh, G. Routledge and Son, 1892.
S. N. Fisher, The Middle East, Routledge & Keegan Paul, 1960.
W. H. Forbis, Fall of the Peacock Throne, Harper & Row NY, 1980.
J. B. Glubb, Syria, Lebanon and Jordan, Thames and Hudson Ltd, 1967.
Helen C. Gordon, Syria As It Is, Methuen and Co., 1939.
Joan Hasler, The Making of Russia, Longmans, Green and Co., 1969.
Philip K. Hitti, A Short History of The Near East, D. Van Nostrand Co., 1966, Canada.
Diana D. Hovanessian, Anthology of Armenian Poetry, Columbia University Press, 1978.
Peter Jay, Song of Songs, Anvil Press, 1975.
Rene R. Khawam (trans.), The Subtle Ruse, East-West Publications, 1980.
O. Khayyam, Rubaiyat, Trans. E. Fitzgerald, Quaritch, 1859.
The Rubaiyat of Omar Khayyam, edited by A. J. Arberry, Dent, 1954.
E. W. Lane, Manners and Customs of the Modern Egyptians, J. Murray, 1860.
Tone Lane, Men are like that.
B. Lewis, Arabs in History, Hutchinson and Co. Ltd, 1950.
B. Lewis, Emergence of Modern Turkey, Oxford University, 1961.
Mahmut Makal, A Village in Anatolia, Vallentine, Mitchell and Co., 1954.
Gavin Maxwell, The Reed Shaken by the Wind, Longmans, Green and Co., 1959.
Nermin Menemiencioglu, Turkish Verse, Penguin Books, 1978.
J. Murray, Sketches of Persia, 1827.
Isaac Myer, Oldest Book in the World, Keegan Paul & French, 1900.
Shaykh Nefzawi, Perfumed Garden, Neville Spearman Ltd, 1963.
Alexander Pallis, In the Days of the Janissaries, Hutchinson and Co., 1951.
A. S. Pirouzian, The Armenian Kitchen, Haybed-Hrad, Armenia, 1963.
Plato, The Republic, Paul Shorey, Heinemann Ltd, 1930.
Pliny the Elder, A Natural History, Trans. H. Rackham, 1950.
Marco Polo, Travels, Trans. Ronald Latham, Penguin Books, 1958.
Paul Ricaut, State of Greek and Armenian Churches, 1679.
Claudia Roden, A Book of Middle Eastern Food, T. Nelson and Sons, 1968.
Claudia Roden, Coffee, Faber and Faber, 1977.
R. T. Rundel Clark, Myth and Symbol in Ancient Egypt, Thames and Hudson, 1995.
J. H. Schofield, The Historical Background of the Bible, Nelson & Sons Ltd, 1938.
Idries Shah, The Pleasantries of the Incredible Mulla Nasrudin, Jonathan Cape Ltd, 1968.
Henry D. Spalding, Encyclopedia of Jewish Humour, Jonathan David, NY 1969.
D. Stacton, The World on the Last Day, Faber & Faber, 1965.
Tacitus, Annals, Trans. A. H. Beesley 1870.
Reay Tannahill, Food in History, Eyre Methuen Ltd, 1973.
J. D. Vehling, Apicius, Dover Publications, NY, 1957.
## footnote
The prevailing concept of the Middle East, whether as a geographical, cultural or political entity, includes the following countries: United Arab Republic, Israel, Syria, Lebanon, Iraq, Turkey, the Caucasian republics of Armenia, Georgia and Azerbaijan, the Arab states of South West Asia and finally Iran.
Zoroaster, a Median reformer, flourished about 60 BC. He preached duality in man, personifying the two opposing principles of good and evil, light and darkness. His teaching became the state religion of Persia and spread beyond. Today there are several hundred thousand Zoroastrians still living in Iran, with 2–3 million more in India (Parsis).
A short history of the Near East, Philip K. Hitti.
Seleucus took Syria, most of Asia Minor and Northern Iran; Antigonus the remainder of Asia Minor, barring Armenia; Antipater Macedonia; Ptolemy Egypt.
Syria, Lebanon and Jordan, J. B. Glubb.
From Syria, the Desert and the Dawn.
Founded by Osman, 1299–1328. A Turkish tribe, one of many originating from Central Asia. They were first Islamised in Persia then settled in western Anatolia where they absorbed their Seljug kinsmen. In time the house of Osman—Ottomans—became the masters of the entire Near East.
Oldest Book in the World, Isaac Myes.
Pliny the Elder in his Natural History described this type of porridge thus 'Soak, leave it for a night to dry. Next day dry it by the fire and then grind it in a mill...they (then) mix three pounds of flax seed, half a pound of coriander seeds and eighth of a pint of salt, previously roasting them all. A Natural History—trans. H. Rackman, 1950.
The World on the Last Day, D. Stacton.
The Itinerary of Benjamin of Tudelo, M. N. Adler, Oxford 1907.
Aspects of Islamic Civilization, A. J. Arbemy.
ibid.
Manti in Turkish; mant'ou in Chinese, Korean and Mongolian; dyushbara in Azerbaijanian; pelmeny in Russian; vareniky in Ukranian.
Travels, Marco Polo.
ibid.
Under the Turk in Constantinople, G. E. Abbott.
The majority of the people of Egypt, Turkey, Iran, as well as Syria and Iraq, still live in small villages. On average 80 per cent of the Middle Easterners are peasants with a significant minority of nomads.
Both the Bible and the Koran have numerous examples of such social and domestic customs illustrated, e.g. 'Jesus answered, He it is, to whom I shall give a sop, when I have dipped it. And when he had dipped the sop, he gave to Judas Iscariot, the son of Simon.' (St. John 13/26)
State of Greek and Armenian Churches, Paul Ricaut, 1679.
ibid.
The exact day and month of any festival varies because the Muslim calender (Hagira) is based on a lunar month cycle which also means that the exact day of a given festival will differ from country to country.
Fall of the Peacock Throne, W. H. Forbis.
Alya in Arabic; tumag in Armenian; kuyruk yag in Turkish.
From Sanskrit pur, phour in Greek, pour in Armenian, firm in Turkish.
'Kitab-al-Tabikh wa islah al-aghdhiyat al-makulat' by Warrag—Bodleian Library, Oxford. Manuscript no. 187.
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"redpajama_set_name": "RedPajamaBook"
}
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Q: how to make C# video source played in C++ window Our client use C++ code and they have a window, they give us this window's handle,
HWND playbackWndHandle
and let us play our video in this window. Our video source is from C# code, a class object inherit from System.Windows.Controls.UserControl.
They let us make the window they give us the parent window and create a child window in it and show our video in the child window.
Here is the way I used but does not work really well, this is C# code, m_UIVideoControl is our video source and parentHwnd is the handler playbackWndHandle they give us.
This one has many bugs when playing the video, I wonder are there any other way to do that? Please let me know, thank you so much for everybody!
public void CreateHostHwnd(IntPtr parentHwnd)
{
// Set up the parameters for the host hwnd.
parameters = new HwndSourceParameters("Video Window", 800,800);
parameters.WindowStyle = WS_VISIBLE | WS_CHILD;
parameters.SetPosition(0, 0);
HwndSourceParameters parameters.ParentWindow = parentHwnd;
// Create the host hwnd for the visuals.
HwndSource myHwndSource = new HwndSource(parameters);
myHwndSource.RootVisual = m_UIVideoControl;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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{"url":"http:\/\/afantasy.ninja\/2017\/10\/08\/color%20notes\/","text":"When studying the CSS framework Bluma, I found that it use a very smart way to determine the text color based on the bg color, utiliizing the color luminance to decide to use black text or white text, which is interesting and I found some interesting facts to note it down.\n\n### Color Luminance\n\nSteven Bradley wrote an great article about color luminance, and here are some details:\n\n1. Our eyes have rods\uff08\u89c6\u6746\u7ec6\u80de\uff09and cones\uff08\u89c6\u9525\u7ec6\u80de\uff09to perceive light\/dark as well as different color information; rods are responsible for seeing in low light and are sensitive to light\/dark, while cones are responsible for distinguishing different colors. Moreover we have S-cones (blue), M-cones (green) and L-cones (red) for short, medium and long wavelength. \u2013 three primary colors.\n2. Luminance is the measurement of the intensity of light that reaches our eye, while brightness and value are only the perception of an object\u2019s luminance. Lightness is the brightness relative to the brightness of a similarly illuminated white.\n3. Our perception of lightness (or brightness) don\u2019t scale linearly with luminance. The perceived luminance is dependent on both light intensity and the specific wavelength of that light (a.k.a type of color).\n4. Every color has its own natural luminance levels.\n5. Saturation also affects luminance. If you reduce the saturation of a pure color to 0% the result is a 50% grey with a 50% value for luminance.\n6. differences between HSL, HSB, and HSV:\n\u2022 HSB\/V is measuring the amount of light\n\u2022 HSL is measuring the amount of white\n\n#### Quantitive Implementation\n\nFrom WCAG\u2019s Definition, the relative luminance is:\n\nthe relative brightness of any point in a colorspace, normalized to 0 for darkest black and 1 for lightest white.\n\nWCAG also provides the formula for luminance:\n\nFor the sRGB colorspace, the relative luminance of a color is defined as L = 0.2126 * R + 0.7152 * G + 0.0722 * B where R, G and B are defined.\n\nas:\n\n\u2022 if RsRGB <= 0.03928 then R = RsRGB\/12.92 else R = ((RsRGB+0.055)\/1.055) ^ 2.4\n\u2022 if GsRGB <= 0.03928 then G = GsRGB\/12.92 else G = ((GsRGB+0.055)\/1.055) ^ 2.4\n\u2022 if BsRGB <= 0.03928 then B = BsRGB\/12.92 else B = ((BsRGB+0.055)\/1.055) ^ 2.4\n\nand RsRGB, GsRGB, and BsRGB are defined as:\n\n\u2022 RsRGB = R8bit \/ 255\n\u2022 GsRGB = G8bit \/ 255\n\u2022 BsRGB = B8bit \/ 255\n\nBluma strictly implements the spec above by having such Sass function:\n\nAnother similar implementation is documented here\n\nWork With Color says:\n\nLuminance on the other hand is a measure to describe the perceived brightness of a color\n\nContrast as in the distance of luminance between two colors\n\nColor decisions need to consider luminance \/ contrast because it is key to usability.\nHSL Color Picker\n\nTo put it simply, the luminance of a color defines whether its brightness. A luminance of 1 means the color is white. On the opposite, a luminance score of 0 means the color is black.\n\nNOTE: WCAG = Web Content Accessibility Guidelines\n\n### This video blows my mind\n\nNotice 3:50 \u2013 that is crazy. Remember that our eyes could be cheated.\n\n### Color Models\n\nColorizer have listed all common color models.\n\n#### RGB\n\nFrom Wikipedia:\n\nThe RGB color model is an addictive color model in which red, green and blue light are added together in various ways to reproduce a broad array of colors.\n\nThe RGB color model is addictive in the sense that the three light beams are added together, and their light spectra add, wavelength by wavelength, to make the final color\u2019s spectrum. This is essentially opposite to the subtractive color model that applies to paints, inks, dyes and other substances whose colors depends on reflecting the light under which we see them.\n\n##### why R,G,B\n\nThe choice of primary colors (red, green, blue) is related to the physiology of the human eye; good primaries are stimuli that maximize the difference between the responses of the cone cells of the human retina to light of different wavelengths, and that thereby make a large color triangle.\n\nThe difference in the signals received from the three kinds allows the brain to differentiate a wide gamut of different colors.\n\nThe color triangle represents the range of colors which could be reproduced by additive mixing of non-negative amounts of three primary colors.\n\n\u201c\u539f\u8272\u201d\u7684\u6307\u5b9a\u5e76\u6ca1\u6709\u552f\u4e00\u7684\u9009\u6cd5\uff0c\u56e0\u4e3a\u5c31\u7406\u8bba\u4e0a\u800c\u8a00\uff0c\u51e1\u662f\u5f7c\u6b64\u4e4b\u95f4\u65e0\u6cd5\u66ff\u4ee3\u7684\u989c\u8272\u90fd\u53ef\u4ee5\u88ab\u9009\u4e3a\u201c\u539f\u8272\u201d\uff0c\u53ea\u662f\u76ee\u524d\u666e\u904d\u8ba4\u5b9a\u201c\u5149\u7684\u4e09\u539f\u8272\u201d\u4e3a\u7ea2\u7eff\u84dd\u3002\n\n##### numeric representations\n\u2022 float\n\u2022 percentage\n\u2022 integer range (0,255) - decimal \/ hexadecimal\n\u2022 larger integer ranges for each primary color (High-end digital image equipment)\n\n[Concept] illuminant\n\nFrom Wikipedia Standard Illuminant:\n\nThe International Commission on Illumination (usually abbreviated CIE for its French name) is th body responsible for publishing all of the well-known standard illuminants.\n\nThere are many standards like Illuminant A, B, C \u2026 The most common used one is called Illuminant D, which:\n\n\u2026 represents phases of daylight, \u2026 the D series of illuminants are constructed to represent natural daylight. They are difficult to produce artificially, but are easy to characterized mathematically.\n\n##### What is white?\n\nWhite Point\n\nA white point is a set of tristimulus values (a set of values of 3 primary colors) or chromaticity coordinates that serve to define the color \u201cwhite\u201d in image capture, encoding, or reproduction. Depending on the application, different definitions of white are needed to give acceptable results.\n\nSPD = Spectral Power Distribution (\u5149\u8c31\u80fd\u91cf\u5206\u5e03??)\n\n\u8272\u5ea6(Colorfulness\/Chroma\/Saturation), \u8272\u76f8(Hue)\n\n#### CMYK\n\nCMYK - Wikipedia\n\nThe CMYK color model (process color, four color) is a subtractive color model, used in color printing, and is also used to describe the printing process itself.\n\n\u2022 C - Cyan\n\u2022 M - Magenta\n\u2022 Y - Yellow\n\u2022 K - Key (black)\n\n#### HSL\n\nHSL\u548cHSV\u8272\u5f69\u7a7a\u95f4\n\nHSL\u548cHSV\u90fd\u662f\u4e00\u79cd\u5c06RGB\u8272\u5f69\u6a21\u578b\u4e2d\u7684\u70b9\u5728\u5706\u67f1\u5750\u6807\u7cfb\u4e2d\u7684\u8868\u793a\u6cd5\u3002\u8fd9\u4e24\u79cd\u8868\u793a\u6cd5\u8bd5\u56fe\u505a\u5230\u6bd4RGB\u57fa\u4e8e\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u7684\u51e0\u4f55\u7ed3\u6784\u66f4\u52a0\u76f4\u89c2\u3002\n\nColorizer \u4f7f\u7528\u4e86\u5706\u9525\u8868\u793a HSL\uff1a\n\n\u2022 Hue\uff08H\uff09\uff0c\u8272\u76f8\uff0c\u662f\u8272\u5f69\u7684\u57fa\u672c\u5c5e\u6027\uff0c\u5373\u5e73\u65f6\u6240\u8bf4\u7684\u989c\u8272\u540d\u79f0\n\u2022 Saturation\uff08S\uff09\uff0c\u9971\u548c\u5ea6\uff0c\u662f\u6307\u8272\u5f69\u7684\u7eaf\u5ea6\uff0c\u8d8a\u9ad8\u8272\u5f69\u8d8a\u7eaf\uff0c\u4f4e\u5219\u9010\u6e10\u53d8\u7070\uff0c\u53d6 0~100% \u7684\u6570\u503c\n\u2022 Lightness\uff08L\uff09\uff0c\u4eae\u5ea6\uff0c\u53d6 0~100%\n\n#### RGB => HSL\n\nAlgorithm in CSS3 Specification\n\nAn JS implementation based on the spec:\n\n### Color Meaning\n\nhttp:\/\/vanseodesign.com\/web-design\/color-meaning\/\n\n\u2026 there is no substantive evidence that support a universal system of color meaning. It\u2019s not that colors themselves have specific meaning, but rather that we have culturally assigned meanings to them.\n\n\u2026 it\u2019s important to understand who your target audience is and how your audience attaches meaning to color.\n\n\u2022 warm colors: red, orange, yellow \u2013 passion, energy, impulsiveness, happiness, coziness and comfort\n\u2022 cool colors: green, blue, violet \u2013 calm, trust, professionalism, also sadness and melancholy","date":"2018-04-19 13:43:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.46225088834762573, \"perplexity\": 4101.68807494174}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125936969.10\/warc\/CC-MAIN-20180419130550-20180419150550-00019.warc.gz\"}"}
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Cemetery/Memorial: Jerusalem War Cemetery. Israel and Palestine (Incl. Gaza) Grave Reference: D. 64.
Additional Info. Son of John and Janet Jack, of 49, Ramsay Rd., Kirkcaldy.
Additional Info. Born in Kirkcaldy, Son of Benjamin Boon Turner Johnston and Cecilia Anderson Simpson.
Posted: Sat Mar 19, 2016 1:13 pm Post subject: Johnston, Peter Simpson Dickson "Terry"
Cemetery/Memorial: Ypres (Menin Gate) Memorial, West-Vlaanderen, Belgium. Grave Reference: Panel 8 and 12.
Additional Info. Son of David Kennedy Kay and Margaret Burgess Kay of Kirkcaldy. Grandson of George and Helen Kennedy Kay.
Cemetery/Memorial: Buzancy Military Cemetery, Aisne, France Grave Reference: I. C. 18.
Additional Info. Eldest Son of James Tulloch and Magdalene Leishman.
Additional Info. Died at Pathhead, Kirkcaldy. Husband of Cecilia Smith Beveridge.
Cemetery/Memorial: Combles Communal Cemetery Extension, Somme, France Grave Reference: IV. A. 11.
Additional Info. Son of Jame Peddie and Mary Cox (Hutchison) Lockhart.
Additional Info. Son of Andrew and Helen (Clunie) McBain.
Cemetery/Memorial: Houplines Communal Cemetery, Extension, Nord, France. Grave Reference: II. D. 19.
Additional Info. Only Son of Mary Turneer (McDowell) Methven of Wemyss Park, Kirkcaldy and the Late James Methven. Killed In Action Near Frelinchien.
Cemetery/Memorial: Archangel Memorial, Russia. Grave Reference: Stone No. 3.
Additional Info. Son of the Late David and Mary (Lang) Michie of East Lynne, Whytehouse Avenue, Kirkcaldy. Died in Action at Borock, Russia.
Cemetery/Memorial: Hotton War Cemetery, Luxembourg, Belgium. Grave Reference: XII. A. 8.
Additional Info. Son of George D. and Janet Barclay Mitchell.
Additional Info. Son of Alexander and Joan Gray Nicol of Kirkcaldy, Fife.
Posted: Sat Mar 19, 2016 4:55 pm Post subject: Oliver, Richard F.
Cemetery/Memorial: Jerusalem War Cemetery. Israel and Palestine (Incl. Gaza) Grave Reference: Y. 82.
Additional Info. Son of Richard and Jeanie Oliver, of 349, High St., Kirkcaldy.
Headstone in Palestine reads 'He Sleeps With the Brave No tears Of His Loved Ones Drop On His Grave'.
Cemetery/Memorial: Knightsbridge War Cemetery, Acroma, Libya. Grave Reference: 10. E. 12.
Additional Info. Son of Thomas and Euphemia Anderson Preece.
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Q: Joomla : 404 pages showing homepage content I have a site in joomla (version 1.5.23), wherein any url which isn't present shows content of homepage (and its 200 ok). How do i get it to go to a 404 page
ex. www.example.com/sdfsdf or www.example.com/blahblah all shows homepage content.
Any help would be appreciated
Regards,
Sushil
A: These methods work. Different explanation, but the same concept.
Method
Original Joomla! doc
If your using Apache check the presence of the following line and all 404 errors will be redirected to your homepage
Code:
ErrorDocument 404 index.php
Solution about a similar problem: Joomla! forum link
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"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/www.cut-the-knot.org\/Curriculum\/Geometry\/GeoGebra\/TangentToTwoCircles.shtml","text":"# How to Construct Common Tangents to Two Circles\n\nGiven circles $A(B)$ and $C(D)$, construct the lines tangent to the two circles.\n\nSolution\n\nBelow is an applet that is supposed to be suggestive of the construction:\n\n22 January 2015, Created with GeoGebra\n\nGiven circles $A(B)$ and $C(D)$, construct the lines tangent to the two circles.\n\nAs is often the case in mathematics, the sought construction of common tangents two to circles is reduced to a simpler construction of tangents from a point to a circle.\n\nTo see how it works, let's introduce notation $R_c$for the radius of circle $c$. Assume (without loss of generality) that $R_{(A)}\\ge R_{(B)}.$ Form an auxiliary circle $(A)'$ with center $A$ and radius $R_{(A)}\\ge R_{(B)}.$\n\nTangents to a circle are perpendicular to the radii at the points of contact. Thus, in the diagram, if $CK$ is tangent to $(A)'$ then $AK\\perp CK.$ If $E$ is the intersection of $AK$ with $A(B)$ and $AK||CF$, with $F$ on $C(D)$, then, first of all $AE$ and $CF$ are radii of $A(B)$ and $C(D),$ respectively. Also, by the construction, $KE=CF,$ $KE||CF,$, and the angle at $K$ is right, making $ACFE$ a rectangle. It follows that the radii $AE$ and $CF$ are perpendicular to $EF$, implying tat the latter is tangent to both circles.\n\nTo construct the inner tangents, reduce first $C(D)$ to a point and simultaneously expand $A(B)$ by $R_{C(D)}.$ The argument is about the same as before.\n\nSeveral cases are possible. If the circles coincide, the number of common tangents is infinite - one per every point on the circle. If one of the circles lies entirely within the other, they have no common tangents. If they touch internally, their single common tangent can be said to play a double role. If the circles intersect, they have two outer tangents. If they touch externally, there are two outer tangents and one tangent at their common point. Otherwise, there are two outer and two inner tangents.\n\nWhen two circles are tangent externally, their common tangents are found to be in an interesting relationship.\n\n### Various Geometric Constructions\n\n\u2022 How to Construct a Radical Axis\n\u2022 Constructions Related To An Inaccessible Point\n\u2022 Inscribing a regular pentagon in a circle - and proving it\n\u2022 The Many Ways to Construct a Triangle and additional triangle facts\n\u2022 Easy Construction of Bicentric Quadrilateral\n\u2022 Easy Construction of Bicentric Quadrilateral II\n\u2022 Star Construction of Shapes of Constant Width\n\u2022 Four Construction Problems\n\u2022 Geometric Construction with the Compass Alone\n\u2022 Construction of n-gon from the midpoints of its sides\n\u2022 Short Construction of the Geometric Mean\n\u2022 Construction of a Polygon from Rotations and their Centers\n\u2022 Squares Inscribed In a Triangle I\n\u2022 Construction of a Cyclic Quadrilateral\n\u2022 Circle of Apollonius\n\u2022 Six Circles with Concurrent Pairwise Radical Axes\n\u2022 Trisect Segment: 2 Circles, 4 Lines\n\u2022 Tangent to Circle in Three Steps\n\u2022 Regular Pentagon Construction by K. Knop","date":"2017-09-22 08:11:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8173341751098633, \"perplexity\": 461.9995269671739}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818688926.38\/warc\/CC-MAIN-20170922074554-20170922094554-00694.warc.gz\"}"}
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The banners covering the corners of the main stand at Ibrox have been taken down.
Supporters were unhappy the iconic glass staircases were hidden from view when the banners first went up back in August.
While many assumed this was to cover construction work, the club's supporter liaison officer revealed this was not the case, prompting many fans to question the board's motivation.
Rangers have now said in a statement they are pursuing extra revenue through sponsorship and advertising, and covered the glass stairwells to "highlight the potential" for companies to possibly use in future.
For the meantime, though, the banners have been removed.
A statement read: "The wrap around banners at both ends of the main Ibrox stand are to be taken down.
"The glass stairwells had been 'dressed' during the Commonwealth Games for promotional purposes and the thinking was to look at a branding exercise which would also highlight the potential for partners/sponsors to utilise the space in the future.
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La crise politique de 2022 au Pakistan débute lorsque le Premier ministre Imran Khan perd sa majorité absolue à l'Assemblée nationale en mars 2022 et se transforme en crise constitutionnelle. Le 3 avril, le vice-président de l'Assemblée nationale refuse de mettre au vote une motion de censure contre le Premier ministre déposée par l'opposition. Dans la foulée, le président de la République Arif Alvi dissout l'Assemblée nationale sur conseil du Premier ministre et convoque des élections anticipées.
Le , la Cour suprême déclare inconstitutionnelle la décision du vice-président de l'Assemblée nationale et restaure cette dernière, grâce à laquelle l'opposition espère prendre le contrôle du gouvernement. Le , Imran Khan devient le premier chef du gouvernement à être démis de ses fonctions par une motion de censure. Le lendemain, le chef de l'opposition Shehbaz Sharif est élu par l'Assemblée pour le remplacer.
Contexte
Victoire du PTI aux législatives de 2018 et formation du gouvernement
Pour les élections législatives de 2018, le Mouvement du Pakistan pour la justice (PTI) fait partie des deux favoris avec le pouvoir sortant de la Ligue musulmane du Pakistan (N). Imran Khan établit d'ailleurs à l'occasion de celles-ci un virage plus conservateur, indiquant par exemple vouloir sauvegarder la loi interdisant le blasphème. Ses opposants lui reprochent de vouloir ainsi s'attirer le vote des conservateurs religieux. En revanche, son programme accorde une place signifiative à l'écologie et il promet des investissements dans l'éducation et la santé.
Durant la campagne, le clan Sharif accuse la puissante armée pakistanaise de comploter contre lui et de favoriser le PTI, alors que certains médias et fonctionnaires notent une répression à l'encontre du parti sortant et dénoncent des censures. Le , il arrive largement en tête des législatives mais sans obtenir de majorité absolue, bien que ses rivaux dénoncent des fraudes électorales. Il entame alors des négociations pour former un gouvernement de coalition avec des petits partis et des indépendants.
Le , le PTI annonce avoir trouvé un accord en vue de former un gouvernement de coalition, ralliant des candidats indépendants, la Ligue musulmane du Pakistan (Q), le Parti baloutche Awami et la Grande alliance démocratique ainsi que le Mouvement Muttahida Qaumi. Imran Khan reçoit le l'investiture de l'Assemblée nationale par favorables, soit quatre de plus que la majorité requise. En arrivant au pouvoir, il interrompt l'alternance traditionnelle entre le Parti du peuple pakistanais et les factions de la Ligue musulmane du Pakistan, notamment celle de Nawaz.
Contestation d'Imran Khan par l'opposition
Fazal-ur-Rehman, dirigeant du parti islamiste JUI-F prend la tête d'une manifestation anti-gouvernementale de partisans en novembre 2019. Le , le Mouvement démocratique pakistanais est fondé par la Ligue musulmane du Pakistan (N) et le Parti du peuple pakistanais (PPP) ainsi que neuf autres partis de l'opposition parmi ses membres. Fazal-ur-Rehman est nommé à la tête de l'alliance, un choix consensuel alors qu'il évite de choisir entre les deux principaux partis.
Le mouvement mène différentes actions à travers les quatre provinces du pays, souvent dirigées par le chef du PPP Bilawal Bhutto Zardari et la vice-présidente de la Ligue Maryam Nawaz Sharif. Son premier rassemblement le 16 octobre 2020 à Gujranwala regroupe jusqu'à sympathisants dans le stade de la ville. Un second rassemblement mené le 18 octobre à Karachi réunit entre personnes selon les autorités et selon le PPP. Le mouvement rassemble ensuite près de personnes à Quetta le 25 octobre, sous la surveillance de . Le 22 novembre, c'est à Peshawar que le mouvement mène sa quatrième action en revendiquant participants. Le 30 novembre, un nouveau rassemblement à Multan est mené par Maryam Nawaz et Asifa Zardari, sœur de Bilawal.
Le mouvement démocratique pakistanais critique le pouvoir exorbitant des militaires au Pakistan, réclame des élections anticipées en 2021 et le départ du gouvernement d'Imran Khan, « choisi par l'armée » selon l'alliance. Il dénonce des fraudes lors des élections législatives de 2018. Si selon les observateurs internationaux, les militaires n'ont pas directement interféré dans le processus de vote, de nombreux analystes estiment qu'ils ont activement soutenu certains partis et isolé leurs détracteurs. Dans une charte en douze points, le mouvement demande notamment l'indépendance du Parlement et de la justice, la liberté de la presse, une réforme électorale et la fin de l'interférence de l'armée.
Imran Khan dénonce le mouvement comme un chantage de ses dirigeants, qui chercheraient selon lui à échapper aux poursuites pour corruption et à obtenir une amnistie, à l'instar de l'ordonnance nationale de réconciliation de 2007. Le clan Sharif, dont Nawaz Sharif, son frère Shehbaz Sharif et sa fille Maryam Nawaz Sharif font ainsi l'objet de poursuites depuis les révélations des Panama Papers, alors que le premier est en « exil » à l'étranger et le second emprisonné.
Effondrement de la coalition au pouvoir
Dans le contexte d'une crise économique persistante, le gouvernement du Premier ministre Imran Khan, élu après les élections de 2018, fait face à des critiques de plus en plus virulentes de la part de ses partenaires de coalition. Le , l'unique député du Jamhoori Wattan annonce quitter la coalition gouvernementale. Le lendemain, quatre des cinq députés du Parti baloutche Awami, un allié clé du parti au pouvoir, annoncent à leur tour rejoindre les rangs de l'opposition. Le 30 mars, les sept élus du Mouvement Muttahida Qaumi et ses deux ministres fédéraux quittent également l'alliance.
Imran Khan accuse l'opposition d'être manipulée par les États-Unis, qui lui reprocherait sa visite à Vladimir Poutine en Russie le , premier jour de l'invasion russe de l'Ukraine en 2022, ainsi que sa neutralité dans le conflit. Il met notamment en avant une prétendue « lettre de menace » envoyée par des diplomates américains indiquant que les relations entre les deux pays dépendront du résultat de la motion de censure. Les États-Unis démentent toute ingérence. L'opposante Maryam Nawaz Sharif dénonce un faux du Premier ministre, qui chercherait à s'attirer les faveurs de l'opinion publique en se posant comme le défenseur des intérêts nationaux.
Déroulement
Tentative de censure du gouvernement
Le 28 mars, 161 députés de l'opposition menés par leur chef à l'Assemblée nationale Shehbaz Sharif déposent une demande de censure contre le gouvernement. Le 3 avril, au début de la session de l'Assemblée nationale, le ministre de l'Information Fawad Chaudhry a pris la parole en déclarant que la loyauté envers l'État était le devoir fondamental de tout citoyen en vertu de l'article 5 de la Constitution. Il a réitéré les affirmations antérieures d'Imran Khan selon lesquelles un complot étranger avait été ourdi pour renverser le gouvernement. Chaudhry a ensuite appelé le vice-président à décider de la constitutionnalité de la motion de censure. Par conséquent, Suri a qualifié la motion de violation de l'article 5 de la Constitution en raison d'un « complot étranger » à l'appui de la motion. Peu de temps après, Khan, dans un discours à la nation, a annoncé qu'il avait conseillé au président Arif Alvi de dissoudre les assemblées à la suite du rejet de la motion de censure à son encontre. Ainsi, le même jour, le Président a dissous l'Assemblée nationale en vertu de l'article 58 de la Constitution.
Le 4 avril, le Secrétariat du gouvernement a publié une notification indiquant qu'Imran Khan avait « cessé d'occuper le poste de Premier ministre du Pakistan avec effet immédiat ». Cependant, selon une notification publiée par le bureau du président le même jour, Imran Khan continuera d'exercer ses fonctions de Premier ministre jusqu'à la nomination d'un premier ministre par intérim.
Élection du ministre en chef du Pendjab
Décision de la Cour suprême
La décision de dissoudre l'assemblée est particulièrement controversée en raison de la mention explicite dans l'article 58 du fait qu'un Premier ministre « contre lequel un avis de résolution de vote de défiance a été donné à l'Assemblée nationale mais n'a pas été voté » n'a pas le pouvoir mentionné ci-dessus de conseiller au Président de révoquer l'Assemblée. Plus tard dans la journée, un banc de trois membres de la Cour suprême comprenant le ainsi que les juges et a déclaré que la juridiction examinerait les actions du vice-président. Le même jour, la coalition d'opposition du Mouvement démocratique pakistanais a tenu une session non-officielle à l'Assemblée nationale après que la chambre a été ajournée et a passé le vote de défiance contre Imran Khan, le déclarant réussi avec 197 voix pour 172 nécessaires.
La Cour suprême examine du 3 au 7 avril un recours déposé par l'opposition. Le 7 avril, elle déclare inconstitutionnelle le rejet de la motion de censure et invalide par conséquent la dissolution, à l'unanimité des cinq juges. La Cour ordonne également à l'Assemblée nationale de se réunir le 9 avril pour voter la motion de censure.
Renversement d'Imran Khan
Imran Khan est finalement renversé par la motion de censure, votée le lendemain peu après minuit par 174 députés, soit deux voix de plus que le minimum requis. Ce vote fait suite à la démission du président et du vice-président de l'Assemblée. Alors que la session chaotique avait débuté la veille à 10 heures du matin, elle a été ajourné à plusieurs reprises, alors que des députés des deux camps se sont échangés des invectives et que des ministres ont retardé le vote par des discours-fleuve en guise d'obstruction parlementaire.
L'ex-premier ministre est cependant toujours populaire auprès de larges pans de la population, et ceux-ci se rassemblent à la demande de l'homme politique.
Conséquences
Formation d'un nouveau gouvernement
L'Assemblée nationale se réunit dès le pour élire un nouveau Premier ministre. Shehbaz Sharif se présente pour succéder à Imran Khan en tant que candidat commun de l'opposition. Face à lui, le ministre des Affaires étrangères d'Imran Khan, Shah Mehmood Qureshi, tente de lui faire barrage. Finalement, les soutiens d'Imran Khan boycottent massivement la session, au cours de laquelle Shehbaz Sharif est élu sans opposition avec 174 voix en sa faveur, sur un total de 342 sièges, soit deux de plus que la majorité absolue.
Notes et références
Avril 2022
2022 au Pakistan
Crise politique
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{Introduction\label{sec:intro}}
The same way that a quantum channel describes the most general transformation mapping an input quantum state to an output quantum state \cite{Kraus1983}, a \textit{quantum supermap} describes the most general transformation mapping an input quantum channel to an output quantum channel \cite{Chiribella2008}.
Interpreting the quantum channel as a transformation between states, the supermap is then a transformation of transformations. For that reason, it is called \textit{higher-order} transformation. Since nothing forbids \textit{a priori} to nest transformations of transformations, one can consider successive nestings to recursively build a whole hierarchy of higher-order transformations \cite{Chiribella2009,Perinotti2016,Bisio2018}.
Fragments of a quantum circuit are a concrete instance of the use of a higher-order hierarchy. A fragment of quantum circuit that `goes around' a channel is a supermap: it takes a channel as input and outputs a channel. This supermap can itself be seen as the input for some super-supermap that will output a channel, and so on. This ensuing hierarchy has been defined under the name \textit{quantum comb formalism} \cite{Chiribella2009}, which has proven to be a valuable tool in the field of quantum information theory.
With a different goal than modeling circuit fragments, supermaps with multiple inputs were subsequently studied. First, the \textit{quantum switch} was proposed as a supermap that takes two channels and outputs them in an order that depends on a control qubit \cite{Chiribella2013}. Soon after, \textit{Process Matrices} (PM) were proposed as a general framework of supermaps that take a fixed number of quantum instruments \cite{Davies1970} and map them to a joint probability for their outcomes \cite{OCB2012}. (In the case when the inputs are channels, that probability is 1.)
Both concepts led to the identification of Indefinite Causal Order (ICO) as a feature of supermaps.
One can then wonder what differentiates the switch from a comb, or the PM from a comb. Especially, why certain maps and hierarchies of maps may feature ICO while others will not.
Motivated by these considerations, the goal of this work is to present a framework that formalizes and characterizes higher-order quantum transformations. The two main questions answered are `given an operator on a set of input and output Hilbert spaces, does it represent (the Choi-Jamio{\l}kowski operator of) a higher order object' and `what is the underlying causal structure(s) of such an object?'. This work extends two previous characterizations, one done using type theory \cite{Perinotti2016,Bisio2018}, and the other using category theory \cite{Kissinger_2019}.
This extension relies on the use of superoperator projectors \cite{Araujo2015}.
These projectors have a twofold advantage: first, they make the characterization more straightforward, as one can answer the first question simply by applying the projector corresponding to a given higher-order object on the operator. Second, they offer an intuitive explanation of the type-theoretic semantics of higher order: the algebraic rules for composing these projectors correspond to the semantic rules for forming new types.
Below, we concisely reformulate ideas from Ref. \cite{Bisio2018} to give the reader an overview of the theory upon which this work is based.
\paragraph{Types.}The formalism of quantum channels and operations \cite{Kraus1983} describes transformations of quantum states into quantum states, but is unable to effectively describe the transformations whose inputs and outputs are transformation themselves \cite{Chiribella2013}. To overcome this issue, Perinotti \cite{Perinotti2016} and Bisio \cite{Bisio2018} extended the concept of a supermap \cite{Chiribella2008,Chiribella2009} into a suitable framework to treat these general transformations of transformations, or higher-order quantum maps.
At the core of their work is the utilization of the Choi-Jamio{\l}kowski (CJ) isomorphism \cite{Jamiolkowski1972,Choi1975}. Define $\Hilb{A}$ to be a Hilbert space of dimension $d_A$ associated to system $A$, $\LinOp{\Hilb{A}}$ to be the space of linear operators on $\Hilb{A}$, and $\LinOpB{\LinOp{\Hilb{A}}}{\LinOp{\Hilb{B}}}$ to be the space of linear maps from $\LinOp{\Hilb{A}}$ to $\LinOp{\Hilb{B}}$.
Let $\ket{\phi^+} = \sum_{i=0}^{d_A-1} \ket{i}^A\otimes \ket{i}^{A'}$ be a(n unnormalized) maximally entangled state on space $\Hilb{A}\otimes\Hilb{A'}$, with $A'$ a copy of $A$, and $\phi^+ \equiv \dyad{\phi^+}$ its density operator representation. Then,
\begin{equation}\label{eq:CJ}
\begin{gathered}
\LinOp{\LinOp{\Hilb{A}},\LinOp{\Hilb{B}}} \ni \mathcal{M} \mapsto M \in \LinOp{\Hilb{A} \otimes \Hilb{B}} \::\\
M \equiv \left[\left( \mathcal{I} \otimes \mathcal{M} \right)\left\{\phi^+\right\}\right]^T \:,
\end{gathered}
\end{equation}
is an isomorphic bijective mapping. The correspondence sends linear maps between operator spaces to operators in tensor product spaces. It has the properties that a Hermitian-preserving (HP) map is mapped to a Hermitian operator and a completely positive (CP) map is mapped to a positive semidefinite operator. To recover the action of the map, the `reverse direction' of the CJ correspondence is used:
\begin{equation}\label{eq:CJ^-1}
\mathcal{M}\left(A\right) = \left( \TrX{A}{M \cdot \left(A \otimes \mathds{1}_B\right) }\right)^T \:.
\end{equation}
Using this correspondence, maps, maps of maps, and so on, can all be represented as CJ operators on some composite space, and this is how we will describe them.
Then it is possible to define \textit{types} of maps. One first defines base types, out of which one defines a hierarchy of higher-order types as maps between types of lower order. The state spaces (in density matrix form) of standard quantum systems of a given finite dimension are standardly taken as base types (but different constructions are also possible). Although two different systems $A$ and $B$ could have state spaces of the same type (e.g., if they are both qubits), in order to avoid complicated notation, when we describe types instantiated on different systems we will use different letters -- by convention the same letters as for the systems -- and will indicate by words, if needed, when the types are the same.
More general types are then defined as sets of \textit{admissible} maps \footnote{A map is \textit{admissible} in this framework under certain axioms, which can be proven to reduce to the requirement of CP preservation and trace-rescaling for some normalization.} that have the same types of input and output spaces. For example, the above map $M$ is in type $A\rightarrow B$, where $A$ and $B$ are base types. However, in the same way one could describe maps whose input and output types $A$ and $B$ are maps themselves, by representing the input and output maps via their CJ operators (in this case, the input or output 'state' space would generally consist, up to a normalization, of a subset of the set of all standard states on the respective Hilbert space).
The formalism is an instance of a \textit{type system}. This ``$\rightarrow$'' connector, here nicknamed \textbf{transformation}, is the key element of the type theory of higher order transformations: each set can be seen as an abstract type, and new types can be defined out of existing ones using the transformation connector as a semantic rule.
Two special cases are of particular interest: the set of \textit{deterministic effects}, where the output type is the trivial one-dimensional system 1, \textit{i.e.} $A \rightarrow 1 \equiv \bar{A}$; and the parallel composition of two types, obtained as the transformation $\left(A \rightarrow \left(B \rightarrow 1 \right)\right)\rightarrow 1 = \overline{A \rightarrow \overline{B}} \equiv A \otimes B $ \cite{Perinotti2016}. The type $A \otimes B$ generalizes the idea of a no signaling channel \cite{Beckman2001,Piani2006} from the joint inputs of $A$ and $B$ to their joint outputs.
Thus, starting from some postulates, a trivial type 1, an elementary type $A$ (which is usually taken as the set of quantum states in density matrix form), and this connector rule, all the higher-order generalizations of the quantum formalism can be defined using types. These in turn yield the constraints on the operators representing transformations of a given type \cite{Perinotti2016, Bisio2018}.
For example, let $A_0$ and $A_1$ be, respectively, the set of input and output quantum states of Alice, who applies some quantum operation in between. Then, one can infer that the set of allowed transformations to which she has access is of type $A_0 \rightarrow A_1$. This simple semantic statement is then translated into constraints to apply on the Hilbert space $\mathcal{L}\left(\mathcal{H}^{A_0} \otimes \mathcal{H}^{A_1} \right)$ and yields (the Choi-Jamio{\l}kowski representation of) the set valid quantum channels for Alice.
A more complex example is obtained by recovering the set of bipartite process matrices \cite{OCB2012}, which corresponds to the type normalized on the local quantum instruments of two parties, say Alice and Bob. Knowing that their local instruments sum up to quantum channels, \textit{i.e.} they belong to types $\left(A_0 \rightarrow A_1 \right)$ and $\left(B_0 \rightarrow B_1\right)$, the set of process matrices is the type that takes a composition of these two types as input and outputs a trivial system. In the semantics, this statement corresponds to type $\left(\left(A_0 \rightarrow A_1 \right)\otimes \left(B_0 \rightarrow B_1\right)\right) \rightarrow 1$, from which the constraints for the characterization directly ensue.
Importantly, the transformation connector ``$\rightarrow$'' is not associative: $A \rightarrow (B \rightarrow C) \neq (A \rightarrow B) \rightarrow C$. This semantic fact is what allows to define the notion of an order with respect to (w.r.t.) a base type. Let $A$ be an instance of the type used as base defined in space $\LinOp{\Hilb{A}}$, and define $B,C,D$ in the same way. Then, $A,B,C,D$ are first-order types (trivial transformation) w.r.t. the base, types $A\rightarrow B$ and $C \rightarrow D$ are two instances of second order (both the same kind of transformations, but defined on different Hilbert spaces), and type $(A\rightarrow B) \rightarrow (C\rightarrow D)$ is third order (a transformation of transformation). Notice that while this last expression is third order, an expression like $((A\rightarrow B) \rightarrow C)\rightarrow D)$ is only second order, as it is but a cascading of second order transformations. With these considerations, it becomes possible to talk about a hierarchy of higher order quantum transformations as in reference \cite{Bisio2018}.
\paragraph{Outline and results.}
In section \ref{sec:Proj_char}, types are put in correspondence with subsets of operators called \textit{state structures} that are characterized by a superoperator projector. Looking at these projectors instead of the state structures, and by extension the types they define, is the idea behind projective characterization. It is then proven that this way of characterizing recovers the type theory.
A convenient aspect of projectors is that they form an algebra, so that manipulations of formulae are simplified by using algebraic properties. This point is developed in Section \ref{sec:Projos_alg}.
It is then observed that this algebra furthermore constitutes a model of linear logic.
In Section \ref{sec:NS}, a new projector encoding no signaling between two systems is introduced.
In addition to its obvious role of making the signaling structure of types apparent, it has useful algebraic properties. As presented in \ref{sec:applications_NSandrelations}, these allow one to write a canonical form of a projector, which is useful to compare any two state structure.
Finally, in Section \ref{sec:applications_iso} an example of the use of projective characterization is given. We recover the result \cite[Prop. 6]{Bisio2018} that quantum combs are quantum networks, which have a single signaling direction.
\section{Projective characterization of the type theory of higher-order quantum transformations\label{sec:Proj_char}}
Central in that work will be the notion of a \textit{trace-normalized positive operator system} in $\LinOp{\Hilb{A}}$. This kind of set is the backbone of the characterization as to each type -- or ``level in the hierarchy of transformations'' -- corresponds such a set (Appendix \ref{sec:examples_motiv} provides a motivating example if needed).
\begin{defi}[Operator system \cite{ChoiEffros1977}] \label{def:OS}
For a given space of operators, an \textbf{operator system} is a self-adjoint subspace that contains the identity.
\end{defi}
We will refer to both an operator system and its positive trace-normalized subset by using the calligraphic font of the letter associated with the subsystem it is defined upon; it should be clear from the context which one is under consideration. For example, $\mathscr{A}$ and $\mathscr{A}'$ are two different operator systems over $\LinOp{\Hilb{A}}$.
By definition, an operator system is closed under complex conjugation and Hermitian conjugation, which means that they are also closed under transposition. Note that $*$-subalgebras are a special case of operator systems that are closed under multiplication.
Since operator systems are linear subspaces, they can be characterized by linear superoperator projectors noted $\Proj{}{A} : \LinOp{\Hilb{A}} \rightarrow \mathscr{A}$, see Ref. \cite{Araujo2015,MPM}. For the purposes of the characterization, it is assumed that all such projectors are self-adjoint and that the composition of any two projector must yield a valid projector. Their characteristics are presented in details in the Appendix \ref{sec:projos}.
A positive and trace normalized operator system can be characterized by the following set of conditions that any of its elements $A$ have to respect:
\begin{subequations}\label{eq:det_struct}
\begin{gather}
A \geq 0 \:, \label{eq:det_struct_pos}\\
\TrX{}{A} = c_A \:, \label{eq:det_struct_norm}\\
\ProjOn{}{A}{A} = A \:. \label{eq:det_struct_proj}
\end{gather}
\end{subequations}
This abstract structure will play a central role in the characterization of higher-order quantum transformations, we name it \textit{state structure}.
\begin{defi}[State structure] \label{def:struct}
A \textbf{state structure} $\mathscr{A} \in \LinOp{\Hilb{A}}$ is a positive operator system that is trace-normalized.
\end{defi}
An example of state structure is the set of quantum states in density matrix form, characterized by
\begin{subequations}\label{eq:state_char}
\begin{gather}
\rho \geq 0 \:, \label{eq:state_char_pos}\\
\TrX{}{\rho} = 1\:, \label{eq:state_char_norm}\\
\mathcal{I}\{\rho\} = \rho \:. \label{eq:state_char_proj}
\end{gather}
\end{subequations}
Here, $\mathcal{I}$ is the \textit{identity mapping}, defined as
\begin{equation}\label{eq:Id}
\forall A \in \LinOp{\Hilb{A}}\::\: \mathcal{I}_A(A) = A \:.
\end{equation}
Here is another example. The elements of a POVM are actually operators summing up (or \textit{resolving}) the single-element state structure defined by
\begin{subequations}\label{eq:effect_char}
\begin{gather}
\mathds{1} \geq 0 \:, \label{eq:effect_char_pos}\\
\TrX{}{\mathds{1}} = d\:, \label{eq:effect_char_norm}\\
\mathcal{D}\{\mathds{1}\} = \mathds{1} \:. \label{eq:effect_char_proj}
\end{gather}
\end{subequations}
$\mathcal{D}$ is the \textit{depolarising superoperator}, obeying
\begin{equation}\label{eq:depolop}
\forall A \in \LinOp{\Hilb{A}}\:: \mathcal{D}_A(A) = \MapX{A}{A} \:,
\end{equation}
which projects onto the span of the identity.
\subsection{Functionals}
First, we characterize the type $\overline{A}$, generalizing the notion of a set of \textit{deterministic effects}. It is the set of operators representing all of the functionals mapping each element of a state structure to the number 1 via the Hilbert-Schmidt inner product $\TrX{}{\overline{A}\cdot A}=1$.
Note that in the following we will always mean deterministic effect when we use the word `effect' alone.
\begin{figure}
\centering
\subfloat[States and effects defining relation \eqref{eq:fctal_def}\label{fig:Tr_A_notA}]{
\centering
\includegraphics[height=0.12\textheight]{Figures/Diagrams/Fig1a.png}
}~\hfill
\subfloat[Diagrammatic representation of how the projectors define a splitting of the operator space\label{diag:AnotA}]{
\includegraphics[height=.12\textheight]{Figures/Sets/Fig1b.png}
}
\caption{Picturing the characterizing equations of states and effects deterministic type structures.}
\label{fig:fctal}
\end{figure}
In a previous work \cite{MPM}, it was noticed that since both the set of states and effects must contain an element proportional to the identity, the two subspaces on which the states and effects are respectively defined must be \textit{quasi-orthogonal} \footnote{The terminology ``quasi-orthogonal'' originated in the study of maximally Abelian subalgebras (MASA), see Ref. \cite{Hiai2014}}.
This means that operators $A$ and $\overline{A}$ must be orthogonal everywhere but at the span of the identity. If $\CompProj{}{A}$ is the projector for $\overline{\mathscr{A}}$ it is related to $\Proj{}{A}$ by $\CompProj{}{A}\equiv \mathcal{I}_A - \Proj{}{A} +\mathcal{D}_A$. This leads to the rephrasing in terms of projectors of the characterization of $\overline{\mathscr{A}}$ from type theory \cite[Lemma 4]{Bisio2018}.
\begin{prop}[Functional]\label{theo:det_fctal}
Let $\mathscr{A}$ be a state structure. Then the set $\bar{\mathscr{A}}$ of operators taking each element of $\mathscr{A}$ to the number 1 through the inner product,
\begin{equation}\label{eq:fctal_def}
\overline{A} \in \overline{\mathscr{A}} \iff \forall A \in \mathscr{A}\: : \:\TrX{}{\overline{A}\cdot A} = 1\:,
\end{equation}
is a state structure characterized by the following conditions:
\begin{subequations}\label{eq:det_fctal}
\begin{gather}
\bar{A} \geq 0 \:, \label{eq:det_fctal_pos}\\
\TrX{}{\bar{A}} = \frac{d_A}{c_A} \equiv c_{\bar{A}}\:,\label{eq:det_fctal_norm}\\
\left\{\mathcal{I}_A - \Proj{}{A} +\mathcal{D}_A\right\}(\bar{A})\equiv \CompProjOn{}{A}{\bar{A}}= \bar{A} \label{eq:det_fctal_proj} \:.
\end{gather}
\end{subequations}
\end{prop}
The proof is provided in Appendix \ref{app:fctal_proof}, see Figure \ref{fig:fctal} for an illustration. Concrete examples of states structures characterized by Prop. \ref{theo:det_fctal} are given in Appendix \ref{sec:examples_single_part}.
This proposition was first obtained as a theorem in our earlier work on Multi-round Process Matrices (MPM) \cite{MPM}.
Quasi-orthogonality implies the following property \cite[Theorem 2.37 iii)]{Hiai2014}:
\begin{equation}\label{eq:QO}
\forall A\in \mathscr{A}, \forall \overline{A}\in \overline{\mathscr{A}}\:,\: \TrX{}{\overline{A} \cdot A} = \frac{1}{d_A}\TrX{}{\overline{A}}\TrX{}{A} \: .
\end{equation}
As both sets have an element proportional to the identity, combining equations \eqref{eq:fctal_def} and \eqref{eq:QO} yields the following.
\begin{equation}\label{eq:fctal_indep}
\TrX{}{\overline{A} \cdot A} = \TrX{}{\overline{A}\cdot \frac{\mathds{1}}{c_{\bar{A}}}}\TrX{}{\frac{\mathds{1}}{c_A} \cdot A} \: .
\end{equation}
\begin{figure}
\centering
\includegraphics[height=0.12\textheight]{Figures/Diagrams/Fig2.png}
\caption{Quasi-orthogonality condition \eqref{eq:QO}\label{fig:Quasiortho}}
\label{fig:QO}
\end{figure}
Figure \ref{fig:QO} provides a graphical interpretation of this relation. This is the physical content of Proposition \ref{theo:det_fctal}: the generalized Born rule is independent of a \textit{particular} choice of deterministic state and deterministic effect; Both the state and the effect can be replaced by the identity -- or any other element of their respective state structures -- without altering the outcome probability of 1. Put differently, even when replacing either the state or the effect by pure noise, \textit{something} has to be measured in the end.
Here, the bar in ``$\CompProj{}{A} \equiv \mathcal{I} - \Proj{}{A} +\mathcal{D}$'' can be seen as an operation at the level of projectors that was applied on $\Proj{}{A}$.
As such, the projector $\CompProj{}{A}$ will be called the \textbf{negation} of $\Proj{}{A}$ because the operation has properties similar to a logic \textit{not}. This is developed in more details below and in Appendix \ref{sec:projos_prop}.
\subsection{Tensor product}
The tensor product of two types as a parallel composition generalizes the notion of \textit{no signaling channel} \cite{Piani2006}. In terms of projectors, the characterization relies on raising the tensor product operation at the level of superoperators in a natural fashion. Let $\Proj{}{A}$ and $\Proj{}{B}$ be respectively projectors on operator spaces $A$ and $B$, then $\Proj{}{A} \otimes \Proj{}{B}$ acts on $\LinOp{\Hilb{A}\otimes \Hilb{B}}$ so that
\begin{multline}
\left(\Proj{}{A} \otimes \Proj{}{B} \right) \left\{\sum_i \; q_i \; \left(A_i \otimes B_i\right)\right\} \equiv\\ \sum_i\;q_i\;\left(\ProjOn{}{A}{A_i}\otimes \ProjOn{}{B}{B_i}\right) \:,
\end{multline}
$\forall q_i \in \mathbb{C}$.
\begin{prop}[No signaling (bipartite) composition]\label{prop:tensor}
Let $\mathscr{A}$ and $\mathscr{B}$ be two state structures as in Eqs. \ref{eq:det_struct}, their no signaling composition $\mathscr{A}\otimes \mathscr{B} \subset \LinOp{\Hilb{A}\otimes \Hilb{B}}$ is the set of all operators $X$ characterised by the following constrains:
\begin{subequations}\label{eq:det_tensor}
\begin{gather}
X\in \mathscr{A}\otimes \mathscr{B} :\notag \\
X \geq 0 \:,\label{eq:det_tensor_pos}\\
\TrX{}{X} = c_Ac_B \label{eq:det_tensor_norm}\:,\\
\left(\Proj{}{A}\otimes\Proj{}{B}\right)\{X\} = X\:. \label{eq:det_tensor_proj}
\end{gather}
\end{subequations}
Consequently, $\mathscr{A}\otimes \mathscr{B}$ is \textit{the affine span} of $\mathscr{A}$ and $\mathscr{B}$.
\end{prop}
See Appendix \ref{app:tensor_proof} for the proof. We have just stated that the affine hull of a tensor product of state structures is a state structure itself \cite[Lemma 5]{Bisio2018}, this proposition will help us in the characterization of the CJ representation of mappings between state structures below. A diagrammatic depiction is given in Figure \ref{fig:tensor}, and examples of the use of the above can be found in Appendix \ref{sec:examples_tensor}.
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{Figures/Diagrams/Fig3.png}
\caption{Illustration of Proposition \ref{prop:tensor} and diagrammatic representation in the case of the tensor product of 2 subsystems: the full wheel represents $\LinOp{\Hilb{A}\otimes \Hilb{B}}$ while its parts represent its tensor factors (`Im' means `Image', so that e.g. $D$ is a shortcut for $\mathrm{Im}\{\mathcal{D}\} = \Span{\mathds{1}}$). By convention, the projectors acting on $\LinOp{\Hilb{A}}$ are always on the left-hand side of tensor products and vice-versa for $B$, so that subscripts can be omitted. Note that the intersections are well defined; for example the line `$A \otimes D$' is indeed the intersection $A \otimes B \cap A \otimes \bar{B}$.}
\label{fig:tensor}
\end{figure}
\subsection{Transformations \label{sec:Proj_char_trans}}
In the type theory of higher-order transformations, every type is an instance of a transformation, so the previous characterization of generalized effects (type $\bar{A}$) and bipartite states (type $A \otimes B$) are but special cases of a more general rule. This rule says that if $A$ and $B$ are types, then $A \rightarrow B$ is itself a type that takes $A$ to $B$. A state of type $A$ is actually a \textit{transformation} of the trivial type -- the number 1 -- into $A$, noted $1\rightarrow A$. Accordingly, the effect of type $\overline{A}$ is the transformation of type $A$ into a trivial system, $A\rightarrow 1$. No signaling composition is the type $A\otimes B \equiv \overline{A \rightarrow \overline{B}}$.
As we have seen, the set of elements of a given type corresponds to a state structure, ultimately determined by its projector. Thus, to type $A \rightarrow B$ corresponds a state structure, noted $\mathscr{A} \rightarrow \mathscr{B}$, which is obtained by combining state structures $\mathscr{A}$ and $\mathscr{B}$ in a certain way. This structure should then be characterized by a composite projector built by combining $\Proj{}{A}$ and $\Proj{}{B}$ using the two previously introduced rules.
Following previous approaches \cite{Kraus1983,Perinotti2016,Bisio2018,Dynamics}, we want the transformation -- or dynamics -- to conserve the total probability as well as probabilistic mixtures, and it should also be defined when acting on a tensor factor of \textit{any} no signaling composite system \cite[Definition 7]{Bisio2018}. In other words, the dynamical law should remain valid even when one is coarse-graining or fine-graining the system. The preservation of probabilistic mixtures, convex linearity, implies that the maps representing such dynamics are linear (see Ref. \cite{Kraus1983,OCB2012,Perinotti2016,Bisio2018} for the proof).
\begin{defi}[Structure-preserving map] \label{def:struc_pres}
Let $\mathcal{M}$ be a linear map from $\LinOp{\Hilb{A}}$ to $\LinOp{\Hilb{B}}$. This map is \textbf{structure-preserving} between $\mathscr{A}$ and $\mathscr{B}$ if it maps any element of state structure $\mathscr{A}$ to one in $\mathscr{B}$, and moreover if it keeps this property when these state structures are embedded in larger systems. That is, for any state structure $\mathscr{C}$, the map $\mathcal{M} \otimes \mathcal{I}_C \in \LinOpB{\LinOp{\Hilb{A}\otimes\Hilb{C}}}{\LinOp{\Hilb{B}\otimes\Hilb{C}}}$ should map any element of $\mathscr{A}\otimes\mathscr{C}$ to one of $\mathscr{B}\otimes\mathscr{C}$.
\end{defi}
Moving on to the characterisation of valid dynamics, let $\mathcal{M} \in \LinOpB{\LinOp{\Hilb{A}}}{\LinOp{\Hilb{B}}}$ be a linear map. We translate the first condition of Definition \ref{def:struc_pres} into the following conditions for all $A\in \mathscr{A}$ \cite{Dynamics} :
\begin{subequations}\label{eq:struc_pres}
\begin{gather}
\MOn{A} \geq 0\:, \label{eq:struc_pres_pos}\\
\TrX{}{\MOn{A}} = c_B\:, \label{eq:struc_pres_resc}\\
\Proj{}{B} \circ \mathcal{M} \circ \Proj{}{A} = \mathcal{M} \circ \Proj{}{A}\:. \label{eq:struc_pres_proj}
\end{gather}
\end{subequations}
The second condition further constrains the positivity preserving condition \eqref{eq:struc_pres_pos} to $\mathcal{M}$ being completely positive (CP) as $\mathscr{C}$ can in general be defined over the whole of $\LinOp{\Hilb{C}}$.
If we denote by $M$ the CJ representation of $\mathcal{M}$, the set of all these maps $M$ is a state structure \cite[Proposition 1]{Bisio2018}.
\begin{prop}[Mapping between state structures]\label{theo:det_map}
Let $\mathcal{M} \in \LinOpB{\LinOp{\Hilb{A}}}{\LinOp{\Hilb{B}}}$ be a structure-preserving map between state structures $\mathscr{A}$ and $\mathscr{B}$. Call $M \in \LinOp{\Hilb{A}\otimes \Hilb{B}}$ the Choi-Jamio{\l}kowski representation of this map, as in Eq. \eqref{eq:CJ}. Then, the set $\{M\}$ of all such maps belongs to the type structure $\overline{\mathscr{A} \otimes \overline{\mathscr{B}^T}}$, which means that it satisfies the following conditions:
\begin{subequations} \label{eq:det_map}
\begin{gather}
M \geq 0 \:,\label{eq:det_map_pos}\\
\TrX{}{M} = c_{\bar{A}}c_B = \frac{c_B}{c_A}d_A \:,\label{eq:det_map_norm}\\
\left(\Proj{}{A}\rightarrow \Proj{}{B^T}\right)\{M\} \equiv \left(\overline{\Proj{}{A}\otimes\CompProj{}{B^T}}\right)\{M\} = M \:.\label{eq:det_map_proj}
\end{gather}
\end{subequations}
\end{prop}
\begin{figure}
\centering
\subfloat[Graphical interpretation of $\mathscr{A}\rightarrow\mathscr{B}$\label{fig:A_to_B}]{
\includegraphics[width=.5\linewidth]{Figures/Diagrams/Fig4a.png}
}~\hfill
\subfloat[Diagram for the subspaces defined by Eq. \eqref{eq:det_map_proj} (blue) and $\Proj{}{A}\otimes \CompProj{}{B}$ (yellow)]{
\includegraphics[width=.40\linewidth]{Figures/Sets/Fig4b.png}
}
\caption{Illustration of Proposition \ref{theo:det_map}}
\label{fig:det_map}
\end{figure}
\paragraph*{Remark.} The above derivation is very close in spirit to Ref. \cite{Dynamics} where the projective characterization of transformations was first made in the context of transformation between process matrices, but the obtained projector was missing two terms \footnote{What the authors had obtained in the context of process matrix to process matrix transformation was $\mathcal{I}_A\otimes \mathcal{I}_B - \mathcal{P}_A \otimes \mathcal{I}_B + \Proj{}{A} \otimes \Proj{}{B}$, where $\Proj{}{A}$ is the projector characterizing the input process matrix state structure and $\Proj{}{B}$ the output. Compared to equation \eqref{eq:det_map_proj}, here written fully without negation,
$\overline{\Proj{}{A}\otimes\CompProj{}{B}} = \mathcal{I}_A\otimes \mathcal{I}_B - \mathcal{P}_A \otimes \mathcal{I}_B + \Proj{}{A} \otimes \Proj{}{B} - \mathcal{P}_A \otimes \mathcal{D}_B + \mathcal{D}_A\otimes \mathcal{D}_B$,
one sees that two terms are missing. In particular, they are the ones that forbids to postselect on a process matrix before preparing a new one that is purely noisy.}. After verification, it appears that this omission has not hindered the other conclusions of this article \cite{ECR_comm}.
Concrete examples of the use of this Proposition can be found in the Appendix \ref{sec:examples_transfo}. This Appendix also provides motivation for the ideas introduced in Section \ref{sec:NS}.
\subsection{The algebra of projectors recovers the type theory}
We started from the type theory of higher-order transforms in which $ \bar{A}, A\rightarrow B, A \otimes B$ are the new types that one can obtain from a base type $A$ under the semantic rules $\{1,(,),\rightarrow\}$.
We observed that types define \textit{state structures}, which are trace-normalized subsets of positive operator systems for which $\overline{\mathscr{A}}, \mathscr{A} \rightarrow \mathscr{B}, \mathscr{A} \otimes \mathscr{B}$ are the new state structures one can obtain from a base $\mathscr{A}$. These state structures are the sets of CJ operators representing higher-order maps.
As these state structures are defined on linear subspaces, most of their properties are encoded by the superoperator projector that defines them. $\CompProj{}{A}, \Proj{}{A}\rightarrow\Proj{}{B}, \Proj{}{A}\otimes \Proj{}{B}$ are the new projectors that one can obtain from a projector $\Proj{}{A}$ according to the algebraic rules $\{1,(,),\rightarrow\}$ (which can be taken as $\{\overline{\:\cdot\:},\otimes,\rightarrow\}$ instead).
Therefore, working with projectors is a handy way to define the characterization constraints implied by type theory.
\section{Abstracting the algebra of projectors\label{sec:Projos_alg}}
We now make a short pause in the characterization to point out that the algebraic rules obeyed by the projectors are not arbitrary. They actually form an algebra, as well as a model of logic.
\subsection{Beyond type theory: algebra of projectors
Type theory is dependent on a base type. In Ref. \cite{Bisio2018}, the first non-trivial type in the hierarchy, called the \textit{elementary type}, is taken as the set of quantum states as in Eqs. \eqref{eq:state_char}.
Within the framework of state structures, one can use other elementary types as a base. The question then is how to classify these nonequivalent theories, and how to compare them.
In terms of subspaces, comparing is straightforward as one can consider the \textit{overlap} between the two subspaces. For this purpose, one needs two new rules. These are the intersection, nicknamed `cap',
\begin{equation}
\Proj{}{A}\cap \Proj{'}{A}\equiv \Proj{}{A}\circ \Proj{'}{A} \:,
\end{equation}
and the union, `cup',
\begin{equation}
\Proj{}{A}\cup \Proj{'}{A}\equiv \Proj{}{A} + \Proj{'}{A} - \Proj{}{A}\cap \Proj{'}{A}\:.
\end{equation}
This definition assumes that all projectors in the algebra must commute with respect to the cap, $\Proj{}{A}\cap \Proj{'}{A} = \Proj{'}{A}\cap \Proj{}{A}$, $\forall \Proj{}{A}, \Proj{'}{A}$. This is indeed a sufficient condition for $\Proj{}{A}\cap \Proj{'}{A}$ to be a projector on operator system as well. More details are given in App. \ref{sec:projos_prop}.
With these, one can prove that an operator system is contained within another by showing either of the following:
\begin{subequations}
\begin{align}
\Proj{}{A} \cap \Proj{'}{A} = \Proj{'}{A} \:;\\
\Proj{}{A} \cup \Proj{'}{A} = \Proj{}{A} \:.
\end{align}
\end{subequations}
In terms of projectors, these conditions will be concisely noted
\begin{equation}
\Proj{'}{A} \subset \Proj{}{A} \:.
\end{equation}
Since the projectors are themselves CPTP maps, inclusion of projectors is indeed sufficient to show inclusion of the corresponding state structures (up to normalization).
As it turns out, the projectors form an algebra under the rules $\{\cap,\cup\}$ (see Appendix \ref{sec:projos_prop} for proof). And because every operator system must contain the identity element, every projector in the algebra is contained between the depolarizing and identity projectors,
\begin{equation}
\mathcal{D} \subset \Proj{}{} \subset \mathcal{I}.
\end{equation}
However, the cap and the cup are new rules that cannot be expressed using the $\rightarrow$ connector. So the downside of the algebra is that it is outside of the type-theoretic framework of references \cite{Perinotti2016,Bisio2018}.
Adding $\overline{\:\cdot\:}$ as an operation in the algebra promotes it into a \textit{Boolean algebra}, \textit{i.e.} an algebra of idempotent elements which possess a \textit{negation}.
$\overline{\:\cdot\:}$ acts as the negation since it makes $\CompProj{}{A}$ complementary to $\Proj{}{A}$, $\Proj{}{A}\cup \CompProj{}{A} = \mathcal{I}_A$ and it is an involution $\overline{\overline{\Proj{}{}}}_A = \Proj{}{A}$ (see Appendix \ref{sec:projo_Boolean} for more details).
\subsection{The algebra of projectors is (almost) Linear Logic}
With the further additions of the tensor and of the transformations operations, the Boolean algebra of projectors is lifted to an abstract structure that happens to follow the rules of linear logic (LL) \cite{GIRARD1987}. This extends an observation made in Ref. \cite{Kissinger_2019} that the logic of higher-order quantum transformations, in our language the projector rules $\{\overline{\:\cdot\:}, \otimes, \rightarrow\}$, form an instance of multiplicative linear logic.
Observe that the transformation between states is equivalent to the reverse transformation between effects,
\begin{equation}\label{eq:transfoduality}
\Proj{}{A} \rightarrow \Proj{}{B} = \CompProj{}{A} \leftarrow \CompProj{}{B} = \overline{\Proj{}{A} \otimes \CompProj{}{B}} \:.
\end{equation}
This is indeed a sign of the algebra being an instance of linear logic: it follows the logic rule that if A implies B, then not B must imply not A.
As such, the $\rightarrow$ corresponds to the \textit{linear implication} of LL, nicknamed \textit{lollipop} and noted ``$\multimap$''.
Negating the input of a transformation allows one to treat it as a composition:
\begin{equation}
\CompProj{}{A} \rightarrow \Proj{}{B} = \overline{\CompProj{}{A} \otimes \CompProj{}{B}} \:.
\end{equation}
This is a ``larger'' composition than the tensor as $\overline{\CompProj{}{A} \otimes \CompProj{}{B}} \neq \Proj{}{A} \otimes \Proj{}{B}$ and
\begin{equation}\label{eq:tensor_in_par}
\Proj{}{A} \otimes \Proj{}{B} \subset \overline{\CompProj{}{A} \otimes \CompProj{}{B}}.
\end{equation}
As such, operation `` $\overline{\:\cdot\:} \rightarrow \:\cdot\:$'' corresponds to the \textit{par}, ``$\parr$'', of LL, which, together with $\otimes$, respectively constitute the \textit{multiplicative disjunction} and \textit{conjunction}. The \textit{additive disjunction} and \textit{conjunction} correspond to the cup and the cap, respectively, and the \textit{linear negation} is the negation. They happen to be because on the one hand they respect the De Morgan laws
\begin{subequations}\label{eq:deMorgan}
\begin{gather}
\overline{\overline{\mathcal{P}}}_A = \Proj{}{A} \:;\\
\overline{\Proj{}{A}\cap\Proj{'}{A}} = \CompProj{}{A} \cup \CompProj{'}{A} \:; \label{eq:deMorgan_add}\\
\overline{\CompProj{}{A} \otimes \Proj{}{B}} = \CompProj{}{A} \rightarrow \CompProj{}{B} \label{eq:deMorgan_mult}\:.
\end{gather}
\end{subequations}
On the other hand, there exist a truth and a falsity for both additive and multiplicative rules, and they are the negation of each other. See Appendix \ref{sec:projo_prop_LL} for the details.
The correspondence with LL is made in Table \ref{tab:LL_comparison}, where the rules we have introduced are put in correspondence with their usual notation of LL.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c||c|c|}
\hline
Name & \multicolumn{2}{c||}{Symbol} & \multicolumn{2}{c|}{Unit} \\
& proj. & LL & proj. & LL \\
\hline
Negation & $\overline{\:\cdot\:}$ & $\cdot^\perp$ & / & / \\
\hline
Additive conjunction & $\cap $ & $ \& $ & $\mathcal{I}$ & {\sffamily T} \\
Additive disjunction & $ \cup $ & $ \oplus $ & $\mathcal{D}$ & 0 \\
Multiplicative conjunction & $ \otimes $ & $ \otimes $ & 1 & 1 \\
Multiplicative disjunction& $\overline{\:\cdot\:} \rightarrow \cdot$ & $ \parr $ & 1 & $\perp$ \\
\hline
Linear Implication & $\rightarrow$ & $\multimap$ & 1 & / \\
\hline
\end{tabular}
\caption{Correspondence of the algebraic rules of projective characterization (proj.) with linear logic (LL)}
\label{tab:LL_comparison}
\end{table}
Nonetheless, the algebra of projectors is still not a full model of linear logic as the \textit{exponential} connectors used to relate additive and multiplicative formulae have not been defined. To be precise, it is a model of \textit{multiplicative additive linear logic (MALL)}.
In addition to that, remark that the multiplicative truth and falsity are the same thing: the trivial projector onto the trivial system, \textit{i.e.} the number 1. There is also the issue that the additive falsity is not absorbant for multiplicative formulae, \textit{i.e.}, $\Proj{}{A} \otimes \mathcal{D}_B \neq \mathcal{D}_{AB}$.
The fact that the algebra of the projectors characterizing quantum higher-order transformations is an instance of MALL can additionally be understood as the consequence of the correspondence between the types of Ref. \cite{Bisio2018} and $*-$autonomous categories as in Ref. \cite{Kissinger_2019}.
\section{No signaling\label{sec:NS}}
In a previous work, the projective characterization was used to prove that the multi-round process matrix is a linear sum of quantum combs \cite[Theorem 2]{MPM}. That is, it allowed to split an MPM, which typically involves several directions of signaling, into a sum of quantum combs, featuring single, fixed directions.
In this section, the notion of no signaling is extended to state structures, in order for this decomposition to be generalized. Quasi-orthogonality will be shown to be the relevant notion for no signaling at the level of projectors, allowing in turn to define the \textit{prec}, an algebraic rule for composing projectors that encodes a one-way signaling constraint.
\subsection{Definition \label{sec:NS_def}}
No signaling from the input to the output can be imposed on a transformation as an extra condition. This translates the physical idea that, if the output is assumed to be in the causal future of the input, no (deterministic) operation done in the future can influence, or signal to, the past \cite{Chiribella_2011}.
The subset obeying this constraint will be noted using a ``$\prec$'' symbol, nicknamed the \textbf{prec}. By convention this connector will be seen as a composition, meaning that to represent a transformation, the input has to be negated. Thus, we want to define the set $\overline{\mathscr{A}} \prec \Alg{B}{} \subset \mathscr{A} \rightarrow \mathscr{B}$.
This condition is expressed as the requirement that the input of the transformation should be independent of the \textit{particular choice} of effect applied at its output:
\begin{multline}\label{eq:a}
M\in \CompAlg{A}\prec \Alg{B}{} \subset \ChanScr{A}{B}: \quad \forall \overline{B}, \overline{B}' \in \overline{\mathscr{B}},\\
\TrX{B}{M\cdot \left(\mathds{1} \otimes \overline{B} \right) } = \TrX{B}{M\cdot \left(\mathds{1} \otimes \overline{B}'\right)}\:.
\end{multline}
As the identity matrix $\mathds{1}$ is always both a valid state and effect (with suitable normalization), this requirement is rephrased as the following constraint,
\begin{equation}\label{eq:caus}
\forall \overline{B} \in \overline{\mathscr{B}},\; \TrX{B}{M\cdot \left(\mathds{1} \otimes \overline{B} \right) } =\TrX{B}{M\cdot \left(\mathds{1} \otimes \frac{\mathds{1}}{c_B}\right)}\:.
\end{equation}
This is depicted in Fig. \ref{fig:notA_prec_B}. When the above is satisfied, we will use it as a definition to say that $M$ is \textit{no signaling} from $B$ to $A$ \cite{Beckman2001,Piani2006}. The same way, one can define the no input-to-output signaling subset, $\CompAlg{A}\succ \Alg{B}{}$, by requiring the condition drawn in Fig. \ref{fig:notA_succ_B}.
\begin{figure}
\centering
\subfloat[$\CompAlg{A}\leftarrow\CompAlg{B}$, which is an equivalent condition to Fig. \ref{fig:A_to_B} because of Eq. \eqref{eq:transfoduality} \label{fig:notB_to_notA}]{\includegraphics[width=.45\linewidth]{Figures/Diagrams/notBTonotA.png}} \hfill
\subfloat[Notion of causality for channels; $\CompAlg{A}\leftarrow\CompAlg{B}$, in the case $\CompProj{}{B} = \mathcal{D}_B$ \label{fig:StrongCausality}]{\includegraphics[width=.45\linewidth]{Figures/Diagrams/nI_to_nI.png}} \\
\subfloat[No signaling from output to input, Eq. \eqref{eq:caus}; Eq. \eqref{eq:nA_prec_B} \label{fig:notA_prec_B}]{\includegraphics[width=.45\linewidth]{Figures/Diagrams/notAprecB.png}} \hfill
\subfloat[Eq. \eqref{eq:caus} is trivial for the case $\CompProj{}{B} = \mathcal{D}_B$\label{fig:StrongCausality_trivial}]{\includegraphics[width=.45\linewidth]{Figures/Diagrams/Strong_nA_prec_B.png}}\\
\subfloat[Diagram for the subspace defined by Eq. \eqref{eq:caus}\label{fig:nA_prec_B_diag}]{\includegraphics[width=.4\linewidth]{Figures/Sets/D_nA_prec_B_bis.png}}\hfill
\subfloat[No signaling from input to output, Eq. \eqref{eq:nA_succ_B}\label{fig:notA_succ_B}]{\includegraphics[width=.45\linewidth]{Figures/Diagrams/notA_succ_B.png}}
\caption{Graphical interpretation of the various conditions that can be imposed on a map, and illustration of Prop. \ref{prop:semi-causal_compo}}
\label{fig:oui}
\end{figure}
\paragraph*{Relation with the usual definition.}This no signaling definition recovers the one used in the respective case of quantum channel formalism. Indeed, the validity condition of transformations between quantum states implies that $\Alg{A}{}=\left\{\rho|\rho\geq 0 \cap \TrX{}{\rho} = 1\right\}$, so that $\CompAlg{A} = \{\mathds{1}_A\}$ and $\CompAlg{B} = \left\{ \mathds{1}_B \right\}$.
Observe that because of Eq. \eqref{eq:transfoduality}, a map $M \in \mathscr{A}\rightarrow \mathscr{B}$ can be equivalently interpreted as $\CompAlg{B} \rightarrow \CompAlg{A}$. Since $M$ must obey $\TrX{B}{M\left(\mathds{1}_A\otimes \overline{B}\right)} \in \CompAlg{A}$ by definition (see Fig. \ref{fig:notB_to_notA}), and since $\CompAlg{A} = \{\mathds{1}_A\}$, we attain the usual $M \in \mathscr{A}\rightarrow \mathscr{B} \iff$
\begin{equation}\label{eq:NS_usual}
\TrX{B}{M\left(\mathds{1}_A\otimes \mathds{1}_B\right)} = \TrX{B}{M} = \mathds{1}_A \:,
\end{equation}
which is used as both a validity and a causality condition in previous works \cite{Chiribella2009,Kissinger_2019} (Fig. \ref{fig:StrongCausality}). The reason is that condition \eqref{eq:caus} becomes tautological in that case (Fig. \ref{fig:StrongCausality_trivial}).
However, this equivalence between a general transformation and a no signaling one is only valid for the case of density matrices and hides the subtle difference between the two definitions in the general case for which being a valid transformation does not guarantee that the output cannot be used to signal to the input.
\subsection{Projective characterization}
Equation \eqref{eq:caus} can be rewritten as
\begin{equation}\label{eq:causal_notA_to_B}
\forall \overline{B} \in \overline{\mathscr{B}},\; \TrX{B}{M\cdot \left(\mathds{1} \otimes \overline{B} \right) } =\frac{1}{d_B}\TrX{B}{M}\TrX{}{\overline{B}}\:.
\end{equation}
This `factorization' of the partial trace is reminiscent of the property \eqref{eq:QO} of quasi-orthogonal subspaces. In Appendix \ref{app:NS=QO}, it is proven that the no signaling requirement defined in the previous section is enforced by a system-wise notion of quasi-orthogonality.
\begin{theo} \label{theo:causal_map}
Let $M$ be a positive operator in $\LinOp{\Hilb{A}\otimes \Hilb{B}}$, let $A$ be one in $\mathscr{A}$ and let $B$ be one in $\mathscr{B}$. Then a necessary and sufficient condition for condition
\begin{equation}\label{eq:partTrAB=TrATrB}
\TrX{A}{M \cdot \left(A \otimes B\right)} = \frac{\TrX{A}{M}\TrX{A}{A}}{d_A} \cdot B \:,
\end{equation}
to hold for all $A$, implying that
\begin{equation}
\TrX{A}{M \cdot \left(A \otimes \mathds{1}\right)} = c_A\TrX{A}{M} \:,
\end{equation}
is that
\begin{equation}\label{eq:nAotimesiB}
\left(\CompProj{}{A} \otimes \mathcal{I}_B \right)\left\{M\right\} = M \:.
\end{equation}
That is to say, that $M$ belongs to $\overline{\mathscr{A}} \otimes \LinOp{\Hilb{B}}$.
\end{theo}
As the above theorem states that no signaling is equivalent to restricting the operator to a subspace, a necessary and sufficient condition for the state structure $\mathscr{A}\rightarrow\mathscr{B}$ to be no signaling from the output to the input is that its projector gets restricted accordingly.
The projector for $\overline{\mathscr{A}} \prec \mathscr{B}$ is thus obtained by taking the intersection of the projectors
\begin{multline}\label{eq:nA_prec_B_proj}
\left(\Proj{}{A}\rightarrow\Proj{}{B}\right)\cap\left(\mathcal{I}_A \otimes \Proj{}{B}\right)\\
= \left(\mathcal{I}_A\otimes\mathcal{I}_B - \Proj{}{A}\otimes\CompProj{}{B} + \mathcal{D}_A \otimes \mathcal{D}_B\right) \cap \left(\mathcal{I}_A \otimes \Proj{}{B}\right)\\
= \mathcal{I}_A \otimes \Proj{}{B} - \Proj{}{A} \otimes \mathcal{D}_B + \mathcal{D}_A \otimes \mathcal{D}_B\\
\equiv \CompProj{}{A} \prec \Proj{}{B} \:.
\end{multline}
As a consequence, $\overline{\mathscr{A}} \prec \mathscr{B}$ is a state structure.
\begin{prop}[One-way signaling composition] \label{prop:semi-causal_compo}
Let $\mathscr{A}$ and $\mathscr{B}$ be two state structures as in Eqs. \eqref{eq:det_struct}, their one-way signaling composition $\mathscr{A}\prec \mathscr{B} \subset \LinOp{\Hilb{A}\otimes\Hilb{B}}$ is the set of all operators $W$ characterized by the following conditions:
\begin{subequations}\label{eq:det_semi-causal_B!<A}
\begin{gather}
W \geq 0 \:,\label{eq:det_semi-causal_B!<A_pos}\\
\TrX{}{W} = c_Ac_B \label{eq:det_semi-causal_B!<A_norm}\:,\\
\left(\Proj{}{A}\prec\Proj{}{B}\right)\{W\} = W\:. \label{eq:det_semi-causal_B!<A_proj}
\end{gather}
\end{subequations}
where
\begin{equation}\label{eq:semi-causal_comp_bis}
\Proj{}{A} \prec \Proj{}{B} \equiv \mathcal{I}_A \otimes \Proj{}{B} - \CompProj{}{A} \otimes \mathcal{D}_B + \mathcal{D}_A \otimes \mathcal{D}_B\;.
\end{equation}
\end{prop}
\begin{proof}
Directly from the above discussion, the conditions are obtained by applying the linear constraint $\mathcal{I}_A \otimes \Proj{}{B}$ to the set $\overline{\mathscr{A}}\rightarrow \mathscr{B} = \overline{\overline{\mathscr{A}}\otimes \overline{\mathscr{B}}} = \mathscr{A} \leftarrow \overline{\mathscr{B}}$ characterized by Proposition \ref{theo:det_map}.
\end{proof}
See Fig. \ref{fig:nA_prec_B_diag} for a diagrammatic depiction of the subspace associated with projector \eqref{eq:det_semi-causal_B!<A_proj}. The main rules for characterizing higher-order theories through their projector rules are summarized in Table \ref{tab:sum_char}.
\begin{table}
\centering
\begin{tabular}{|l|l|c|c|}
\hline
Name & Characterization & Proj. rule & Rule nickname \\
\hline
State & Def. \ref{def:struct}; Eqs. \eqref{eq:det_struct} & $\Proj{}{A}$ & / \\
Effect & Prop. \ref{theo:det_fctal}; Eqs. \eqref{eq:det_fctal} & $\CompProj{}{A}$ & Negation \\
\hline
No Sign. & Prop. \ref{prop:tensor}; Eqs. \eqref{eq:det_tensor} & $\CompProj{}{A} \otimes \Proj{}{B}$ & Tensor \\
2-ways Sign. & Prop. \ref{theo:det_map}; Eqs. \eqref{eq:det_map} & $\Proj{}{A} \rightarrow \Proj{}{B}$ & Transformation \\
1-way Sign. & Prop. \ref{prop:semi-causal_compo}; Eqs. \eqref{eq:det_semi-causal_B!<A} & $\CompProj{}{A} \prec \Proj{}{B}$ & Prec\\
\hline
\end{tabular}
\caption{Summary of the characterization rules for state structure of states, effects, and the three compositions \label{tab:sum_char}}
\end{table}
\subsection{Relating the compositions}
Using the algebra, the three bipartite rules $\otimes, \prec, \rightarrow$ can be related together by the following relations:
\begin{subequations}\label{eq:relations}
\begin{align}
\CompProj{}{A} \otimes \Proj{}{B} &= (\CompProj{}{A} \prec \Proj{}{B}) \cap (\CompProj{}{A} \succ \Proj{}{B})\:, \label{eq:1}\\
\Proj{}{A} \rightarrow \Proj{}{B} &= (\CompProj{}{A} \prec \Proj{}{B}) \cup (\CompProj{}{A} \succ \Proj{}{B}).
\end{align}
\end{subequations}
See Appendix \ref{sec:projo_prec} for the proof.
These have a concrete physical interpretation.
In a first time, consider the set $\CompAlg{A} \otimes \mathscr{B}$. Its elements are transformations from $\Alg{A}{}$ to $\Alg{B}{}$ as each $M \in \CompAlg{A} \otimes \mathscr{B}$ satisfies
\begin{equation}\label{eq:A_to_B}
\TrX{A}{M\cdot (A\otimes \mathds{1})} \in \mathscr{B} \:,
\end{equation}
because $\CompProj{}{A} \otimes \Proj{}{B} \subset \Proj{}{A}\rightarrow \Proj{}{B}$.
In addition to that, the right-hand side of Eq. \eqref{eq:1} puts two extra conditions on $\CompAlg{A} \otimes \mathscr{B}$ that are similar to Eq. \eqref{eq:caus}; it requires that $\left(\CompProj{}{A} \otimes \Proj{}{B}\right)\{M\} = M \iff \forall A \in \Alg{A}{}, \forall \overline{B} \in \overline{\mathscr{B}}$,
\begin{subequations}\label{eq:A_to_B_one_way}
\begin{align}
&\TrX{A}{M\cdot \left(A \otimes \mathds{1}\right) } =\TrX{A}{M\cdot \left(\frac{\mathds{1}}{c_{\bar{A}}} \otimes \mathds{1}\right)}\:; \label{eq:nA_prec_B}\\
& \TrX{B}{M\cdot \left(\mathds{1} \otimes \overline{B} \right) } =\TrX{B}{M\cdot \left(\mathds{1} \otimes \frac{\mathds{1}}{c_B}\right)}\:. \label{eq:nA_succ_B}
\end{align}
\end{subequations}
Therefore, the above pair of conditions reveals the set $\CompAlg{A} \otimes \mathscr{B}$ as the set of transformations that are compatible with no signaling from system A to system B \textbf{and} from B to A at the same time; it is no signaling in both directions, hence the name.
The same way, the $\Alg{A}{} \rightarrow \mathscr{B}$ set is the set of transformations that respect no signaling from A to B \textbf{or} B to A. Thus, it may allow linear combinations of signaling in both directions and consequently to indefinite causal order.
The sets $\CompAlg{A}\prec \Alg{B}{}$ and $\CompAlg{A}\succ \Alg{B}{}$ lie in between, as they permit signaling in only one direction. Their elements indeed obey condition \eqref{eq:1} but only one of the conditions \eqref{eq:A_to_B_one_way}.
This discussion underlies the following chain of inclusions:
\begin{subequations}\label{eq:inclusions}
\begin{gather}
\CompProj{}{A} \otimes \Proj{}{B} \subset \CompProj{}{A} \prec \Proj{}{B} \subset \Proj{}{A}\rightarrow \Proj{}{B}\:,\\
\CompProj{}{A} \otimes \Proj{}{B} \subset \CompProj{}{A} \succ \Proj{}{B} \subset \Proj{}{A}\rightarrow \Proj{}{B} \:,
\end{gather}
\end{subequations}
See Figure \ref{fig:compos} for a diagrammatic interpretation.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Figures/Sets/Compositions.png}
\caption{Inclusions between the various ways of composing projectors}
\label{fig:compos}
\end{figure}
\paragraph*{Defining the tensor product as no signaling.}Naturally, Eqs. \eqref{eq:A_to_B_one_way} are also valid for bipartite states: $\left(\Proj{}{A} \otimes \Proj{}{B}\right)\{W\} = W \iff \forall \overline{A} \in \CompAlg{A}{}, \forall \overline{B} \in \overline{\mathscr{B}}$,
\begin{subequations}\label{eq:A_otimes_B_def}
\begin{align}
&\TrX{A}{\left(\overline{A} \otimes \mathds{1}\right)\cdot W } =\TrX{A}{ \left(\frac{\mathds{1}}{c_{A}} \otimes \mathds{1}\right) \cdot W}\:; \\
& \TrX{B}{\left(\mathds{1} \otimes \overline{B} \right) \cdot B } =\TrX{B}{ \left(\mathds{1} \otimes \frac{\mathds{1}}{c_B}\right)\cdot W}\:.
\end{align}
\end{subequations}
It should be noted that the definition of the no signaling composition $\otimes$ can be done solely by requiring these two equations to be valid simultaneously; there is no need to additionally require that $W$ is a transformation (\textit{i.e.} that it belongs to $\CompAlg{A}\rightarrow\Alg{B}{}=\Alg{A}{}\rightarrow\CompAlg{B}$).
This observation have a two corollaries: on the one hand, it provides a `transformation-free' definition of the tensor product based only on the validity of Theorem \ref{theo:causal_map}. On the other hand, it justifies its nickname of `no signaling composition'.
\paragraph*{Remark.} While the interpretation of the \textit{transformation} (``$\rightarrow$'') in linear logic is the linear implication (``$\multimap$''), the ``$\prec$'' connector has no studied counterpart in linear logic. Still, the fact that it is a multiplicative connector that commutes with negation makes its appearance in the algebra look natural. Exploring the properties of the prec in LL is left for future work.
\paragraph*{Categorical perspectives.} In terms of categories, higher-order theories of transformations built using two-way signaling composition can be seen as a category with objects in $\Alg{A}{}$ and morphisms in $\Alg{A}{} \rightarrow \Alg{B}{}$, thereafter noted $\mathcal{C}(\Alg{A}{},\Alg{A}{}\rightarrow \Alg{B}{})$. These are $*-$autonomous categories, meaning that they possess two monoidal structures ($\otimes$ and $\overline{\cdot}\rightarrow\cdot$) and that negation commutes with neither of them \cite{Kissinger_2019}.
The one-way signaling composition provides a decomposition of these $*-$autonomous categories into the union of two closed compact categories, $\mathcal{C}(\Alg{A}{},\CompAlg{A}\prec \Alg{B}{})$ and $\mathcal{C}(\Alg{A}{},\CompAlg{A}\succ \Alg{B}{})$. These induce the following chains of wide subcategories:
\begin{align}
&\mathcal{C}(\Alg{A}{},\overline{\Alg{A}{}}\otimes \Alg{B}{}) \subset \mathcal{C}(\Alg{A}{},\overline{\Alg{A}{}}\prec \Alg{B}{})\subset\mathcal{C}(\Alg{A}{},\Alg{A}{}\rightarrow \Alg{B}{})\:,\\
&\mathcal{C}(\Alg{A}{},\overline{\Alg{A}{}}\otimes \Alg{B}{}) \subset \mathcal{C}(\Alg{A}{},\overline{\Alg{A}{}}\succ \Alg{B}{})\subset\mathcal{C}(\Alg{A}{},\Alg{A}{}\rightarrow \Alg{B}{})\:.
\end{align}
Surely, this decomposition calls for a more systematic functorial treatment: the category of higher-order transformations actually features not two but three monoidal structures, with one of them ($\prec$) being compact and closed.
\subsection{The algebra of projectors recovers no signaling}
We started from the projective characterization of higher-order transformations in which $\CompProj{}{A}, \Proj{}{A}\rightarrow\Proj{}{B}, \Proj{}{A}\otimes \Proj{}{B}$ are the new projectors one can obtain from a projector $\Proj{}{A}$ under the algebraic rules $\{\overline{\:\cdot\:},\otimes,\rightarrow\}$.
We observed that the notion of no signaling could be generalized for the case of state structures. In particular, it was noticed that the notion could be encoded by a new rule $\prec$ in the algebra of projectors, introduced under the name prec.
Bipartite state structures obtained by combining $\CompAlg{A}$ and $\Alg{B}{}$ fall into four classes, essentially determined by conditions \eqref{eq:A_to_B}, \eqref{eq:nA_prec_B}, and \eqref{eq:nA_succ_B}. All of these classes must satisfy the first condition, which amounts to being a valid transformation and to which corresponds the projector $\Proj{}{A}\rightarrow \Proj{}{B}$.
If in addition condition \eqref{eq:nA_prec_B} (respectively, \eqref{eq:nA_succ_B}) is satisfied, the set is restricted to transformation that is no signaling from the output to the input (resp., from the input to the output), to which corresponds the projector $\CompProj{}{A}\prec \Proj{}{B}$ (resp., $\CompProj{}{A}\succ \Proj{}{B}$).
If both conditions are satisfied, the set is no signaling, to which corresponds the projector $\CompProj{}{A} \otimes \Proj{}{B}$.
Therefore, working with projectors provides a handy way of assessing the signaling structure within a state structure.
\section{Implications of the formalism}
With the formalization of the projective characterization and the introduction of the one-way signaling subsets using the prec, we now have the tools to address signaling in theories of higher-order quantum transformations. In this section, we are interested only in the qualitative aspects of state structures, meaning that we will not be referring to operators, but only sets and their projectors. As a consequence, the calligraphic notation will be dropped for clarity, and capital letters will refer to state structure or projectors depending on the context.
\subsection{Algebraic manipulations using the prec: two birds with one stone \label{sec:applications_NSandrelations}}
On purely algebraic grounds, the transformation $\rightarrow$ has two defining properties that make it inconvenient to work with.
The first is not commuting with negation:
\begin{equation}
\overline{A\rightarrow B} \neq \overline{A} \rightarrow \overline{B} \:.
\end{equation}
While in the categorical treatment this is a manifestation of the $*-$autonomous character of the category of higher-order transformations \cite{Kissinger_2019}, it leads to difficulties for relating seemingly similar sets together using the algebra alone. Two expressions involving the same states and effects may be incomparable because one features a negation on several subsystems at once while the other does not. For example, $\overline{A\rightarrow B}= A \otimes \overline{B}$ cannot be compared with $\overline{A} \rightarrow \overline{B} = \overline{\overline{A} \otimes B}$ without explicitly doing computations to show set inclusions.
The second inconvenient property is that $\rightarrow$ is not associative:
\begin{equation}
A \rightarrow (B \rightarrow C) \neq (A \rightarrow B) \rightarrow C \:.
\end{equation}
As we have seen, this is what allows one to define the notion of an \textit{order} with respect to a base type in the first place. Let $A_0, A_1, A_2, A_3,\ldots$ be some base type $A$ defined on different spaces $\LinOp{\Hilb{A_0}}, \LinOp{\Hilb{A_1}}, \ldots$, then $A_0$ is a type of first order, $A_0\rightarrow A_1$ is second order, $(A_0\rightarrow A_1)\rightarrow (A_2 \rightarrow A_3)$ is third order and so on...
However, a naive application of the notion of order may hide the equivalence between some theories. For example, consider $((A_0 \rightarrow A_1) \rightarrow A_2) \rightarrow A_3$ and $(A_1\rightarrow A_2) \rightarrow (A_0 \rightarrow A_3)$. Some computations will reveal that they both feature $\overline{A}_0 \otimes A_1 \otimes \overline{A}_2 \otimes A_3$ as their no signaling subset (this notion is more precisely defined below), and accordingly they should be comparable. But they are not of the same order: the former is a succession of three transformations of a state $A$, whereas the latter is a transformation of higher-order state $A_1 \rightarrow A_2$ into $A_0 \rightarrow A_3$. Looking at the formula alone, one would conclude that there is no use in comparing them because they do not feature the same base state structures. Yet, in some cases they happen to be the same. In this example, if the base types $A$'s are the quantum states, then the two expressions are two ways of describing the same thing, the set of \textit{quantum 2-combs} \cite{Chiribella2009}.
Surprisingly, the prec does not lead to these issues, as it commutes with the negation:
\begin{equation}
\overline{A \prec B} = \overline{A} \prec \overline{B} \:.
\end{equation}
And, like the tensor, it is associative:
\begin{equation}
(A \prec B) \prec C = A \prec (B \prec C) = A \prec B \prec C \:.
\end{equation}
As the algebra obeys the De Morgan relations (Eqs. \eqref{eq:deMorgan}) and that the tensor and transformation split into precs (Eqs. \eqref{eq:relations}), it becomes possible to express any expression $\Gamma$ involving $n$ projectors $A_1,A_2,\ldots,A_n$ composed with $\otimes$ and $\rightarrow$ as the union and intersection of types built using the prec.
To make this point precise, we first define the following.
\begin{defi}[No signaling subset]
For a projector expression $\Gamma$ that involves $n$ projectors $A_1,A_2,\ldots,A_n$ composed together using $\otimes$ and $\rightarrow$, the \textbf{no signaling subset} is the subset built using only negations over single projector and tensor products.
It has the form
\begin{equation}\label{eq:no-sig_subset}
\Tilde{A}_1 \otimes \Tilde{A}_2 \otimes \ldots \otimes \Tilde{A}_n \subset \Gamma \:,
\end{equation}
where the $\Tilde{A}_i$ notation means that projector $A_i$ is potentially negated, depending on $\Gamma$.
\end{defi}
Using the definition \eqref{eq:det_map_proj}, any $\rightarrow$ can be turned into a mix of tensor and negation, $\:\cdot\:\rightarrow\:\cdot\: = \overline{\:\cdot\:\otimes \overline{\:\cdot\:}}$. Then, because of Eq. \eqref{eq:tensor_in_par}, one can always find the \textit{no signaling subset} of an expression by distributing the negations.
A rule of thumb for finding the no signaling subset is to `count the number of bars' above each projectors in an expression. Since negation is an involution, an odd (respectively, even) amount will indicate that the projector is (not) negated in the no signaling subset.
For example, to find the no signaling subset of $\left(A_0 \otimes \left(A_1 \rightarrow A_2 \right)\right)\rightarrow A_3$, one first expresses it using negations and tensors products, $\left(A_0 \otimes \left(A_1 \rightarrow A_2 \right)\right)\rightarrow A_3 = \overline{A_0 \otimes \overline{A_1 \otimes \overline{A_2}} \otimes \overline{A_3}}$, then by inspection $A_0$ and $A_2$ have an odd number of negations above them (one and three, respectively) so they will be negated, whereas $A_1$ and $A_3$ have an even number (two and two) so they will not be. On that account, the no signaling subset of $\left(A_0 \otimes \left(A_1 \rightarrow A_2 \right)\right)\rightarrow A_3$ is $\overline{A_0} \otimes A_1 \otimes \overline{A_2} \otimes A_3$.
Now, in the algebra, the cap is distributive over the cup,
\begin{equation}
(A\cup A') \cap A'' = (A\cap A'') \cup (A' \cap A'') \:,
\end{equation}
but the converse is not true.
Furthermore, the cap and the cup are both distributive over the prec (and in this case the converse holds, see App. \ref{sec:projo_prec_dist} for proofs),
\begin{subequations}
\begin{gather}
(A\cup A') \prec B = (A \prec B) \cup (A' \prec B) \:,\\
A \prec (B\cap B') = (A \prec B) \cap (A\prec B') \:.
\end{gather}
\end{subequations}
Where the above are also satisfied when caps and cups are switched, $(\cap \leftrightarrow \cup)$. Consequently, there is an order in distributivity; expressions involving these three connectors can always be rewritten so that they are put into a union of intersections of one-way signaling compositions. As these three operations are associative, the intermediate parenthesis can be dropped everywhere.
Finally, by combining Eqs. \eqref{eq:relations} with the De Morgan rule \eqref{eq:deMorgan_add} and using the fact that prec commutes with negation, it should be clear that in the decomposition, the projectors on each single subsystem are the same as the ones appearing in the no signaling subset.
For these reasons, any expression $\Gamma$ involving $n$ projectors $A_1,A_2,\ldots,A_n$ composed using $\otimes$ and $\rightarrow$ can be brought to a \textit{canonical} form.
\begin{defi}[Canonical form]\label{def:canonical_form}
Let $\Gamma$ be a projector expression that involves $n$ projectors $A_1,A_2,\ldots,A_n$ composed together using $\otimes$ and $\rightarrow$, and whose no signaling subset is given by $\Tilde{A}_1 \otimes \Tilde{A}_2 \otimes \ldots \otimes \Tilde{A}_n \subset \Gamma$.
Then, its \textbf{canonical form} is the decomposition
\begin{equation}
\Gamma=\bigcup_{i=1}^{x}\left(\bigcap_{j=1}^{y_i} \Tilde{A}_{\sigma_{ij}(1)} \prec \Tilde{A}_{\sigma_{ij}(2)} \prec \ldots \prec \Tilde{A}_{\sigma_{ij}(n)} \right) ,
\end{equation}
in which there are $x$ unions of expressions labeled by index $i$, and each expression involves $y_i$ intersections of sub-expressions labelled by index $j$. $\sigma_{ij}$ is an element of the permutation group so that each sub-expression is a permutation of $\Tilde{A}_1 \prec \Tilde{A}_2 \prec \ldots \prec \Tilde{A}_n$ (indices $i$ and $j$ do not necessarily run over the full permutation group).
\end{defi}
This implies that any expression can be brought down to the union of several first-order expressions (which further decompose into intersections if needed)). This way of rewriting is essentially unique for \textit{any} state structure, lifting any ambiguity induced by the negation or non-associativity.
In the first example discussed above, the expressions reduce to $\overline{A\rightarrow B}= (A \prec \overline{B}) \cap (A \succ \overline{B})$ and $\overline{A} \rightarrow \overline{B} = (A \prec \overline{B}) \cup (A \succ \overline{B})$. Now one can meaningfully compare them: they are both built from a combination of the same two one-way signaling transformations, but one set is the intersection and the other is the union, we thus conclude that $\overline{A\rightarrow B}\subset \overline{A}\rightarrow \overline{B}$.
Moreover, the prec connector and the decomposition it induces also allows for an identification of the possible signaling directions these two sets may feature. Indeed, since the prec is exactly the no signaling condition, the above canonical form is a breakdown of a general expression into several ordered expressions built using only the prec, then combined first by requiring no signaling between some subsystems, i.e. using cap, and then by allowing signaling in several directions, i.e. using cup.
Summarizing, the algebraic fact that the prec is associative and commutes with the negation allows one to write an unambiguous `canonical' form useful to compare theories.
This gives a tool to determine whether two higher-order theories are equivalent by sole inspection of their projectors.
In addition, the physical fact that the prec is a one-way signaling composition, combined with the algebraic rules, allows one to split projectors in several terms with a fixed signaling direction between its base state structures.
This implies that one can know the possible signaling structure that a higher-order theory may feature by sole inspection of its projector.
\subsection{When quantum combs are isomorphic to quantum networks \label{sec:applications_iso}}
The next part of the results is concerned with the relationship between two different ways of building higher-order objects using the developed formalism of projectors. This can be seen as an example of how to use it in order to swiftly recover results of reference \cite{Bisio2018}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Figures/Sets/Accidental_Isomorphism.png}
\caption{Isomorphisms when the projectors are $\mathcal{I}$, \textit{i.e.} in the case of transformations between quantum states}
\label{fig:Accident}
\end{figure}
Following reference \cite{Chiribella2009}, a \textbf{network} is defined as the causally ordered (i.e. one-way signaling) succession of `nodes' of the same state structure. A `1-network of base $A$' will be the set $A$ itself, thereafter noted $A_0$ for clarity, then the `2-network' will be the set $A_0 \prec A_1$, the `3-network' will be $A_0 \prec A_1 \prec A_2$, etc.
A common occurrence of this structure is the quantum network, where the base object is a quantum channel such as $A\equiv I_{A_0} \rightarrow I_{A_1}$. This quantum network, here associated with some party that will be called Alice, represents the successive operations of that party. If Alice has a single node quantum network, it means that Alice acts once on the system $A_0$ with a quantum channel and outputs a quantum state in $A_1$. If she has a network with two nodes, she will act a first time on $A_0$ and output a first state at $A_1$, then a second time on $A_2$, now potentially using any size of ancillary qudit as a memory register she preserved from her first operation, and output in $A_3$. And so on, as defined recursively.
Another way of building a higher-order state structure is by doing a \textbf{comb}, which consists of recursively transforming into a base type: the `1-comb of base $A$' is again $A_0$, then the $2-$comb is $A_0 \rightarrow A_1$, the $3-$comb is $(A_0 \rightarrow A_1) \rightarrow A_2$, etc up to the $n-$comb defined as $(...(A_0\rightarrow A_1) \rightarrow ...)\rightarrow A_n$. Note that we reserve the name ``quantum comb'' to the specific case of combs based on quantum channels.
When the two constructions were introduced in reference \cite{Chiribella2009}, it was proven that a quantum network \textit{is} a quantum comb.
When treated using the formalism developed here, there is a stark contradiction: All quantum combs are built using the transformation, $\rightarrow$, which permits two-ways signaling. How could it be that they are all equivalent to networks -- that is, objects featuring a single direction of signaling? Especially, why is the 1-comb (built using the two-ways signaling transformation) equivalent to the quantum channel (which is causal)? Why does the quantum 2-comb reduce to a two-node quantum network, i.e., a map acting on two quantum states, when by definition it should be a supermap? And why does it reduce to a succession of two operations that have a well-defined direction of signaling between parties, and not, as its projector suggests, a nesting of two two-way signaling compositions, resulting in four possible signaling directions?
\paragraph{Transformations between quantum states.}
From the definition (see App. \ref{sec:projo_prec_iso}), it can be proven that the expression for the prec simplifies when either side corresponds to the identity projector or the depolarizing one:
\begin{subequations}\label{eq:isomorphisms}
\begin{align}
\overline{A} \prec B = A \rightarrow B \:&\iff\: A=I \:\text{or}\: B=I\:; \label{eq:isomorphisms_transfo}\\
\overline{A} \prec B = \overline{A} \otimes B \: &\iff \quad A=D \:\text{or}\: B=D\:. \label{eq:isomorphisms_tensor}
\end{align}
\end{subequations}
These relations are depicted in Figure \ref{fig:Accident} for the case where both $A$ and $B$ are density operators. This should be compared with the general case of Figure \ref{fig:compos}.
Eq. \eqref{eq:isomorphisms_transfo} is abstractly the reason why the quantum 1-comb, or quantum channel, happens to have no signaling from its output to its input. As previously discussed in Section \ref{sec:NS_def}, this is the case where the definition of no signaling overlaps with the one of transformation like in equation \eqref{eq:NS_usual}, and the usual definition of no signaling is recovered.
Because of that, a network of quantum channels is then an alternating network of effects and states as associativity can be used.
Eq. \eqref{eq:isomorphisms_tensor} is for example the reason why the single partite process matrix reduces to an effect and a state in tensor product \cite{OCB2012}.
\paragraph{Equivalence of quantum combs and networks.}
Now, all the ideas needed to understand why quantum combs are causal networks are in place. Formally, it was proven \cite{Perinotti2016,Bisio2018} that:\\
1) The comb based on quantum states (for clarity, let $A_i = I_{A_i}$):
\begin{equation}\label{eq:states_comb}
(\ldots((A_0\rightarrow A_1) \rightarrow A_2 ) \rightarrow \ldots)\rightarrow A_{2n-1} \:,
\end{equation}
is equivalent to the quantum comb (notice the reordering of the terms),
\begin{multline}\label{eq:channel_comb}
(\ldots((A_{n-1} \rightarrow A_n)\rightarrow (A_{n-2} \rightarrow A_{n+1}))\rightarrow \ldots)\\
\rightarrow (A_0 \rightarrow A_{2n-1}) \:.
\end{multline}
2) Operators $M$ in these structures obey the causality condition
\begin{equation}\label{eq:causality_cond}
\TrX{A_{2i+1}}{M} = \frac{1}{d_{A_{2i}}}\TrX{A_{2i+1}A_{2i}}{M} \otimes \mathds{1}_{A_{2i}} \quad \forall i.
\end{equation}
According to the discussion in Section \ref{sec:NS_def}, the causality conditions can be recast into a single projector,
\begin{equation}\label{eq:causality_cond_proj}
\left(A_0 \rightarrow A_1\right) \prec \ldots \prec \left(A_{2n-2} \rightarrow A_{2n-1}\right)\:.
\end{equation}
Because of the isomorphism \eqref{eq:isomorphisms_transfo} and associativity of the prec, this is equivalent to
\begin{equation}
\overline{A_0} \prec A_1 \prec \ldots \prec \overline{A_{2n-2}} \rightarrow A_{2n-1}\:.
\end{equation}
For $n=1$, these three equations \eqref{eq:states_comb}, \eqref{eq:channel_comb} and \eqref{eq:causality_cond_proj} are trivially equivalent. For $n=2$, we recover the second example of the last section: Statement 1) implies that the quantum 2-comb $(A_1\rightarrow A_2) \rightarrow (A_0 \rightarrow A_3)$ is equivalent to the quantum-state-based 4-comb $(((A_0 \rightarrow A_1) \rightarrow A_2) \rightarrow A_3$, although, in general, the latter is a subset of the former. Statement 2) implies that, in the quantum case, the two have a fixed signaling direction, although they are built using the 2-way signaling transformation.
With the prec rules and the isomorphisms of equations \eqref{eq:isomorphisms}, the proofs of both statements are obtained from algebraic manipulations. First, for the state-based comb, the transformation is isomorphic to a prec, $A_0 \rightarrow A_1 = \overline{A_0}\prec A_1$, then $((A_0 \rightarrow A_1)\rightarrow A_2 ) \rightarrow A_3 = \overline{(\overline{(\overline{A_0}\prec A_1)} \prec A_2 )} \prec A_3 = \overline{A_0} \prec A_1 \prec \overline{A_2} \prec A_3$, because the negation is distributed over the prec, and this connector is associative.
Next, the equivalence between the combs, statement 1), is obtained using the same properties: $(A_1\rightarrow A_2) \rightarrow (A_0 \rightarrow A_3) = (\overline{A_1}\prec A_2) \rightarrow (\overline{A_0} \prec A_3) = \overline{(\overline{A_1}\prec A_2) \otimes (A_0 \prec \overline{A_3})} = \overline{(\overline{A_1}\prec A_2) \otimes (A_0 \otimes \overline{A_3})}$, some reordering and the use of the tensor associativity yield $\overline{A_0 \otimes ((\overline{A_1}\prec A_2) \otimes \overline{A_3})} = \overline{A_0 \otimes (\overline{A_1}\prec A_2 \prec \overline{A_3})} = \overline{A_0 \prec\overline{A_1}\prec A_2 \prec \overline{A_3}} = \overline{A_0} \prec A_1 \prec \overline{A_2} \prec A_3$.
Finally, the proofs can be generalized \textit{mutatis mutandis} for any number of nodes, see Appendix \ref{sec:comb=net} for the details.
We therefore have
\begin{multline}
(\ldots((A_0\rightarrow A_1) \rightarrow A_2 ) \rightarrow \ldots)\rightarrow A_{2n-1} =\\ \overline{A_0} \prec A_1 \prec \overline{A_2} \prec \ldots \prec A_{2n-1} \:,
\end{multline}
so the statement 2) that this comb has a single signaling direction is now directly apparent as the structure has a single `prec chain' in canonical form.
We also have that
\begin{multline}
(\ldots(A_{n-1} \rightarrow A_n)\rightarrow \ldots) \rightarrow (A_0 \rightarrow A_{2n-1}) =\\ \overline{A_0} \prec A_1 \prec \overline{A_2} \prec \ldots \prec A_{2n-1}\:.
\end{multline}
Hence isomorphisms between $\prec$ and $\rightarrow$ in the case of theories whose base structure are the quantum states explain this apparent counter-logical equivalence of a second-order (with respect to quantum states) theory with a third-order one.
A different example of the consequences of this result in the case of MPM can be found in Appendix \ref{sec:examples_dynamics_constr}. It also features another example of the use of the canonical form.
\section{Discussion}
In this work, a correspondence was developed between several characterizations of the theory of higher-order quantum transformations.
Starting at the most abstract layer, its formulation in terms of types, we added a new layer, its formulation in terms of superoperator projectors, to bridge to a less abstract layer, its formulation in terms of sets of CJ operators -- here called state structures.
Correspondences between the characterizations can be summed up as follow: Given types $A$ and $B$, types $\bar{A}, A\otimes B$, and $A\rightarrow B$ can be constructed using the semantic rules $\{1,(,),\rightarrow\}$.
To these types correspond projectors: given projectors $\Proj{}{A}$ and $\Proj{}{B}$, projectors $\CompProj{}{A}, \Proj{}{A}\otimes \Proj{}{B}$ and $\Proj{}{A}\rightarrow \Proj{}{B}$ can be constructed using the algebraic rules $\{\overline{\:\cdot\:},\otimes\}$ as in equations \eqref{eq:det_fctal_proj}, \eqref{eq:det_tensor_proj}, and \eqref{eq:det_map_proj}.
To these types and projectors correspond state structures: given state structures $\Alg{A}{}$ and $\Alg{B}{}$ as in equation \eqref{def:struct}, the state structures $\CompAlg{A}, \CompAlg{A} \otimes \Alg{B}{}$, and $\Alg{A}{}\rightarrow\Alg{B}{}$ can be constructed by requiring that, respectively, conditions \eqref{eq:fctal_def}, \eqref{eq:A_otimes_B_def}, and \eqref{eq:A_to_B} hold.
The main novelty is the prec connector, Prop. \ref{prop:semi-causal_compo}, giving the state structure $\CompAlg{A}\prec \Alg{B}{}$ of maps which are no signaling from the output to the input.
This allows to speak about the possible signaling structure of the maps characterized by the projective framework.
In addition to that, the properties of the prec give an unambiguous way to compare different theories as they allow to expand them as unions and intersection of expressions involving only this prec connector.
In the mean time, we identified the features of the algebra of projectors. We showed in particular that it was a Boolean algebra under the rules $\{\overline{\:\cdot\:}, \cap, \cup\}$, which is subsequently extended into a model of linear logic under the rules $\{\overline{\:\cdot\:}, \cap, \cup, \otimes, \overline{\:\cdot\:} \rightarrow \:\cdot\: \}$.
The introduced prec fits naturally as a multiplicative rule in-between $\otimes$ and $\overline{\:\cdot\:} \rightarrow \:\cdot\:$, which moreover has the property of commuting with the negation. It remains to investigate how the role of the prec is understood in terms of linear logic, as well as in terms of categories. To the authors' knowledge, the prec connector appears as a genuinely new abstract mathematical concept discovered in a physically motivated problem.
Finally, we rederived the proof of equivalence between combs based on quantum states, quantum combs, and networks based on quantum channels \cite{Bisio2018} as an example of the use of the projective characterization.
This equivalence explains in particular why these structures feature a definite signaling direction, which comes with a clear physical implementation.
The question of whether these are the only isomorphic constructions is left open for future work.
Another related but more general question that can be asked in terms of projectors is to determine for which base state structures a given construction based on the transformation reduces to a construction based on the one-way signaling composition. In other words, given an abstract way of constructing higher-order theories, for which base state structures does it reduce to a network?
Several aspects of the theory of higher-order transformations have not been explored here and remain open for future work.
The composition of types in the sense of reference \cite{Bisio2018} has only been partially addressed. Knowing the projector characterizing a transformation is sufficient to know what will be the projector characterizing its set of outputs. However, if the input state structure is now restricted to a subset with a different projector, is there a rule to apply on the projector of the transformation in order to get the projector of the possibly now restricted output set?
For example, an $n$-comb takes $(n-1)$-combs as inputs and outputs a 1-comb. One can ask what happens when the inputs are restricted to the no signaling subset: when $(n-1)$ 1-combs are plugged into an $n-$comb instead, is the output set still the full set of 1-combs?
The notion of causal separability \cite{OCB2012,Giarmatzi2016,Wechs2019,MPM} still is to be defined for general higher-order transformations. The link between the prec and single signaling direction, as well as the canonical form, should give a basis for the definition. Nonetheless, there remains to tackle the question of dynamical causal orders, which is a probabilistic notion, and therefore not naturally captured by the formalism of projectors.
The issue of realizability has not been explored. That is, given a state structure, is it possible to realize each of its elements in a lab experiment? Is there a systematic way to relate these abstract mathematical objects to a circuit realization, as is the case for quantum combs \cite{Chiribella2009} and for time-delocalized subsystems \cite{Oreshkov_2019,Wechs2022}? One may expect that the canonical decomposition of general state structures into the union of several one-way signaling structures could prove useful for answering this question.
\textbf{Note added:} After completion of this work and during the preparation of the manuscript, we became aware of an independent work by Simmons and Kissinger \cite{https://doi.org/10.48550/arxiv.2205.11219}, which also investigates the signaling structure of higher-order quantum transformations. Some of the results presented here were also found in this other work using a different formalism.
\begin{acknowledgments}
T.H. is grateful to the organizers and participants of the 2021 Sejny Summer Institute for listening to his rambling interventions about nesting supermaps -- in particular to Pablo Arnault, Titouan Carette, Nat{\'a}lia M{\'o}ller, Eleftherios Tselentis, and Augustin Vanrietvelde for technical discussions. T. H. is especially indebted to Titouan Carette for pointing out the connection with linear logic. In addition, T.H. would like to thank Esteban Castro-Ruiz and Joseph Cunningham for help and comments, as well as Alexandra Elbakyan for providing access to the scientific literature.
T. H. benefits from the support of the French Community of Belgium within the framework of the financing of a FRIA grant.
O. O. is a Research Associate of the Fonds de la Recherche Scientifique (F.R.S.–FNRS).
This publication was made possible through the support of the ID\# 61466 grant from the John Templeton Foundation, as part of the "The Quantum Information Structure of Spacetime (QISS)" Project (qiss.fr). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
This work was supported by the Program of Concerted Research Actions (ARC) of the Universit\'{e} libre de Bruxelles.
Illustrations were drawn using draw.io (\url{https://www.diagrams.net/}).
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,277
|
301 D Street is a $12.1 million project that is an exciting new addition to the burgeoning Washington District in West Sacramento. Taking an existing single story, abandoned warehouse, this project will become a vibrant and thriving place to call home. It features 40 unique apartments including 10 spectacular lofts and a 700 Sq Ft rooftop deck. Designed by the award-winning San Diego-based De Bartolo + Rimanic Design Studio, construction started in August 2018 and is expected to be ready for move in late 2019. 301 D Street embraces the neighborhood's glorious past while building on a community focused on the future.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,065
|
import dict from 'ember-metal/dictionary';
export default dict(null);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,822
|
Q: Magento 1.9.2.4: Custom Options price not changing I got this issue where Price is not updated when checking the custom options (Fixed Price: +400).
I got this error:
TypeError: Product.OptionsPrice is not a constructor
TypeError: optionsPrice is undefined
Related questions:
Custom Options not updating price - JS error
Custom Option Price Not Updating
Custom Option Price is not changing
Update product custom option price on product price change
A: It happened to me after upgrading to magento 1.9.3.
In my custom theme was missing product_options.js call.
You need to add
<action method="addJs"><script>varien/product_options.js</script></action>
Check catalog.xml and configurableswatches.xml in rwd theme for reference.
You will also need to copy
\skin\frontend\rwd\default\js\configurableswatches\configurable-swatch-prices.js
to
\skin\frontend\yourTheme\default\js\configurableswatches\
Hope can help
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,682
|
{"url":"https:\/\/web2.0calc.com\/questions\/altitudes-of-triangles","text":"+0\n\n# altitudes of triangles\n\n0\n38\n2\n\nAltitudes $\\overline{AD}$ and $\\overline{BE}$ of $\\triangle ABC$ intersect at $H$. If $\\angle BAC = 54^\\circ$ and $\\angle ABC = 52^\\circ$, then what is $\\angle AHB$?\n\nNov 17, 2019\n\n#1\n0\n\nthe figure doesn not show any other angles if u draw it out\n\nam i missing somehting??\n\nNov 17, 2019\n#2\n+2490\n+2\n\nNote that all angles in a triangle sum to 180.\n\nNotice\u00a0$$\\Delta ABE$$, Angle BEA is 90, angle A is 54, so angle ABE is 180 - 90 - 54 = 36.\n\nNotice\u00a0$$\\Delta BDA$$, Angle HDB is 90, angle B is 52, so angle\u00a0DAB\u00a0is 180 - 90 - 52 = 38\n\nNotice\u00a0$$\\Delta ABH$$, Angle\u00a0ABE\u00a0is 36, angle\u00a0DAB is 38, so angle AHB\u00a0is 180 - 36 - 38 = 106\n\nNov 17, 2019","date":"2019-12-12 01:58:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7824569940567017, \"perplexity\": 949.0439423500231}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540534443.68\/warc\/CC-MAIN-20191212000437-20191212024437-00282.warc.gz\"}"}
| null | null |
package se.dolkow.tangiblexml;
import android.support.annotation.NonNull;
import java.lang.annotation.ElementType;
import java.lang.annotation.Retention;
import java.lang.annotation.RetentionPolicy;
import java.lang.annotation.Target;
@Retention(RetentionPolicy.RUNTIME)
@Target(ElementType.FIELD)
public @interface TangibleField {
/**
* Relative xml path from its parent node.
*/
@NonNull String value();
/**
* We will throw exceptions if a required field is missing.
* For lists, "required" means that we require at least one element.
* Default is true.
*/
boolean required() default true;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,786
|
Q: AsyncTask Implementation using callback interface - Handle multiple call responses in single callback interface In my Android application, I am making multiple API calls from one Activity. I am using following approach: Implementing interface in Activity and call that interface function from Async class.
public interface AsyncResponse {
public void processFinish(JSONObject sb);
}
public class FetchData extends AsyncTask<String, Integer, JSONObject> {
HttpURLConnection urlConnection;
String url;
String method;
String payload = null;
AsyncResponse delegate = null;
public FetchData(AsyncResponse delegate, String url, String method) {
this.delegate = delegate;
this.url = url;
this.method = method;
}
public FetchData(AsyncResponse delegate, String url, String method, JSONObject payload) {
this(delegate, url, method);
this.payload = payload.toString();
}
@Override
protected JSONObject doInBackground(String... args) {
BufferedReader reader = null;
try {
URL url = new URL(this.url);
// Open HTTP connection
urlConnection = (HttpURLConnection) url.openConnection();
// HTTP method GET/POST/PUT/DELETE
urlConnection.setRequestMethod(this.method);
// handle issues
int statusCode = urlConnection.getResponseCode();
// Get the response
InputStream inputStream = urlConnection.getInputStream();
if(inputStream == null) {
// Nothing to do
return null;
}
reader = new BufferedReader(new InputStreamReader(inputStream, "UTF-8"));
String inputLine = null;
StringBuffer response = new StringBuffer();
while ((inputLine = reader.readLine()) != null) {
response.append(inputLine + "\n");
}
return new JSONObject(response.toString());
} catch(Exception e) {
try {
// Return error response
} catch(Exception e1) {
System.out.println(e1);
return null;
}
} finally {
if (urlConnection != null) {
urlConnection.disconnect();
}
if (reader != null) {
try {
reader.close();
} catch (final IOException e) {
Log.e("PlaceholderFragment", "Error closing stream", e);
}
}
}
}
@Override
protected void onPostExecute(JSONObject result) {
super.onPostExecute(result);
this.delegate.processFinish(result);
}
}
public class AsyncTasks extends AppCompatActivity implements AsyncResponse {
TextView view = null;
int a = 1;
Utility utility = Utility.getInstance();
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_async_tasks);
new FetchData(this, "<url 1>", "GET").executeOnExecutor(THREAD_POOL_EXECUTOR);
new FetchData(this, "<url 2>", "GET").executeOnExecutor(THREAD_POOL_EXECUTOR);
}
@Override
public void processFinish(JSONObject data) {
utility.showDialog(this, data.toString());
}
}
Here how to handle response of 2nd GET call in processFinish() interface function? What is the best approach?
A: public interface AsyncResponse {
//Add requestCode to identify request.
public void processFinish(JSONObject sb, int requestCode);
}
public class FetchData extends AsyncTask<String, Integer, JSONObject> {
HttpURLConnection urlConnection;
String url;
String method;
String payload = null;
AsyncResponse delegate = null;
int requestCode;
public FetchData(String url, String method) {
this(url, method, null);
}
public FetchData(String url, String method, JSONObject payload) {
this.url = url;
this.method = method;
if(payload!=null){
this.payload = payload.toString();
}
}
//You can set AsyncResponse and RequestCode in constructor also.
public FetchData setListener(AsyncResponse asyncResponse, int requestCode){
this.delegate = asyncResponse;
this.requestCode = requestCode;
return this;
}
@Override
protected JSONObject doInBackground(String... args) {
....
}
@Override
protected void onPostExecute(JSONObject result) {
super.onPostExecute(result);
if(delegate!=null){
//post result with given requestCode
this.delegate.processFinish(result, requestCode);
}
}
}
public class AsyncTasks extends AppCompatActivity implements AsyncResponse {
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_async_tasks);
// Make first call with request code as 1
new FetchData("<url 1>", "GET").setListener(this, 1).executeOnExecutor(THREAD_POOL_EXECUTOR);
// Make second call with request code as 2
new FetchData("<url 2>", "GET").setListener(this, 2).executeOnExecutor(THREAD_POOL_EXECUTOR);
}
@Override
public void processFinish(JSONObject data, int requestCode) {
switch(requestCode){
case 1:
//perform task on 1st call finish
break;
case 2:
utility.showDialog(this, data.toString());
//perform task on 2nd call finish
break;
}
}
}
A: maybe i'm a bit late.
But i was facing the same problem as you and this switch case in the validated answer was burning my eyes so here you can find a way of implementing it for as many requests as you want using anonymous AsyncResponse.
In you fetch data when you are passing your AsyncTasks class to manage the callBack by implementing the processFinish Method, change it with a new AsyncResponse and implement in the method.
Like This:
public interface AsyncResponse {
//Add requestCode to identify request.
public void processFinish(JSONObject sb, int requestCode);
}
public class FetchData extends AsyncTask<String, Integer, JSONObject> {
HttpURLConnection urlConnection;
String url;
String method;
String payload = null;
AsyncResponse delegate = null;
int requestCode;
public FetchData(String url, String method) {
this(url, method, null);
}
public FetchData(String url, String method, JSONObject payload) {
this.url = url;
this.method = method;
if(payload!=null){
this.payload = payload.toString();
}
}
//You can set AsyncResponse and RequestCode in constructor also.
public FetchData setListener(AsyncResponse asyncResponse, int requestCode){
this.delegate = asyncResponse;
this.requestCode = requestCode;
return this;
}
@Override
protected JSONObject doInBackground(String... args) {
....
}
@Override
protected void onPostExecute(JSONObject result) {
super.onPostExecute(result);
if(delegate!=null){
//post result with given requestCode
this.delegate.processFinish(result, requestCode);
}
}
}
public class AsyncTasks extends AppCompatActivity{
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_async_tasks);
// Make first call with request code as 1. Use anonym callBack implementation
new FetchData("<url 1>", "GET").setListener(new AsyncResponse () {
@Override
public void processFinish(JSONObject sb, int requestCode) {
//perform task on 1st call finish
}
}, 1).executeOnExecutor(THREAD_POOL_EXECUTOR);
// Make second call with request code as 2
new FetchData("<url 2>", "GET").setListener(new AsyncResponse () {
@Override
public void processFinish(JSONObject sb, int requestCode) {
//perform task on 2st call finish
}
}, 2).executeOnExecutor(THREAD_POOL_EXECUTOR);
// Make as many call to FetchData and implement an anonym AsyncResponse each time
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,617
|
Q: FirebaseMessagingService onMessageReceived() not executing I just started out on Firebase Cloud Messaging, and i set up all dependecies, Firebase service and also its service tags in Manifest.
When sending a message from console, the target users are detected, but the app does not receive a notification.
So i set up a break point on onMessageReceived(). but it did not get fired up.
Below is my firebase Messaging service
public class MyFirebaseMessagingService extends FirebaseMessagingService {
private static final String TAG = "MyFirebaseMessagingServ";
@Override
public void onMessageReceived( RemoteMessage remoteMessage) {
super.onMessageReceived(remoteMessage);
String notificationBody = "";
String notificationTitle = "";
String notificationData = "";
try{
notificationData = remoteMessage.getData().toString();
notificationTitle = remoteMessage.getNotification().getTitle();
notificationBody = remoteMessage.getNotification().getBody();
}
catch (NullPointerException e){
Log.e(TAG, "onMessageReceived: NullPointerException: " + e.getMessage() );
}
NotificationHelper.displayNotification(getApplicationContext(), notificationTitle,
notificationBody);
Log.d(TAG, "onMessageReceived: data: " + notificationData);
Log.d(TAG, "onMessageReceived: Notification body : " + notificationBody);
Log.d(TAG, "onMessageReceived: notification title: " + notificationTitle);
}
Android Manifest
<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
package="com.sys.systec">
<uses-permission android:name="android.permission.INTERNET"></uses-permission>
<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE"/>
<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE"/>
<uses-permission android:name="android.permission.CAMERA"/>
<application
android:allowBackup="true"
android:icon="@mipmap/ic_launcher"
android:label="@string/app_name"
android:roundIcon="@mipmap/ic_launcher_round"
android:supportsRtl="true"
android:theme="@style/AppTheme"
tools:ignore="GoogleAppIndexingWarning">
<activity android:name=".LoginActivity">
<intent-filter>
<action android:name="android.intent.action.MAIN"/>
<category android:name="android.intent.category.LAUNCHER"/>
</intent-filter>
</activity>
...
<service android:name=".utility.MyFirebaseMessagingService"
android:stopWithTask="false"
android:exported="false">
<intent-filter>
<action android:name="com.google.firebase.MESSAGING_EVENT" />
</intent-filter>
</service>
</application>
A: if you are using firebase terminal you will face the issue so you can use notification API (use data message)
API:
https://fcm.googleapis.com/fcm/send
Header:
"Content-Type": "application/json",
"Authorization": "key=<Server_key>"
Body:
{
"to": "<Device FCM token>",
"notification": {
"title": "Demo tittle",
"body": "Demo body",
"mutable_content": true,
},
"data": {
"url": "<media image url>",
"dl": "<deeplink action>"
}
}
Refer:
URL: https://firebase.google.com/docs/cloud-messaging/http-server-ref. https://firebase.google.com/docs/cloud-messaging/downstream.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,184
|
Losing command and control…and living with it!
Buyers GuideLatest News
Published: May 1st, 2019
- David Rubinstein
There's a saying that goes, "Developers don't want to miss the boat, and operations don't want to sink the ship." But for organizations to create value streams to observe their operations and eliminate waste while driving efficiencies, they might have to yield some control.
ConnectALL's Lance Knight recalled his time as a Novell network administrator, who required requests in triplicate—from the requester, his boss and HIS boss —before granting access to files. "That was a whole other time of IT command and control that kind of took over for a while, and you've got to get rid of that from an IT operations perspective, and think about how you're trying to support the business, not control it as much.
"IT thinks of themselves as process enforcers, rather than the enablers," he continued. "If you didn't groom the backlog right, we're not doing anything with it. That kind of stuff. I remember being that guy, that IT enforcer. You guys are spending the wrong time on what you're supposed to do on this PC out on the shop floor, so I'm going to turn it off at your lunch hour from now on. Right? An enforcer. They're used to having that power. They have the passwords, security. 'I have the admin passwords for the HR system, I know what all you guys make, I am special!' "
Gently down the value stream
Creating a value stream map
How does your solution help organizations on their value stream journey?
A guide to VSM tools
Plutora's chief marketing officer Bob Davis said, "As you go from command and control, old-school process to the newfangled world of Agile and smaller bites released more frequently, you have to be able to collaborate. And one of the things developers like to do less of is collaborate. If the system can provide the KPIs, if the system can allow them to collaborate silently, in effect, by alerting dependent systems, dependent development processes, etc., automatically, because the system is plugged in and understands the relationships and notifies or alerts the relevant parties … anything that's happening, to the extent that can be automated and served up into a system, the better you are.
"I think that that's the promise," Davis continued, "and as we go down the future and say OK, what happens next, you start to get better machine learning and the processes get even more intelligent relative to how to weave in security, how to weave in compliance, in a more automated kind of parallel process way, without having that 'oh shit' moment where you go, 'I didnt do that.' Any of those things that are made possible are real advances in the world of software development, and that's what value stream promises."
The key question is, how do you bridge the gap from command and control and the highly autonomous, self-led teams? In other words, 'How do you get to Amazon?'
According to Plutora's Jeff Keyes, the answer is value stream management. "You have to go through the process of understanding, here's our value stream from beginning to end. Then you start to break that down, you have to start integrating your tools and bringing them together. Third, you've got to add a layer of orchestration across it so you can incorporate these things, because that reduces your time to delivery. Fourth, as you're measuring the time that things are going, now that you're orchestrating it, you can start to see a path of, 'well, these checks, I can automate these, because that will improve my delivery performance,' and it brings the team in so that they're bought in. Do you still have command and control? Not in the same fashion, but they're acting as coaches in compliance and ensuring that the right things happen, even in these automated pipelines. What about audit? How do you make that happen? Well because it's all happening there, and all that data is rolling back into a value stream management platform, audit is easy. You can verify that the right things happen."
Tasktop's Carmen DeArdo said control gets baked into our objectives and our incentives. "I work with people whose complete incentive was around stability of production. They had no skin in the game to go faster but you'd think they could think a little more broadly about, OK, well, we'll just never release another feature. We won't be in business very long, but we'll protect production, maybe. That's what led to DevOps. Everybody has to have skin in the game to be aligned with goals. It's not just deliver business value, but protect it and improve it. The product model is better at protecting because it elevates risk to be a first-class citizen rather than everything being subservient to features, and it's the same with process."
Subscribe to SDTimes
About David Rubinstein
David Rubinstein is editor-in-chief of SD Times.
View all posts by David Rubinstein
NativeScript 6.0 released with full support for Vue.js
7 deadly mistakes to avoid while testing your mobile application
Securing Microservices: The API gateway, authentication and authorization
IBM: 'Red Hat will still be Red Hat' after $34 billion acquisition
Measurements, metrics and KPIs: Three keys to Value Stream Management
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,529
|
\subsection{Quantitative evaluation of the benefits of stochastic embedding}
We first measure performance on the
verification task, where the network is used to compute $p(m|x_1,x_2)$ for 10k pairs of test images, half of which are matches and half are not.
The average precision (AP) of this task is reported in the top half of Table \ref{tab:maintask}. HIB\xspace shows improved performance, especially for corrupted test images. For example, in the
$N=2$ digit case, when using $D=2$ dimensions, point embeddings achieve 88.0\% AP on corrupt test images, while hedged instance embeddings improves to 90.7\% with $C=1$ Gaussian, and 91.2\% with $C=2$ Gaussians.
We next measure performance on the KNN identification task. The bottom half of Table \ref{tab:maintask} reports the results. In the KNN experiment, each clean image is identified by comparing to the test gallery which is either comprised of clean or corrupted images based on its $K=5$ nearest neighbors. The identification is considered correct only if a majority of the neighbors have the correct class. Again, the proposed stochastic embeddings generally outperform point embeddings, with the greatest advantage for the corrupted input samples.
For example, in the
$N=2$ digit case, when using $D=2$ dimensions, point embeddings achieve 58.3\% AP on corrupt test images, while HIB\xspace improves to 76.0\% with $C=1$ Gaussian, and 75.7\% with $C=2$ Gaussians. Appendix \ref{asec:knn_plurality} contains additional KNN experiment results.
\begin{table*}
\centering
\scriptsize
\setlength\tabcolsep{1.5pt}
\begin{tabular}{@{}lrrrcrrrcrrrcrrr@{}}\toprule
& \multicolumn{3}{c}{$N=2$, $D=2$ } & \phantom{abc}
& \multicolumn{3}{c}{$N=2$, $D=3$ } & \phantom{abc}
& \multicolumn{3}{c}{$N=3$, $D=2$} & \phantom{abc}
& \multicolumn{3}{c}{$N=3$, $D=3$}\\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12} \cmidrule{14-16}
& point & MoG-1& MoG-2 && point & MoG-1& MoG-2 && point & MoG-1& MoG-2 && point & MoG-1& MoG-2 \\ \midrule
\textbf{Verification AP}\\
\hspace{3mm}clean & 0.987 & 0.989 & 0.990&&
0.996 & 0.996 & 0.996 && 0.978& 0.981 & 0.976 && 0.987 & 0.989 & 0.991\\
\hspace{3mm}corrupt & 0.880 & 0.907 & 0.912&& 0.913 & 0.926 & 0.932 && 0.886& 0.899 & 0.904 && 0.901 & 0.922 &0.925\\
\textbf{KNN Accuracy}\\
\hspace{3mm}clean & 0.871& 0.879 & 0.888 && 0.942 & 0.953 & 0.939 &&0.554& 0.591 & 0.540 && 0.795 & 0.770 & 0.766\\
\hspace{3mm}corrupt & 0.583& 0.760 & 0.757 && 0.874 & 0.909&0.885 && 0.274& 0.350 & 0.351 && 0.522& 0.555 & 0.598\\
\bottomrule
\end{tabular}
\caption{Accuracy of pairwise verification and KNN identification tasks for point embeddings, and our hedged embeddings with a single Gaussian component (MoG-1) and two components (MoG-2).
We report results for images with $N$ digits and using $D$ embedding dimensions.
}
\label{tab:maintask}
\vspace{2em}
\scriptsize
\setlength\tabcolsep{4pt}
\begin{tabular}{@{}lccccccccccc@{}}\toprule
& \multicolumn{2}{c}{$N=2$, $D=2$} & \phantom{a}
& \multicolumn{2}{c}{$N=2$, $D=3$} & \phantom{a}
& \multicolumn{2}{c}{$N=3$, $D=2$} & \phantom{a}
& \multicolumn{2}{c}{$N=3$, $D=3$}\\
\cmidrule{2-3} \cmidrule{5-6} \cmidrule{8-9} \cmidrule{11-12}
& MoG-1& MoG-2 && MoG-1& MoG-2 && MoG-1& MoG-2 && MoG-1& MoG-2 \\ \midrule
\textbf{AP Correlation}\\
\hspace{3mm}clean & 0.74 & 0.43&& 0.68 & 0.48&& 0.63 & 0.28 && 0.51 &0.39 \\
\hspace{3mm}corrupt & 0.81 & 0.79 &&0.86 & 0.87&& 0.82 & 0.76 && 0.85 &0.79 \\
\textbf{KNN Correlation}\\
\hspace{3mm}clean & 0.71 & 0.57 && 0.72 & 0.47 && 0.76 &0.29 && 0.74& 0.54 \\
\hspace{3mm}corrupt & 0.47 & 0.43 && 0.55 & 0.52&& 0.49 & 0.50 && 0.67 & 0.34 \\
\bottomrule
\end{tabular}
\caption{Correlations between each input image's measure of uncertainty, $\eta(x)$, and AP and KNN performances. High correlation coefficients suggest a close relationship. }
\label{tab:uncertaintymeasuresmog}
\end{table*}
\section{Discussion and Future Work}
Hedged instance embedding is a stochastic embedding that captures the uncertainty of the mapping of an image to a latent embedding space, by spreading density across plausible locations.
This results in improved performance on various tasks,
such as verification and identification, especially for ambiguous corrupted input.
It also allows for a simple way to estimate the uncertainty of the embedding
that is correlated with performance on downstream tasks.
There are many possible directions for future work, including experimenting with higher-dimensional embeddings, and harder datasets.
It would also be interesting to consider the ``open world'' (or ``unknown unknowns'') scenario, in which the test set may contain examples of novel classes,
such as
digit combinations that were not in the training set
(see e.g., \citet{Lakkaraju2017,Gunther2017}).
This is likely to result in uncertainty about where to embed the input which is different from the uncertainty induced by occlusion,
since uncertainty due to open world is {\em epistemic} (due to lack of knowledge of a class),
whereas uncertainty due to occlusion is {\em aleatoric} (intrinsic, due to lack of information in the input),
as explained in \citet{NIPS2017_7141}.
Preliminary experiments suggest that $\eta(x)$ correlates well with detecting occluded inputs, but does not work as well for novel classes.
We leave more detailed modeling of epistemic uncertainty as future work.
\section{N-digit MNIST\xspace Dataset}
\label{sec:data}
We present N-digit MNIST\xspace, \url{https://github.com/google/n-digit-mnist}, a new dataset based upon MNIST \citep{lecun1998mnist} that has an exponentially large number of classes on the number of digits $N$,
for which embedding-style classification methods are well-suited.
The dataset is created by horizontally concatenating $N$ MNIST digit images.
While constructing new classes, we respect the training and test splits.
For example, a test image from 2-digit MNIST of a ``54'' does not have its ``5'' or its ``4'' shared with any image from the training set (in all positions).
\begin{table}[h]
\footnotesize
\centering
\begin{tabular}{*{8}{c}}\toprule
\vspace{-1em} \\
Number && Total & Training & Unseen Test & Seen Test & Training&Test \\
Digits && Classes & Classes& Classes & Classes & Images & Images\\
\vspace{-1em} \\
\cline{1-1} \cline{3-8}
\vspace{-1em} \\
2 && 100 & 70 & 30& 70 & $100\,000$& $10\,000$ \\
3 && 1000 & 700 & 100& 100 & $100\,000$& $10\,000$ \\
\bottomrule
\end{tabular}
\caption{Summary of N-digit MNIST\xspace dataset for $N=2,3$.
}
\label{tab:ndigitmnist}
\end{table}
We employ 2- and 3-digit MNIST~(\Tabref{tab:ndigitmnist}) in our experiments.
This dataset is meant to provide
a test bed for easier and more efficient evaluation of embedding algorithms than with larger and more realistic datasets.
N-digit MNIST\xspace is more suitable for embedding evaluation than other synthetic datasets due to the exponentially increasing number of classes as well as the factorizability aspect: each digit position corresponds to \emph{e.g.}\xspace a face attribute for face datasets. For 3-digit MNIST, there are 1000 total classes, but for testing, we decided to enforce a minimal number of samples per class. To keep the dataset size manageable, we limited the number of classes in the test set to 100 seen and unseen classes, sub-sampled from the 700 and 300 possible classes, respectively.
\eat{
We can draw an analogy between individual digits and face attributes: each class is comprised of elements (e.g., digits and line strokes) that appear in other classes in different configurations, just as faces are composed of parts like noses, ears, and eyes that are similar to those belonging to other people.
}
We inject uncertainty into the underlying tasks by randomly occluding (with black fill-in) regions of images at training and test time.
Specifically, the corruption operation is done independently on each digit of number samples in the dataset. A random-sized rectangular patch is identified from a random location of each $28\times 28$ digit image. The patch side lengths and heights are sampled $L\sim\text{Unif}(0,28)$, and then the top left patch corner coordinates are sampled so that the occluded rectangle fits within the image.
During training, we set independent binary random flags $B$ for every digit determining whether to perform the occlusion at all; the occlusion chance is set to $20\%$.
For testing, we prepare twin datasets, clean and corrupt, with digit images that are either not corrupted with occlusion at all or always occluded, respectively.
\subsection{Experimental details}
We conduct all our experiments on a new dataset we created called
N-digit MNIST\xspace,
which consists of images composed of $N$ adjacent MNIST digits,
which may be randomly occluded (partially or fully).
See \cref{sec:data} for details.
During training, we occlude 20\% of the digits independently. A single image can have multiple corrupted digits.
During testing, we consider both clean (unoccluded) and
corrupted (occluded) images, and report results separately.
We use images with $N=2$ and $N=3$ digits.
In the Appendix, we provide details on the open source version of this data that is intended for other researchers and to ensure reproducibility.
Since our dataset is fairly simple, we use a shallow CNN model to compute the embedding function. Specifically, it consists of 2 convolutional layers,
with $5\times 5$ filters,
each followed by max pooling, and then a fully connected layer mapping to $D$ dimensions.
We focus on the case where $D=2$ or $D=3$.
When we use more dimensions, we find that all methods (both stochastic and deterministic) perform almost perfectly (upper 90\%), so there are no interesting differences to report.
We leave exploration of more challenging datasets, and higher dimensional embeddings, to future work.
Our networks are built with TensorFlow \citep{tensorflow2015whitepaper}. For each task, the input is an image of size $28\times (N\times 28)$, where N is the number of concatenated digit images.
We use a batch size of 128 and 500k training iterations.
Each model is trained from scratch with random weight initialization.
The KL-divergence hyperparameter $\beta$ is set to $10^{-4}$ throughout the experiments.
We report effects of changing $\beta$ in appendix~\ref{sec:kl_regularization}.
\section{Experiments}
\label{sec:experiments}
In this section, we report our experimental results, where we compare our stochastic embeddings to point embeddings.
We consider two main tasks: the verification task
(\emph{i.e.}\xspace, determining if two input images correspond to the same class or not),
and the identification task
(\emph{i.e.}\xspace, predicting the label for an input image).
For the latter, we use a K-nearest neighbors approach with $K=5$.
We compare performance of three methods:
a baseline deterministic embedding method,
our stochastic embedding method with a Gaussian embedding,
and
our stochastic embedding method with a mixture of Gaussians embedding.
We also conduct a qualitative comparison of the embeddings of each method.
\input{details}
\input{qualitative}
\input{accuracy}
\input{uncertainty}
\section{Extra Results}
In this section, we include some extra results which would not fit in the main paper.
\Figref{fig:2DEmbeddingsMOG} shows some 2D embeddings of 2-digit images using a MoG representation, with $C=2$ or $C=4$ clusters per embedding.
\Figref{fig:3DEmbeddings} shows some 3D embeddings of 3-digit images using a single Gaussian.
\begin{table*}
\centering
\scriptsize
\setlength\tabcolsep{1.5pt}
\hfill
\begin{subfigure}{0.45\columnwidth}
\centering
\begin{tabular}{@{}lrrr@{}}\toprule
& \multicolumn{3}{c}{$N=3$, $D=4$ } \\
\cmidrule{2-4}
& point & MoG-1& MoG-2 \\ \midrule
\textbf{Verification AP}\\
\hspace{3mm}clean & 0.996 & 0.996 & 0.995\\
\hspace{3mm}corrupt & 0.942 & 0.944 & 0.948\\
\textbf{KNN Accuracy}\\
\hspace{3mm}clean & 0.914& 0.917 & 0.922\\
\hspace{3mm}corrupt & 0.803& 0.816 & 0.809\\
\bottomrule
\end{tabular}
\caption{Task accuracies for pairwise verification and KNN identification.
}
\label{tab:N2D4embeddingaccuracy}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\columnwidth}
\centering
\begin{tabular}{@{}lcc@{}}\toprule
& \multicolumn{2}{c}{$N=3$, $D=4$} \\
\cmidrule{2-3}
& MoG-1& MoG-2 \\ \midrule
\textbf{AP Correlation}\\
\hspace{3mm}clean & 0.35 & 0.33 \\
\hspace{3mm}corrupt & 0.79 & 0.68 \\
\textbf{KNN Correlation}\\
\hspace{3mm}clean & 0.31 & 0.32 \\
\hspace{3mm}corrupt & 0.42 & 0.35 \\
\bottomrule
\end{tabular}
\caption{Correlations between uncertainty, $\eta(x)$, and task performances.}
\label{tab:N2D4uncertaintymeasuresmog}
\end{subfigure}
\hfill
\caption{Task performance and uncertainty measure correlations with task performance for 4D embeddings with 3-digit MNIST.}
\label{tab:43}
\end{table*}
\subsection{4D MNIST Embeddings}
\label{sec:extraexperiments4D}
In Table \ref{tab:43}, we report the results of 4D embeddings with 3-digit MNIST. Here, the task performance begins to saturate, with verification accuracy on clean input images above 0.99. However, we again observe that HIB provides a slight performance improvement over point embeddings for corrupt images, and task performance correlates with uncertainty.
\begin{figure}[h!]
\centering
\hfill
\begin{subfigure}{0.30\columnwidth}
\includegraphics[width=0.99\columnwidth]{figure/petnet}
\caption{Average precision.}
\end{subfigure}
\hfill
\begin{subfigure}{0.30\columnwidth}
\includegraphics[width=0.99\columnwidth]{figure/petnet_pdf_low_uncertainty}
\caption{Low uncertainty PDF.}
\end{subfigure}
\hfill
\begin{subfigure}{0.30\columnwidth}
\includegraphics[width=0.99\columnwidth]{figure/petnet_pdf_high_uncertainty}
\caption{High uncertainty PDF.}
\end{subfigure}
\hfill
\caption{Correlation between the uncertainty measure $\eta(x)$ and balanced accuracy on the pet test set for 20D single Gaussian embeddings. Uncertainty increases along the horizontal axis. (b) and (c) show match score distributions of pairs of the same and different pets for lowest and highest uncertainty bins. There is clearly more confusion for the highly uncertain samples.}
\label{fig:soft_contrastive_experiments}
\end{figure}
\subsection{HIB Embeddings for Pet Identification}
\label{sec:extraexperimentsPets}
We applied HIB to learn instance embeddings for identifying pets, using an internal dataset of nearly 1M cat and dog images with known identity, including 117913 different pets. We trained an HIB with 20D embeddings and a single Gaussian component (diagonal covariance). The CNN portion of the model is a MobileNet \citep{mobilenet} with a width multiplier of 0.25. No artificial corruption is applied to the images, as there is sufficient uncertainty from sources such as occlusion, natural variations in lighting, and the pose of the animals.
We evaluate the verification task on a held out test set of 8576 pet images, from which all pairs were analyzed. On this set, point embeddings achieve 0.777 balanced accuracy. Binning the uncertainty measures and measuring correlation with the verification task (similarly as with N-digit MNIST), we find the correlation between performance and task accuracy to be 0.995. This experiment shows that HIB scales to real-world problems with real-world sources of corruption, and provides evidence that HIB understands which embeddings are uncertain.
\section{Embedding into 6-Dimensional Space}
\label{sec:higher_dim}
We report results on higher dimensional embeddings in Table~\ref{tab:higher_dims}. We have set $\beta=10^{-6}$ for this experiment. Compared to lower-dimensional ($D=2$ or 3) results in Tables~\ref{tab:maintask} and \ref{tab:uncertaintymeasuresmog}, we observe general increase in main task performances; for example, KNN accuracy for $N=3$ cases improve from 0.598 ($D=3$) to 0.888 ($D=6$). Unlike for $D=3$, $D=6$ exhibits slightly worse performances for probabilistic embeddings than for point embeddings.
\begin{table*}
\centering
\scriptsize
\setlength\tabcolsep{1.5pt}
\hfill
\begin{subfigure}{0.45\columnwidth}
\centering
\begin{tabular}{@{}llrrr@{}}\toprule
&& \multicolumn{3}{c}{$N=3$, $D=6$}\\
\cmidrule{3-5}
&& point & MoG-1& MoG-2 \\ \midrule
\textbf{Verification AP}\\
\hspace{3mm}clean && 0.999 & 0.998 & 0.998\\
\hspace{3mm}corrupt && 0.956 & 0.946 & 0.954\\
\textbf{KNN Accuracy}\\
\hspace{3mm}clean && 0.970 & 0.960 & 0.965\\
\hspace{3mm}corrupt && 0.918 & 0.879 & 0.888\\
\bottomrule
\end{tabular}
\caption{Main task performances.}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\columnwidth}
\centering
\begin{tabular}{@{}llrr@{}}\toprule
&& \multicolumn{2}{c}{$N=3$, $D=6$}\\
\cmidrule{3-4}
&& MoG-1& MoG-2 \\ \midrule
\textbf{AP Correlation}\\
\hspace{3mm}clean && 0.240 & 0.294 \\
\hspace{3mm}corrupt && 0.778 & 0.406 \\
\textbf{KNN Correlation}\\
\hspace{3mm}clean && 0.431 & 0.158 \\
\hspace{3mm}corrupt && 0.322 & 0.446 \\
\bottomrule
\end{tabular}
\caption{Uncertainty quality.}
\end{subfigure}
\hfill
\caption{Main task performances and uncertainty quality for 6-dimensional embedding of 3-digit MNIST images.}
\label{tab:higher_dims}
\end{table*}
\section{Introduction}
An instance embedding is a mapping $f$ from
an input $x$, such as an image,
to a vector representation, $z \in {\mathbb{R}}^D$,
such that ``similar'' inputs are mapped to nearby points in space.
Embeddings are a versatile representation that support various downstream tasks, including image retrieval ~\citep{babenko2014neural} and face recognition \citep{facenet}.
Instance embeddings are often treated deterministically,
\emph{i.e.}\xspace, $z=f(x)$ is a point in ${\mathbb{R}}^D$. We refer to this approach as a \emph{point embedding}. One drawback of this representation is the difficulty of modeling aleatoric uncertainty~\citep{NIPS2017_7141}, \emph{i.e.}\xspace uncertainty induced by the input. In the case of images this can be caused by occlusion, blurriness, low-contrast and other factors.
To illustrate this, consider the example in \Figref{fig:teaser-deterministic}.
On the left, we show an image composed
of two adjacent MNIST digits, the first of which is highly occluded.
The right digit is clearly a 7,
but the left digit could be a 1, or a 4.
One way to express this uncertainty about which choice to make is to map the input to a region of space, representing the inherent uncertainty of ``where it belongs''.
We propose a new method, called
\emph{hedged instance embedding}\xspace (\emph{HIB}\xspace),
which achieves this goal.
Each embedding is represented as a random variable, $Z \sim p(z|x)\in{\mathbb{R}}^D$. The embedding effectively spreads probability mass across locations in space, depending on the level of uncertainty. For example in \Figref{fig:teaser-probabilistic}, the corrupted image is mapped to a two-component mixture of Gaussians covering both the ``17'' and ``47'' clusters. We propose a training scheme for the HIB\xspace with a learnable-margin contrastive loss and the variational information bottleneck (VIB) principle~\citep{VIB,Achille2018jmlr}.
To evaluate our method, we propose a novel open-source dataset,
N-digit MNIST\xspace\footnote{\url{https://github.com/google/n-digit-mnist}}. Using this dataset, we show that HIB exhibits several desirable properties compared to point embeddings: (1) downstream task performance (\emph{e.g.}\xspace recognition and verification) improves for uncertain inputs; (2) the embedding space exhibits enhanced structural regularity; and (3) a per-exemplar uncertainty measure that predicts when the output of the system is reliable.
\begin{figure}
\centering
\begin{subfigure}{0.45\columnwidth}
\centering
\includegraphics[width=0.9\columnwidth]{figure/teaser-deterministic}
\caption{\label{fig:teaser-deterministic}Point embedding.}
\end{subfigure}
\begin{subfigure}{0.45\columnwidth}
\centering
\includegraphics[width=0.9\columnwidth]{figure/teaser-stochastic}
\caption{\label{fig:teaser-probabilistic}Stochastic embedding.}
\end{subfigure}
\vspace{0em}
\caption{\label{fig:teaser}Unlike point embeddings, stochastic embeddings may hedge their bets across the space. When both ``17'' and ``47'' are plausible, our 2-component Gaussian mixture embedding has the power to spread probability mass on clusters with clean ``17'' and ``47'' images. By contrast, the point embedding will choose to be close to one or the other of these points (or somewhere between).}
\end{figure}
\section{KL Divergence Regularization}
\label{sec:kl_regularization}
The training objective (Equation~\ref{eq:vib-metric}) contains a regularization hyperparameter $\beta\geq 0$ controlling the weight of the KL divergence regularization that controls, from information theoretic perspective, bits of information encoded in the latent representation.
In the main experiments, we consistently use $\beta=10^{-4}$.
In this Appendix, we explore the effect of this regularization by considering other values of $\beta$.
See Figure~\ref{fig:3DKL} for visualization of embeddings according to $\beta$. We observe that increasing KL term weight induces overall increase in variances of embeddings. For quantitative analysis on the impact of $\beta$ on main task performance and uncertainty quality, see Table~\ref{tab:kl_divergence} where we compare $\beta=0$ and $\beta=10^{-4}$ cases. We observe mild improvements in main task performances when the KL divergence regularization is used. For example, KNN accuracy improves from 0.685 to 0.730 by including the KL term for $N=3$, $D=3$ case. Improvement in uncertainty quality, measured in terms of Kendall's tau correlation, is more pronounced. For example, KNN accuracy and uncertainty are nearly uncorrelated when $\beta=0$ under the $N=3$, $D=3$ setting (-0.080 for clean and 0.183 for corrupt inputs), while they are well-correlated when $\beta=10^{-4}$ (0.685 for clean and 0.549 for corrupt inputs). KL divergence helps generalization, as seen by the main task performance boost, and improves the uncertainty measure by increasing overall variances of embeddings, as seen in Figure~\ref{fig:3DKL}.
\begin{table*}
\centering
\scriptsize
\setlength\tabcolsep{1.5pt}
\hfill
\begin{subfigure}{0.45\columnwidth}
\centering
\begin{tabular}{@{}llrrcrr@{}}\toprule
&& \multicolumn{2}{c}{$N=2$, $D=2$ } & \phantom{abc}
& \multicolumn{2}{c}{$N=3$, $D=3$}\\
\cmidrule{3-4} \cmidrule{6-7}
\hspace{3mm}$\beta$ && $0$ & $10^{-4}$ && $0$ & $10^{-4}$ \\ \midrule
\textbf{Verification AP}\\
\hspace{3mm}clean && 0.986 & 0.988 && 0.993 & 0.992 \\
\hspace{3mm}corrupt && 0.900 & 0.906 && 0.932 & 0.932 \\
\textbf{KNN Accuracy}\\
\hspace{3mm}clean && 0.867 & 0.891 && 0.858 & 0.861 \\
\hspace{3mm}corrupt && 0.729 & 0.781 && 0.685 & 0.730 \\
\bottomrule
\end{tabular}
\caption{Task performances.}
\label{tab:kl_divergence_accuracy}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\columnwidth}
\centering
\begin{tabular}{@{}llrrcrr@{}}\toprule
&& \multicolumn{2}{c}{$N=2$, $D=2$ } & \phantom{abc}
& \multicolumn{2}{c}{$N=3$, $D=3$}\\
\cmidrule{3-4} \cmidrule{6-7}
\hspace{3mm}$\beta$ && $0$ & $10^{-4}$ && $0$ & $10^{-4}$ \\ \midrule
\textbf{AP Correlation}\\
\hspace{3mm}clean && 0.680 & 0.766 && 0.425 & 0.630 \\
\hspace{3mm}corrupt && 0.864 & 0.836 && 0.677 & 0.764 \\
\textbf{KNN Correlation}\\
\hspace{3mm}clean && 0.748 & 0.800 && -0.080 & 0.685 \\
\hspace{3mm}corrupt && 0.636 & 0.609 && 0.183 & 0.549 \\
\bottomrule
\end{tabular}
\caption{Uncertainty correlations.}
\label{tab:kl_divergence_uncertainty}
\end{subfigure}
\hfill
\caption{Results for single Gaussian embeddings with and without KL divergence term.
We report results for images with $N$ digits and using $D$ embedding dimensions.
}
\label{tab:kl_divergence}
\end{table*}
\section{KNN with Plurality Voting and Corrupt Image Gallery}
\label{asec:knn_plurality}
In this section we discuss several improvements and variations to the KNN baseline described in Section \ref{sec:experiments}. Other classifiers more sophisticated than KNN could be trained atop the embeddings. However, our goal is not necessarily to achieve the best possible classification accuracy, but rather to study the interaction between HIB and point embeddings and we selected KNN because of its simplicity. First, in the original KNN experiments, a query image's classification is considered correct only if a majority of the nearest neighbors are the correct class. However, this requirement is unnecessarily strict, and instead, the query image's class here is the \textit{plurality} class of the neighbors. Ties are broken by selecting the class the closest matching sample.
Secondly, in the original experiment, the test gallery is either comprised of corrupted or clean images, and the probe images are clean. Perhaps a more relevant situation is the test where the gallery is clean, but the probe images are corrupt. In fact, we find that this experimental condition is actually more difficult because when a digit is completely occluded in the probe, one can at best guess on the of possibly several classes that contain the non-occluded digits in the correct place positions.
The results are reported in Tables \ref{tab:knn_plurality} and \ref{tab:knn_plurality_corr} where ``point'' again refers to learning point embeddings with contrastive loss, and ``MoG-1'' refers to our hedged embeddings with a single Gaussian component. We observe the following. First, by comparing Table \ref{tab:knn_plurality} with Table \ref{tab:maintask}, we see that using a plurality-based KNN generally improves the task accuracy. Further, a corrupted probe is indeed more difficult than a clean probe with a corrupted gallery. For KNN classification tasks that involve corrupted input either in the gallery or in the probe, we see that HIB improves over point embeddings. Finally, we observe from Table \ref{tab:knn_plurality_corr} that the correlations between the measure of uncertainty $\eta(x)$ and KNN performance are highest for the most difficult task -- classifying a corrupt probe image.
\begin{table*}
\centering
\scriptsize
\setlength\tabcolsep{1.5pt}
\begin{tabular}{@{}llrrcrrcrrcrr@{}}\toprule
& & \multicolumn{2}{c}{$N=2$, $D=2$ } & \phantom{abc}
& \multicolumn{2}{c}{$N=2$, $D=3$ } & \phantom{abc}
& \multicolumn{2}{c}{$N=3$, $D=2$} & \phantom{abc}
& \multicolumn{2}{c}{$N=3$, $D=3$}\\
\cmidrule{3-4} \cmidrule{6-7} \cmidrule{9-10} \cmidrule{12-13}
& & point & MoG-1 && point & MoG-1 && point & MoG-1 && point & MoG-1 \\ \midrule
{\textbf{Gallery}} & {\hspace{3mm}\textbf{Probe}\hspace{3mm}}\\
\hspace{0mm}clean&\hspace{3mm}clean\hspace{3mm}& 0.882& 0.879 && 0.950 & 0.952 &&0.658& 0.650 && 0.873 & 0.873 \\
\hspace{0mm}clean&{\hspace{3mm}corrupt{\hspace{3mm}\phantom{a}}}& 0.435& 0.499 && 0.516 & 0.578 && 0.293& 0.318 && 0.447& 0.499 \\
\hspace{0mm}corrupt&\hspace{3mm}clean\hspace{3mm}& 0.762& 0.816 && 0.922 & 0.943 && 0.495& 0.540 && 0.776& 0.812\\
\bottomrule
\end{tabular}
\caption{Accuracy of pairwise verification and KNN identification tasks for point embeddings, and our hedged embeddings with a single Gaussian component (MoG-1).
We report results for images with $N$ digits and using $D$ embedding dimensions.
}
\label{tab:knn_plurality}
\vspace{2em}
\scriptsize
\setlength\tabcolsep{4pt}
\begin{tabular}{@{}llccccccc@{}}\toprule
& & \multicolumn{1}{c}{$N=2$, $D=2$} & \phantom{a}
& \multicolumn{1}{c}{$N=2$, $D=3$} & \phantom{a}
& \multicolumn{1}{c}{$N=3$, $D=2$} & \phantom{a}
& \multicolumn{1}{c}{$N=3$, $D=3$}\\
\cmidrule{3-3} \cmidrule{5-5} \cmidrule{7-7} \cmidrule{9-9}
& & MoG-1 && MoG-1 && MoG-1 && MoG-1 \\ \midrule
{\textbf{Gallery}} & {\hspace{3mm}\textbf{Probe}}\\
\hspace{0mm}clean&\hspace{3mm}clean& 0.71 && 0.61 &&0.72 && 0.60 \\
\hspace{0mm}clean&\hspace{3mm}corrupt& 0.94 && 0.89 && 0.86 && 0.90 \\
\hspace{0mm}corrupt&\hspace{3mm}clean& 0.46&& 0.52 && 0.29&& 0.45\\
\bottomrule
\end{tabular}
\caption{Correlations between each input image's measure of uncertainty, $\eta(x)$, and AP and KNN performances. High correlation coefficients suggest a close relationship.}
\label{tab:knn_plurality_corr}
\end{table*}
\section{Organization of the Latent Embedding Space}
\label{sec:latentSpace}
\begin{figure}[t!]
\begin{center}
\begin{subfigure}{0.48\columnwidth}
\centering
\includegraphics[width=0.48\columnwidth]{figure/2DMean/180919P002_seen_cc1_classmeans.png}
\includegraphics[width=0.48\columnwidth]{figure/2DMean/180919P002_seen_cc10_classmeans.png}
\caption{\label{fig:2DCentroidOrganizea} Point embedding centroids.}
\end{subfigure}
\begin{subfigure}{0.48\columnwidth}
\centering
\includegraphics[width=0.48\columnwidth]{figure/2DMean/180919P016_seen_cc1_classmeans.png}
\includegraphics[width=0.48\columnwidth]{figure/2DMean/180919P016_seen_cc10_classmeans.png}
\caption{\label{fig:2DCentroidOrganizeb} Hedged embedding centroids.}
\end{subfigure}
\end{center}
\caption{Uncertain embeddings self-organize. Class centroids for 2D point embeddings of 2-digit MNIST are shown, colored here by ones digit (left) and tens digit (right). The hedged instance embeddings have a structure where the classes self-organize in an axis aligned manner because of the diagonal covariance matrix used in the model. }
\label{fig:2DCentroidOrganize}
\vspace{2em}
\centering
\includegraphics[width=0.96\columnwidth]{figure/1DEmbeddingOrder.png}
\vspace{1em}
\caption{Embedding 2-digit MNIST onto the number line. \textbf{Top:} An ordering of class centroids produced with {hedged instance embedding}\xspace with Gaussian embeddings. \textbf{Bottom:} An ordering produced with point embedding. Centroids are colored according to least significant (ones) digit. The {hedged instance embedding}\xspace more often places classes that share attributes (in this case, a ones digit or tens digit). }
\label{fig:one-d-embedding}
\end{figure}
As {hedged instance embedding}\xspace training progresses, it is advantageous for any subset of classes that may be confused with one another to be situated in the embedding space such that a given input image's embedding can strategically place probability mass. We observe this impacts the organization of the underlying space. For example, in the 2D embeddings shown in \Figref{fig:2DCentroidOrganize}, the class centers of mass for each class are roughly axis-aligned so that classes that share a tens' digit vary by x-coordinate, and classes that share a least significant (ones) digit vary by y-coordinate.
To further explore this idea, we embed 2-digit MNIST into a single dimension, to see how the classes get embedded
along the number line. For {hedged instance embedding}\xspace, a single Gaussian embedding was chosen as the representation. We conjectured that because {hedged instance embedding}\xspace reduces objective loss by placing groups of confusing categories nearby one another, the resulting embedding space would be organized to encourage classes that share a tens or ones digit to be nearby.
\Cref{fig:one-d-embedding} shows an example embedding learned by the two methods.
We assess the embedding space as follows.
First the centroids for each of the 100 classes are derived from the test set embeddings. After sorting the classes, a count is made of adjacent class pairs that share a ones or tens digit, with the maximum possible being 99. The hedged embeddings outscored the point embeddings on each of the the four trials, with scores ranging from 76 to 80 versus scores of 42 to 74. Similarly, consider a \textit{run} as a series of consecutive class pairs that share a ones or tens digit. The average run contains 4.6 classes with from hedged embeddings, and only 3.0 for point embeddings,
as illustrated in \Figref{fig:one-d-embedding}.
\subsubsection*{Acknowledgments}
We are grateful to Alex Alemi and Josh Dillon, who were very helpful with discussions and suggestions related to VAEs and Variational Information Bottleneck.
\section{Methods}
\label{sec:method}
In this section, we describe our method in detail.
\subsection{Point embeddings}
\label{sec:learnable-margin}
Standard point embedding methods try to compute embeddings
such that $z_1=f(x_1)$ and $z_2=f(x_2)$ are ``close'' in the embedding space if $x_1$ and $x_2$ are ``similar'' in the ambient space.
To obtain such a mapping, we must decide on the definition of ``closeness'' as well as a training objective, as we explain below.
\eat{
HIB\xspace extends the point embedding $f:x\mapsto z$ trained with the pairwise contrastive loss. This subsection introduces a probabilistic modification on the contrastive loss called \emph{soft contrastive loss} that has a soft learnable margin. It bridges the usual contrastive loss and our final formulation.
}
\paragraph{Contrastive loss}
Contrastive loss~\citep{contrastiveloss}
is designed to encourage a small Euclidean distance between a similar pair, and large distance of margin $M>0$ for a dissimilar pair. The loss is
\begin{align}
\mathcal{L}_{\text{con}}=
\begin{cases}
||z_1-z_2||_2^2 & \text{if match}\\
\max(M-||z_1-z_2||_2, 0)^2 & \text{if non-match }
\end{cases}
\end{align}
where $z_i=f(x_i)$. The hyperparameter $M$ is usually set heuristically or
based on validation-set performance.
\paragraph{Soft contrastive loss}
A probabilistic alternative to contrastive loss, which we will use in our experiments is defined here.
It represents the probability that a pair of points is matching:
\begin{align}
\label{eq:matchprob}
p(m|z_1,z_2) := \sigma( -a||z_1-z_2||_2 + b)
\end{align}
with scalar parameters $a>0$ and $b\in{\mathbb{R}}$, and the sigmoid function $\sigma(t)=\frac{1}{1+e^{-t}}$. This formulation calibrates Euclidean distances into a probabilistic expression for similarity. Instead of setting a hard threshold like $M$, $a$ and $b$ together comprise a soft threshold on the Euclidean distance. We will later let $a$ and $b$ be trained from data.
Having defined the match probability $p(m|z_1,z_2)$, we formulate the contrastive loss as a binary classification loss based on the softmax cross-entropy (negative log-likelihood loss). More precisely, for an embedding pair $(z_1,z_2)$ the loss is defined as
\begin{align}
\label{eq:matchloss}
\mathcal{L}_{\text{softcon}}=
-\log p(m=\hat{m}|z_1,z_2)=
\begin{cases}
-\log p(m|z_1,z_2) & \text{if }\hat{m}=1,\\
-\log \left(1-p(m|z_1,z_2)\right) & \text{if }\hat{m}=0,
\end{cases}
\end{align}
where $\hat{m}$ is the indicator function with value $1$ for ground-truth match and $0$ otherwise.
Although some prior work has explored
this soft contrastive loss (\emph{e.g.}\xspace \citet{bertinetto2016fully,Orekondy2018UnderstandingAC}),
it does not seem to be widely used.
However, in our experiments, it performs strictly better than the hard margin version, as explained in \Cref{sec:soft_contrastive_experimental}.
\subsection{Stochastic embeddings}
\label{sec:stochastic-embedding}
In HIB\xspace, we treat embeddings as stochastic mappings $x\mapsto Z$, and write the distribution as $Z\sim p(z|x)$.
In the sections below, we show how to learn and use this mapping.
\paragraph{Match probability for probabilistic embeddings}
The probability of two inputs matching,
given in \Eqref{eq:matchprob},
can easily be extended to stochastic embeddings, as follows:
\begin{align}
p(m|x_1,x_2) = \int p(m|z_1,z_2)\, p(z_1|x_1)\, p(z_2|x_2)\,\text{d}z_1\text{d}z_2.
\end{align}
We approximate this integral via Monte-Carlo sampling from $z_1^{(k_1)}\sim p(z_1|x_1)$ and $z_2^{(k_2)}\sim p(z_2|x_2)$:
\begin{align}
p(m|x_1,x_2) \approx
\frac{1}{K^2}
\sum_{k_1=1}^K \sum_{k_2=1}^{K} p\left(m|z_1^{(k_1)},z_2^{(k_2)}\right).
\label{eq:mc}
\end{align}
In practice, we get good results using $K=8$ samples per input image. Now we discuss the computation of $p(z|x)$.
\paragraph{Single Gaussian embedding}
The simplest setting is to let $p(z|x)$ be a $D$-dimensional Gaussian with mean $\mu(x)$ and diagonal covariance $\Sigma(x)$,
where $\mu$ and $\Sigma$ are computed via a deep neural network with a shared ``body'' and $2D$ total outputs.
Given a Gaussian representation, we can draw $K$ samples $z^{(1)},\cdots,z^{(K)}\overset{\text{iid}}{\sim}p(z|x)$,
which we can use to approximate the match probability.
Furthermore,
we can use the reparametrization trick~\citep{vae}
to rewrite the samples as
$z^{(k)}=\text{diag}\left(\sqrt{\Sigma(x)}\right)\cdot\epsilon^{(k)}+\mu(x)$,
where $\epsilon^{(1)},\cdots,\epsilon^{(K)}\overset{\text{iid}}{\sim}N(0,I)$. This enables easy backpropagation during training.
\paragraph{Mixture of Gaussians (MoG) embedding}
We can obtain a more flexible representation of uncertainty by using a mixture of $C$ Gaussians to represent our embeddings,
\emph{i.e.}\xspace $p(z|x) = \sum_{c=1}^C \mathcal{N}(z;\mu(x,c),\Sigma(x,c))$.
To enhance computational efficiency, the $2C$ mappings $\{(\mu(x,c), \Sigma(x,c))\}_{c=1}^C$ share a common CNN stump and are branched with one linear layer per branch.
When approximating \Eqref{eq:mc}, we use stratified sampling, \emph{i.e.}\xspace we sample the same number of samples from each Gaussian component.
\paragraph{Computational considerations}
\begin{figure*}
\centering
\includegraphics[width=1\columnwidth]{figure/architecture-v3}
\vspace{0em}
\caption{Computational graph for computing $p(m|x_1,x_2)$
using HIB\xspace with Gaussian embeddings.}
\label{fig:architecture}
\end{figure*}
The overall pipeline for computing the match probability is shown in \Figref{fig:architecture}.
If we use a single Gaussian embedding,
the cost (time complexity) of computing the stochastic representation is essentially the same as for point embedding methods, due to the use of a shared network for computing $\mu(x)$ and $\Sigma(x)$.
Also, the space requirement is only 2$\times$ more. (This is an important consideration for many embedding-based methods.)
\subsection{VIB training objective}
\label{sec:training-vib}
For training our stochastic embedding, we combine two ingredients: soft contrastive loss in \Eqref{eq:matchloss} and the VIB principle~\cite{VIB,Achille2018jmlr}. We start with a summary of the original VIB formulation, and then describe its extension to our setting.
\paragraph{Variational Information Bottleneck (VIB)}
A discriminative model $p(y|x)$ is trained under the information bottleneck principle \citep{Tishby99} by maximizing the following objective:
\begin{align}
\label{eq:vib}
I(z,y) - \beta I(z,x)
\end{align}
where $I$ is the mutual information, and $\beta>0$ is a hyperparameter
which controls the tradeoff between the sufficiency of $z$ for predicting $y$, and the minimality (size) of the representation.
Intuitively, this objective lets the latent encoding $z$ capture the salient parts of $x$ (salient for predicting $y$),
while disallowing it to ``memorise'' other parts of the input
which are irrelevant.
Computing the mutual information is generally computationally intractable,
but it is possible to use a tractable variational approximation as shown in \citet{VIB,Achille2018jmlr}.
In particular,
under the Markov assumption that $p(z|x,y)=p(z|x)$ we arrive at a lower bound on \Eqref{eq:vib} for every training data point $(x,y)$ as follows:
\begin{align}
\label{eq:vib-loss}
-\mathcal{L}_{\text{VIB}}:=\mathbb{E}_{z\sim p(z|x)}\left[\log q(y|z)\right] - \beta\cdot \text{KL}(p(z|x)||r(z))
\end{align}
where $p(z|x)$ is the latent distribution for $x$,
$q(y|z)$ is the decoder (classifier), and $r(z)$ is an approximate marginal term that is typically set to the unit Gaussian $\mathcal{N}(0,I)$.
In \citet{VIB},
this approach was shown (experimentally)
to be more robust to adversarial image perturbations
than deterministic classifiers.
It has also been shown to provide a useful way to
detect out-of-domain inputs \citep{Alemi2018uai}.
Hence we use it as the foundation for our approach.
\paragraph{VIB for learning stochastic embeddings}
We now apply the above method to learn our stochastic embedding.
In particular,
we train a discriminative model based on matching or mismatching pairs of inputs $(x_1,x_2)$, by minimizing the following loss:
\begin{align}
\label{eq:vib-metric}
\mathcal{L}_{\text{VIBEmb}}:=& -\mathbb{E}_{z_1\sim p(z_1|x_1),z_2\sim p(z_2|x_2)}\left[\log p(m=\hat{m}|z_1,z_2)\right] \nonumber \\
&+ \beta\cdot
\left[\text{KL}(p(z_1|x_1)||r(z_1))
+ \text{KL}(p(z_2|x_2)||r(z_2))
\right]
\end{align}
where the first term is given by
the negative log likelihood loss with respect to the ground truth match $\hat{m}$ (this is identical to \Eqref{eq:matchloss}, the soft contrastive loss), and the second term is the KL regularization term, $r(z)=\mathcal{N}(z;0,I)$. The full derivation is in \cref{sec:VIBderivation}.
We optimize this loss with respect to the embedding function
$(\mu(x),\Sigma(x))$, as well as with respect to the $a$ and $b$ terms in the match probability in \Eqref{eq:matchprob}.
Note that most pairs are not matching, so the $m=1$ class is rare.
To handle this, we encourage a balance of $m=0$ and $m=1$ pair samples within each SGD minibatch by using two streams of input sample images. One samples images from the training set at random and the other selects images from specific class labels, and then these are randomly shuffled to produce the final batch. As a result, each minibatch has plenty of positive pairs even when there are a large number of classes.
\subsection{Uncertainty measure}
\label{subsec:uncertainty-measure}
One useful property of our method is that the embedding is a distribution and encodes the level of uncertainty for given inputs.
As a scalar uncertainty measure, we propose the \emph{self-mismatch} probability as follows:
\begin{align}
\eta(x) := 1-p(m|x,x)\geq 0
\label{eqn:uncertainty}
\end{align}
Intuitively, the embedding for an ambiguous input will span diverse semantic classes (as in \Figref{fig:teaser-probabilistic}).
$\eta(x)$ quantifies this by measuring the chance two samples of the embedding $z_1, z_2 \overset{\text{iid}}{\sim} p(z|x)$ belong to different semantic classes (i.e., the event $m=0$ happens).
We compute $\eta(x)$ using the Monte-Carlo estimation in \Eqref{eq:mc}.
Prior works~\citep{vilnis2014word,bojchevski2017deep} have computed uncertainty for Gaussian embeddings based on volumetric measures like trace or determinant of covariance matrix.
Unlike those measures, $\eta(x)$ can be computed for \emph{any} distribution from which one can sample, including multi-modal distributions like mixture of Gaussians.
\subsection{Qualitative evaluation of the representation}
\Cref{fig:2DEmbeddings} shows HIB 2D Gaussian embeddings for the clean and corrupt subsets of the test set.
We can easily see that the corrupt images generally have larger (\emph{i.e.}\xspace, less certain) embeddings. In the Appendix, \Figref{fig:2DEmbeddingsMOG} shows a similar result when using a 2D MoG representation,
and \Cref{fig:3DEmbeddings} shows a similar result for 3D Gaussian embeddings.
\Cref{fig:embeddingimages} illustrates the embeddings for several test set images, overlaid with an indication of each class' centroid. Hedged embeddings capture the uncertainty that may exist across complex subsets of the class label space, by learning a layout of the embedding space such that classes that may be confused are able to receive density from the underlying hedged embedding distribution.
We observe enhanced spatial regularity when using HIB\xspace. Classes with a common least or most significant digit roughly align parallel to the $x$ or $y$ axis.
This is because of the diagonal structure of the embedding covariance matrix. By controlling the parametrization of the covariance matrix, one may apply varying degrees and types of structures over the embedding space (\emph{e.g.}\xspace diagonally aligned embeddings).
See \cref{sec:latentSpace} for more analysis of the learned latent space.
\section{Related Work}
In this section, we mention the most closely related work
from the fields of deep learning and probabilistic modeling.
\paragraph{Probabilistic DNNs}
Several works have considered the problem of estimating the uncertainty of a regression or classification model, $p(y|x)$, when given ambiguous inputs.
\eat{
Lakshminarayanan2017nips,
}
One of the simplest and most widely used techniques is known as Monte Carlo dropout \citep{gal2016dropout}.
In this approach, different random components of the hidden activations are ``masked out'' and a distribution over the outputs $f(x)$ is computed.
By contrast, we compute a parametric representation of the uncertainty and use Monte Carlo to approximate the probability of two points matching.
Monte Carlo dropout is not directly applicable in our setting as the randomness is attached to model parameters and is independent of input; it is designed to measure model uncertainty (epistemic uncertainty).
On the other hand, we measure input uncertainty where the embedding distribution is conditioned on the input.
Our model is designed to measure input uncertainty (aleatoric uncertainty).
\paragraph{VAEs and VIB}
A variational autoencoder~(VAE,~\citet{vae}) is a latent variable model of the form $p(x,z) = p(z) p(x|z)$, in which the generative decoder $p(x|z)$ and an encoder network, $q(z|x)$ are trained jointly so as to maximize the evidence lower bound.
By contrast, we compute a discriminative model on pairs of inputs to maximize a lower bound on the match probability.
The variational information bottleneck (VIB) method \citep{VIB,Achille2018jmlr} uses a variational approximation similar to the VAE to approximate the information bottleneck objective \citep{Tishby99}.
We build on this as explained in \ref{sec:training-vib}.
\eat{
and follow-up works~\citep{cvae,Vamp} train it by first constructing the generative process $Z\rightarrow X$ and variationally appoximating the posterior $p(z|x)$. Variational information bottleneck~(VIB,~\citet{VIB}) first treats the embedding $p(x|z)$ as a stochastic mapping, and variationally approximates the subsequent decoder $Z\rightarrow Y$ (\emph{e.g.}\xspace a classifier). HIB\xspace also starts by treating $p(x|z)$ as a stochastic mapping, but the decoder executes a matching task: $(Z_1,Z_2)\rightarrow M$, where $M$ is a binary random variable indicating match. HIB\xspace is constructed under the VIB principle with the decoder for matching task.
Another possibility to obtain uncertainty estimates is Monte-Carlo dropout at test time~\citep{gal2016dropout}, which we have not used. One can also resort to a more efficient surrogate using knowledge distillation~\citep{ddn_wip}.
}
\if 0
Reparametrization of many traditionally non-differentiable distribution parameters\cite{figurnov2018implicit}
CVAE - latent model with class info together. Roughly related. \cite{cvae}
VAE - first efficient application to generative model for a large-scale dataset (MNIST) - variational approximation of latent posterior and reparametrization trick.\cite{vae,doersch2016tutorial}
VampPrior - Allows for more structured priors. \cite{Vamp}
VIB+VampPrior \cite{alemi2018information}
Standard neural networks already predict uncertainty - only need to calibrate better.
\cite{guo2017calibration}
Aleatoric versus epistemic uncertainty \cite{kendall2017uncertainties}.
Dropout \cite{gal2016dropout}
Training time dropout \citep{zheng2017discriminatively}
Colourization\cite{baig2017multiple, deshpande2016learning}, future prediction \cite{lee2017desire}, caption generation, ...
Multiple choice learning (min-loss) \cite{LeeNIPS2016}
GAN\cite{goodfellow2014generative}
GAN based diversity. \cite{ghosh2017multi}
Improving GANs to avoid mode collapse \cite{salimans2016improved}.
Caption generation with GAN \cite{shetty2017speaking}
MPI - Apratim (CVPR\cite{bhattacharyya2018accurate}, NIPS)
Our work is related.
\fi
\paragraph{Point embeddings}
Instance embeddings are often trained with metric learning objectives, such as contrastive~\citep{contrastiveloss} and triplet~\citep{facenet} losses.
Although these methods work well,
they require careful sampling schemes~\citep{wu2017sampling,movshovitz2017no}. Many other alternatives have attempted to decouple the dependency on sampling, including softmax cross-entropy loss coupled with the centre loss~\citep{Wan_2018_CVPR}, or a clustering-based loss~\citep{song2017deep}, and have improved the embedding quality.
In HIB\xspace, we use a soft contrastive loss,
as explained in \cref{sec:learnable-margin}.
\paragraph{Probabilistic embeddings}
The idea of probabilistic embeddings is not new.
For example, \citet{vilnis2014word} proposed Gaussian embeddings to represent levels of specificity of word embeddings (\emph{e.g.}\xspace ``Bach'' is more specific than ``composer''). The closeness of the two Gaussians is based on their KL-divergence, and uncertainty is computed from the spread of Gaussian (determinant of covariance matrix).
See also \citet{karaletsos2015bayesian,bojchevski2017deep} for related work.
\citet{Neelakantan2014emnlp} proposed to represent each word using multiple prototypes,
using a ``best of $K$'' loss when training.
HIB\xspace, on the other hand, measures closeness based on
a quantity related to
the expected Euclidean distance, and measures uncertainty using the
\emph{self-mismatch} probability.
\if 0
When people do embedding with variational latent posterior approximation, people couple it with clustering, where the number of clusters is predefined \cite{VariationalEmbedding}
Embedding to improve uncertainty estimates \cite{Qian_2018_CVPR} -- but not really represent uncertainty
\fi
\section{Soft Contrastive Loss versus Contrastive Loss}
\label{sec:soft_contrastive_experimental}
\begin{figure}[h!]
\centering
\hfill
\begin{subfigure}{0.35\columnwidth}
\includegraphics[width=0.99\columnwidth]{figure/soft_contrastive_plots/test_seen_ap}
\caption{Average precision.}
\end{subfigure}
\hfill
\begin{subfigure}{0.35\columnwidth}
\includegraphics[width=0.99\columnwidth]{figure/soft_contrastive_plots/test_seen_knn}
\caption{K nearest neighbour accuracy.}
\end{subfigure}
\hfill
\hspace{1.5em}
\caption{Soft contrastive versus vanilla contrastive loss for point embeddings.}
\label{fig:soft_contrastive_experiments}
\end{figure}
As a building block for the HIB\xspace, soft contrastive loss has been proposed in \S\ref{sec:learnable-margin}. Soft contrastive loss has a conceptual advantage over the vanilla contrastive loss that the margin hyperparameter $M$ does not have to be hand-tuned. Here we verify that soft contrastive loss outperforms the vanilla version over a range of $M$ values.
\Figref{fig:soft_contrastive_experiments} shows the verification (average precision) and identification (KNN accuracy) performance of embedding 2-digit MNIST samples. In both evaluations, soft contrastive loss performance is upper bounding the vanilla contrastive case. This new formulation removes one hyperparameter from the learning process, while not sacrificing performance.
\subsection{Known unknowns}
\begin{figure}[t!]
\begin{center}
\begin{subfigure}{0.24\columnwidth}
\centering
\includegraphics[width=0.96\columnwidth]{figure/uncertaintycorrelation/180919P006cleanseenAP.png}
\caption{\label{fig:correlationa}AP for clean test}
\end{subfigure}
\begin{subfigure}{0.24\columnwidth}
\centering
\includegraphics[width=0.96\columnwidth]{figure/uncertaintycorrelation/180919P006corruptseenAP.png}
\caption{\label{fig:correlationb}AP for corrupt test}
\end{subfigure}
\begin{subfigure}{0.24\columnwidth}
\centering
\includegraphics[width=0.96\columnwidth]{figure/uncertaintycorrelation/180919P006cleanseenknn.png}
\caption{\label{fig:correlationc}KNN for clean test}
\end{subfigure}
\begin{subfigure}{0.24\columnwidth}
\centering
\includegraphics[width=0.96\columnwidth]{figure/uncertaintycorrelation/180919P006corruptseenknn.png}
\caption{\label{fig:correlationd}KNN for corrupt test}
\end{subfigure}
\end{center}
\caption{Correlations between the uncertainty measure $\eta(x)$ and AP and KNN accuracy on the test set for the $N=3$, $D=3$ case
using single Gaussian embeddings.
Uncertainty increases along the horizontal axis. We observe that accuracy generally decreases as uncertainty increases.
}
\label{fig:uncertaintycorrelations}
\end{figure}
In this section, we address the task of estimating when an input can be reliably recognized or not, which has important practical applications.
To do this, we use the measure of uncertainty $\eta(x)$
defined in \Eqref{eqn:uncertainty}.
We measure the utility of $\eta(x)$ for the identification task as follows.
For the test set, we sort all test input examples according to $\eta(x)$, and bin examples into 20 bins ranging from the lowest to highest range of uncertainty.
We then measure the KNN classification accuracy for the examples falling in each bin.
To measure the utility of $\eta(x)$ for the verification task,
we take random pairs of samples, $(x_1,x_2)$,
and compute the mean of their uncertainties,
$\eta(x_1,x_2) =\frac{1}{2}(\eta(x_1)+\eta(x_2))$.
We then distribute the test pairs to 20 equal-sized bins according to their uncertainty levels,
and compute the probability of a match for each pair.
To cope with the severe class imbalance (most pairs don't match),
we measure performance for each bin using average precision (AP).
Then, again, the Kendall's tau is applied to measure the uncertainty-performance correlation.
\Cref{fig:uncertaintycorrelations} plots the AP and KNN accuracy vs the uncertainty bin index, for both clean and corrupted inputs.
We see that when the performance drops off, the model's uncertainty measure increases, as desired.
To quantify this, we compute the correlation between the performance metric and the uncertainty metric.
Instead of the standard linear correlations (Pearson correlation coefficient), we use Kendall's tau correlation~\citep{kendall1938new} that measures the degree of monotonicity between the performance and the uncertainty level (bin index), inverting the sign so that positive correlation aligns with our goal.
The results of different models are shown in \cref{tab:uncertaintymeasuresmog}. Because the uncertainty measure includes a sampling step, we repeat each evaluation 10 times, and report the mean results, with standard deviations ranging from 0.02 to 0.12. In general, the measure $\eta(x)$ correlates with the task performance.
As a baseline for point embeddings in KNN, we explored using the distance to the nearest neighbor as a proxy for uncertainty, but found that it performed poorly. The HIB uncertainty metric correlates with task accuracy even in within the subset of clean (uncorrupted) input images having no corrupted digits, indicating that HIB's understanding of uncertainty goes beyond simply detecting which images are corrupted.
\section{Derivation of VIB Objective for Stochastic Embedding}
\label{asec:derivation}
\label{sec:VIBderivation}
Our goal is to train a discriminative model for match prediction on a pair of variables, $p(m|f(x_1), f(x_2))$,
as opposed to predicting a class label, $p(y|x)$.
Our VIB loss (\Eqref{eq:vib-metric}) follows straightforwardly from the original VIB, with two additional independence assumptions.
In particular,
we assume that the samples in the pair are independent,
so $p(x_1,x_2)=p(x_1)p(x_2)$.
We also assume the embeddings do not depend on the other input in the pair,
$p(z_1,z_2|x_1,x_2)=p(z_1|x_1)p(z_2|x_2)$.
With these two assumptions,
the VIB objective is given by the following:
\begin{align}
\label{eq:vib-objective-match}
I((z_1,z_2),m)-\beta I((z_1,z_2),(x_1,x_2)).
\end{align}
We variationally bound the first term using the approximation $q(m|z_1,z_2)$ of $p(m|z_1,z_2)$ as follows
\begin{align}
I((z_1,z_2),m)
&=\int p(m,z_1,z_2)\log \frac{p(m,z_1,z_2)}{p(m)p(z_1,z_2)} \,\text{d}m \,\text{d}z_1\,\text{d}z_2 \\
&=\int p(m,z_1,z_2)\log \frac{p(m|z_1,z_2)}{p(m)} \,\text{d}m \,\text{d}z_1\,\text{d}z_2 \\
&=\int p(m,z_1,z_2)\log \frac{q(m|z_1,z_2)}{p(m)} \,\text{d}m \,\text{d}z_1\,\text{d}z_2 + \text{KL}(p(m|z_1,z_2)\,||\,q(m|z_1,z_2)) \\
&\geq\int p(m,z_1,z_2)\log \frac{q(m|z_1,z_2)}{p(m)} \,\text{d}m \,\text{d}z_1\,\text{d}z_2 \\
&=\int p(m,z_1,z_2)\log q(m|z_1,z_2) \,\text{d}m \,\text{d}z_1\,\text{d}z_2 + H(m) \\
&\geq\int p(m,z_1,z_2)\log q(m|z_1,z_2) \,\text{d}m \,\text{d}z_1\,\text{d}z_2 \\
&=\int p(m|x_1,x_2)p(z_1|x_1)p(z_2|x_2)p(x_1)p(x_2)\log q(m|z_1,z_2) \,\text{d}m \,\text{d}z_1\,\text{d}z_2\,\text{d}x_1\,\text{d}x_2.
\end{align}
The inequalities follow from the non-negativity of KL-divergence $\text{KL}(\cdot)$ and entropy $H(\cdot)$. The final equality follows from our assumptions above.
The second term is variationally bounded using approximation $r(z_i)$ of $p(z_i|x_i)$ as follows:
\begin{align}
I((z_1,z_2),(x_1,x_2))
&= \int p(z_1,z_2,x_1,x_2) \log \frac{p(z_1,z_2|x_1,x_2)}{p(z_1,z_2)}\,\text{d}z_1\,\text{d}z_2\,\text{d}x_1\,\text{d}x_2 \\
&= \int p(z_1,z_2,x_1,x_2) \log \frac{p(z_1,z_2|x_1,x_2)}{r(z_1)r(z_2)}\,\text{d}z_1\,\text{d}z_2\,\text{d}x_1\,\text{d}x_2 \nonumber \\
& \quad\quad - KL(p(z_1)\,||\,r(z_1)) - KL(p(z_2)\,||\,r(z_2)) \\
&\leq \int p(z_1,z_2,x_1,x_2) \log \frac{p(z_1,z_2|x_1,x_2)}{r(z_1)r(z_2)}\,\text{d}z_1\,\text{d}z_2\,\text{d}x_1\,\text{d}x_2 \\
&= \int p(z_1,z_2|x_1,x_2)p(x_1,x_2) \log \frac{p(z_1,z_2|x_1,x_2)}{r(z_1)r(z_2)}\,\text{d}z_1\,\text{d}z_2\,\text{d}x_1\,\text{d}x_2 \\
&= \int p(z_1|x_1)p(x_1) \log \frac{p(z_1|x_1)}{r(z_1)}\,\text{d}z_1\,\text{d}x_1 \nonumber\\
&\quad\quad +\int p(z_2|x_2)p(x_2) \log \frac{p(z_2|x_2)}{r(z_2)}\,\text{d}z_2\,\text{d}x_2.
\end{align}
The inequality, again, follows from the non-negativity of KL-divergence, and the last equality follows from our additional independence assumptions.
Combining the two bounds, the VIB objective (\Eqref{eq:vib-objective-match}) for a fixed input pair $(x_1,x_2)$ is bounded from below by
\begin{align}
&\int p(z_1|x_1)p(z_2|x_2)\log q(m|z_1,z_2) \,\text{d}z_1\,\text{d}z_2 \\
&\quad\quad - \beta \left( \text{KL}(p(z_1|x_1)\,||\,r(z_1)) + \text{KL}(p(z_2|x_2)\,||\,r(z_2)) \right).
\end{align}
The negative of the above expression is identical to the loss $\mathcal{L}_{\text{VIBEmb}}$ in \Eqref{eq:vib-metric}.
\if 0
\begin{align}
&\int p(x_1)p(x_2)p(m|x_1,x_2)p(z_1|x_1)p(z_2|x_2)\log q(m|z_1,z_2) \,\text{d}m \,\text{d}z_1\,\text{d}z_2\,\text{d}x_1\,\text{d}x_2 \\
&\quad\quad - \beta\int p(x_1) p(z_1|x_1) \log \frac{p(z_1|x_1)}{r(z_1)}\,\text{d}z_1\,\text{d}x_1 \nonumber\\
&\quad\quad -\beta\int p(x_2) p(z_2|x_2) \log \frac{p(z_2|x_2)}{r(z_2)}\,\text{d}z_2\,\text{d}x_2.
\end{align}
\fi
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 621
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Prose of life
Work, career, business
Unusual holidays, or Is it possible to have a feast without salami?
It is impossible to imagine a festive table without sausage salami. Sliced in tiny transparent pieces and beautifully laid out on a dish, it is the quintessence of a snack table both at Russian holidays (along with brawn and herring under a fur coat) and at the US (along with fresh Italian loaf, olives and Antipasto salad).
In the Soviet Union, salami-type sausage was usually called smoked or smoked. In fact, salami - sausage is not smoked, but dried, that is, it is prepared by drying, although some salami are slightly smoked, but only to add flavor. This sausage appeared in Italy, among the poor peasantry, as a way of preserving meat for the long winter months. The word "salami" comes from the Italian "fat" - salt, because salt played and plays a significant role in the production of salami.
Main ingredients salami, besides salt, are meat (pork, beef, venison, game, even horse meat or donkey in Italy), fat, spices and seasonings and wine or brandy. The meat is turned through the meat grinder with the addition of fat, the sausage "dough" is kneaded, into which all the other ingredients are added, after which the sausage casing is stuffed with this dough. Then sausage is hung for drying. In the process of drying, it develops special bacteria that increase the acidity inside the casing, and thus the meat inside is "cooked" and becomes edible. After the sausage is well-worn, it is subjected to additional drying, during which a characteristic white patina appears on the skin. This plaque is a penicillin mold that protects sausage from damage.
In Italy, salami is usually called the region where it is produced: Milan, Tuscan, etc. The city of San Francisco is recognized as the capital of salami in America, where Italian settlers have kept the secrets of salami production to the present day.
The tradition of slicing salami into thin slices gave the name to an interesting technique that has no direct relation to the sausage, but is related to the robbery of banks. It is called "Salami method", implemented by programmers and looks like this.
As a programmer, you have access to programs that control customer accounts. You give the computer a command to withdraw three cents from each account each month and transfer this money to your account. The clientele of your bank is, say, a million people (you work in such a small bank). Thus, "cutting off a thin piece" from each account, you get 3 thousand dollars every month. At the same time, no one notices, because these three cents are withdrawn from interest, and who among the bank's customers can figure out how these interest are calculated? It's funny that one of these crimes was revealed because the bank held a month-long competition on the topic: "Who often puts money into his account?". The criminal programmer won a tour to Hawaii and a subpoena ...
But back to the holiday. Salami Day is celebrated by eating this delicious product in any form: with bread and butter in the form of sandwiches, as a filling for calzone or pizza, as part of the Antipasto salad, you can add it to the pasta sauce, you can shake old things and cook chocolate »Sausage for dessert.
Usually there are at least 5 different types of salami on the table, as well as cheese (mostly Italian), fresh white bread (better than homemade cakes), Italian chianti wine and olives. The rest is optional.
Watch the video: Traditional Holiday Feasts (January 2020).
← What role do windows play in a person's life?
Diamonds in Arkansas? Treasure hunters. →
Does airbag always save?
Where do dropouts win? In language
Nina Simon: how thanks to "Chanel № 5" appeared a hit for all times?
How did the Monopoly game fight fascism? "Silk Road" war
Appliances and the Internet
Feng shui and the unknown
Laws and Security
Alien among his own. How to cope with someone else's child if you were asked to temporarily look after him?
Why do you need to manage time and how best to do it?
Six shades of vulgarity. №6. What is the history of the hit about summer from "Arrhythmia"?
Children's dreams: do they disappear without a trace? Dream seen in a dream
Lithuanian cuisine: what is it? Zeppelins, Varya, Kugelis ...
February 23 is a memorable date?
Are two eyes always better than one?
Copyrights © 2020 Academy of Life
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 794
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Dokpesi, Adeniran reject outcome of PDP convention
Two Chairmanship aspirants of the Peoples Democratic Party , PDP, Chief Raymond Dokpesi and Prof. Tunde Adeniran have rejected the outcome of the party's national convention and the process that led to the emergence of the key officers.
The aspirants announced their positions in separate interviews with newsmen at the venue of the convention, on Saturday in Abuja.
Dokpesi, the media magnate said he rejected the elections following alleged distribution of a so-called unity list to delegates, describing it as a charade that may destroy the party.
Dokpesi said that the process of voting had been rigged, following the distribution of a unity list containing names of candidates believed to have won elections into various National leadership positions in the party.
He explained that the 21 names of candidates contained in the list appeared on the ballot papers as number one and in the voting booths as number one.
The list has Uche Secondus, a former chairman of the party as the new chairman.
Dokpesi said complaints were lodged to the chairman of the PDP electoral committee Gabriel Suswam who confirmed that he had seen the list with some delegates but he was overwhelmed.
Dokpesi said it was unfortunate that a party which was just getting out of a major leadership crisis would be involved in acts of impunity and election malpractice.
He called on the party leadership to urgently rectify the anomaly before it becomes another major challenge in the party, ahead of the 2019 general elections.
Similarly, Taiwo Akeju, Director, Media and Publicity of the Adeniran Campaign, called for the cancellation of the entire exercise.
He accused Governors Nyesom Wike of Rivers and Ayodele Fayose of Ekiti of being the masterminds of the unity list distributed to the delegates.
"We reject the entire electoral process. The election has been grossly compromised to achieve a predetermined end.
"The illegal unity list is prepared by governors Wike and Ayodele Fayose to foist on the entire delegates" he said.
He, however, called on the Board of Trustees (BOT) of the party to takeover the affairs of the party until proper election would be conducted, since the National Caretaker Committee has been dissolved.
The News Agency of Nigeria (NAN) reports that the rejection of the unity list by the frontline chairmanship candidates, was done before the final collation of the elections results. (NAN)
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Pressure: 30.5"Hg
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| 1,402
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{"url":"https:\/\/math.stackexchange.com\/questions\/2899461\/rigorous-proof-of-lim-limits-n-to-infty-int-0n-fn-t-dt-int-0-inft","text":"# Rigorous proof of $\\lim\\limits_{n \\to \\infty } \\int_0^n f(n,t) dt = \\int_0^\\infty \\lim\\limits_{n \\to \\infty } f(n,t) dt$\n\nI faced with this equality $$\\lim _{n \\to \\infty } \\int_0^n \\left( {1 - \\frac{m}{n}}\\right) ^n \\log(m)dm= \\int_0^\\infty {{e^{ - m}}} \\log mdm.$$ I know a rigorous proof for $\\displaystyle \\lim_{n\\to\\infty} \\left(1+\\frac {x}{n}\\right)^n=\\exp x$ but the problem is a general effect of limit on both integrand and upper bound and my question is proving the equality in the title and not just the example above; that is I can't rigorously prove why $$\\lim _{n \\to \\infty } \\int_0^n f(n,t) dt = \\int_0^\\infty \\lim_{n \\to \\infty } f(n,t) dt.$$\n\nUnfortunately, I don't know more than undergraduate real analysis. A simpler and clear proof would be much appreciated.\n\n\u2022 Something like this is probably easiest to handle using Lebesgue theory, e.g. dominated convergence theorem. Something simple like uniform convergence is going to struggle with the singularity, I think. \u2013\u00a0Ian Aug 30 '18 at 12:59\n\u2022 I thought that the Dominated convergence theorem only applied to integration over fixed sets? \u2013\u00a0MSobak Aug 30 '18 at 13:03\n\u2022 @Sobi $\\int_A f(x) dx = \\int_D f(x) \\chi_A(x) dx$ if $A \\subset D$. Here, $D=[0,\\infty),A=[0,n]$. \u2013\u00a0Ian Aug 30 '18 at 13:05\n\u2022 @Ian Nice! Never thought of that! \u2013\u00a0MSobak Aug 30 '18 at 13:07\n\u2022 @Ian unfortunately I don't know more than undergraduate real analysis which doesn't cover Lebesgue theory.. \u2013\u00a0user231343 Aug 30 '18 at 13:10\n\nWe can fashion a dominated convergence theorem of the kind that the old masters, pre measure-theory, would have known. I'll present an argument for your particular case, but it will be clear that there is a general theorem here.\n\nSketch of main ideas: First make all the domains of integration $(0,\\infty).$ Just set\n\n$$f_n(x) = (1-x\/n)^n\\log x \\,\\chi_{(0,n)}(x),\\,\\,f(x)=e^{-x}\\log x.$$\n\nYour problem is then to show $\\int_0^\\infty f_n \\to \\int_0^\\infty f.$\n\nNote that $|f_n(x)| \\le |f(x)|$ for all $x\\in (0,\\infty).$ Furthermore, $\\int_0^\\infty |f| <\\infty.$ And very importantly, $f_n \\to f$ uniformly on any $(a,b)$ with $0<a<b<\\infty.$ (If you haven't seem the last result don't despair; it's provable with the tools of undergraduate real analysis, give it a try.)\n\nWe then proceed:\n\n$$|\\int_0^\\infty f-\\int_0^\\infty f_n| =|\\int_0^\\infty (f-f_n) |\\le \\int_0^\\infty|f-f_n|$$ $$= \\int_0^a |f-f_n| + \\int_a^b |f-f_n| + \\int_b^\\infty |f-f_n|$$ $$\\le \\int_0^a 2|f| + \\int_a^b |f-f_n| + \\int_b^\\infty 2|f|.$$\n\nWe can choose $a,b$ so that the first and third integrals are as small as we like. Uniform convergence shows the second integral $\\to 0.$ We're in a good spot now. It will lead to the result you're after.\n\n\u2022 From what I understood, am I right to say that the necessary and sufficient condition for interchange of lim and integral (for fixed upper and lower bound of integration) is that $f_n$ converges uniformly to $f$ for all $x \\in (a,b)$? \u2013\u00a0user231343 Aug 30 '18 at 16:06\n\u2022 It's a sufficient condition; it's not necessary. \u2013\u00a0zhw. Aug 30 '18 at 16:13\n\nAssuming everything has sufficient decay (i.e. the integrals converges), your question clearly depends on the growth of the area $$\\int_{0}^{n}(f(n,t)-\\lim_{m\\to\\infty}f(m,t))dt.$$ Your equality will hold if an only if the above goes to $0$ as $n\\to \\infty$. For instance, this fails for $f(n,t)=e^t+\\frac{1}{n}$.\n\nTo prove your particular example (with the domain $[1,\\infty)$), just divide the range of the above integration as $[1,n^{1\/4}]$ and $[n^{1\/4},n]$ and show that each goes to zero. (In the first range the difference $(f(n,t)-\\lim_{m\\to\\infty}f(m,t))$ decays rapidly and in the second range $f(n,t)$ and $\\lim_{m\\to\\infty}f(m,t)$ decay individually.)\n\n\u2022 Why does $(1-n^{-3\/4})^n$ decay faster than $1\/n$? \u2013\u00a0Ian Aug 30 '18 at 15:27\n\u2022 In fact $m(n)=n^{1\/4}$ seems rather arbitrary...all you really need is $(1-m(n)\/n)^n=o(1\/n)$ and $m(n)^2\/n=o(1)$, so it is enough to have $m=o(n^{1\/2})$ and $m(n)=\\Omega(1)$. \u2013\u00a0Ian Aug 30 '18 at 15:40\n\nApply Monotone Convergence Theorem https:\/\/en.wikipedia.org\/wiki\/Monotone_convergence_theorem: Since the successive terms $(1-\\frac{m}{n})^{n}$ increases as proved in the following: (Using $\\log(x) \\leq x-1$) $$f(x) = (1-\\frac{m}{x})^x \\\\ \\log(f(x)) = x \\log(1-\\frac{m}{x})\\\\ f'(x) = f(x) (\\log(1-\\frac{m}{x}) + \\frac{x}{1-\\frac{m}{x}} \\frac{m}{x^2}) \\\\ f'(x) = f(x) (\\log(1-\\frac{m}{x}) + \\frac{m}{x-m}) \\\\ f'(x) = f(x) (-\\log(\\frac{x}{x-m}) + \\frac{m}{x-m}) \\\\ f'(x) \\geq f(x) (-(\\frac{x}{x-m}-1) + \\frac{m}{x-m}) \\\\ \\geq 0 \\\\$$","date":"2020-02-18 19:13:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9133177995681763, \"perplexity\": 361.68377796690515}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875143805.13\/warc\/CC-MAIN-20200218180919-20200218210919-00419.warc.gz\"}"}
| null | null |
package org.funytest.common.exception;
public class CheckFailException extends Exception {
private static final long serialVersionUID = 1L;
/* fail 信息 */
public String failMessage;
public CheckFailException(String failMessage) {
this.failMessage = failMessage;
}
public CheckFailException(String expect, String actual, String msg){
StringBuffer buffer = new StringBuffer();
buffer.append("===========[CHECK FAIL]===========\r\n");
buffer.append("msg is : \r\n").append(msg);
buffer.append("detail is :\r\n");
buffer.append("expect = ").append(expect).append("\r\n");
buffer.append("actual = ").append(expect).append("\r\n");
buffer.append("==================================\r\n");
this.failMessage = buffer.toString();
}
public String getFailMessage() {
return failMessage;
}
public void setFailMessage(String failMessage) {
this.failMessage = failMessage;
}
}
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La Catedral Archibasílica Papal del Santísimo Salvador del Mundo, y de los Santos Juan Bautista y Juan Evangelista en Letrán, más conocida como Archibasílica de San Juan de Letrán, es la catedral de la diócesis de Roma, donde se encuentra la sede episcopal del obispo de Roma (el papa). Está dedicada a Cristo Salvador, sin embargo es más conocida con el nombre de San Juan, por estar dedicada a los dos santos principales que llevan este nombre.
Junto al palacio anexo y algunos otros edificios cercanos, goza del estatus de extraterritorialidad dentro del Estado italiano, por lo que es propiedad de la Santa Sede. La basílica es una de las iglesias que se deben visitar en el peregrinaje de las siete iglesias de Roma para alcanzar la indulgencia plenaria en Año Santo.
En 1980 fue incluida en la lista del Patrimonio de la Humanidad en Europa por la Unesco, con el número de identificación 91-002. Desde el mismo año, además, se puede acceder a través de la estación del metro de Roma San Giovanni.
Descripción
El nombre oficial es Archibasilica Sanctissimi Salvatoris, es la más antigua y la de rango más alto entre las cuatro basílicas mayores o papales de Roma, y tiene el título honorífico de «Omnium urbis et orbis ecclesiarum mater et caput» (madre y cabeza de todas las iglesias de la ciudad de Roma y de toda la tierra), por ser la sede episcopal del primado de todos los obispos, el papa. Fue consagrada por el papa San Silvestre en el año 324.
Las otras tres basílicas mayores, todas caracterizadas por tener una puerta santa y un altar papal, son:
La Basílica de San Pedro del Vaticano
La Basílica de San Pablo Extramuros
La Basílica de Santa María la Mayor
La Archibasílica surge en el en tierras de los Lateranos, noble familia romana caída en desgracia bajo Nerón, cuya propiedad pasó por tanto al dominio imperial. El palacio pasó a manos de Constantino I cuando se casó con su segunda mujer, Fausta, hermana de Majencio, y era conocido con el nombre de Domus Faustae. Por tanto, Constantino era su propietario cuando ganó la batalla del Puente Milvio (contra Majencio), en el 312.
La tradición cristiana indica que los terrenos y la residencia de los Lateranos fueron donados al obispo de Roma (la fecha de la donación no es segura pero debería ser durante el pontificado del papa Melquíades), en señal de gratitud del emperador a Cristo, que apareciéndosele durante el sueño, le había hecho vencer en la batalla del Puente Milvio.
El baptisterio de esta basílica es un edificio independiente de planta octogonal, y tiene la forma típica de los baptisterios de los primeros siglos, cuando el bautismo se hacía por inmersión. Por tanto, cuenta con una piscina en la cual el neófito se sumergía para salir por el lado opuesto.
Anexo a la archibasílica hay un claustro con jardines y arquerías, y un palacio (el Palacio de Letrán), propiedad del papa. Antiguamente, todo este complejo lateranense fue la sede del gobierno eclesiástico, hasta el tiempo en que la corte pontificia se mudó a Aviñón (Francia), periodo conocido como Cautiverio de Babilonia. Al regresar los papas a Roma, se establecieron en la colina vaticana, donde actualmente está la Santa Sede.
Cerca de esta basílica está el edificio que alberga la Escalera Santa, cuyos escalones, traídos de Tierra Santa, son según la tradición los mismos que subió Cristo en el palacio de Poncio Pilato. No se permite subirlos de pie. Los devotos los suben de rodillas.
La actual basílica es de estilo barroco, fruto de una radical transformación de Francesco Borromini en el ; de época anterior se conservan los magníficos mosaicos del ábside, el ciborio gótico y el pavimento de estilo cosmatesco. En lo alto de la fachada se encuentran estatuas de Cristo, los santos Juanes (el Evangelista y el Bautista) y los Apóstoles. La fachada fue reformada en el , siguiendo el estilo de la de San Pedro, por el arquitecto Alessandro Galilei.
Ya en el interior, destacan las monumentales estatuas de los doce apóstoles de la nave central. Bajo el altar mayor está enterrado el papa Martín V, bajo cuyo pontificado se abrió por primera vez la Puerta Santa en esta basílica. El ara de este altar es una losa que, según la tradición, es la misma que usaban san Pedro y los primeros papas al celebrar la misa. Sobre el altar hay un baldaquino con un relicario en el que se conservan las cabezas de san Pedro y san Pablo. En el fondo del ábside está la cátedra, el trono episcopal del obispo de Roma, hecho de mármol y mosaicos.
El papa suele celebrar ciertas ceremonias litúrgicas en este lugar (por ejemplo, la misa de la Cena del Jueves Santo, y la misa de la fiesta del Corpus Christi; esta última tiene lugar en el atrio, a partir del cual parte la procesión eucarística).
El canónigo de honor de San Juan de Letrán es el presidente de la República Francesa, según una tradición que se remonta al , cuando el jefe del Estado era un rey. Nicolas Sarkozy tomó posesión del cargo en una ceremonia el 20 de diciembre de 2007.
En el calendario católico, el día 9 de noviembre se celebra la fiesta de la dedicación de esta basílica mayor.
El actual maestro de capilla es monseñor Marco Frisina.
Papas sepultados en San Juan de Letrán
Véase también
Palacio de Letrán
Basílica mayor
Baptisterio de Letrán
Peregrinaje de las siete iglesias de Roma
Referencias
Enlaces externos
Arcibasilica Papale di San Giovanni IIn Laterano (en italiano)
Visita virtual a la archibasílica de San Juan de Letrán (en inglés e italiano)
Interactive Nolli Map Website
Áreas de Roma con extraterritorialidad a favor de la Santa Sede
Catedrales de Lacio
Juan Laterano
Juan Laterano
Catedrales de Italia del siglo XVII
Monumentos de Roma (ciudad)
Arquitectura paleocristiana en Italia
Obras de Francesco Borromini
Juan
Basílicas mayores
Letran
Letran
Letran
Edificios y estructuras terminadas en el siglo IV
Iglesias del siglo IV
Diócesis de Roma
Arquitectura de Italia del siglo IV
|
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| 2,968
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Complete Sales Analysis December 2016 – Two & Three Wheeler
automobilesIndian Automobile Sales Report December 2016
Sales of Two Wheelers and Three Wheelers stood at 10.79 lakh units in December, 2016 against 13.76 lakh units during the same time last year resulting in a contraction of 21.6%.
YTD sales stood at 149.88 lakh units against 141.47 lakh resulting resulting in a growth of 5.9%.
Bajaj Auto Sales At 33 Month Low, Rural Demand Down 20% In Addition To Fall In Collections
Hero MotoCorp Bitten By Demonetization Bug, Sales Down 34% or 1.7 Lakh Units
Honda, the world's largest motorcycle manufacturer dominating the Southeast Asian countries made an investment in ride-hailing app Grab – a ridesharing rival to Uber. Grab includes motorbike taxis on demand, alongside private cars and licensed taxis in six countries in Southeast Asia (Go-Jek, a startup recently valued at $1.3 billion, leads the growing motorbike taxi market in Indonesia). Grab has over 24 million app downloads and a pool of more than 500,000 drivers.
Honda will use Grab customer data to develop new products and services and also put information technology to work to ease urban gridlock.
This comes at a time when new business models like the sharing economy threatens Honda's dominance in the motorcycle market. Also as sales of motorcycle takes a beating due to tough competition in India, Honda is left with a big question on how will be it able to shape its future growth.
To take on its partner from the past Hero MotoCorp, Honda launched its second motorcycle with BS-IV compliant engine – the Honda CB Unicorn 160.
Honda is confident of moving its entire product range to BS-IV compliant much before the deadline of April 1, 2016.
Honda also introduced the Honda Navi In Adventure And Chrome Edition. A crossover between a motorcycle and scooter, Navi stands out against anything available in the two segment market currently. Read here on how India and Thailand powers Honda's revenue.
Honda reported sales of 231,654 units in December, 2016 against 306,779 units during the same time last year resulting in a contraction of 24.5%.
MSI senior vice-president for sales and marketing YS Guleria said:
Selling close to 5 million units, we grew 50% faster than industry in 2016. While the industry sales fell to an 80-month low in December on demonetisation blues, we grew on a year-to year basis 11%or 0.48 million additional units in the year to 4,988,512 units up from 4,508,222 units in 2015, while the industry sales (domestic and exports) grew around 7%.
With demonetisation impact ebbing and upcoming festive season, coupled with expectations of a good Budget, we are cautiously optimistic about the March quarter and hopes to close 2016-17 with a double-digit growth. Despite continuing pressure of noteban, we grew over 50 percent faster than industry in 2016 with volume growing by 0.48 million units to 4,988,512 units. The fall, due to the noteban, would have been higher had it not been for a 66% jump in exports to 26,602 units in the month.
Royal Enfield Reports Overall Sales Growth At 42%, Exports Double To 1082 Units
Yet to disclose sales for December.
Suzuki launched the Hayabusa in the latest colours for 2017. The bike would be imported as CKD (Completely Knocked Down) Kits and assembled at the SMIPL factory in Gurgaon.
Suzuki has partnered with PayTM and HDFC Bank to tide the demonetization drive. Customers would receive Rs. 3,000/-cashback from PayTM on delivery while they would have to pay Rs. 20,000/- during the booking. While HDFC Bank will provide 100% finance for any Government of HDFC employee.
Mr. Satoshi Uchida, Managing Director, SMIPL, said:
We aim to limit the impact of demonetisation by providing our customers a convenient purchase alternative that is not dependent on availability of cash in hand. We will continue to stand with our patrons and ensure a delightful customer experience every time they visit our outlets.
For those customers who have income proof, but do not have an account with HDFC, there is a limited period Low Down Payment Scheme with 90% finance for any Suzuki two wheeler product.
TVS Motors Sales Fall By 17,120 Units, Both Domestic & Export Sales Take A Beating
Yamaha reported sales of 49,775 units in December, 2016 against 38,833 units during the same time last year.
YTD sales stood at 6.10 lakh units against 4.75 lakh units.
Roy Kurian, Vice President, Sales & Marketing, Yamaha Motor India Sales said:
The year 2016 has been a landmark year. Yamaha as a company managed to perform strongly in all aspects and maintained a steady sales growth across the year. This year Yamaha crossed one lakh sales figure first in India consecutively in two months (Sept & Oct).
The continuous growth numbers are a sign of Yamaha's robust business plans and strategic customer engagement programs. The New Year holds tremendous opportunity for further growth as Yamaha will enhance its product portfolio with the launch of new and exciting models and will intensify its network expansion plan across the country. As a young brand fostering style and excitement along with innovation, Yamaha will continue to grow with a target of achieving 1 million sales in 2017.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,223
|
Cobalt Iron Launches Compass Enterprise Object Search
Cobalt Iron, a provider of SaaS (Software as a Service)-based enterprise data protection, has enhanced the search capabilities of its Compass enterprise SaaS backup platform with a new Enterprise Object Search (EOS) feature.
Free for all Compass users, EOS allows the entire enterprise backup landscape to be searched at the object level, including paths, directories, and filenames, to identify and locate important files more quickly.
"The ability to locate particular objects or files across an enterprise is becoming increasingly important in meeting a variety of ever-expanding regulatory, compliance, and e-discovery requirements," said Paul Linder, product development manager at Cobalt Iron. "Unfortunately, most products on the market are restricted to searches against a single backup server. By allowing users to garner insights from their entire backup data repositories in a single request, Compass EOS streamlines operations for our customers while extending our market leadership in comprehensive data governance."
To search a backup server's files, administrators have traditionally been required to attempt a restore of a backup or run a query backup command in the command line interface, which is time-consuming and resource-intensive, needing to be repeated for every backup server across the enterprise. With its ability to search across all backup servers in a single request, without having to utilize computing resources to accommodate an entire restore, Compass EOS allows users to determine if and when an object was backed up, the types of objects that exist, and where in the backup landscape they are stored.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,619
|
\section{Introduction}
Brain rhythms contribute in every aspect of brain function from sensory and cognitive processing, and memory to motor control
\cite{Buzsaki_2006}. Origin and physiological functions of brain rhythms are a topic problem in neuroscience. Brain rhythms are also related to many unusual phenomena observed in the brain.
Interactions between billions of neurons give rise to phase transitions, self-organization, and critical phenomena \cite{Chialvo_2006,Chialvo_2010}. Phase transitions were observed, for example, in human bimanual coordination \cite{Kelso_1984,Kelso_1985,Kelso_1986,Kelso_1987,Kelso_2010} and in living neural networks stimulated by electric fields \cite{Eckmann_2007}.
There are evidences that epileptic seizures, alpha and gamma oscillations, and the ultraslow oscillations of BOLD fMRI patterns emerge as a result of non-equilibrium phase transitions. Neural avalanches are one more example of critical collective phenomena observed in the brain \cite{Beggs_2003,Chialvo_2010}.
Various resonance phenomena were also observed in the brain. Experimental investigations of CA1 neuronal networks from mammalian brain demonstrated that stochastic resonance can enhance effects of intrinsic 4-10 Hz hippocampal theta and 40 Hz gamma oscillations \citep{Gluckman_1996}. Recently, using a functional imaging technique, Sasaki \emph{et al.} \cite{Sasaki_2006} revealed that the majority of rat CA1 neurons act collectively like a band-pass filter. Damped oscillations and the Berger effect are also related to brain rhythms. The Berger effect manifests itself in activation of alpha waves on the electroencephalogram when the eyes are closed and diminution of alpha waves when they are opened \cite{Hari_1997}.
In the present paper, we study collective dynamics of neural networks composed by excitatory and inhibitory neurons in the presence of noise.
Based on exact analytical calculations and numerical simulations, we show that spontaneous emergence of network oscillations occurs as a dynamical (non-equilibrium) phase transition
at a critical level of noise.
The transition manifests itself in slowing down of the relaxation of a perturbed neural activity to a steady state, a strong enhancement of stochastic fluctuations of activities of neural populations and an increase of
the linear response function to afferent periodic stimuli at the frequency of neural oscillations. We show that near to the critical boundary, neural networks act as damped harmonic oscillators or band-pass filters that pass frequencies within a certain range and attenuate frequencies outside that range.
\section{Cortical circuit model}
We use a cortical circuit model \cite{Goltsev_2010} composed of $N_e$ pyramidal cells (excitatory neurons) and $N_i$ interneurons (inhibitory neurons) that form a sparsely connected network. The probability that there is a synaptic connection between two neurons is $c/N$ where $N=N_e+N_i$ is the total number of neurons and $c$ is the mean degree. This network has the structure of a directed
classical random graph (or Erd\H{o}s-R\'{e}nyi graph) with the Poisson degree distribution $P_{n}(c)=c^{n}e^{-c}/n!$ where $n$ is the number of presynaptic neurons.
Neurons receive sporadic inputs from a remote part of the cortex and synaptic noise.
Neurons fire with
a constant firing frequency $\nu$ that is the same for both excitatory and inhibitory neurons.
The total input $V_m$ to a neuron with index $m$, $m=1,2,\dots, N$,
is the sum of random spikes from
noise, excitatory and inhibitory neurons,
\begin{equation}
V_m(t)=\sum_{n=1}^{N} k_n(t)a_{nm}J_{nm}+\xi(t),
\label{input}
\end{equation}
where $k_n(t)$ is the number of spikes that arrive from presynaptic neuron $n$ during the time interval $[t-\tau,t]$, $\tau$ is the integration time. Below we will consider the case $\tau \nu \leq 1$ when the number of spikes $k_n(t)$ is 1 or 0. If we assume that the emissions times of spikes of different neurons are uncorrelated, then the parameter $\tau \nu$ has a meaning of the probability that a postsynaptic neuron receives a spike from an active presynaptic neuron during time $\tau$.
Furthermore, $a_{nm}$ is the
adjacency matrix, i.e., $a_{nm}=1$ if there is a direct edge from neuron
$n$ to neuron $m$, otherwise $a_{nm}=0$. $J_{nm}$ is the efficacy
of the synapse connecting neuron $n$ with neuron $m$. $J_{nm}$ is positive if presynaptic neuron $n$ is excitatory and it is negative if the neuron is inhibitory. $\xi(t)$ is the number of random spikes from noise that neuron $m$
receives during the time interval $[t-\tau,t]$. We use the Gaussian distribution for $\xi(t)$,
\begin{equation}
G(\xi)=A\exp \Big[-\frac{(\xi-\langle n\rangle)^2}{2\sigma^2}\Big],
\label{noise}
\end{equation}
where A is the normalization constant, $\sigma^2$ is the variance, $\langle n\rangle$ is the mean number of random spikes
determined by the mean rate $\omega_{rs}$, $\langle n\rangle = \omega_{rs}\tau$. Note that noise in our model is actually shot noise. According to Schottky's theorem, the intensity of this noise is proportional to $\langle n\rangle$.
We consider stochastic neurons. Their
response on input is a stochastic process that occurs with a certain rate.
Two rules determine dynamics of stochastic neurons \citep{Goltsev_2010}:
\begin{enumerate}
\item If the total input $V_m(t)$
at an inactive excitatory or inhibitory neuron $m$ at time $t$ is at
least a certain threshold $\Omega$ (i.e., $V_m(t)\geq \Omega$), then this
neuron is activated at a rate $\mu_{e}$ or $\mu_{i}$, respectively, and fires with a cyclic frequency $\nu$.
\item Active excitatory (inhibitory) neuron $m$ is inactivated at a
rate $\mu_{e}$ ($\mu_{i}$) if $V_m(t)< \Omega$.
\end{enumerate}
We assume that $1/\mu_{e}$ and $1/\mu_{i}$ are of the order of the first spike
latencies of excitatory and inhibitory neurons, respectively. We introduce the
ratio
\begin{equation}
\alpha \equiv \mu_{i}/\mu_{e}
\label{alpfa}
\end{equation}
that plays an important role in our model, as it will be shown below.
The advantage of this model with stochastic neurons is that it can be solved analytically.
In numerical simulations, we studied
sparsely connected networks of size $N=10^3-10^5$ and applied the following algorithm. We divided time $t$ into intervals of width $\Delta t=\tau$. At each time step, for each neuron, we calculated the input Eq.~(\ref{input}), taking into account that each active presynaptic neuron contributes with a spike with probability $\tau \nu$. The number of random spikes from noise in this input is generated according to the Gaussian distribution, Eq.~(\ref{noise}). Then, with the probability $\tau \mu_a$, $a=e,i$, we updated the states of all neurons using the stochastic rules formulated above.
We used the following parameters:
the fraction of excitatory neurons is $g_e =N_e/N=75 \%$, the fraction of inhibitory neurons is $g_i=N_i/N=25 \%$,
the mean number of connections $c=1000$ (750 excitatory and 250 inhibitory connections), the threshold $\Omega=30$, and the variance of noise $\sigma^2 =10$. Following \cite{Amit_1997}, we chose $J_{ie}=J_{ii}\equiv J_i$, $J_{ee}=J_{ei}\equiv J_e$, and $J_{i}=-3J_{e}$.
These parameters agree with anatomical estimates for cortex. In cortex, the fraction $g_i$ of inhibitory neurons is between $0.15$ and $0.3$, the mean number of synaptic connections $c$
is about $7000$. The threshold $\Omega$ is between $15$ and $30$ in
neural networks {\it in vivo} \citep{Eckmann_2007} and about $30-400$ in the brain. The level of noise $\langle n\rangle$ was varied in the interval $0-150$ spikes per integration time $\tau$. We also assumed that, for simplicity, $\tau \nu =1$ and $\tau \mu_e =0.1$.
Dynamical behavior of the model
is described by the fractions $\rho_e(t)$ and $\rho_i(t)$ of active
excitatory and inhibitory neurons, respectively, at time $t$. We will call them
`activities' of the neural populations.
Using the rules of the stochastic dynamics formulated above and assuming that activities are changed slightly during the integration time $\tau$, in the infinite size limit $N\rightarrow\infty$, we find a rate equation \citep{Goltsev_2010},
\begin{equation}
\frac{d\rho_a (t)}{\mu_a dt}=f_{a}(t)(1{-}\rho_a(t))-\rho_{a}(t)+\Psi_a(\rho_{e}(t),\rho_{i}(t)).
\label{eq:10}
\end{equation}
for $a=e,i$. The function $\Psi_a(\rho_e,\rho_i)$ is the probability
that at time $t$ the input to a randomly chosen excitatory or inhibitory neuron
is at least the threshold $\Omega$. For the model under consideration $\Psi_i(\rho_e,\rho_i)=\Psi_e(\rho_e,\rho_i)\equiv
\Psi(\rho_e,\rho_i)$, where
\begin{equation}
\Psi(\rho_e,\rho_i)=\sum_{k=0}^{\infty}\sum_{l= 0}^{\infty}\sum_{\xi = -\infty}^{\infty}\Theta(J_{e}k {+}J_{i}l{+}\xi{-}\Omega) G(\xi) P_{k}(g_{e}\rho_{e} \widetilde{c})P_{l}(g_{i}\rho_{i} \widetilde{c}).
\label{eq:14}
\end{equation}
Here $\Theta(x)$ is the Heaviside step function, the parameter $\widetilde{c}$ is defined as
$\widetilde{c} \equiv c\nu\tau$, and
$P_k(g_e\rho_e \widetilde{c})$ and $P_l(g_i\rho_i \widetilde{c})$ are the probabilities that
a randomly chosen neuron receives $k$ spikes from active presynaptic excitatory and $l$ spikes from inhibitory neurons, respectively, during the time window $\tau$ at given activities $\rho_e$ and $\rho_i$. The functions $f_{e}(t)$ and $f_{i}(t)$ represent a rate of spontaneous activation of excitatory and inhibitory neurons, respectively, by stimulus, for example, an electric field.
The rate equation (\ref{eq:10}) is similar to the Wilson-Cowan equations \citep{Wilson_1972,Wilson_1973}, see also \citep{Goltsev_2010}. Equation (\ref{eq:10}) is asymptotically exact in the limit $N\rightarrow\infty$.
\begin{figure}
\includegraphics[height=.3\textheight]{fig_Goltsev_Granada_2012}
\caption{Schematic phase diagram of the cortical model and critical and resonance phenomena near the critical boundary of the non-equilibrium phase transition to sustained network oscillations. }
\label{fig-overview}
\end{figure}
Steady states of the neural populations can be found from Eq.~(\ref{eq:10}), supposing $d\rho_{a}/dt = 0$ in the limit $t\rightarrow \infty$.
If $\rho_e(t)$ and $\rho_i(t)$ at time $t$ are close to steady state activities $\rho_e(\infty)$ and $\rho_i(\infty)$, then Eq.~(\ref{eq:10}) enables us to describe relaxation of $\rho_a(t)$ to the steady state. We introduce
\begin{equation}
\delta\rho_a(t)\equiv \rho_a(t)-\rho_a(\infty)
= Re (A_a e^{-\gamma t})
\label{relaxation}
\end{equation}
where $A_a$ is a complex amplitude.
Using the standard perturbation theory, we solve Eq.~(\ref{eq:10}) in the first order in $\delta\rho_a(t)$. We find
\begin{equation}
\gamma_{\pm}=\frac{1}{2}(B_1+B_2)\pm \frac{1}{2}\Bigl[(B_1-B_2)^2+4\alpha D_{ei}D_{ie}\Bigr]^{1/2},
\label{eq-gamma}
\end{equation}
where we introduced parameters $B_1=1-D_{ee}$, $B_2=\alpha(1-D_{ii})$,
$D_{ab}=d\Psi_a(\rho_e,\rho_i)/d\rho_b$ for $a,b=e,i$.
respectively.
Derivatives $D_{ab}$ are
determined by the activities $\rho_e(\infty)$ and $\rho_i(\infty)$ from the non-linear equation Eq.~(\ref{eq:10}) when $d\rho_{a}/dt = 0$ \citep{Goltsev_2010}.
The real and imaginary parts of the complex rate $\gamma$ ($\gamma_r\equiv Re(\gamma_{-})$ and $\gamma_i\equiv Im(\gamma_{-})$) determine the relaxation rate and the angular frequency of damped oscillations, respectively. Notice that the period of the oscillations equals $2\pi/ \gamma_i$.
Analyzing behavior of $\gamma_r$ and
$\gamma_i$ in dependence on $\alpha$ and $\langle n\rangle$,
we obtain the phase diagram in Fig.~\ref{fig-overview}.
One can see that there are three regions. There is a region I (small noise level and/or large $\alpha$) where the relaxation of the neural activity to a steady state is exponential ($\gamma_r >0$ and $\gamma_i=0$). In region II,
the neural activity relaxes in a form of damped
oscillations ($\gamma_r >0$ and $\gamma_i \neq 0$).
In region III, network oscillations are sustained.
A similar phase diagram was found in \citep{Goltsev_2010} for a simpler model.
If $\alpha$
is above a critical value $\alpha_t$, that corresponds to the $\alpha$-coordinate of the top point of the region III in Fig.~\ref{fig-overview}, then
with increasing the noise level $\langle n\rangle$, the activities $\rho_e$ and $\rho_i$ in the steady state undergo a first-order phase transition
at a critical noise level $n_c$.
A similar discontinuous transition was observed in living neural networks {\it in vitro} when living neural networks were stimulated by an electric field \cite{Eckmann_2007}. Neuronal avalanches are precursors of this phase transition. Activation (or inactivation) of one neuron can trigger avalanche process of activation (or inactivation) of a cluster of neurons.
In cortex, neuronal avalanches have been observed experimentally \cite{Beggs_2003}, see the review \cite{Chialvo_2010}.
If the parameter $\alpha <\alpha_t$,
sustained networks oscillations appear in a certain `optimal' range of the noise level $\langle n\rangle$ between two critical points.
Weak noise can not stimulate network oscillations. Too strong noise over-activates neural networks and only damped oscillations can occur.
The critical boundary of region with the sustained oscillations
is determined by the condition that the relaxation rate $\gamma_r$ is zero,
\begin{equation}
\gamma_r =Re(\gamma_{-})=0,
\label{gamma-r}
\end{equation}
where the complex frequency $\gamma_{-}$ is given by Eq.~(\ref{eq-gamma}).
For the parameters given above and $\tau=10$ ms,
frequencies of the oscillations lie in the range of brain waves (1-- 100 Hz).
\section{Linear response function and band-pass filter behavior}
\label{response function}
Now we study critical phenomena that precede the non-equilibrium phase transition from asynchronious dynamics to sustained oscillations. For this purpose we calculate the linear response of the neural network to a time-dependent stimulus $f_{e}(t)$ and $f_{i}(t)$ in Eq.~(\ref{eq:10}) for region I and II. Here we are not studying a response in region III that needs a special consideration.
A response of the neural population $a=e,i$ to a weak stimulus $f_{a}(t)$
is determined by the linear response function $\chi_{ab}(t-t')$,
\begin{equation}
\Delta\rho_a(t){\equiv} \rho_a(t) {-} \rho_a(\infty){=}\!\!\sum_{b=e,i} \!\!\int_{-\infty}^{t}\chi_{ab}(t{-}t') f_b(t')dt'.
\label{t-response}
\end{equation}
Solving Eq.~(\ref{eq:10})
in the linear-response regime, we find that in the regions I and II the neural network behaves as a damped oscillator driven by a force $F_e (t)$,
\begin{equation}
\frac{d^2 \Delta\rho_e(t)}{dt^2}+2\zeta \omega_0 \frac{d \Delta\rho_e(t)}{dt}+\omega_{0}^2 \Delta\rho_e(t)=F_{e}(t),
\label{d-oscillations}
\end{equation}
(see, for example, in \citep{Kubo_book}). Here we introduced the damping ratio $\zeta=\gamma_r /\omega_0$ and a frequency $\omega_0 =(\gamma_{r}^2 +\gamma_{i}^2)^{1/2}$.
In region I, the network is critically damped because $\zeta =1$ and it is underdamped in region II, where $\zeta < 1$.
In the case $f_{e}(t)\neq 0$ and $f_{i}(t)=0$, the force $F_{e}(t)$ equals $F_{e}(t)=(1-\rho_{e}(\infty))(B_2 f_{e}(t)+ d f_{e}(t)/dt)$. The parameter $B_2$ was defined above.
Solving Eq.~(\ref{d-oscillations}) leads to a response function,
\begin{equation}
\chi_{ee}(t-t')=X_e e^{-\gamma_r (t-t')}\sin\Big[\gamma_{i} (t-t')+\Phi_e \Big].
\label{eq:y}
\end{equation}
where $X_e{=}(1{-}\rho_e(\infty))[1{+}(B_2 {-} \gamma_r)^{2}/\gamma_{i}^{2}]^{1/2}$
and $\Phi_e {=} \tan^{-1}[\gamma_i/(B_2 {-} \gamma_r)]$ (one finds a similar result for $X_i$ and $\Phi_i$ of inhibitory neurons). If $\gamma_r > 0$, then Eq.~(\ref{eq:y}) shows loss of memory in the neural network with increasing time interval $t{-}t'$. If $\gamma_r$ tends to zero, the memory becomes long-range.
The Fourier transform $\widetilde{\chi}_{ee}(\omega)$ of the linear response function is
\begin{equation}
\widetilde{\chi}_{ee}(\omega)=\frac{(1-\rho_e)(i\omega+B_2)}{\omega_{0}^2-\omega^2+2i\zeta \omega_0 \omega}.
\label{Fourier}
\end{equation}
Equation~(\ref{Fourier}) shows that at $\zeta < 1$,
the neural network acts as a band-pass filter.
The spectral intensity as a function of $\omega$ has a maximum at a resonance frequency $\omega_r \approx \omega_{0}\sqrt{1-2\zeta^2}$ at $\zeta < 1/\sqrt{2}$.
The maximum value $\|\widetilde{\chi}_{ee}(\omega_r)\|^2$
depends on the noise level $\langle n \rangle$.
When approaching the critical point, $\gamma_r \rightarrow 0$,
the value $\|\widetilde{\chi}_{ee}(\omega_r)\|^2$
diverges as $\widetilde{\chi}_{ee}(\omega_r) \propto 1/\gamma_{r}^2 \rightarrow\infty$, while the angular frequency of damped oscillations $\gamma_i$ tends to the frequency of stable network oscillations. This behavior signals that, in this regime, in the presence of noise, a neuronal network can amplify periodic signals. This amplification may be a mechanism of stochastic resonance observed in brain \cite{Moss_2004}.
The band pass filter behavior described by Eq.~(\ref{Fourier}) seems to be supported by
measurements of response of rat CA1 neurons to afferent stimulation \emph{in vitro} \citep{Sasaki_2006}. These measurements
revealed that the majority of rat CA1 neurons act collectively like a band-pass filter and fire synchronously in response to a limited range of presynaptic firing rates ($20-–40$ Hz) that are in the range of gamma oscillations in the rat hippocampus \citep{Csicsvari_2003}.
One can also note that, a long time ago,
a number of characteristics of a band-pass filter behavior and a resonance response on sin wave trains already have been observed in EEG recordings of alpha activity \citep{Tweel_1964}.
Based on Eq.~(\ref{Fourier}), we suggest that band-pass filter behavior observed in \citet{Sasaki_2006} and \citet{Tweel_1964} is a manifestation of the critical phenomena near to the transition to neural network oscillations.
\section{Stochastic fluctuations of neuronal activity}
\label{fluctuations}
EEG measurements demonstrate that brain activity always contains a stochastic component. In this section we will show that stochastic fluctuations are enhanced when a neural network is close to the critical point of the non-equilibrium phase transition.
For characterizing stochastic fluctuations, we introduce the autocorrelation function
\begin{equation}
C_{ab}(t)=
\frac{1}{T}\int_{0}^{T}\delta\rho_a(t_{1})\delta\rho_b(t_{1}+t) dt_1,
\label{correl-1}
\end{equation}
where $\delta \rho_a(t)=\rho_a(t)-\overline{\rho}_a$ describes fluctuations of activity $\rho_a(t)$ of population $a$, $a=e,i$, around the mean value $\overline{\rho}_a$ (see, for example, Ref.~\citep{Gardiner_2002}).
$C_{ab}(t)$ is a measure of correlations between values of $\delta \rho_a (t_1)$ and $\delta \rho_b(t_1 +t)$ at two different instants separated by a lag $t$ and averaged over an arbitrary large time window $T$.
The Wiener-Khintchine theorem states that the power density spectrum of the fluctuations is the Fourier transform of the autocorrelation function.
For calculating the autocorrelation function, one uses the standard method \citep{Gardiner_2002,Thomas_1982}. In the deterministic equation (\ref{eq:10}), we assume that $f_a(t)$ is a stochastic force that satisfies conditions $\langle f_a(t) \rangle=0$ and $\langle f_a(t) f_b(t')\rangle=f_{0}^2 \delta(t-t')\delta_{a,b}$.
If fluctuations are small, the autocorrelation function may be found in the linear response theory \citep{Gardiner_2002,Thomas_1982}. Assuming, for simplicity, $f_i(t)=0$, we obtain Eq.~(\ref{t-response}) that leads to
\begin{equation}
C_{ee}(t)=2\pi f_{0}^2\int_{-\infty}^{\infty}e^{i \omega t}\|\widetilde{\chi}_{ee}(\omega)\|^2 d\omega,
\label{correl-2}
\end{equation}
where the linear response function $\widetilde{\chi}_{ee}(\omega)$ is given by Eq.~(\ref{Fourier}).
In the region of damped oscillations, the autocorrelation function $C_{ee}(t)$ has a form
\begin{equation}
C_{ee}(t)=A_e e^{-\gamma_r |t|}\cos\Big(\gamma_{i} |t|+\Phi_e\Big).
\label{correl-4}
\end{equation}
The parameter $A_e$ and the phase $\Psi_e$ behave as $A_e \propto 1 /\gamma_r$ and $\Phi_e \propto \gamma_r/\gamma_i$ at small $\gamma_r$. For inhibitory neurons we obtain a similar behavior. Thus, stochastic fluctuations
of activities of excitatory and inhibitory neural populations
are enhanced when approaching the critical point $\gamma_r=0$, Eq.~(\ref{gamma-r}), of the emergence of network oscillations (see Fig.~\ref{fig-overview}). However, the linear-response approximation is not valid when fluctuations become sufficiently large. This occurs near to the non-equilibrium phase transition and non-perturbative methods are required for calculating $C_{ab}(t)$.
\section{Conclusion}
\label{conclusion}
In the present paper, using a cortical model with stochastic neurons, we have showed that, in neuronal networks, spontaneous appearance of sustained network oscillations occurs as a non-equilibrium phase transition. The critical point is determined by the level of noise, structure of the neural network, the balance between excitatory and inhibitory neurons, and other parameters. We have found critical and resonance phenomena that precede the transition. The important property of this transition is that, at the critical point, the relaxation time of the neuronal activity to a steady state becomes infinite in the infinite size limit.
An increase of the response of neural networks to periodic afferent stimulations and a strong enhancement of stochastic fluctuations of activities of neural populations are also the critical phenomena that precede the transition.
Note, that these phenomena
are general properties of second-order phase transitions observed in physical, chemical and biological systems (see, for example, \citet{Stanley_book,Haken_1983,Kelso_2010}). These critical phenomena have been observed near the non-equilibrium phase transition in human hand movements \cite{Kelso_1984,Kelso_1985,Kelso_1986,Kelso_1987,Kelso_2010}. The noise-induced nonequilibrium phase transition found in \cite{Toral1994} is one more example of a phase transition with similar critical phenomena.
Furthermore, we have demonstrated that near to the critical point, neuronal networks behave as damped harmonic oscillators or band-pass filters in agreement
with band-pass filter behavior observed \emph{in vitro} in networks
of CA1 neurons in mammalian brain \citep{Sasaki_2006}. We suggest that band-pass filter behavior is a manifestation of critical phenomena near to the transition to network oscillations.
We have also demonstrated that, in the cortical model, stochastic neural activity generated by a stochastic force is similar to spontaneous alpha activity
observed in EEG recordings of both a normal man
and human epileptic seizures of petit mal activity \citep{Babloyantz_1986}.
\begin{theacknowledgments}
This work was partially supported by the following projects PTDC:
SAU-NEU/ 103904/2008, FIS/108476/2008, MAT/114515/2009, and PEst-C/CTM/LA0025/2011.
K.~E.~Lee was supported by FCT under grant SFRH/BPD/ 71883/2010,
M.~A.~L. was supported by FCT under Grant No. SFRH/BD/ 68743/2010.
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,398
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<?php
namespace frontend\models;
use Yii;
/**
* This is the model class for table "employees".
*
* @property integer $id
* @property string $family_name
* @property string $first_name
* @property string $login
* @property string $password
*/
class Employees extends \yii\db\ActiveRecord
{
/**
* @inheritdoc
*/
public static function tableName()
{
return 'employees';
}
/**
* @inheritdoc
*/
public function rules()
{
return [
[['family_name', 'first_name', 'login', 'password'], 'required'],
[['family_name', 'first_name', 'login', 'password'], 'string', 'max' => 255]
];
}
/**
* @inheritdoc
*/
public function attributeLabels()
{
return [
'id' => 'ID',
'family_name' => 'Family Name',
'first_name' => 'First Name',
'login' => 'Login',
'password' => 'Password',
];
}
public function getEmployees()
{
return $this->hasMany(Employees::className(), ['id' => 'employee_id'])
->viaTable('employees_reports', ['report_id' => '']);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,754
|
\section{Introduction}
Lattice materials are cellular structures obtained by tessellating a unit cell comprising a few beams. Such lattice materials exhibit the characteristic of pass and stop bands determining frequency intervals over which wave motion can or can not occur, respectively \cite{RSS03,KL15, LB19}. This unique directional behavior complements the stop-pass band pattern and makes the application of 2D periodic structures as directional mechanical filters \cite{RSS03}. For models on special lattices, e.g. graphene, interesting physical and spectral properties have been observed such as the presence of special conical points in the dispersion relation, where its different sheets touch to form a two-sided conical singularity \cite{SWF06, KP07, ZEKBSA17,BHJ19,BHJZ21}.
The analysis of wave motion in periodic systems such as lattice materials and vibrations in harmonic atomic lattices are traced back to early studies of string vibration and later by Brillouin \cite{B53}. Under certain simplification assumptions, modeling variety of natural and engineered tessellated lattices can generally be studied under beam theories \footnote{most inclusive classical beam models are the Euler-Bernouli and Timoshenko beam theories.}. Under Euler-Bernouli beam model, each beam is described by an energy functional which involves four degrees of freedom for every infinitesimal element along the beam: axial, lateral (2 degrees of freedom) and angular displacements. At a joint, these four functions, supported on the beams involved, must be related via matching conditions that take into account the physics of a joint, see \cite{BL04, GLL17, BE21} for more details. In the special case of the planar frames, the operator decomposes into a direct sum of two operators, one coupling out-of-plane to angular (torsional) displacements and the other coupling in-plane with axial displacements \cite{BE21}.
From more theoretical point of view, recently the analysis of Hamiltonians corresponding to these symplectic structures has gained interest by mathematicians working on differential operators on metric graphs, see e.g. \cite{KKU15, GM20, BE21} and references therein. Along this line, early studies on derivation of dispersion relation (or variety) of second order {S}chr{\"o}dinger operator defined on a periodic graph, splits Hamiltonian into two essentially unrelated parts: the analysis on a single edge, and the spectral analysis on the combinatorial graph, the former being independent of the graph structure, and the latter independent of the potential \cite{KP07}. However, contrary to {S}chr{\"o}dinger type operator on graph, vertex conditions for beam Hamiltonian encode geometry by its dependence on the angles at which the edges are met. As a result, extension of the existing theory to the latter operator on periodic lattices is not trivially accessible.
The main focus of the current work is the extension of the reported results in \cite{KP07} to the fourth order operator $\mathcal{H} = d^4/dx^4 + q(x)$ with self-adjoint vertex conditions and a real periodic symmetric potential on graphene and lattices in geometric neighborhood of it, see Figure \ref{fig:fundDomain}. This is done by considering the analysis of the operator $\mathcal{H}$ on a single edge, and then the spectral analysis of $\mathcal{H}$ on the combinatorial graph. The spectrum of the self-adjoint operator $\mathcal{H}^{\text{per}} = d^4/dx^4 + q_0(x)$ on the real line with a real periodic potential (known as Hill operator for the second-order operator) has a band-gap structure and bounded below. In contrast to the Hill operator, the edges of the spectral bands may belong to not only the periodic or anti-periodic spectra of $\mathcal{H}$ on $(0,1)$, but also the set of resonances \cite{BK10}. However the latter case may happen at most for finitely many bands \cite{BK10}. The resonances are the branch points of the Lyapunov function, which is an analytic function on a two-sheeted Riemann surface and depends on the monodromy matrix of $\mathcal{H}$. The Lyapunov function characterizes the spectrum $\sigma(\mathcal{H})$ and multiplicities of its points. We refer interested reader to Section 2.2 of this work and \cite{BK05,BK10,BK12} for detailed discussions.
Before stating the structure of this paper, we briefly summarize the main results. In Theorem \ref{detProp2}, we obtain the dispersion relation of $\mathcal{H}$ on graphene where it is shown that the absolutely continuous spectrum coincides with $\sigma(\mathcal{H})$ as a set and the singular continuous spectrum is empty. However the pure point spectrum is non-empty and coincides with the set of eigenvalues of $\mathcal{H}$ on $(0,1)$ with Dirichlet boundary conditions and zero second derivative boundary conditions on both endpoints. Theorem \ref{grapheneSpectrum} describes these spectral properties of $\mathcal{H}$ on graphene. In Theorem \ref{DiracPointsThm}, we prove a representation of the set of Dirac points (conical singularities) of the dispersion relation in terms of the two branches of the Lyapunov function. Then in Theorem \ref{ReducibleFermisurface}, we characterize reducible and irreducible Fermi surfaces. Investigation on the role of angle-dependent vertex conditions is done through perturbation analysis, where we present existence and stability of Dirac points under perturbed angles.
\begin{figure}[ht]
\centering
\includegraphics[width=0.325\textwidth]{W0}
\caption{The hexagonal lattice $G$ and a fundamental domain $W$
together with its set of vertices $V(W) = \{\mathfrak{v}_1,\mathfrak{v}_2\}$ and set of edges
$E(W) = \{e_1,e_2,e_3\}$.}
\label{fig:fundDomain}
\end{figure}
The paper is structured as follows: in Section \ref{sec:Preliminaries}, we summarize preliminary background starting with discussion on the parametrization of the beam deformation, energy functional, quadratic form and Hamiltonian on planar frames. This discussion is continued with a brief review of spectral properties of the fourth order periodic operator $\mathcal{H}^{per}$ on the real line. In Section \ref{sec:SHLH}, we give a characterization of hexagonal elastic lattice's Hamiltonian on graphene and its perturbations outside the Dirichlet spectrum. Section \ref{sec:GH} is devoted to the derivation of the dispersion relation, Dirac points, and spectral structure for the graphene lattice. Extension of the results for perturbed angles is the topic of Section \ref{sec:APH}. Section \ref{sec:Outlook} contains additional remarks and potential future extensions.
\section{Preliminaries}
\label{sec:Preliminaries}
In this section we will briefly review existing results in the literature to build necessary background for understanding the forthcoming materials. More specifically, in the first part self-adjoint beam operator $\mathcal{H}$ on graph $G$ along with corresponding vertex conditions is defined. Next, we briefly discuss spectral results for similar type of periodic operator but defined on the real line, $\mathcal{H}^{\text{per}}$, known as Hill operator in the second-order case. We summarize these result from \cite{BL04, BK05, BK10} in Theorem \ref{summaryRefResults}, which will be repeatedly referred in the forthcoming sections. Reader familiar with these materials can safely skip this preliminary discussions and start with results in Section \ref{sec:SHLH}.
\subsection{Elastic Planar Graphs}
Under Euler-Bernouli beam model, each beam is described by an energy functional which involves four degrees of freedom for every infinitesimal element along the beam: axial, lateral (2 degrees of freedom) and angular displacements. A central importance here is how to derive vertex matching conditions which are at the same time mathematically general and physically sound for application purposes. By restricting to one-degree of freedom, namely lateral displacement, vertex conditions for planar graphs have been derived by assuming that the deformed lattice will remain locally planar at vertex, i.e. existence of the tangent plane at that vertex \cite{KKU15}. It is shown that the resulting scalar-valued operator is self-adjoint. Extension of these results to generally three-dimensional graphs is developed in \cite{BE21}. This has been done by introducing the notion of rigidity at the vertex, on which matching conditions can be derived out of the first principle. Interestingly, the remaining vertex conditions which make vector-valued operator self-adjoint, have connection to the engineering world, namely satisfying equilibrium of force and moments at vertex. Further extensions of these result to semi-rigid type joint have been recently proposed in \cite{SBE21} where discontinuity of the displacement and rotation fields are admissible at a vertex. In a special case of planar frames, the operator decomposes into a direct sum of two operators, one coupling out-of-plane to angular (torsional) displacements and the other coupling in-plane with axial displacements. However achieving this level of physically sound models means that the operator is no longer scalar valued and contains at-least two degrees of freedom (for planar graphs) coupled at the joints. In this work we follow results of the scalar valued operator from \cite{KKU15} with the benefit of revealing some solid theoretical results regarding spectra of the corresponding Hamiltonian on periodic hexagonal lattice.
\subsubsection{\textbf{Energy Functional on Planar Lattice}}
\label{sec:frames_description}
A beam frame is a collection of beams connected at joints.
We describe a beam frame as a geometric graph $\Gamma = (V,E)$,
where $V$ denotes the set of vertices and $E$ the set of edges. The
vertices $\mathfrak{v} \in V$ correspond to joints and edges $e \in E$ are the
beams. Each edge $e$ is a collection of the following information:
origin and terminus vertices $\mathfrak{v}_{e}^{o},\mathfrak{v}_{e}^{t}\in V$, length
$\ell_e$ and the local basis
$\{\vec{i}_e, \vec{j}_e, \vec{k}_e\}$. For special planar graphs $\vec{k}_e = \vec k$ for all edges $e \in E$, and thereby the graph $\Gamma$ can be embedded in $\mathbb{R}^2$. Describing the
vertices $V$ as points in $\mathbb{R}^2$ also fixes the length $\ell_e$ and
the axial direction $\vec{i}_e$ (from origin to terminus). However the choice
of $\vec{j}_e$ in the plane orthogonal to $\vec{i}_e$ still needs to
be specified externally. The distinction between origin and terminus, and thus the direction of
$\vec i_e$ is unimportant in analysis but should be fixed for
consistency. It is important to use the same beam basis when writing
out joint conditions at both ends of the beam. We will use the
\textit{incidence indicator} $s_{\mathfrak{v}}^e$ which is defined to be $1$
when $\mathfrak{v}$ is the origin of $e$, $-1$ if it is the terminus of $e$ and
$0$ otherwise. In the context of the kinematic Bernoulli assumptions for beam frame without pre-stress and external force, the total strain energy of the
beam frame is expressed as
\begin{equation}
\label{eq:energyFunc}
\mathcal{U}:= \frac{1}{2} \sum_{e \in E} \int_e \big(
a_e |u''_e(x)|^2 + q(x) |u_e(x)|^2 \big) dx.
\end{equation}
Above, parameter $a_e$ is positive and fixed over edge $e$ representing bending stiffness about the local axis $\vec j_e$ and $q \in L^2(e)$ is real-valued function.
\begin{assum}
The potential term $q \in L^2(e)$ is satisfying the evenness (symmetry) property
\begin{equation}
\label{eq:evenPotential}
q(x) = q(1-x).
\end{equation}
The evenness assumption \eqref{eq:evenPotential} is made not just for mathematical convenience, this condition is required if one considers operators invariant with respect to all symmetries of the periodic lattice.
\end{assum}
\subsubsection{\textbf{Quadratic and Operator Forms}}
\label{sec:main_results}
We now give a formal mathematical description of the Euler--Bernoulli
strain energy form.
\begin{thm}\emph{\textbf{(sesqulinear form \cite{KKU15,BE21})}}
\label{primaryCond}
Energy functional~\eqref{eq:energyFunc} of the planar beam lattice with
free rigid joints is the quadratic form corresponding to the
positive closed sesquilinear form
\begin{align}
\label{varEnergyForm}
\mathcal{Q}\left[u,\widetilde u\right] :=
\sum_{e \in E} \int_e \big( a_e u_e''(x)\cc{\widetilde u_e''(x)} + q(x) u_e(x) \cc{\widetilde u_e(x)} \big)dx
\end{align}
densely defined on the Hilbert space ${L^2(\Gamma)} := \mathop{\mathsmaller\bigoplus}_{e \in E} L^2(e)$ with the domain of $\mathcal{Q}$ consisting of the vectors $\mathop{\mathsmaller\bigoplus}_{e \in E} H^2(e)$ that satisfy at every vertex $\mathfrak{v} \in V$ \textit{rigid joint}
conditions, namely for all $e \sim \mathfrak{v}$
\begin{subequations}
\begin{gather}
\label{dispRigid11}
u_1(\mathfrak{v}) = \cdots = u_{n_\mathfrak{v}}(\mathfrak{v}) \\
\label{rotationRigid11}
(\vec j_2 \cdot \vec i_e) u_1'(\mathfrak{v})
+ (\vec j_e \cdot \vec i_1) u_2'(\mathfrak{v})
+ (\vec j_1 \cdot \vec i_2) u_e'(\mathfrak{v})
= 0
\end{gather}
\end{subequations}
\end{thm}
Above, all functions are evaluated at the vertex $\mathfrak{v}$ and all derivatives are taken in direction $\vec i_e$. We remark here that condition \eqref{rotationRigid11} guarantees to preserve the local structure of a planar frame at each vertex, see e.g. \cite{BL04, KKU15}. The following theorem characterizes the Hamiltonian of the frame as a self-adjoint differential operator on the metric graph.
\begin{thm}\emph{\textbf{(operator form \cite{KKU15})}}
\label{MainTheorem}
Energy form~\eqref{varEnergyForm} on a beam frame with free rigid
joints corresponds to the self-adjoint operator
$\mathcal{H} \colon {L^2(\Gamma)} \to {L^2(\Gamma)}$ acting as
\begin{equation}
\label{eq:diffSystem}
v_e
\mapsto
a_e u_e'''' + q u_e
\end{equation}
on every edge $e\in E$ of the graph. The domain of the operator $\mathcal{H}$
consists of the functions from $\mathop{\mathsmaller\bigoplus}_{e \in E} H^4(e)$
that satisfy at each vertex $\mathfrak{v} \in V$:
(i) primary conditions
\begin{subequations}
\begin{gather}
\label{dispRigid1}
u_1(\mathfrak{v}) = \cdots = u_{n_\mathfrak{v}}(\mathfrak{v}) \\
\label{rotationRigid1}
(\vec j_2 \cdot \vec i_e) u_1'(\mathfrak{v})
+ (\vec j_e \cdot \vec i_1) u_2'(\mathfrak{v})
+ (\vec j_1 \cdot \vec i_2) u_e'(\mathfrak{v})
= 0
\end{gather}
\end{subequations}
(ii) conjugate conditions, namely for $e_\ell, e_{\ell'} \sim \mathfrak{v}$ such that $\vec i_\ell \times \vec i_{\ell'} \not = 0$
\begin{subequations}
\begin{gather}
\label{secondBcThmDisp}
\sum_{e \sim \mathfrak{v}} s_\mathfrak{v}^e a_e u_e'''(\mathfrak{v})= \vec0 \\
\label{eq:secondBcThmRota}
\sum_{e \sim \mathfrak{v}} s_\mathfrak{v}^e a_e (\vec i_\ell \cdot \vec j_e) u_e'' (\mathfrak{v}) = 0 \qquad \& \qquad \sum_{e \sim \mathfrak{v}} s_\mathfrak{v}^e a_e (\vec i_{\ell'} \cdot \vec j_e) u_e''(\mathfrak{v}) = 0
\end{gather}
\end{subequations}
\end{thm}
Thus the defined operator $\mathcal{H}$ is unbounded and self-adjoint
in the Hilbert space $L^2(\Gamma)$. Due to the condition \eqref{eq:evenPotential} on the potential, the Hamiltonian $\mathcal{H}$ is invariant with respect to all symmetries
of the hexagonal lattice $\Gamma$, in particular with respect to the $\mathbb{Z}^2$-shifts, which will play a crucial role in our considerations, see \cite{KP07} for a detailed discussion on the role of symmetry of the potential.
\subsection{Periodic fourth order Operator on the Real Line}\label{4thorderHill}
Next we will summarize existing results on the spectrum of fourth-order operator with periodic potential on the real line. There are key differences compare to the second-order (Hill's) operator which are essential for us to develop our results. The reader familiar with the aforementioned discussions can skip this subsection and directly jump to Theorem \ref{summaryRefResults}. Consider the self-adjoint operator $\mathcal{H}^{\text{per}} := d^4/dx^4 + q_0(x)$, acting on $L^2(\mathbb{R})$, where the real 1-periodic potential $q_0(x)$ belongs to the real space
\begin{equation*}
L_0^2(\mathbb{T}) := \Big\{q_0 \in L^2(\mathbb{T}) : \int_{0}^{1} q_0(x) dx = 0\Big\},
\end{equation*}
where $\mathbb{T} = \mathbb{R} / \mathbb{Z}$. Introduce the fundamental solutions $\{g_k(x)\}_{k=1}^4$ of the eigenvalue problem
\begin{equation}
\label{eq:rigrnValRealPer}
\mathcal{H}^{\text{per}} u(x) = \lambda u(x), \qquad (x,\lambda) \in \mathbb{R} \times \mathbb{C}
\end{equation}
satisfying for $j, k \in \{1,\ldots,4\}$ the conditions
\begin{equation}\label{fundsolconditions}
g_k^{(j-1)}(0) = \delta_{j k},
\end{equation}
where $\delta_{jk}$ is the Kronecker delta function and $g^{(k)} = d^kg/dx^k$. The monodromy matrix has the form $M(\lambda) := \mathcal{M}(1,\lambda)$ in which for $x \in \mathbb{R}$
\begin{equation}
\label{eq:monodMatrix}
\mathcal{M}(x,\lambda) := \{\mathcal{M}_{j,k}(x,\lambda)\}_{j,k=1}^4 = \{g_k^{(j-1)}(x)\}_{j,k=1}^4
\end{equation}
and it shifts by the period along the solutions of \eqref{eq:rigrnValRealPer}. It is well-known that the monodromy matrix $M(\lambda)$ is entire on $\lambda$ and its eigenvalue $\tau \in \mathbb{C}$, i.e. root of algebraic polynomial $D(\tau, \lambda) := \det(M(\lambda) - \tau \mathbb{I}_4)$, is called a multiplier. According to the Lyapunov Theorem if for some $\lambda \in \mathbb{C}$, $\tau(\lambda)$ is a multiplier, then $\tau^{-1}(\lambda)$ is a multiplier of same multiplicity. Moreover, each $M(\lambda), \lambda \in \mathbb{C}$ has exactly four multipliers $\tau^{\pm1}_{1}(\lambda), \tau^{\pm 1}_2(\lambda)$\cite{BK05}. If we let $D_{\pm}(\lambda) = \frac{1}{4}D(\pm1,\lambda)$, then zeros of $D_+(\lambda)$ and $D_-(\lambda)$ are the eigenvalues of the periodic and anti-periodic problem respectively for \eqref{eq:rigrnValRealPer}. Denote by $\lambda_0^+, \lambda_{2n}^{\pm}$ and $\lambda_{2n-1}^\pm$ with $n \in \mathbb{N}$, the sequence of zeros of $D_+$ and $D_-$ (counted with multiplicity) respectively such that $\lambda_0^+ \leq \lambda_2^- \leq \lambda_2^+ \leq \lambda_4^- \leq \lambda_4^+ \leq \cdots$ and $\lambda_1^- \leq \lambda_1^+ \leq \lambda_3^- \leq \lambda_3^+ \leq \lambda_5^- \leq \cdots$. It is well known that the spectrum of $\mathcal{H}^{\text{per}}$ is purely absolutely continuous and consists of non-degenerate intervals\cite{BK05,BK10}. These intervals are separated by the gaps $G_n = (E_n^-,E_n^+)$, $n \in \mathbb{N}$, with positive length. We introduce the functions
\begin{equation}
T_1(\lambda) := \frac{1}{4} \mathrm{tr}\big(M(\lambda)\big), \qquad T_2 := \frac{1}{2} \big(\mathrm{tr}\big(M^2(\lambda)\big)+1\big) - \mathrm{tr}^2\big(M(\lambda)\big).
\end{equation}
The functions $T_1(\lambda) $, $T_2(\lambda)$ are entire, real on $\mathbb{R}$ and
\begin{equation}
D(\tau,\cdot) = \big(\tau^2 - 2(T_1-T_2^{1/2})\tau+1\big)\big(\tau^2 - 2(T_1+T_2^{1/2})\tau+1\big).
\end{equation}
For the special case of the zero potential, i.e. $q_0 \equiv 0$, the corresponding functions have the form
\begin{equation}
T_1^0(\lambda) = \frac{1}{2}\big(\cosh( \lambda^{1/4}) + \cos( \lambda^{1/4})\big), \qquad
T_2^0(\lambda) = \frac{1}{4}\big(\cosh( \lambda^{1/4}) - \cos( \lambda^{1/4})\big)^2,
\end{equation}
with $\arg (\lambda^{1/4}) \in (-\frac{\pi}{4}, \frac{\pi}{4}]$.
Let $\{r_0^-, r_n^\pm\}_{n \in \mathbb{N}}$ be the sequence of zeros of $T_2(\lambda)$ in $\mathbb{C}$ (counted with multiplicity) such that $r_0^-$ is the maximal real zero, and $\cdots \leq \Re r_{n+1}^+ \leq \Re r_n^+ \leq \cdots \leq \Re r_1^+$.
If $r_n^{+} \in \mathbb{C}_{+}$, then $r_n^{-} = \overline{r_n^{+}} \in \mathbb{C}_{+}$, and if $r_n^{+} \in \mathbb{R}$, then $r_n^{-} \leq r_n^{+} \leq \Re r_m^{-}$ for $m = 1,\ldots, n$.
Under extra mild conditions, then it has been shown that $r_n^\pm = -4(n \pi)^4 + \mathcal{O}(n^2)$ as $n \rightarrow \infty$. Let $\cdots \leq r_{n_j}^- \leq r_{n_j}^+ \leq \cdots \leq r_{n_1}^- \leq r_{n_1}^+ \leq r_0^-$ be the subsequence of the real zeros of $T_2(\lambda)$, then $T_2(\lambda) < 0$ for any $\lambda \in R_j^0 := (r_{n_{j+1}}^+, r_{n_j}^-)$ for $j \in \mathbb{N}$.
\begin{defn}
Following characterizations are in order
\begin{itemize}
\item A zero of the function $T_2(\lambda)$ is called a resonance of operator $\mathcal{H}^{\text{per}}$.
\item The interval $R_j^0 \subset \mathbb{R}$ is called a resonance gap.
\end{itemize}
\end{defn}
Denote by $R^0 := \cup R_j^0$ and $\eta^0$ which joins the points $r_n^+, \overline{r_n^+}$ and does not cross $R^0$. To deal with the roots of the function $T_2(\lambda)$, the Riemann surface $\mathcal{R}$ is constructed by taking two replicas of the $\lambda$-plane cut along $R^0$ and $\cup \eta_n$ and they are called sheets $\mathcal{R}_1$ and $\mathcal{R}_2$ respectively. As a result, there exists a unique analytic continuation of the function $T_2^{1/2}(\lambda)$. Let introduce Lyapunov function by
\begin{equation}
\Delta(\xi) = T_1(\xi) + T_2^{1/2}(\xi)
\end{equation}
with $ \xi \in \mathcal{R}$. Let $\Delta(\xi) = \Delta_1(\lambda)$ on $\mathcal{R}_1$, and $\Delta(\xi) = \Delta_2(\lambda)$ on the second sheet $\mathcal{R}_2$. Then
\begin{subequations}
\label{eq:Delta12}
\begin{gather}
\Delta_1(\lambda) = T_1(\lambda) + T_2^{1/2}(\lambda)\\
\Delta_2(\lambda) = T_1(\lambda) - T_2^{1/2}(\lambda)
\end{gather}
\end{subequations}
For $q_0 \in L_0^2(\mathbb{T})$, the function $\Delta(\lambda) = T_1(\lambda) + T_2^{1/2}(\lambda)$ is analytic on the two sheeted Riemann surface $\mathcal{R}$ and the branches $\Delta_k$ of $\Delta$ have the forms
\begin{equation}
\Delta_k(\lambda) =
\frac{1}{2}\big(\tau_k(\lambda)+\tau_k^{-1}(\lambda)\big)
\end{equation}
for $ \lambda \in \mathcal{R}_k$ with $k = 1,2$. For the special case $q_0 \equiv 0$, the corresponding functions are characterized by
\begin{equation}
\Delta_1^0(\lambda) = \cosh(\lambda^{1/2}), \qquad \Delta_2^0(\lambda) = \cos(\lambda^{1/2})
\end{equation}
For the operator $\mathcal{H}^{\text{per}}$ the Lyapunov function $\Delta_1$ is increasing and $\Delta_2$ is bounded on the real line at high energy-level (large $\lambda$ values). The Lyapunov function for the operator $\mathcal{H}^{\text{per}}$ defines the band structure of the spectrum, but it is an analytic function on a 2-sheeted Riemann surface. The qualitative behavior of the Lyapunov function for identically vanishing and small potentials are shown in Figure \ref{fig:bandStructure}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.975\textwidth]{D3}
\caption{The function $\Delta$ for the zero potential and a small potential $q_0$.}
\label{fig:bandStructure}
\end{figure}
\begin{remark}
In the case of Hill's operator, the monodromy matrix has exactly 2 eigenvalues $\tau$ and $\tau^{-1}$. The Lyapunov function $\frac{1}{2}(\tau+\tau^{-1})$ is an entire function of the spectral parameter. It defines the band structure of the spectrum, see \cite{K16} for detailed discussions.
\end{remark}
\begin{thm}\emph{\textbf{(spectra of $\mathcal{H}^{\text{per}}$ \cite{BL04, BK05, BK10})}}
\label{summaryRefResults}
Let $\Delta_1(\lambda)$ and $\Delta_2(\lambda)$ as defined in \eqref{eq:Delta12}, then for eigenvalue problem \eqref{eq:rigrnValRealPer} following results hold
\begin{itemize}
\item[(i)] The spectrum $\sigma(\mathcal{H}^{\text{per}})$ of $\mathcal{H}^{\text{per}}$ is purely absolutely continuous.
\item[(ii)] $\lambda \in \sigma(\mathcal{H}^{\text{per}})$ iff $\Delta_k(\lambda) \in [-1,1]$ for some $k=1,2$. If $\lambda \in \sigma(\mathcal{H}^{\text{per}})$, then $T_2(\lambda) \geq 0$.
\item[(iii)] There exists an integer $n_0 \in \mathbb{N}_0$ such that for all $n \geq n_0$
\begin{equation}
\lambda_n^- \leq \lambda_n^+ \leq \lambda_{n+1}^- \leq \lambda_{n+1}^+ \leq \lambda_{n+2}^- \leq \lambda_{n+2}^+ \leq \cdots
\end{equation}
where the intervals $[\lambda_{n}^+,\lambda_{n+1}^-]$ are spectral bands of multiplicity 2 in $(\lambda_{n}^-,\lambda_{n+1}^-)$, and the intervals $(\lambda_{n}^-,\lambda_{n}^+)$ are gaps.
\item[(iv)] Each gap $G_n =(E_n^-,E_n^+)$ for $n \in \mathbb{N}$ is a bounded interval and $E_n^\pm$ are either periodic (anti-periodic) eigenvalues or resonance point, namely, real branch point of $\Delta_k$ for some $k = 1,2$ which is a zero of $T_2(\lambda)$.
\item[(v)] Any $\lambda \in \sigma(\mathcal{H}^{\text{per}})$ on an interval $S \subset \mathbb{R}$ has multiplicity 4 iff $-1 < \Delta_k(\lambda) < 1$ for all $k=1,2$ and $\lambda \in S$, except for finite number of points.
\item[(vi)] Any $\lambda \in \sigma(\mathcal{H}^{\text{per}})$ on an interval $S \subset \mathbb{R}$ has multiplicity 2 iff $-1 < \Delta_1(\lambda) < 1$, $\Delta_2(\lambda) \in \mathbb{R} \setminus [-1,1]$ or $-1 < \Delta_2(\lambda) < 1$, $\Delta_1(\lambda) \in \mathbb{R} \setminus [-1,1]$ for all $\lambda \in S$, except for finite number of points.
\item[(vii)] Let $\Delta_k$ be real analytic on some interval $I \subset \mathbb{R}$ and $-1<\Delta_k(\lambda) < 1$ for any $\lambda \in I$ for some $k = 1,2$. Then $\Delta_k'(\lambda) \not=0$ for $\lambda \in I$ (monotonicity).
\item[(viii)] The dispersion relation for $\mathcal{H}^{\text{per}}$ is given by
\begin{equation}
\Delta_{1,2}(\lambda) := T_1(\lambda) \pm T_2^{1/2}(\lambda) = \cos(\theta),
\end{equation}
where $\theta$ is the one-dimensional quasimomentum.
\end{itemize}
\end{thm}
\section{Spectra of Hexagonal Lattice Hamiltonian}
\label{sec:SHLH}
In this section our aim is to adapt and characterize the spectrum $\sigma(\mathcal{H})$ of operator $\mathcal{H}$ defined in Theorem \ref{MainTheorem} on (graphene like) hexagonal lattices. Due to positiveness and self-adjointness of this operator, its spectrum is real and positive. Let $\lambda \in \sigma(\mathcal{H})$ with $\lambda > 0$ be an eigenvalue of $\mathcal{H}$ with associated eigenfunction $(u_e)_{e\in E} \in \mathcal{D}(\mathcal{H})$. Note that since $a_e$ in \eqref{eq:diffSystem} is a positive constant and identical over the hexagonal lattice, we assume it is identically one. Then $u_e(x)$ satisfies on each edge $e \in E$
\begin{equation}
\mathcal{H} u_e(x) = u_e''''(x) + q(x) u_e(x) =
\lambda u_e(x)
\end{equation}
Let
$\delta_0 :=2\pi/3$ and define angles
\begin{equation}
\label{eq:anglePerturb}
\delta_c\up{\varepsilon} := \delta_0 + c \varepsilon
\end{equation}
For $c_1 \in [-1,1]$ to be an arbitrary parameter and $c_2 := -(1+c_1)$, the eigenfunction $(u_e)_{e\in E} \in \mathcal{D}(\mathcal{H})$ corresponding to the $\varepsilon$-perturbed lattice at each vertex $\mathfrak{v}$ satisfy (see Theorem \ref{MainTheorem}) primary vertex conditions
\begin{subequations}
\label{eq:PCond}
\begin{gather}
\label{eq:P1}
u_1(\mathfrak{v}) = u_2(\mathfrak{v}) = u_3(\mathfrak{v}) \\
\label{eq:P2}
\sin(\delta_{1}\up{\varepsilon}) u_1'(\mathfrak{v}) +\sin(\delta_{c_1}\up{\varepsilon}) u_2'(\mathfrak{v}) +\sin(\delta_{c_2}\up{\varepsilon}) u_3'(\mathfrak{v}) = 0.
\end{gather}
\end{subequations}
along with their conjugate ones
\begin{subequations}
\label{eq:SCond}
\begin{gather}
\label{eq:S1}
\sin^{-1}(\delta_{1}\up{\varepsilon})u_1''(\mathfrak{v}) = \sin^{-1}(\delta_{c_1}\up{\varepsilon})u_2''(\mathfrak{v}) = \sin^{-1}(\delta_{c_2}\up{\varepsilon})u_3''(\mathfrak{v}), \\
\label{eq:S2}
u_1'''(\mathfrak{v}) + u_2'''(\mathfrak{v}) + u_3'''(\mathfrak{v}) = 0.
\end{gather}
\end{subequations}
We stress out here that the result in \eqref{eq:S1} is obtained by setting $\ell, \ell' = 2,3$ in \eqref{eq:secondBcThmRota} and using the fact that $\vec i_3 \cdot \vec j_2 = - \vec i_2 \cdot \vec j_3$. Moreover for graphene and its $\varepsilon$-perturbed angles, conditions above are well-defined, we refer reader to \cite{KKU15} for discussion about special cases e.g. when $\delta_1\up{\varepsilon} = \pi$.
Density of states is determined by the dispersion relation, and thus when the latter is known, the former can be determined as well \cite{K16}. Thereby, we apply now the standard Floquet-Bloch theory with respect to the $\mathbb{Z}^2$-action that we specified before. This reduces the study of the Hamiltonian $\mathcal{H}$ to the study of the family of Bloch Hamiltonians $\mathcal{H}^{\Theta}$ acting in $L^2(W)$ for the values of the \textit{quasimomentum} $\Theta$ in the (first) Brillouin zone $[-\pi, \pi]^2$. Here the Bloch Hamiltonian $\mathcal{H}^{\Theta}$ acts the same way $\mathcal{H}$ does, but it is applied to a different space of functions. Each function $u = \{u_e\}_{e \in E}$ in the domain of $H^{\Theta}$ must belong to the Sobolev space $H^4(e)$ on each edge $e$ and satisfy the vertex conditions \eqref{eq:PCond}-\eqref{eq:SCond}, as well as the cyclic conditions (Floquet-Bloch conditions)
\begin{equation}
\label{eq:FBThm}
\begin{split}
u(x + n_1\vec{b}_1 + n_2\vec{b}_2) = e^{i\vec n\cdot\Theta}u(x) = e^{i(n_1\theta_1+n_2\theta_2)}u(x)
\end{split}
\end{equation}
for any $x$ in the fundamental domain $W$, vector $\vec n = (n_1,n_2) \in \mathbb{Z}^2$, and quasimomentum $\Theta = (\theta_1,\theta_2) \in [-\pi, \pi]^2$, see Figure \ref{fig:fundDomain}. Due to the condition \eqref{eq:FBThm}, function $u$ is uniquely determined by its restriction to the fundamental domain $W$. Then conditions \eqref{eq:PCond}-\eqref{eq:SCond} at the central vertex, i.e. $x = 0$ become
\begin{subequations}
\begin{gather}
\label{eq:cond1At0}
u_1(0) = u_2(0) = u_3(0) =:A \\
\label{eq:cond2At0}
\sin(\delta_{1}\up{\varepsilon}) u_1'(0) +\sin(\delta_{c_1}\up{\varepsilon}) u_2'(0) +\sin(\delta_{c_2}\up{\varepsilon}) u_3'(0) = 0\\
\label{eq:cond3At0}
\sin^{-1}(\delta_{1}\up{\varepsilon})u_1''(0) = \sin^{-1}(\delta_{c_1}\up{\varepsilon})u_2''(0) = \sin^{-1}(\delta_{c_2}\up{\varepsilon})u_3''(0)=: B \\
\label{eq:cond4At0}
u_1'''(0) + u_2'''(0) + u_3'''(0) = 0.
\end{gather}
\end{subequations}
Similarly at other end vertex of edge $e_1$, i.e. $x = 1$ we have
\begin{subequations}
\begin{gather}
\label{eq:cond1At1}
u_1(1) = u_2(1)e^{i \theta_1} = u_3(1)e^{i \theta_2} =:C \\
\label{eq:cond2At1}
\sin(\delta_{1}\up{\varepsilon}) u_1'(1) +\sin(\delta_{c_1}\up{\varepsilon}) u_2'(1)e^{i \theta_1} +\sin(\delta_{c_2}\up{\varepsilon}) u_3'(1)e^{i \theta_2} = 0\\
\label{eq:cond3At1}
\sin^{-1}(\delta_{1}\up{\varepsilon})u_1''(0) = \sin^{-1}(\delta_{c_1}\up{\varepsilon})u_2''(0)e^{i \theta_1} = \sin^{-1}(\delta_{c_2}\up{\varepsilon})u_3''(0) e^{i \theta_2} =: D \\
\label{eq:cond4At1}
u_1'''(1) + u_2'''(1)e^{i \theta_1} + u_3'''(1)e^{i \theta_2}= 0.
\end{gather}
\end{subequations}
By standard arguments, $\mathcal{H}^{\Theta}$ has purely discrete spectrum $\sigma(\mathcal{H}^{\Theta}) = \{\lambda_k(\Theta)\}_{k \in \mathbb{N}}$. The graph of the multiple valued function $\Theta \mapsto \{\lambda_k(\Theta)\}$ is known as the \textit{dispersion relation}, or \textit{Bloch variety} of the operator $\mathcal{H}$. It is known \cite{K16} that the range of this function is the spectrum of $\mathcal{H}$:
\begin{equation}
\sigma(\mathcal{H}) = \bigcup_{\Theta \in [-\pi,\pi]^2}\sigma(\mathcal{H}^{\Theta})
\end{equation}
Our goal now is the determination of the spectrum of $\mathcal{H}^{\Theta}$
and thus the dispersion relation of $\mathcal{H}$. In order to determine this spectrum, we have to solve the eigenvalue problems
\begin{equation}\label{2ndOrderOp}
\mathcal{H}^{\Theta}u(x) = \lambda u(x)
\end{equation}
for $\lambda \in \mathbb{R}$ and non-trivial functions $u_e(x) \in L_e^2(W)$ with the above boundary conditions. Let us denote by $\Sigma^D$ the spectrum of the operator
\begin{equation}\label{eigenvalueEquation}
\mathcal{H} u(x) = au''''(x) +q(x)u(x)
\end{equation}
on interval $(0,1)$ with boundary conditions
\begin{equation}
\label{eq:SigmaDboundary}
u(0) = 0, \quad u''(0) = 0, \quad
u(1) = 0, \quad u''(1) = 0.
\end{equation}
If $\lambda \notin \Sigma^D$, then there exist four linearly independent solutions $\phi_1, \phi_2, \phi_3$ and $ \phi_4$ (depending on $\lambda$) of \eqref{eigenvalueEquation}
on $(0,1)$ such that
\begin{equation}
\label{eq:phiIndSol}
\begin{split}
\phi_1(0) = 1, \qquad \phi_1''(0) = 0, \qquad \phi_1(1) = 0, \qquad \phi_1''(1) = 0, \\
\phi_2(0) = 0, \qquad \phi_2''(0) = 1, \qquad \phi_2(1) = 0, \qquad \phi_2''(1) = 0, \\
\phi_3(0) = 0, \qquad \phi_3''(0) = 0, \qquad \phi_3(1) = 1, \qquad \phi_3''(1) = 0, \\
\phi_4(0) = 0, \qquad \phi_4''(0) = 0, \qquad \phi_4(1) = 0, \qquad \phi_4''(1) = 1.
\end{split}
\end{equation}
For example, if $q \equiv 0$ and $\lambda > 0$, then we have $\lambda \not \in \Sigma^D$ if and only if $\lambda^{1/4} \not \in \pi \mathbb{Z}$. If $\lambda \not \in \Sigma^D$, then
\begin{equation*}
\begin{split}
\phi_1(x) = \frac{1}{2}&\big(\cos(\lambda^{1/4} x)+\cosh(\lambda^{1/4} x)+\cot(\lambda^{1/4} )\cos(\lambda^{1/4} x)-\coth(\lambda^{1/4})\cosh(\lambda^{1/4} x)\big)
\end{split}
\end{equation*}
and so on. We will assume that the functions $\phi_k$ are lifted to each of the edges in $W$,
using the identifications of these edges with the segment $[0, 1]$ described above. Abusing notations, we will use the same names $\phi_k$ for the lifted functions.
For $\lambda \not \in \Sigma^D$
one can use \eqref{eq:phiIndSol} to represent any solution $u$ of \eqref{2ndOrderOp} from the domain of $\mathcal{H}\up{\Theta}$ on each edge in $W$ as follows:
\begin{equation}
\label{eq:funcVs}
\begin{split}
&u_1(x) = A \phi_1(x) + B \sin(\delta_1\up{\varepsilon}) \phi_2(x) + C \phi_3(x)e^{-i\theta_0} +
D \sin(\delta_1\up{\varepsilon}) \phi_4(x)e^{-i\theta_0} \\
&u_2(x) = A \phi_1(x) + B \sin(\delta_{c_1}\up{\varepsilon}) \phi_2(x) + C \phi_3(x) e^{-i\theta_1} +
D \sin(\delta_{c_1}\up{\varepsilon}) \phi_4(x) e^{-i \theta_1}\\
&u_3(x) = A \phi_1(x) + B \sin(\delta_{c_2}\up{\varepsilon}) \phi_2(x) + C \phi_3(x) e^{-i\theta_2} +
D \sin(\delta_{c_2}\up{\varepsilon}) \phi_4(x) e^{-i \theta_2}
\end{split}
\end{equation}
with $\theta_0 = 0$. Next, let us introduce (Wronskian) operator $\mathcal{W} : L^2[0,1] \times L^2[0,1] \rightarrow \mathbb{C}$, defined as
\begin{equation}
\mathcal{W}_x(u_1,u_2) := u_1'''(x)u_2(x) - u_1''(x)u_2'(x) + u_1'(x)u_2''(x) - u_1(x)u_2'''(x)
\end{equation}
for $x \in [0,1]$.
Then for fourth-order Hamiltonian $\mathcal{H}$ we get
\begin{equation}
\label{eq:WronskianSym}
u_2(x) \mathcal{H} u_1(x) - u_1(x)\mathcal{H} u_2(x) = \big( \mathcal{W}_1(u_1,u_2) - \mathcal{W}_0(u_1,u_2)\big)'.
\end{equation}
If $u_1$ and $u_2$ solves $\mathcal{H} u =\lambda u$, then $\mathcal{W}_1(u_1,u_2) - \mathcal{W}_0(u_1,u_2)$ is a constant. In the next lemma we use $\mathcal{W}$ to show some identities of $\phi_k$ at the end points.
\begin{lem}
\label{symResultPhi}
Applying symmetry property of operator $\mathcal{H}$ acting on interval $(0,1)$ we get
\begin{equation*}
\begin{split}
\phi_3'(1) = -\phi_1'(0), \quad \phi_3'(0) = -\phi_1'(1), \quad \phi_3'''(1) = -\phi_1'''(0), \quad
\phi_3'''(0) = -\phi_1'''(1), \quad \phi_2'''(0) = \phi_1'(0), \\
\phi_4'(1) = -\phi_2'(0), \quad \phi_4'(0) = -\phi_2'(1), \quad \phi_4'''(1) = -\phi_2'''(0), \quad
\phi_4'''(0) = -\phi_2'''(1), \quad \phi_2'''(1) = \phi_1'(1).
\end{split}
\end{equation*}
\end{lem}
\begin{proof}[\normalfont \textbf{Proof of Lemma~\ref{symResultPhi}}]
Proof of this lemma is based on \eqref{eq:WronskianSym}. In fact, for $n,m \in \{1,2,3,4\}$ and $n \not=m$, let $\phi_n(x)$ and $ \phi_m(x)$ be two independent solutions of eigenvalue problem
\begin{equation}
\mathcal{H} u(x)= u''''(x) + q(x)u(x) = \lambda u(x)
\end{equation}
on $(0,1)$ satisfying boundary conditions \eqref{eq:phiIndSol}. Now observe that
\begin{equation}\label{Weq1}
\phi_m(x) \mathcal{H}\phi_n(x) - \phi_n(x) \mathcal{H} \phi_m(x) = \phi_m(x)\lambda \phi_n(x) - \phi_n(x) \lambda \phi_m(x) = 0.
\end{equation}
However by \eqref{eq:WronskianSym}
\begin{equation}\label{Weq2}
\phi_m(x) \mathcal{H}\phi_n(x) - \phi_n(x) \mathcal{H} \phi_m(x) = \big( \mathcal{W}_1(\phi_n,\phi_m) - \mathcal{W}_0(\phi_n,\phi_m)\big)'.
\end{equation}
For a constant $\text{c}$, \eqref{Weq1} and \eqref{Weq2} then imply that $ \mathcal{W}_1(\phi_n,\phi_m) - \mathcal{W}_0(\phi_n,\phi_m) = \text{c}$. For any choice of $n \not= m$, observe that the boundary conditions in \eqref{eq:phiIndSol} implies $c = 0$, i.e.
\begin{equation}
\mathcal{W}_1(\phi_n,\phi_m) = \mathcal{W}_0(\phi_n,\phi_m).
\end{equation}
Finally, applying properties of $\phi_n$ from \eqref{eq:phiIndSol}, one concludes the desired result. As an example setting $(n,m)=(1,3)$ and using the property that the only non-zero terms are $\phi_1(0)$ and $\phi_3(1)$, then
\begin{equation*}
\phi_3'''(0) = - \phi_1'''(0).
\end{equation*}
Similar conclusions can be made to derive the desired relations stated in the lemma.
\end{proof}
\begin{defn}
For $k \in \mathbb{N}_0 = \mathbb{N} \cup \{0\}$ and arbitrary $\varepsilon > 0$, let
\begin{equation}
\label{eq:sThetaFunc}
S_k\up{\varepsilon}(\Theta) :=\sin^{-k}(\delta_0)\big(\sin^k(\delta_{1}\up{\varepsilon}) + \sin^k(\delta_{c_1}\up{\varepsilon})e^{-i\theta_1} + \sin^k(\delta_{c_2}\up{\varepsilon})e^{-i\theta_2} \big),
\end{equation}
where $\Theta \in [-\pi,\pi]^2$, $\delta_0 = 2\pi/3$, $c_1 \in [-1,1]$ and $c_2 = -(1+c_1)$.
\end{defn}
Let us introduce scaled version of $\tilde B := \sin(\delta_0) B$ and $\tilde D := \sin(\delta_0) D$ stated in \eqref{eq:cond3At0} and \eqref{eq:cond3At1} respectively. Application of function $u_i$s defined in \eqref{eq:funcVs} in vertex conditions \eqref{eq:cond2At0}, \eqref{eq:cond4At0} and \eqref{eq:cond2At1}, \eqref{eq:cond4At1} reduces the problem to find the vector $\vec \xi := (A~\tilde B ~C ~\tilde D)^T$ satisfying
\begin{equation}
\label{eq:linearSystem1}
\mathbb{M}_\varepsilon \vec \xi =
\begin{pmatrix}
\hspace{4mm}A_0(\varepsilon) & -A_1(\varepsilon) \\
-\widetilde A_1(\varepsilon) & \hspace{2.5mm}A_0(\varepsilon)
\end{pmatrix}
\vec \xi = 0.
\end{equation}
The block components of matrix $\mathbb{M}_{\varepsilon}$ are written in terms of quasimomentum and solutions $\phi_k$s and have forms
\begin{equation*}
A_0(\varepsilon) :=
\begin{pmatrix}
S_1\up{\varepsilon}(0) \phi_1'(0) & S_2\up{\varepsilon}(0) \phi_2'(0)\\
S_0\up{\varepsilon}(0) \phi_1'''(0) & S_1\up{\varepsilon}(0) \phi_2'''(0)
\end{pmatrix}
,\quad
A_1(\varepsilon) :=
\begin{pmatrix}
S_1\up{\varepsilon}(\Theta) \phi_1'(1) & S_2\up{\varepsilon}(\Theta) \phi_2'(1)\\
S_0\up{\varepsilon}(\Theta) \phi_1'''(1) & S_1\up{\varepsilon}(\Theta) \phi_2'''(1)
\end{pmatrix}
\end{equation*}
and
\begin{equation*}
\widetilde A_1(\varepsilon) := -
\begin{pmatrix}
\overline{S_1\up{\varepsilon}(\Theta)} \phi_1'(1) & \overline{S_2\up{\varepsilon}(\Theta)} \phi_2'(1)\\
\overline{S_0\up{\varepsilon}(\Theta)} \phi_1'''(1) &\overline{S_1\up{\varepsilon}(\Theta)} \phi_2'''(1)
\end{pmatrix}.
\end{equation*}
Clearly a non-trivial solution exists if matrix $\mathbb{M}_\varepsilon(\lambda)$ is singular, stated formally as the following proposition.
\begin{prop}
\label{detProp}
If $\lambda \not \in \Sigma^D$, then $\lambda$ is in spectrum of the hexagonal elastic lattice's Hamiltonian $\mathcal{H}$ if and only if there is $\Theta \in [-\pi,\pi]^2$ such that
\begin{equation}
\label{eq:MepsCond}
\det\big(\mathbb{M}_{\varepsilon} (\lambda)\big) = 0.
\end{equation}
\end{prop}
The result in Proposition \ref{detProp} can be (numerically) investigated directly, but we will split the discussions into two parts. In the following section we will state theoretical results for the case $\varepsilon = 0$, namely graphene lattice. In Section \ref{sec:APH}, extension of these results will be presented for perturbed angles by applying tools from perturbation analysis.
\section{Graphene Hamiltonian}
\label{sec:GH}
In this section we discuss the outcome of results from the previous section for the special case of $\varepsilon=0$. In this case for any $k \in \mathbb{N}_0$ let
\begin{equation}
\label{eq:s0}
s_0(\Theta) := S_k\up{0}(\Theta) = 1 + e^{-i\theta_1} + e^{-i\theta_2}.
\end{equation}
Application of $s_0(\Theta)$ reduces the block matrix components defined in \eqref{eq:linearSystem1} into splitted forms
\begin{equation}
A_0(\lambda)= s_0(0) \Phi_0(0), \qquad A_1(\lambda)= -s_0(\Theta) \Phi_0(1), \qquad \widetilde A_1(\lambda)= -\overline{s_0(\Theta)} \Phi_0(1),
\end{equation}
in which matrices $\Phi_0(0)$ and $\Phi_0(1)$ are of the form
\begin{equation}
\label{eq:simplifiedAs}
\Phi_0(0) :=
\begin{pmatrix}
\phi_1'(0) & \phi_2'(0)\\
\phi_1'''(0) & \phi_2'''(0)
\end{pmatrix}
,\qquad
\Phi_0(1) :=
\begin{pmatrix}
\phi_1'(1) &\phi_2'(1)\\
\phi_1'''(1) & \phi_2'''(1)
\end{pmatrix}.
\end{equation}
\begin{lem}
\label{A1NonSingular}
The matrix $\Phi_0(1)$ defined in \eqref{eq:simplifiedAs} is non-singular.
\end{lem}
\begin{proof}[\normalfont \textbf{Proof of Lemma~\ref{A1NonSingular}}]
By contradiction, let's assume $\Phi_0(1)$ is singular, which by application of relations in Lemma \eqref{symResultPhi} reduces to the condition
\begin{equation}
\label{eq:detPhi}
\det(\Phi_0(1)) = \phi_1'(1)\phi_2'''(1) - \phi_2'(1)\phi_1'''(1) = \phi_3'(0)\phi_4'''(0) - \phi_4'(0)\phi_3'''(0) = 0.
\end{equation}
Using the fact that $ \phi_4'''(0) = \phi_3'(0)$, \eqref{eq:detPhi} implies (at least) one of the following conditions is true:
\begin{itemize}
\item[(i)] $\phi_3'(0) = 0 \quad \& \quad \phi_3'''(0) = 0$,
\item[(ii)] $\phi_3'(0) = 0 \quad \& \quad \phi_4'(0) = 0$,
\item[(iii)] $\phi_3'(0) \neq 0\quad \& \quad\phi_3'''(0) \neq 0\quad \& \quad\phi_4'(0) \neq 0$.
\end{itemize}
Recall fundamental solutions $g_k$ from Subsection \ref{4thorderHill}. Then using the fact that $\phi_3$ can be represented as a linear combination of them in the form
\begin{equation}
\label{eq:repG}
\phi_3(x) = b_1 g_1(x) + b_2 g_2(x) + b_3 g_3(x) + b_4 g_4(x)
\end{equation}
along with the conditions $\phi_3(0) = 0$, $\phi_3''(0) = 0$ implies that item (i) turns to $\phi_3 \equiv 0$, which is a contradiction. A similar discussion holds to show that item (ii) above results in $\phi_4 \equiv 0$. Now, considering the last case above, let us introduce
\begin{equation}
r := \frac{\phi_3'(0)}{\phi_3'''(0)} = \frac{\phi_4'(0)}{\phi_3'(0)}.
\end{equation}
Then obviously by our assumption $r \not= 0$. Utilizing representation \eqref{eq:repG} and a similar representation for $\phi_4$, one gets
\begin{align}
\phi_3(x) = \phi_3'(0)g_2(x) + \frac{\phi_3'(0)}{r}g_4(x), \qquad
\phi_4(x) = r\phi_3'(0)g_2(x) + \phi_3'(0)g_4(x).
\end{align}
Comparing these two representations implies that $\phi_4 = r\phi_3$, which is a contradiction, since by our assumption $\phi_3$ and $\phi_4$ are linearly independent solutions. This proves the desired claim of the non-singularity of matrix $\Phi_0(1)$.
\end{proof}
Applying the fact that $s_0(0) = 3$ along with Lemma \ref{A1NonSingular} reduces condition \eqref{eq:MepsCond} in Proposition \ref{detProp} to
\begin{equation}
\label{eq:linearSystem2}
\det\Big( \Lambda_0^2(\lambda) - \frac{|s_0(\Theta)|^2}{9} \mathbb{I}_2 \Big) = 0,
\end{equation}
where $ \Lambda_0(\lambda) := \Phi_0^{-1}(1)\Phi_0(0)$. Then for the graphene lattice we proved the following result.
\begin{prop}
\label{detLem}
If $\lambda \not \in \Sigma^D$, then $\lambda$ is in spectrum of the hexagonal elastic lattice's Hamiltonian $\mathcal{H}$ if and only if there is $\Theta \in [-\pi,\pi]^2$ such that
\begin{equation}
\det\Big(\Lambda_0(\lambda) -\frac{|s_0(\Theta)|}{3} \mathbb{I}_2\Big) \det\Big(\Lambda_0(\lambda) +\frac{|s_0(\Theta)|}{3} \mathbb{I}_2\Big) = 0.
\end{equation}
In other words $|s_0(\Theta)|/3$ is a root of the characteristic polynomial for $\Lambda_0(\lambda)$ or $-\Lambda_0(\lambda)$ matrices, i.e. a root of
\begin{equation}\label{polydispersion}
\mathcal{P}(z;\lambda) = \big(z^2 - \mathrm{tr}(\Lambda_0(\lambda))z + \det(\Lambda_0(\lambda)\big)\big)\big(z^2 + \mathrm{tr}(\Lambda_0(\lambda))z + \det(\Lambda_0(\lambda))\big).
\end{equation}
\end{prop}
Proposition \ref{detLem}, in particular, says that in order to find the spectrum of $\mathcal{H}$, we
need to calculate the range of $|s_0(\Theta)|$ on $[-\pi,\pi]^2$. This function is identical to the one reported for the second order {S}chr{\"o}dinger operator on graphene \cite{KP07}. In summary, $|s_0(\Theta)|$ has range $[0,3]$, its maximum is attained at $(0,0)$ and minimum at $\pm (\delta_0,-\delta_0)$. These properties are based on a simple observation that
\begin{equation}
|s_0(\Theta)|^2 = |1+e^{i \theta_1} + e^{i\theta_2}|^2
\end{equation}
with range $[0,9]$. See Figure \ref{Res0Bars1} (left) for a plot of the level curves of this function.
\subsection{Dispersion Relation via Fundamental Solutions}
Next, we interpret the functions $\phi_k$ and hence matrix $\Lambda_0$ in terms of the original potential $q_0$ on $[0,1]$. To this end, let us extend $q_0$ periodically to the real line and consider operator $\mathcal{H}^{\text{per}}$ on $\mathbb{R}$, defined in preliminary section as
\begin{equation}
\label{perH}
\mathcal{H}^{\text{per}}u(x) = u''''(x) + q_0(x) u(x)
\end{equation}
with the periodic potential extended from $q_0$. Note that with the abuse of notation we maintain the notation $q_0$ for the extended potential. Fundamental solutions $\{g_k(x)\}_{k=1}^4$ of $\mathcal{H}^{\text{per}}$ satisfy for $j, k \in \{1,\ldots,4\}$ conditions
\begin{equation}
g_k^{(j-1)}(0) = \delta_{j k}.
\end{equation}
Thereby, monodromy matrix $M(\lambda)$ defined through \eqref{eq:monodMatrix} shifts by the period along the solutions of \eqref{perH}, i.e.
\begin{equation*}
\begin{pmatrix}
u(1) \\
u'(1) \\
u''(1) \\
u'''(1)
\end{pmatrix} =
\begin{pmatrix}
g_1(1) & g_2(1) & g_3(1) & g_4(1) \\
g_1'(1) & g_2'(1) & g_3'(1) & g_4'(1) \\
g_1''(1) & g_2''(1) & g_3''(1) & g_4''(1) \\
g_1'''(1) & g_2'''(1) & g_3'''(1) & g_4'''(1)
\end{pmatrix}
\begin{pmatrix}
u(0) \\
u'(0) \\
u''(0) \\
u'''(0)
\end{pmatrix}
\end{equation*}
The $4\times4$ matrix valued function $\lambda \mapsto M(\lambda)$ is entire, see the preliminaries Section \ref{sec:Preliminaries} and references therein for more detailed discussions. Since our goal is to obtain the dispersion relation of the operator $\mathcal{H}$, next we derive relations among $g_k$ and $\phi_k$. For simplicity let us introduce the following notation:
\begin{equation}\label{detNotation}
\mathcal{D}(f,g) := f'(0)g'''(1) - g'(1)f'''(0).
\end{equation}
\begin{lem}\label{phitoFundSol}
Fundamental solutions $\{g_k(x)\}_{k=1}^4$ of $\mathcal{H}^{\text{per}}$ can be represented in terms of the functions $\phi_1$ and $\phi_2$ introduced in \eqref{eq:phiIndSol} as:
\begin{align*}
g_1(x) &= \phi_1(x) + \frac{1}{\det(\Phi_0(1))}\big(\mathcal{D}(\phi_1,\phi_2)\phi_3(x) - \mathcal{D}(\phi_1,\phi_1) \phi_4(x)\big),\\
g_3(x) &= \phi_2(x) + \frac{1}{\det(\Phi_0(1))}\big(\mathcal{D}(\phi_2,\phi_2)\phi_3(x) + \mathcal{D}(\phi_1,\phi_2) \phi_4(x)\big),
\end{align*}
and moreover
\begin{align*}
g_2(x) &= \frac{-1}{\det(\Phi_0(1))}\big(\phi_1'(1) \phi_3(x) - \phi_1'''(1) \phi_4(x)\big),\\
g_4(x) &= \frac{1}{\det(\Phi_0(1))}\big(\phi_2'(1) \phi_3(x) - \phi_1'(1) \phi_4(x)\big).
\end{align*}
\end{lem}
\begin{proof}[\normalfont \textbf{Proof of Lemma~\ref{phitoFundSol}}]
Starting with the property that $\{\phi_k\}_{k=1}^4$ and $\{g_k\}_{k=1}^4$ solve the eigenvalue problem
\begin{equation}
u''''(x) + q(x)u(x) = \lambda u(x)
\end{equation}
and the fact that these are linearly independent sets of solutions, then each $g_k$ can be represented in the form
\begin{equation}
g_k(x) = a_k\phi_1(x) + b_k\phi_2(x) + c_k\phi_3(x) + d_k\phi_4(x).
\end{equation}
Applying properties of $\phi_k$ given in \eqref{eq:phiIndSol}, we observe that coefficients corresponding to $g_1$ satisfy
\begin{align*}
g_1(0) = 0 \quad \Rightarrow \quad a_1 = 1, \hspace{20mm}
g_1''(0) = 0 \quad \Rightarrow \quad b_1 = 0.
\end{align*}
Moreover, the remaining conditions result in
\begin{equation}
g_1'(0) = 0 \quad \Rightarrow \quad g_1'(0) = \phi_1'(0) + c_1\phi_3'(0) + d_1\phi_4'(0)
= \phi_1'(0) - c_1\phi_1'(1) - d_1\phi_2'(1) = 0
\end{equation}
and
\begin{equation}
g_1'''(0) = 0 \quad \Rightarrow \quad g_1'''(0) = \phi_1'''(0) + c_1\phi_3'''(0) + d_1\phi_4'''(0)
= \phi_1'(0) - c_1\phi_1'''(1) - d_1\phi_2'''(1) = 0.
\end{equation}
Solving for $c_1$ and $d_1$, we get
\begin{equation}
c_1 = \frac{\mathcal{D}(\phi_1,\phi_2)}{\det(\Phi_0(1))}, \qquad
d_1 = - \frac{\mathcal{D}(\phi_1,\phi_1)}{\det(\Phi_0(1))}.
\end{equation}
Similar discussions can be followed to obtain the coefficients corresponding to remaining $g_k$. This finishes the proof.
\end{proof}
Symmetry of the potential $q_0$ brings additional properties on the fundamental solutions which are summarized in the following lemma.
\begin{lem}
\label{CondofFundSol}
Under symmetry property of potential $q_0$, the fundamental solutions satisfy
\begin{alignat*}{3}
g_1''(1) &= g_2'''(1), \qquad &&g_1'(1) = g_3'''(1), \qquad &&g_1(1) = g_4'''(1) \\
g_2'(1) &= g_4'''(1), &&g_2(1) = g_4''(1),&&g_3(1) = g_4'(1)
\end{alignat*}
\end{lem}
\begin{proof}[\normalfont \textbf{Proof of Lemma~\ref{CondofFundSol}}]
Establishing this result is similar to the proof of Lemma \ref{symResultPhi} along with an application of symmetry of potential.
\end{proof}
Next let us introduce matrix $\mathbb{G}_0(\lambda)$
\begin{equation}
\mathbb{G}_0(\lambda) :=
\begin{pmatrix}
g_1(1) & g_3(1) \\
g_1''(1) & g_3''(1)
\end{pmatrix}.
\end{equation}
This matrix can be interpreted as an extension of the (scalar-valued) discriminant function $D(\lambda) = g_1(1)+g_2'(1)$ for the eigenvalue problem corresponding to the second order Schr{\"o}dinger operator \cite{KP07}. Putting all the observations above together allows us to derive the dispersion relation of $\mathcal{H}$ stated formally in the following theorem.
\begin{thm}\emph{\textbf{(dispersion relation)}}
\label{detProp2}
The dispersion relation of the hexagonal elastic lattice's Hamiltonian $\mathcal{H}$ consists of the variety
\begin{equation}
\label{eq:DispRelation}
\det\Big(\mathbb{G}_0^2(\lambda) - \frac{|s_0(\Theta)|^2}{9} \mathbb{I}_2 \Big) = 0
\end{equation}
and the collection of flat branches $\lambda \in \Sigma^D$.
\end{thm}
\begin{proof}[\normalfont \textbf{Proof of Theorem~\ref{detProp2}}]
Recalling the notation (\ref{detNotation}), then applying Lemma \ref{phitoFundSol} and \eqref{eq:phiIndSol}, the following identities are in order:
\begin{align*}
g_1(1) + g_4'''(1) &= +2\frac{\mathcal{D}(\phi_1,\phi_2)}{\det(\Phi_0(1))}, \qquad
g_3(1) + g_4'(1) = +2\frac{\mathcal{D}(\phi_2,\phi_2)}{\det(\Phi_0(1))},\\
g_1''(1) + g_2'''(1) &= -2\frac{\mathcal{D}(\phi_1,\phi_1)}{\det(\Phi_0(1))}, \qquad
g_3''(1) + g_2'(1) = -2\frac{\mathcal{D}(\phi_2,\phi_1)}{\det(\Phi_0(1))}.
\end{align*}
Since $\phi_2'''(0) = \phi_1'(0)$ and $\phi_2'''(1) = \phi_1'(1)$, observe that right-hand sides of the above equations are the entries of $2\Lambda_0(\lambda)$ introduced in \eqref{eq:linearSystem2}. Therefore using Lemma \ref{CondofFundSol} one gets
\begin{equation*}
2\Lambda_0(\lambda) = \begin{pmatrix}
g_1(1)+g_4'''(1) & g_3(1)+g_4'(1)\\
g_1''(1)+g_2'''(1) & g_3''(1)+g_2'(1)
\end{pmatrix} =
\begin{pmatrix}
2g_1(1) & 2g_3(1) \\
2g_1''(1) & 2g_3''(1)
\end{pmatrix} = 2\mathbb{G}_0(\lambda),
\end{equation*}
Combining the results from Proposition \ref{detLem} and Lemma \ref{LemmasigmaD} establishes the claimed result.
\end{proof}
For specific purposes, e.g. reducibility of Fermi surface, it may be desirable to rephrase \eqref{eq:DispRelation} in terms of characteristic polynomials.
\begin{remark}
$\lambda$ is in the spectrum of the graphene Hamiltonian $\mathcal{H}$ if and only if $\lambda \in \Sigma^{D}$ or $|s_0(\Theta)|/3$ is a root of the characteristic polynomial for $\mathbb{G}_0(\lambda)$ or $-\mathbb{G}_0(\lambda)$, i.e. $\lambda \in \Sigma^{D}$ or is a root of
\begin{equation}
\mathcal{P}(z;\lambda) = \Big(z^2 - \mathrm{tr}\big(\mathbb{G}_0(\lambda)\big)z + \det\big(\mathbb{G}_0(\lambda)\big)\Big) \Big(z^2 + \mathrm{tr}\big(\mathbb{G}_0(\lambda)\big)z + \det\big(\mathbb{G}_0(\lambda)\big)\Big)
\end{equation}
by equation \eqref{polydispersion}.
\end{remark}
Noting that $\phi_2'''(1) = \phi_1'(1)$ we can also write the dispersion relation as follows: $\lambda$ is in the Floquet spectrum of $\mathcal{H}$ if and only if
\begin{equation}
\Big(\Delta_1(\lambda) \pm \frac{|s_0(\Theta)|}{3} \Big)\Big(\Delta_2(\lambda) \pm \frac{|s_0(\Theta)|}{3} \Big) = 0
\end{equation}
or $\lambda \in \Sigma^D$, where $\Delta_{1,2}(\lambda)$ were defined in \eqref{eq:Delta12} and
\begin{equation}
\label{eq:T1T2Def}
T_1 = \frac{ \mathrm{tr}(\mathbb{G}_0)}{2}, \qquad T_2 = \frac{ \mathrm{tr}^2(\mathbb{G}_0)}{4} - \det(\mathbb{G}_0).
\end{equation}
So far, we have been avoiding points of the Dirichlet spectrum $\Sigma^D$ of a single
edge. We will now deal with exactly these points. The idea is to explicitly construct corresponding eigenfunctions as discussed in \cite{KP07}.
\begin{lem}
\label{LemmasigmaD}
Each point $\lambda \in \Sigma^D$ is an eigenvalue of infinite multiplicity of the hexagonal elastic lattice's Hamiltonian $\mathcal{H}$ on graphene. The corresponding eigenspace is generated by simple
loop states, i.e. by eigenfunctions which are supported on a single hexagon and vanish at the vertices.
\end{lem}
\begin{proof}[\normalfont \textbf{Proof of Lemma~\ref{LemmasigmaD}}]
Let us first show that each $\lambda \in \Sigma^D$ is an eigenvalue. Let $u$ be an eigenfunction of the operator $d^4/dx^4 + q_0(x)$ with the eigenvalue $\lambda$ and (Dirichlet type) boundary conditions on $[0,1]$ as stated in \eqref{eq:SigmaDboundary}. Note that $u(1-x)$ is also an eigenfunction with the same eigenvalue, since $q_0(x)$ is even. If $u(x)$ is neither even nor odd, then $u(x) - u(1-x)$ is an odd eigenfunction. For an odd eigenfunction, repeating it on each of the six edges of a hexagon and letting the eigenfunction to be zero on any other hexagon, we get an eigenfunction of the operator $\mathcal{H}$. If $u$ is an even eigenfunction, then repeating it around the hexagon with an alternating sign and letting the eigenfunction to be zero on any other hexagon, we get an eigenfunction of the operator $\mathcal{H}$. Therefore $\lambda \in \sigma_{\text{pp}}(\mathcal{H})$. We get the rest of the proof by following the arguments of Lemma 3.5 in \cite{KP07}.
\end{proof}
\begin{remark}
Compared to {S}chr{\"o}dinger, the Dirichlet type boundary conditions for fourth-order operator may be a place to be cautious. Naturally, one may select vanishing boundary conditions in quadratic form as Dirichlet ones (this choice holds for second-order operator). However, here we defined $\Sigma^D$ as \eqref{eq:SigmaDboundary} to accommodate Floquet vertex conditions in \eqref{eq:cond1At0}, \eqref{eq:cond3At0} and so on. We refer interested reader to the Section \ref{sec:Outlook} for further discussion along this line.
\end{remark}
\begin{exmp}\label{freeOperator}
Let us consider the free operator, i.e. $q \equiv 0$. Setting $\mu := \sqrt[4]{\lambda}$ and using the convention
\begin{equation}
C_{\mu}^{\pm}(x) = \cosh(\mu x) \pm \cos(\mu x), \qquad S_\mu^{\pm}(x) = \sinh(\mu x) \pm \sin(\mu x),
\end{equation}
then the fundamental solutions take the forms
\begin{equation}
g_1(x) = \frac{1}{2} C_{\mu}^{+}(x), \quad g_2(x) = \frac{1}{2\mu} S_{\mu}^{+}(x), \quad
g_3(x) = \frac{1}{2\mu^2} C_{\mu}^{-}(x), \quad g_4(x) = \frac{1}{2\mu^3} S_{\mu}^{-}(x),
\end{equation}
and hence
\begin{equation*}
\mathbb{G}_0(\lambda) =
\begin{pmatrix}
g_1(1) & g_3(1)\\
g_1''(1) & g_3''(1)
\end{pmatrix} = \frac{1}{2}
\begin{pmatrix}
\hspace{4mm}C_{\mu}^{+}(1) & \mu^{-2}C_{\mu}^{-}(1) \\
\mu^2 C_{\mu}^{-}(1) & \hspace{6mm}C_{\mu}^{+}(1)
\end{pmatrix}.
\end{equation*}
This then implies
\begin{align*}
\det\Big(\mathbb{G}_0(\lambda) \pm \frac{|s_0(\Theta)|}{3} \mathbb{I}_2 \Big) =
\Big(\frac{|s_0(\Theta)|}{3}\Big)^2 \pm \mathrm{tr}(\mathbb{G}_0)\Big(\frac{|s_0(\Theta)|}{3}\Big) + \det(\mathbb{G}_0).
\end{align*}
Thereby, the dispersion relation is equivalent to
\begin{equation}
\Big(\cos(\lambda^{1/4})\pm\frac{|s_0(\Theta)|}{3}\Big)\Big(\cosh(\lambda^{1/4})\pm\frac{|s_0(\Theta)|}{3}\Big) = 0
\end{equation}
Since $\cosh(x) \geq 1$ and by taking to account $|s_0(\Theta)| \leq 3$, the only root of the second factor happens at $\Theta = (0,0)$ and $\lambda = 0$ which also solve the first phrase. Therefore the dispersion relation for $q_0 \equiv 0$ reduces to
\begin{equation}
\label{eq:dispSurfZeroPot}
\cos(\lambda^{1/4}) = \pm \frac{|s_0(\Theta)|}{3}.
\end{equation}
\end{exmp}
\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{M00}
\caption{Dispersion relation for zero potential case, see \eqref{eq:dispSurfZeroPot}.}
\label{fig:zeroPotDispersion}
\end{figure}
\begin{remark}
The dispersion relation of the second order Schr\"{o}dinger operator with the vanishing potential, i.e. $\mathcal{H}\up{s} u(x) = -u''(x) $ on graphene has the form
\begin{equation}
\cos(\lambda^{1/2}) = \pm \frac{|s_0(\Theta)|}{3},
\end{equation}
which is interestingly very similar to \eqref{eq:dispSurfZeroPot}. Therefore Example \ref{freeOperator} shows that the dispersion relation of the graphene Hamiltonian $\mathcal{H}$ coincides with the one for the second order Schr\"{o}dinger operator on graphene $\mathcal{H}\up{s}$ if the eigenvalue problems $\mathcal{H}\up{s} u = \lambda u$ and $\mathcal{H} u = \lambda^{1/2} u$ are considered. Figure \ref{fig:zeroPotDispersion} shows the plot of first two spectral sheets of the dispersion relation.
\end{remark}
\subsection{The Spectra of Graphene Hamiltonian}
This section is devoted on full description of spectra, conical singularities and Fermi surfaces corresponding to $\mathcal{H}$ defined on graphene.
\begin{lem}
\label{Dspecsubsetpapspec}
As a set, $\Sigma^D$ belongs to the union of periodic and anti-periodic spectra of $\mathcal{H}^{\text{per}}$.
\end{lem}
\begin{proof}[\normalfont \textbf{Proof of Lemma~\ref{Dspecsubsetpapspec}}]
Let $\lambda \in \Sigma^D$. Since the potential $q_0$ is even, if $u(x)$ is an eigenfunction, then $u(1-x)$ is also an eigenfunction. Therefore we can assume $u$ to be either even or odd. In case $u$ is odd, it satisfies the periodic boundary conditions, i.e.
\begin{equation}\label{perbc}
u(0)=u(1), \quad u'(0)=u'(1), \quad u''(0)=u''(1), \quad u'''(0)=u'''(1).
\end{equation}
On the other-hand for even $u$, it satisfies the anti-periodic boundary conditions
\begin{equation}\label{aperbc}
u(0)=-u(1), \quad u'(0)=-u'(1), \quad u''(0)=-u''(1), \quad u'''(0)=-u'''(1).
\end{equation}
\end{proof}
We can now completely describe the spectral structure of the graphene operator $\mathcal{H}$.
\begin{thm}
\label{grapheneSpectrum}
\emph{\textbf{(spectral description)}}
\begin{itemize}
\item[(i)] The singular continuous spectrum $\sigma_{\text{sc}}(\mathcal{H})$ is empty.
\item[(ii)] The absolutely continuous spectrum $\sigma_{\text{ac}}(\mathcal{H})$ has band-gap structure and coincides as a set with the spectrum $\sigma(\mathcal{H}^{\text{per}})$ of the 4-th order operator $\mathcal{H}^{\text{per}}$ with potential $q_0$ periodically extended from $[0,1]$. Moreover, the absolutely continuous spectrum $\sigma_{\text{ac}}(\mathcal{H})$ has the representation
\begin{equation}\label{abscontspectrum}
\sigma_{\text{ac}}(\mathcal{H}) = \big\{ \lambda \in \mathbb{R} ~|~ \Delta_k(\lambda)=[-1,1] \text{ for some } k=1,2 \big\},
\end{equation}
where $\Delta_{1,2}(\lambda) := \frac{1}{2}\big(\mathrm{tr}(\mathbb{G}_0(\lambda)) \pm \big( \mathrm{tr}^2(\mathbb{G}_0(\lambda)) - 4\det(\mathbb{G}_0(\lambda))\big)^{1/2}\big)$
\item[(iii)] The pure point spectrum $\sigma_{\text{pp}}(\mathcal{H})$ coincides with $\Sigma^D$ as a set and eventually belongs to the union of the edges of the spectral bands of $\sigma_{\text{ac}}(\mathcal{H})$.
\end{itemize}
\end{thm}
\begin{proof}[\normalfont \textbf{Proof of Theorem~\ref{grapheneSpectrum}}]
Proof of the items above is based on the developed tools in this paper along with already-established results in our references. For item (i) observe that the singular continuous spectrum is empty, since $\mathcal{H}$ is a self-adjoint elliptic operator (see e.g. Corollary 6.11 in \cite{K16}). Proof of (ii) is based on Theorem \ref{detProp2}, as we know that any $\lambda \notin \Sigma^D$ belongs to $\sigma(\mathcal{H})$ if and only if $|s_0(\Theta)|/3$ is a root of the characteristic polynomial for $D(\lambda)$ or $-D(\lambda)$, i.e. a root of
\begin{equation*}
\mathcal{P}(z;\lambda) := \big(z^2 - \mathrm{tr}(\mathbb{G}_0(\lambda))z + \det(\mathbb{G}_0(\lambda))\big) \big(z^2 + \mathrm{tr}(\mathbb{G}_0D(\lambda))z + \det(\mathbb{G}_0(\lambda))\big).
\end{equation*}
Since the range of $|s_0(\Theta)|$ is $[0,3]$, then $\mathcal{P}(|s_0(\Theta)|/3;\lambda)=0$ if and only if $\Delta_1 \in [-1,1]$ or $\Delta_2 \in [-1,1]$. This observation along with Proposition \ref{detLem}
provide the desired representation \eqref{abscontspectrum}. According to the Thomas' analytic continuation argument, eigenvalues correspond to the constant branches of the dispersion relation \cite{KP07,RS78,T73}. Since the dispersion surfaces
\begin{equation}
\big\{(\Theta,\lambda) \in \mathbb{R}^3 ~|~ \Delta_{k}(\lambda) = \pm |s_0(\Theta)|/3 \text{ for some } k=1,2 \big\}
\end{equation}
have no constant branches outside $\Sigma^D$, we get $\sigma_{\text{pp}}(\mathcal{H}) \subseteq \Sigma^D$ and hence
\begin{equation}
\sigma_{\text{ac}}(\mathcal{H}) = \{ \lambda \in \mathbb{R} ~|~ \Delta_k(\lambda) \in [-1,1] \text{ for some } k=1,2\}.
\end{equation}
Note that \eqref{abscontspectrum} also represents $\sigma(\mathcal{H}^{\text{per}}) = \sigma_{\text{ac}}(\mathcal{H}^{\text{per}})$ by item (ii) in Theorem \ref{summaryRefResults}. So, the absolutely continuous spectrum $\sigma_{\text{ac}}(\mathcal{H})$ has band-gap structure and coincides as a set with the spectrum $\sigma(\mathcal{H}^{\text{per}})$ of operator $\mathcal{H}^{\text{per}}$ with potential $q_0$ periodically extended from $[0,1]$. Finally for item (iii), we observed that $\sigma_{\text{pp}}(\mathcal{H}) \subseteq \Sigma^D$ and in Lemma \ref{LemmasigmaD} we showed that $\Sigma^D \subseteq \sigma_{\text{pp}}(\mathcal{H})$. Then Lemma \ref{Dspecsubsetpapspec} implies that $\sigma_{\text{pp}}(\mathcal{H}) \subset \Sigma^{\text{p}}\cup\Sigma^{\text{ap}}$, where $\Sigma^{\text{p}}$ and $\Sigma^{\text{ap}}$ denote the periodic and anti-periodic spectra of \eqref{eigenvalueEquation}, i.e. with the boundary conditions \eqref{perbc} and \eqref{aperbc} respectively. However, from item (iii) in Theorem \ref{summaryRefResults} there exists $n_0\in \mathbb{N}$ such that for all $n \geq n_0$ the edges of the n-th spectral band are the n-th periodic and anti-periodic eigenvalues. This concludes the proof.
\end{proof}
Next theorem proves existence of Dirac points, also called diabilical points, in the dispersion relation of $\mathcal{H}$, where its different sheets touch to form a conical singularity.
\begin{thm}
\label{DiracPointsThm}
\emph{\textbf{(Dirac points)}}
The set of Dirac points of $\mathcal{H}$ in the (first) Brillouin zone is
\begin{equation*}
\begin{split}
\big\{(\Theta,\lambda) \in \mathbb{R}^3 ~|~ \Theta = \pm(2\pi/3,- 2\pi/3) \text{, } &T_2(\lambda-\varepsilon,\lambda+\varepsilon) \subset [0,\infty) \text{ and } \\
&\Delta_k(\lambda)=0 \text{ for some } \varepsilon > 0, k\in\{1,2\}\big\}.
\end{split}
\end{equation*}
\end{thm}
\begin{proof}[\normalfont \textbf{Proof of Theorem~\ref{DiracPointsThm}}]
If $T_2(\lambda-\varepsilon,\lambda+\varepsilon) \not \subset [0,\infty)$, then $\lambda$ can not belong to the interior of a spectral band. If it is an edge of a band, it can not be a Dirac point, since it may provide a one-sided conical singularity. Observe that function $|s_0(\Theta)|$ on $[-\pi,\pi]^2$ has vanishing conical singularities at points $\pm (2\pi/3,-2\pi/3)$. From item (vii) in Theorem \ref{summaryRefResults} we know for $k=1,2$ and $\lambda$ so that $\Delta_k(\lambda) \in [-1,1]$, then $\Delta_k$ is analytic and has non-zero derivative in the neighborhood of $\lambda$ restricted to the interior of the corresponding band. Therefore $\Delta_k$ is monotonic in any spectral band around any $\lambda$ satisfying $\Delta_k(\lambda)=0$, so using the dispersion relation of $\mathcal{H}$ we get the set of Dirac points.
\end{proof}
\begin{remark}
\label{DiracPics}
One can classify the Dirac points $(\pm\Theta^{*},\lambda^*)$ with $\Theta^{*}:=(2\pi/3,- 2\pi/3)$ of the dispersion relation as follows:
\begin{itemize}[leftmargin=*]
\item If $\lambda^*$ is not a resonance point (i.e. $T_2(\lambda^*) \neq 0$) and $\Delta_k(\lambda^*) = 0$ for some $k \in \{1,2\}$, then the dispersion relation around each of the singularities $(\pm\Theta^{*},\lambda^*)$ consists of two cones located in opposite directions in $\lambda^*$-axis with the common vertex singularity $(\pm\Theta^{*},\lambda^*)$. See Figure \ref{fig:DiracTypes} (left). This is the case for large $\lambda^*$, i.e. high energy level scheme.
\item If $\Delta_1(\lambda^*) = \Delta_2(\lambda^*) = 0$ and there exists $\delta >0$ so that $|T_2(\lambda)| < 1 $ for all $\lambda \in [\lambda^*-\delta,\lambda^*+\delta]$, and $T_1(\lambda^*-\lambda) \not = T_1(\lambda^*+\lambda)$ for $\lambda \in (0,\delta)$, then dispersion relation around each of the singularities $(\pm\Theta^{*},\lambda^*)$ consists of four cones, two of them located in opposite directions than the other two on $\lambda$-axis with the common vertex singularity at $(\pm\Theta^{*},\lambda^*)$. See Figure \ref{fig:DiracTypes} (right). Note that if $T_1(\lambda^*-\lambda) = T_1(\lambda^*+\lambda)$ for $\lambda \in (0,\delta)$, then the pairs of cones which are in the same directions coincide, so we get the first item above.
\end{itemize}
\begin{figure}[ht]
\centering
\includegraphics[width=0.75\textwidth]{DD2}
\caption{Behaviour of functions $\Delta_1$ and $\Delta_2$ near Dirac point $\lambda^*$. The circular windows schematically show the dispersion relation in a neighborhood of $(\pm\Theta^{*},\lambda^*)$, see Remark \ref{DiracPics} for details.}
\label{fig:DiracTypes}
\end{figure}
\end{remark}
Next result of this section is about irreducibility of Fermi surfaces corresponding to graphene Hamiltonian $\mathcal{H}$ at high-energy levels. Depending on potential, reducibility of this surface may happen for uncountably many (low) energies. This is unlike special cases e.g. 2D and 3D discrete Laplacian plus a periodic potential, continuous Laplacian with special type of potential, and more general graph operators, where the underlying graph is planar with two vertices per period, in which irreducibility happens for all but finitely many energies \cite{LS20,FWS21}. Reducibility is required for the existence of embedded eigenvalues engendered by local defect, except for the anomalous situations when an eigenvalue has compact support \cite{KV06}. In summary, Fermi surface of a 2-periodic operator at an energy $\lambda$ is the set of wavevectors $(\theta_1,\theta_2)$ admissible by the operator at that energy. For periodic graph Hamiltonian, the dispersion function is a Laurent polynomial in the Floquet variables $(z_1,z_2) = (e^{i\theta_1}, e^{i\theta_2})$. When the dispersion function can be factored, for each fixed energy, as a product of two or more Laurant polynomials in $(\theta_1,\theta_2)$, each irreducible component contributes a sequence of special bands and gaps. We refer reader to the work \cite{FWS21} and references there for detailed discussions. By referring to Theorem \ref{detProp2}, the dispersion relation (Fermi surface) of $\mathcal{H}$ is equivalent to the fact that $|s_0(\Theta)|^2/9$ is an eigenvalue of $\mathbb{G}_0^2(\lambda)$, i.e. it is a root of polynomial
\begin{equation*}
z^2 - \mathrm{tr}(\mathbb{G}_0^2(\lambda))z + \det(\mathbb{G}_0^2(\lambda)).
\end{equation*}
The roots of this quadratic polynomial have forms
\begin{equation*}
\frac{|s_0(\Theta)|^2}{9} = \frac{\mathrm{tr}(\mathbb{G}_0^2(\lambda))}{2} \pm \frac{1}{2} \big(\mathrm{tr}^2(\mathbb{G}_0^2(\lambda)) - 4\det(\mathbb{G}_0^2(\lambda))\big)^{1/2}.
\end{equation*}
Now observe that
\begin{align*}
\frac{|s_0(\Theta)|^2}{9} &=
\frac{\mathrm{tr}^2(\mathbb{G}_0(\lambda))}{2} - \det(\mathbb{G}_0(\lambda)) \pm \frac{1}{2}\big(\mathrm{tr}(\mathbb{G}_0(\lambda))\big(\mathrm{tr}^2(\mathbb{G}_0(\lambda)) - 4\det(\mathbb{G}_0(\lambda))\big)^{1/2}\big)\\
&=- \det(\mathbb{G}_0(\lambda)) + \frac{1}{2} \mathrm{tr}(\mathbb{G}_0(\lambda))
\big(\mathrm{tr}(\mathbb{G}_0(\lambda)) \pm \big(\mathrm{tr}^2(\mathbb{G}_0(\lambda)) - 4\det(\mathbb{G}_0(\lambda))\big)^{1/2} \big).
\end{align*}
Application of $T_1(\lambda)$ and $T_2(\lambda)$ from \eqref{eq:T1T2Def} in $\Delta_k$, implies that
\begin{align*}
\frac{|s_0(\Theta)|^2}{9}
= T_1^2(\lambda) + T_2(\lambda) \pm 2T_1(\lambda)T_2^{1/2}(\lambda)
= \Delta_{1,2}^2(\lambda).
\end{align*}
So we proved the following result on reducibility of the Fermi surface of $\mathcal{H}$.
\begin{thm}\label{ReducibleFermisurface}\emph{\textbf{(Fermi surfaces)}}
The relation \eqref{eq:DispRelation} has representation
\begin{equation*}
\big(P(z_1,z_2) P(z_1^{-1},z_2^{-1}) - 9\Delta_1^2(\lambda)\big)\big(P(z_1,z_2) P(z_1^{-1},z_2^{-1}) - 9\Delta_2^2(\lambda)\big) = 0,
\end{equation*}
where $P(\omega_1,\omega_2) := 1+\omega_1+\omega_2$ and $z_1 = e^{i\theta_1}$ and $z_2 = e^{i\theta_2}$. Moreover, letting $\mathcal{S}_1 := \{ \lambda \in \mathbb{R} ~|~ \Delta_1(\lambda) \in [-1,1] \}$ and $\mathcal{S}_2 := \{ \lambda \in \mathbb{R} ~|~ \Delta_2(\lambda) \in [-1,1] \}$, then Fermi surface with the energy level $\lambda \not \in \Sigma^D$ is
\begin{itemize}
\item reducible if $\lambda \in (\mathcal{S}_1\cap \mathcal{S}_2)$,
\item irreducible if $\lambda \in (\mathcal{S}_1\setminus \mathcal{S}_2)\cup(\mathcal{S}_2\setminus \mathcal{S}_1)$,
\item absent if $\lambda \in \mathbb{R}\setminus (\mathcal{S}_1\cup \mathcal{S}_2)$.
\end{itemize}
\end{thm}
\textbf{A Remark on Choices of Brillouin Zone:}
There exists some room on the choice of fundamental domain $W$ for hexagonal lattices. For the selected one in Figure \ref{fig:fundDomain}, space and quasimomentum (conjugate) basis with respect to global coordinate system are of the form
\begin{equation}
\vec b_1 = \frac{1}{2}
\begin{pmatrix}
3\\
\sqrt{3}
\end{pmatrix},
\qquad
\vec b_2 =
\begin{pmatrix}
0\\
\sqrt{3}
\end{pmatrix},
\qquad
\vec b_1^* = \frac{2}{3}
\begin{pmatrix}
1\\
0
\end{pmatrix},
\qquad
\vec b_2^* = \frac{1}{3}
\begin{pmatrix}
-1\\
\sqrt{3}
\end{pmatrix}.
\end{equation}
The dual basis then satisfies
\begin{equation}
\label{eq:orthon}
\vec b_n^* \cdot \vec b_m = \delta_{nm}
\end{equation}
and vectors $2\pi \vec b_1^* $ and $2\pi\vec b_2^* $ span hexagonal lattice as well denoted by $\Gamma^*$. Now the orthonormality condition \eqref{eq:orthon} implies
\begin{equation}
n_1 \theta_1 + n_2 \theta_2 = \big(\theta_1 \vec b_1^* + \theta_2 \vec b_2^*\big) \cdot \big(n_1 \vec b_1 + n_2 \vec b_2\big).
\end{equation}
The two choices of Brillouin zone using coordinates $\Theta = (\theta_1, \theta_2)$ with respect to dual basis vectors $\vec b_1^*, \vec b_2^*$ are shown in Figure \ref{fig:Brillouin}. In the literature it is more common to represent these Brillouin zones in corresponding Cartesian coordinates $\vec \kappa = (k_1, k_2)^T$ given by $\vec \kappa = B^*\Theta$, where $B^*$ is the transformation matrix with columns formed by the dual basis vectors, i.e.
\begin{equation}
B^* =
\begin{pmatrix}
\vec b_1^* ~
\vec b_2^*
\end{pmatrix}
= \frac{1}{3}
\begin{pmatrix}
2 & -1\\
0 & \sqrt{3}
\end{pmatrix}.
\end{equation}
\begin{figure}[ht]
\centering
\includegraphics[width=0.75\textwidth]{T1}
\caption{Choices of Brillouin zone, contour plot of second sheet of the dispersion surface in Left: coordinates $\theta_1, \theta_2$ (drawn as if they were Cartesian) and Right: coordinates $k_1$, $k_2$ (which are Cartesian). }
\label{fig:Brillouin}
\end{figure}
As it is shown in Figure \ref{fig:Brillouin}(right), the resulting Brillouin zones will be symmetric in the new coordinates system $\vec \kappa$. One arrives at the first picture by using $\theta_1$ and $\theta_2$ as parameters for the dispersion relation ranging within $[-\pi, \pi]^2$ and then plots the result using $k_1$ and $k_2$ as Cartesian coordinates. Although these two representations are equal, for symmetry discussion it may be more preferable to work with $\vec \kappa$ coordinate system, while for our case we followed the Brillouin zone in $\Theta$ coordinates due to simpler presentation of the vertex conditions, see \eqref{eq:cond1At1}-\eqref{eq:cond4At1}. Interested reader is encouraged to look at the work \cite{BC18} for detailed discussions.
\section{Perturbed Hamiltonian}
\label{sec:APH}
In this section we will apply tools from perturbation theory to characterize dispersion relation for the case in which edges meet at generally different angels, see Figure \ref{fig:anglePrturb} for schematic fundamental domains. Restricted to fundamental domain $W$, this is equivalent to find $(\lambda, \Theta) \in \mathbb{R} \times [-\pi ,\pi]^2$ so that $\det(\mathbb{M}_\varepsilon(\lambda)) = 0$ as stated in Proposition \ref{detProp}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{W6}
\caption{Fundamental domains for angle perturbed hexagonal lattices, see \eqref{eq:anglePerturb}. The middle picture shows the graphene lattice in which edges are met with equal angles at each vertex.}
\label{fig:anglePrturb}
\end{figure}
First observe that for angle $\delta_{c}\up{\varepsilon}$, see \eqref{eq:anglePerturb}, an expansion of sine function has the form
\begin{equation}
\sin(\delta_{c}\up{\varepsilon}) = \sin(\delta_0) + \varepsilon c \cos(\delta_0) + \mathcal{O}(\varepsilon^2)
\end{equation}
as $\varepsilon$ goes to zero. A similar result holds as
\begin{equation}
\sin^2(\delta_{c}\up{\varepsilon}) = \sin^2(\delta_0) + 2\varepsilon c \cos^2(\delta_0) + \mathcal{O}(\varepsilon^2).
\end{equation}
Let us introduce
\begin{equation}
\label{eq:s1}
s_1(\Theta) := \cot(\delta_0) (1 + c_1 e^{-i \theta_1} + c_2 e^{-i \theta_2}).
\end{equation}
Then up to order $\mathcal{O}(\varepsilon^2)$ accuracy, $\mathbb{M}_\varepsilon(\lambda)$ has expansion of the form
\begin{equation*}
\mathbb{M}_{\varepsilon} := \mathbb{M}_0 + \varepsilon \mathbb{M}_1 + O(\varepsilon^2),
\end{equation*}
in which the two matrices have block structures as
\begin{equation*}
\mathbb{M}_0 := \begin{pmatrix}
s_{0}(0)\Phi_0(0) & - s_{0}(\Theta)\Phi_0(1)\\
- \overline{s_{0}(\Theta)}\Phi_0(1) & s_{0}(0)\Phi_0(0)
\end{pmatrix}, \qquad \mathbb{M}_1:= \begin{pmatrix}
s_1(0)\Phi_{1}(0) & - s_1(\Theta)\Phi_{1}(1)\\
- \overline{s_1(\Theta)}\Phi_{1}(1) & s_1(0)\Phi_{1}(0)
\end{pmatrix}
\end{equation*}
with $2 \times 2$ blocks
\begin{align*}
\Phi_0(0) &:= \begin{pmatrix}
\phi_1'(0) & \phi_2'(0)\\
\phi_1'''(0) & \phi_2'''(0)
\end{pmatrix}, \qquad \Phi_0(1) := \begin{pmatrix}
\phi_1'(1) & \phi_2'(1)\\
\phi_1'''(1) & \phi_2'''(1)
\end{pmatrix},\\
\Phi_1(0) &:= \begin{pmatrix}
\phi_1'(0) & 2\phi_2'(0)\\
0 & \phi_2'''(0)
\end{pmatrix}, \qquad \Phi_1(1) := \begin{pmatrix}
\phi_1'(1) & 2\phi_2'(1)\\
0 & \phi_2'''(1)
\end{pmatrix}.
\end{align*}
Applying the fact that $\Phi_0(1)$ is non-singular, see Lemma \ref{A1NonSingular}, we introduce
\begin{equation}
\Lambda_0(0) := \Phi_0^{-1}(1) \Phi_0(0), \qquad \Lambda_1(1) := \Phi_0^{-1}(1) \Phi_1(1).
\end{equation}
Then up to $\mathcal{O}(\varepsilon^2)$ error, the perturbed matrix $\mathbb{M}_\varepsilon$ can be explicitly written as
\begin{equation}
\label{eq:Meps}
\mathbb{M}_\varepsilon(\lambda) =
\begin{pmatrix}
3 \Lambda_0(0) & -s_0(\Theta) \\
-\overline{s_0(\Theta)}& 3 \Lambda_0(0)
\end{pmatrix}
+ \varepsilon
\begin{pmatrix}
0 & - s_1(\Theta) \Lambda_1(1) \\
- \overline{s_1(\Theta)} \Lambda_1(1) & 0
\end{pmatrix}.
\end{equation}
As stated in Theorem \ref{detProp2}, equality $\mathbb{G}_0(\lambda) = \Lambda_0(0)$ holds with components of $\mathbb{G}_0(\lambda)$ in terms of the fundamental solutions
\begin{equation}
\mathbb{G}_0(\lambda) =
\begin{pmatrix}
g_1(1) & g_3(1)\\
g_1''(1) & g_3''(1)
\end{pmatrix}.
\end{equation}
Denoting by $\widetilde{\mathcal{D}}(f,g) := f(1)g''(1) - g(1)f''(1)$, we represent functions $\phi_1$ and $\phi_2$ in terms of the fundamental solutions.
\begin{lem}\label{FundSoltoPhi}
Functions $\phi_1$ and $\phi_2$ have representations
\begin{align*}
\phi_1(x) &= g_1(x) + \widetilde{\mathcal{D}}^{-1}(g_2,g_4) \big(\widetilde{\mathcal{D}}(g_4,g_1) g_2(x) + \widetilde{\mathcal{D}}(g_1,g_2)g_4(x)\big),\\
\phi_2(x) &= g_3(x) + \widetilde{\mathcal{D}}^{-1}(g_2,g_4)\big(\widetilde{\mathcal{D}}(g_4,g_3) g_2(x) + \widetilde{\mathcal{D}}(g_3,g_2)g_4(x)\big).
\end{align*}
\end{lem}
Application of Lemma \ref{FundSoltoPhi} along with characterization $\phi_2'''(1) = \phi_1'(1)$ yields representation of $\Lambda_1(1)$ in \eqref{eq:Meps} in terms of the fundamental solutions. From now on we call this representation $\mathbbm{G}_1(\lambda)$ matrix. One way to calculate determinant of $\mathbbm{M}_\varepsilon(\lambda)$ is to apply results on analysis of perturbed matrices, e.g. see \cite{K95} and references there. However, we calculate this quantity directly up to $\mathcal{O}(\varepsilon^2)$ order, which under heavy simplification of the terms turns to be
\begin{equation}
\label{eq:DetMeps}
\det(\mathbbm{M}_\varepsilon) =d_0 + \varepsilon d_1+ \mathcal{O}(\varepsilon^2).
\end{equation}
The $d_0$ is equal to determinant of $\mathbb{M}_0$ matrix
\begin{equation}
\begin{split}
d_0 = \det(\mathbbm{M}_0) = \frac{|s_0(\Theta)|^4}{81} - \frac{|s_0(\Theta)|^2}{9} \mathrm{tr}(\mathbbm{G}_0^2) + \det(\mathbbm{G}_0^2)
\end{split}.
\end{equation}
Moreover, the $\varepsilon$-contribution term is
\begin{equation}
d_1 = -4\frac{|s_0(\Theta)|^2}{9}\Re\big(s_0(\Theta) \overline{ s_1(\Theta)}\big) G(\lambda)
\end{equation}
with the purely $\lambda$-dependent function
\begin{equation}
G(\lambda) = -\frac{1}{2} \big\{(1-(\mathbbm{G}_0^2)_{22})(\mathbbm{G}_1)_{11} + (1-(\mathbbm{G}_0^2)_{11}) (\mathbbm{G}_1)_{22} +
(\mathbbm{G}_0^2)_{21}(\mathbbm{G}_1)_{12} +
(\mathbbm{G}_0^2)_{12}(\mathbbm{G}_1)_{21} \big\}.
\end{equation}
Therby up to $\varepsilon^2$ accuracy, zeros of perturbed determinant \eqref{eq:DetMeps} is equivalent to the fact that $|s_0(\Theta)|^2/9$ is a root of polynomial
\begin{equation}
\label{eq:poly1}
\mathcal{P}(z) = z^4 -\big(\mathrm{tr}(\mathbbm{G}_0^2)+4\varepsilon \Re(s_0(\Theta) \overline{ s_1(\Theta)}) \xi(\lambda)\big)z^2 + \det(\mathbbm{G}_0^2).
\end{equation}
Notice that a fourth-order polynomial of form $z^4 -az^2 + b$ can be factorized as
\begin{equation}
z^4 -az^2 + b = (z^2+\tilde a z+ \tilde b)(z^2-\tilde a z+ \tilde b)
\end{equation}
in which $\tilde a= (a+2b^{1/2})^{1/2}$ and $\tilde b = b^{1/2}$. This realization along with the form \eqref{eq:poly1} implies that $\pm|s_0(\Theta)|/3$ are roots of $\mathcal{P}(z) = \mathcal{P}_1(z)\mathcal{P}_2(z)$, where
\begin{equation}
\begin{split}
\mathcal{P}_{1,2}(z) = z^2 \pm\big(\mathrm{tr}(\mathbbm{G}_0^2)+2\text{det}^{1/2}(\mathbbm{G}_0^2)
&+4\varepsilon \Re(s_0(\Theta) \overline{s_1(\Theta)})G(\lambda) \big)^{1/2}z \\+
\big(\mathrm{tr}(\mathbbm{G}_0^2)&+4\varepsilon \Re(s_0(\Theta) \overline{ s_1(\Theta)})G(\lambda)\big)^{1/2}
\end{split}.
\end{equation}
Without loss of generality, let's assume that $|s_0(\Theta)|/3$ is a root of $\mathcal{P}_2$, i.e.
\begin{equation}
\begin{split}
\frac{2}{3} |s_0(\Theta)| = &\big(\mathrm{tr}(\mathbbm{G}_0^2)+2\text{det}^{1/2}(\mathbbm{G}_0^2)
+4\varepsilon \Re(s_0(\Theta) \overline{s_1(\Theta)})G(\lambda) \big)^{1/2} \pm \\
&\big(\mathrm{tr}(\mathbbm{G}_0^2)-2\text{det}^{1/2}(\mathbbm{G}_0^2)
+4\varepsilon \Re(s_0(\Theta) \overline{ s_1(\Theta)})G(\lambda) \big)^{1/2}.
\end{split}
\end{equation}
Now applying the fact that
\begin{equation}
\mathrm{tr}(\mathbbm{G}_0^2) = \mathrm{tr}^2(\mathbbm{G}_0) - 2\text{det}(\mathbbm{G}_0)
\end{equation}
along with equality $\text{det}^{1/2}(\mathbbm{G}_0^2) = \text{det}(\mathbbm{G}_0) $ implies that
\begin{equation}
\begin{split}
\frac{|s_0(\Theta)| }{3} = &\big( \frac{1}{4}\mathrm{tr}^2(\mathbbm{G}_0)
+\varepsilon\Re(s_0(\Theta) \overline{s_1(\Theta)})G(\lambda) \big)^{1/2} \pm \\
&\big(\frac{1}{4} \mathrm{tr}^2(\mathbbm{G}_0)-\text{det}(\mathbbm{G}_0)
+\varepsilon \Re(s_0(\Theta) \overline{s_1(\Theta)})G(\lambda) \big)^{1/2}.
\end{split}
\end{equation}
However using the definitions of $T_1$ and $T_2$ in \eqref{eq:T1T2Def}, we introduce $\varepsilon$-extension of these functions as
\begin{equation}\label{eq:T1T2eps1}
T_1\up{\varepsilon} := \big(T_1^2(\lambda) +\varepsilon\Re(s_0(\Theta) \overline{ s_1(\Theta)}) G(\lambda)\big)^{1/2}
\end{equation}
and
\begin{equation}
T_2\up{\varepsilon} := T_2(\lambda) +\varepsilon \Re(s_0(\Theta) \overline{ s_1(\Theta)}) G(\lambda).
\end{equation}
Finding the roots of quadratic polynomials $\mathcal{P}_{1,2}$ is then reduces to condition $|s_0(\Theta)|/3$ satisfying
\begin{equation}
\label{eq:T1T2eps2}
\pm \frac{|s_0(\Theta)|}{3} = T_1\up{\varepsilon}+ (T_2\up{\varepsilon})^{1/2} \quad \text{or} \quad
\pm \frac{|s_0(\Theta)|}{3} = T_1\up{\varepsilon} - (T_2\up{\varepsilon})^{1/2}.
\end{equation}
Thus we proved an $\varepsilon$-extended dispersion relation for perturbed Hamiltonian as stated below.
\begin{thm}\label{anglePertDispThm}\emph{\textbf{(perturbed dispersion)}}
The dispersion relation for perturbed graphene Hamiltonian up to accuracy $\mathcal{O}(\varepsilon^2)$ satisfies
\begin{equation}
\label{eq:D1D2eps}
\Big(\Delta_{1}\up{\varepsilon}(\lambda,\Theta) \pm \frac{|s_0(\Theta)|}{3}\Big)\Big( \Delta_{2}\up{\varepsilon}(\lambda,\Theta) \pm \frac{|s_0(\Theta)|}{3} \Big)= 0,
\end{equation}
where $\Delta_{1,2}\up{\varepsilon} := T_1\up{\varepsilon} \pm (T_2\up{\varepsilon})^{1/2}$.
\end{thm}
We stress out here that for the case $\varepsilon = 0$, results above are consistent with the ones stated for graphene Hamiltonian. One of the interesting futures of Theorem \ref{anglePertDispThm} is to answer whether singular Dirac points will be preserved under $\varepsilon$-perturbed geometry. To answer this we first characterize the behaviour of $\Theta$-dependent function $ \Re(s_0(\Theta) \overline{ s_1(\Theta)})$ in perturbed part.
\begin{lem}
\label{ReS0S1}
Function $ \Re(s_0(\Theta) \overline{s_1(\Theta)})$ is $2\pi \mathbb{Z}^2$ periodic, its magnitude is bounded by $2(1+|c_1|)$ and zeros are at $(0,0)$ and $\pm(2\pi/3, -2\pi/3)$.
\end{lem}
\begin{proof}[\normalfont \textbf{Proof of Lemma~\ref{ReS0S1}}]
Recalling the definitions of $s_0(\Theta)$ and $s_1(\Theta)$ from \eqref{eq:s0} and \eqref{eq:s1} respectively
\begin{equation}
s_0(\theta) \overline{ s_1(\Theta)} = -\cot(\delta_0)(1+e^{-i\theta_1} + e^{-i\theta_2}) (1+c_1e^{i\theta_1} +c_2e^{i\theta_2}).
\end{equation}
By representation of exponential terms using Euler-formula
\begin{equation}
\Re(s_0(\Theta) \overline{ s_1(\Theta)}) = -\cot(\delta_0) \big((1+c_1) \cos(\theta_1) + (1+c_2) \cos(\theta_2) + (c_1+c_2) \cos(\theta_2- \theta_1) \big),
\end{equation}
which after further simplification and application of identity $1+c_1+c_2 = 0$ will reduce to
\begin{equation}
\label{eq:s0Bars1}
\Re(s_0(\Theta)\overline{ s_1(\Theta)}) = -\cot(\delta_0) \big( \cos(\theta_2- \theta_1) + c_1 \cos(\theta_2) + c_2 \cos(\theta_1) \big).
\end{equation}
Applying the fact that $\cos(\delta_0) = \cos(2\delta_0)$, then $(0,0)$ and $\pm(2\pi/3, -2\pi/3)$ are zeros of the functions. Finally, setting $c_2 = -1-c_1$ above we get
\begin{equation}
|\Re(s_0(\Theta) \overline{ s_1(\Theta)}) | \leq | \cos(\theta_2- \theta_1) -\cos(\theta_1) + c_1\big(\cos(\theta_2)-\cos(\theta_1)\big)| \leq 2(1+|c_1|)
\end{equation}
as desired.
\end{proof}
Figure \ref{Res0Bars1}(right) shows the behaviour of function $\Re(s_0(\Theta)\overline{s_1(\Theta)})$ for a fixed value of $c_1$ parameter.
\begin{figure}[ht]
\centering
\includegraphics[width=0.85\textwidth]{Res0Bars1}
\caption{Plots of $|s_0(\Theta)|$ and $\Re(s_0(\Theta) \overline{ s_1(\Theta)})$. Highlighted rectangle shows the first Brillouin zone.}
\label{Res0Bars1}
\end{figure}
\begin{cor}\label{DiracPerturb}\emph{\textbf{(Dirac points)}}
Singular Dirac points of graphene remain under angle-perturbed Hamiltonian.
\end{cor}
\begin{proof}[\normalfont \textbf{Proof of Corollary~\ref{DiracPerturb}}]
Let $(\Theta^*, \lambda^*)$ with $\Theta^* := \pm(2\pi/3, -2\pi/3)$ be a Dirac point for graphene Hamiltonian. Then, result from Lemma \ref{ReS0S1} implies that function $ \Re(s_0(\Theta) \overline{s_1(\Theta)})$ vanishes on quasimomentum $\Theta^*$. Thereby, there is no spectral gap at energy $\lambda^*$ for the perturbed Hamiltonian as well. Regarding singularity at this point, let define $\varepsilon$-dependent function
\begin{equation}
D_\varepsilon(\lambda,\Theta) := \pm \frac{|s_0(\Theta)|}{3} - T_1\up{\varepsilon} - (T_2\up{\varepsilon})^{1/2}
\end{equation}
and similarly for $\Delta_2\up{\varepsilon}$. Applying continuity property of function $D_\varepsilon(\lambda,\Theta)$ with respect to $\Theta$, then there exist $\varepsilon$-dependent neighborhood $\mathcal{N}_{\lambda,\Theta}\up{\varepsilon} := \mathcal{N}_{\lambda^*}(\lambda) \times \mathcal{N}_{\Theta^*}\up{\varepsilon}(\Theta)$ containing $(\lambda^*, \Theta^*)$ so that $D_\varepsilon(\lambda,\Theta)$ is well defined for all $(\lambda,\Theta) \in \mathcal{N}_{\lambda,\Theta}\up{\varepsilon} $. For $\lambda \in \mathcal{N}_{\lambda,\Theta}\up{\varepsilon} \setminus\{\lambda^*\}$ and the case $T_2(\lambda) > 0$, then application of inverse function theorem implies that solution set for $D_\varepsilon(\lambda,\Theta) = 0$ is a simple closed loop (distorted ellipse) in quasimomentum $\mathcal{N}_{\Theta^*}\up{\varepsilon}(\Theta)$. Moreover, observe that singularity of function $D_\varepsilon(\lambda,\Theta) $ only occurs at $\Theta^*$ due to $|s_0(\Theta)|$. For the case $T_2(\lambda) = 0$, function $D_\varepsilon(\lambda,\Theta)$ is only well-defined for $\mathcal{N}_{\Theta^*}\up{\varepsilon}(\Theta) \cap \{\Theta : \Re(s_0(\Theta) \overline{ s_1(\Theta)})G(\lambda) \geq 0\} $. Similar discussion implies that solution set for $D_\varepsilon(\lambda,\Theta) = 0$ is a simple connected curve (not closed) in quasimomentum $\mathcal{N}_{\Theta^*}\up{\varepsilon}(\Theta)$. In this case, dispersion relation is lost locally for $\Theta$ such that $\Re(s_0(\Theta) \overline{ s_1(\Theta)})G(\lambda) < 0$. In all two cases, the gap remain closed at Dirac point, however only one-side differentiability exists for the latter case.
\end{proof}
\begin{remark}
Here we stress out that for the case $T_2(\lambda) = 0$ explained in the proof of Corollary \ref{DiracPerturb}, concern is only about $\lambda \not = \lambda^*$ as for $\Theta^*$ the $\varepsilon$-term vanishes. Moreover, Corollary \ref{DiracPerturb} guarantees that Dirac points appear at $\Theta^* = \pm(2\pi/3, -2\pi/3)$ quasimomenta, but with possible shift in the energy space.
\end{remark}
Investigation on presence of a pure point spectrum has been an active research area. Changing the geometry of medium (e.g. working with 2D periodic graph instead of a real line), imposing perturbation through potential, and applying different Hamiltonian model are among few ways to guarantee presence of a pure point spectrum, e.g. see \cite{K03,KP07,H18,L21}. As stated in Theorem \ref{grapheneSpectrum} pure point spectrum for graphene is non-empty. This has been proved by explicit construction of even (or odd) eigenfunctions with support on single hexagon. However, existence of pure point spectrum will fail for perturbed Hamiltonian.
\begin{thm}\label{PerturbedgrapheneSpectrum}\emph{\textbf{(the spectral description)}}
The spectrum of the perturbed Hamiltonian is purely absolutely continuous.
\end{thm}
\begin{proof}[\normalfont \textbf{Proof of Theorem~\ref{PerturbedgrapheneSpectrum}}]
The singular continuous spectrum is empty, since the Hamiltonian is a self-adjoint elliptic operator like the unperturbed case (see proof of Theorem \ref{grapheneSpectrum}). Next, let's show the absence of the pure point spectrum unlike the graphene case. Using the dispersion relation, we get $\sigma_{\text{pp}}(\mathcal{H}) \subset \Sigma^{D}$ like we did in the unperturbed case. Now let us assume $\sigma_{\text{pp}}(\mathcal{H}) \neq \emptyset$. Then the corresponding eigenfunction $u$ restricted to any edge should either be identically zero or solve $d^4u(x)/dx^4 + q(x)u(x) = \lambda u(x)$ with the boundary conditions $u(0)=u(1)=u''(0)=u''(1)=0$ on that edge. Therefore restriction of an eigenfunction to any edge on its support should be an eigenfunction of the operator $d^4/dx^4 + q(x)$ for the same eigenvalue $\lambda$ on $[0,1]$ interval with the boundary conditions $u(0)=u(1)=u''(0)=u''(1)=0$. Note that $u$ should also satisfy the vertex conditions.
If $u$ is compactly supported, then the vertex conditions on the vertices of the boundary of the support of $u$ imply $\varepsilon = c_1\varepsilon = c_2\varepsilon$. Recall that $1 + c_1 + c_2 = 0$, so $\varepsilon = 0$ is the only solution, which is the unperturbed case. For the non-compactly supported $u \in H^4(\Gamma)$, same discussion holds to show that vertex conditions can not be met at any vertex. Therefore the pure point spectrum is also empty. From the dispersion relation we get that the spectrum is non-empty, so we get the desired result that the spectrum is purely absolutely continuous.
\end{proof}
\begin{remark}
Applying the result in Lemma \ref{ReS0S1} and the proof of Corollary \ref{DiracPerturb} some arguments can be made to quantify shift of dispersion relation \eqref{eq:D1D2eps} for perturbed Hamiltonian compared to graphene case at any $\lambda$. More precisely, for $T_2 > 0$, expansion of $T_1\up{\varepsilon}$ and $T_2\up{\varepsilon}$ in \eqref{eq:T1T2eps2} implies that
\begin{equation}
\label{eq:qusiMomeps}
\pm\frac{|s_0(\Theta)|}{3} = \Delta_1(\lambda) \Big\{1+ \varepsilon \Re(s_0(\Theta) \overline{ s_1(\Theta)})G(\lambda)T_1^{-1}(\lambda)T_2^{-1/2}(\lambda) \Big\} + \mathcal{O}(\varepsilon^2)
\end{equation}
and similarly for $\Delta_2$ with sign changes. Now for fixed value of $\lambda$, the shift with respect to graphene, i.e. case $\varepsilon = 0$, in quasimomentum can be found by solving \eqref{eq:qusiMomeps}.
\end{remark}
Finally, in the following section we give a partial list of topics which may be interesting to the reader for future extension of current work.
\section{Outlook}
\label{sec:Outlook}
The viability of the frame model as a structure composed of one-dimensional segments needs to be verified mathematically, as a limit of a three-dimensional structure as the beam widths go to zero. There is a significant mathematical literature on this question for second-order operators (see, for example \cite{Z02,Gri_incol08,Post_book12}), with a variety of operators arising in the limit. This variety will increase in the case of fourth-order equations, and may be expected to incorporate masses concentrating at joints and other cases of applied interest. Moreover, validity of Euler-Bernouli beam theory specially at high-energy level maybe a place to be questioned. Unlike this, richer Timoshenko model no longer assumes the cross-sections remain orthogonal to the deformed axis and therefore incorporates more degrees of freedom \cite{MPZ02,GR15,Mei19}. Of applied interest would be to extend the current results to the latter model.
In this work we focused on Euler-Bernouli beam theory and its restriction to scalar-valued lateral displacement $v(x)$. In the work \cite{BE21} it is shown that for planar graphs, more accurate way to present the operator is by including angular displacement field $\eta(x)$ as well. This then shifts our problem to a vector-valued operator and more complicated vertex conditions. We refer to recent work \cite{SBE21} for analysis in this line and potential future work for interesting three dimensional periodic graphs.
An interesting problem is to employ two-scale analysis for understanding the homogenized behavior and spectra of Hamiltonian on periodic lattices with more complex fundamental domain, e.g. see \cite{K16,ZP16,GMO18,CLM20,KM21} and references there. Of similar interest is generalization of our result to multi-layer quantum graph models equipped with beam Hamiltonian. In the work \cite{FWS21}, it is shown that for {S}chr{\"o}dinger operator dispersion relation of wave vector and energy is a polynomial in the dispersion relation of the single layer. This leads to the reducibility of the algebraic Fermi surface, at any energy, into several components. For the beam Hamiltonian, it has been shown that in the special case of planar frames, the operator decomposes into a direct sum of two operators, one coupling out-of-plane displacement to angular displacement and the other coupling in-plane displacement with axial displacement \cite{BE21}. Understanding the interaction of these decoupled systems on multi-layer graphs may be interesting from both theory and applied perspectives.
\bibliographystyle{abbrv}
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{"url":"https:\/\/honglangwang.wordpress.com\/tag\/math\/","text":"You are currently browsing the tag archive for the \u2018Math\u2019 tag.\n\nhttp:\/\/terrytao.wordpress.com\/2010\/08\/19\/lindenstrauss-ngo-smirnov-villani\/\u00a0(a post about the winners in icm2010, including this area)\n\nQ6: Do \u201cImaginary Numbers\u201d Really Exist?\n\nAn \u201cimaginary number\u201d is a multiple of a quantity called \u201ci\u201d which is defined by the property that i squared equals -1. This is puzzling to most people, because it is hard to imagine any number having a negative square. The result: it is tempting to believe that i doesn\u2019t really exist, but is just a convenient mathematical fiction.\n\nThis isn\u2019t the case. Imaginary numbers do exist. Despite their name, they are not really imaginary at all. (The name dates back to when they were first introduced, before their existence was really understood. At that point in time, people were imagining what it would be like to have a number system that contained square roots of negative numbers, hence the name \u201cimaginary\u201d. Eventually it was realized that such a number system does in fact exist, but by then the name had stuck.)\n\nBefore discussing why imaginary numbers exist, it\u2019s helpful to think about why we\u2019re even asking the question. Why is it so hard to accept that there could be numbers with negative squares? One has to come to terms with the things that seem so puzzling and confusing about this concept and see that they are not really so unreasonable after all, before one can move on to accept the existence of imaginary numbers. Having done that, we can move on to seeing why they exist, and what relevance they have.\n\nTherefore, we will address the following questions (you may select any of the items below to see the explanation):\n\n\u2022 Imaginary Numbers: More Reasonable than they First Appear\n\u2022 Imaginary Numbers: How To Show They Exist\n\u2022 Imaginary Numbers: Relevance to the Real World\n\u2022\n\nAnother very insightful article is A Visual, Intuitive Guide to Imaginary Numbers.\n\nWhy some theorems important? There are at least several reasons that come to mind:\n\n${\\bullet }$ Some theorems are important because of their intrinsic nature. They may not have applications, but they just are beautiful. Or they have a interesting proof.\n\n${\\bullet }$ Some theorems solve open problems. Of course any such theorem is automatically important, since the field has already decided that the question is interesting.\n\n${\\bullet }$ Some theorems create whole new directions for mathematics and theory. These are sometimes\u2014not always\u2014relatively easy theorems to prove. But they may be very hard theorems to realize they should be proved. Their importance is that they show us that something new is possible.\n\n${\\bullet }$ Some theorems are important because they introduce new proof techniques. Or contain a new lemma that is more useful than the theorem proved.\n\n${\\bullet }$ Some theorems are important because of their \u201cpromise.\u201d This is a subjective reason\u2014a theorem may be important because people feel it could be even more important. Here, both the relation to group equations and the constraints-on-interval-graphs view make us feel Klyachko Car Crash Theorem has some hidden possibilities.\n\nAnd there is also a paper written by Terry Tao on what\u2019s good mathematics.\n\nhttp:\/\/www.springer.com\/librarians\/e-content\/ebooks?SGWID=0-40791-12-784104-0\n\n### Bayesian Computation with R\n\nI have noticed this concept before. Since I am just new in Probability field, so you should forgive me that I just noticed this academic area several months ago and did not realize the importance of it. Today I attended the regular colloquium of my department and the speaker, Zbigniew J. Jurek, gave a lecture about The Random Integral Representation Conjecture. In this talk, he mentioned free probability. Moreover, he also joked that free statistics will come into being.\n\nI also fount a useful link about the survey of free probability. I hope it will be useful for you. Terry Tao also have a post about this.\n\n# Math Sex Jokes\n\nAre you 2x? Because I want to integrate you from 10 to 13!\n\nI derived your mom last night.\nIt was f prime.\n\nHow is sex like math?\n1. Half the time I get an odd result.\n2. If my hands aren\u2019t enough, I end up using my head.\n3. I always wonder how the person next to me is doing on his work.\n4. My average at each is pretty dismal.\n\nWhat is 69 and 69?\nDinner for four..\n\nWhat is 6.9?\nGood sex interrupted by a period.\n\nQ: If you go to bed 8 hours before you have to wake up, and your wife wants to have 2 hours of sex, how much sleep will you get?\nA: 7 hours, 57 minutes \u2013 who cares what she wants!\n\nAt this moment 5 million are having sex, 2 million are in gun fights, 91 million at a party, and one sad loser is reading this joke\n\nA graduate student of mathematics who used to come to the university on foot every day arrives one day on a fancy new bicycle. \u201cWhere did you get the bike from?\u201d his friends want to know.\u201dIt\u2019s a `thank you\u2019 present\u201d, he explains, \u201cfrom that freshman girl I\u2019ve been tutoring. But the story is kind of weird\u2026\u201d \u201cTell us!\u201d \u201cWell\u201d, he starts, \u201cyesterday she called me on the phone and told me that she had passed her math final and that she wanted to drop by to thank me in person. As usual, she arrived at my place riding her bicycle. But when I had let her in, she suddenly took all her clothes off, lay down on my bed, smiled at me, and said: `You can get from me whatever you desire!'\u201d\n\nOne of his friends remarks: \u201cYou made a really smart choice when you took the bicycle.\u201d\n\n\u201cYeah\u201d, another friend adds, \u201cjust imagine how silly you would have looked in girls clothes \u2013 and they wouldn\u2019t have fit you anyway!\u201d\n\nQ: How are math and sex the same?\nA: I don\u2019t get either one.\n\nA mathematician and an engineer agreed to take part in a psychological test. They sat on one side of a room and waited not knowing what to expect. A door opened on the other side and a naked woman came in the room and stood on the far side. They were then instructed that every time they heard a beep they could move half the remaining distance to the woman. They heard a beep and the engineer jumped up and moved halfway across the room while the mathematician continued to sit, looking disgusted and bored. When the mathematician didn\u2019t move after the second beep he was asked why. \u201cBecause I know I will never reach the woman.\u201d The engineer was asked why he chose to move and replied, \u201cBecause I know that very soon I will be close enough for all practical purposes!\u201d\n\nA physicist, a mathematician and a computer scientist discuss what is better: a wife or a girlfriend. The physicist: \u201cA girlfriend. You still have freedom to experiment.\u201d The mathematician: \u201cA wife. You have security.\u201d The computer scientist: \u201cBoth. When I\u2019m not with my wife, she thinks I\u2019m with my girlfriend. With my girlfriend it\u2019s vice versa. And I can be with my computer without anyone disturbing me\u2026\u201d\n\nWhy does 1+1=1?\n1 male + 1 female = 1 baby\n\nQ: If you have two friends and six women, how many women do each of your friends get?\nA: None.\n\nQ. How do you teach a blond math?\nA. Subtract her clothes, divide her legs, and square root her.\n\nBefore I root you, are you over 18?\n\n\u201cWhat happened to your girlfriend, that really cute math student?\u201d\n\u201cShe no longer is my girlfriend. I caught her cheating on me.\u201d\n\u201cI don\u2019t believe that she cheated on you!\u201d\n\u201cWell, a couple of nights ago I called her on the phone, and she told me that she was in bed wrestling with three unknowns\u2026\u201d\n\nSex is like math:\nSubtract the clothes,\nDivide the legs,\nand pray to God you don\u2019t Multiply!\n\nGeneral philosophy of probability theory\nProbability is central to science, more than any other part of math. It enters statistics, physics, biology, and even medicine as we will see when and if we discuss tomography. This is the broad view.\nThere is also a narrow view \u2013 one needs to understand it before one can effectively apply it and it has many subtleties. Possibly this is due to the fact that probability, stochasticity, or randomness, may not actually exist! I think it mostly exists in our uncertainty about the world. The real world seems to be deterministic (of course one can never test this hypothesis). It is chaotic and one uses probabilistic models to study it mainly because we don\u2019t know the initial conditions. Einstein said that \u201dgod does not play dice\u201d. My own view is that the world may be deterministic, but I like to think I have free will. I believe that probability should be regarded only as a model of reality.\n\nFrom the notes of Lawrence A. Shepp\n\nToday I just found a nice list from xi\u2019an\u2019s blog of Top 15 papers for his graduate students\u2019 reading:\n\n1. B. Efron (1979) Bootstrap methods: another look at the jacknife Annals of Statistics\n2. R. Tibshirani (1996) Regression shrinkage and selection via the lasso J. Royal Statistical Society\n3. A.P. Dempster, N.M. Laird and D.B. Rubin (1977) Maximum likelihood from incomplete data via the EM algorithm J. Royal Statistical Society\n4. Y. Benjamini & Y. Hochberg (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. Royal Statistical Society\n5. W.K.Hastings (1970) Monte Carlo sampling methods using Markov chains and their applications, Biometrika\n6. J. Neyman & E.S. Pearson (1933) On the problem of the most efficient test of statistical hypotheses Philosophical Trans. Royal Statistical Society London\n7. D.R. Cox (1972) Regression models and life-table J. Royal Statistical Society\n8. A. Gelfand & A.F.M. Smith (1990) Sampling-based approaches to calculating marginal densities J. American Statistical Assoc.\n9. C. Stein (1981) Estimation of the mean of a multivariate normal distribution Annals of Statistics\n10. J.O. Berger & T. Sellke (1987) Testing a point null hypothesis: the irreconciability of p-values and evidence J. American Statistical Assoc\n\nWhich ones should I now add? First, Steve Fienberg pointed out to me the reading list he wrote in 2005 for the iSBA Bulletin. Out of which I must select a few ones:\n\n1. A. Birnbaum (1962) On the Foundations of Statistical Inference J. American Statistical Assoc.\n2. D.V. Lindley & A.F.M. Smith (1972) Bayes Estimates for the Linear Model\u00a0 J. Royal Statistical Society\n3. J.W.Tukey (1962) The future of data analysis. Annals of Mathematical Statistics\n4. L. Savage (1976) On Rereading R.A. Fisher Annals of Statistics\n\nAnd then from other readers, including Andrew, I must also pick:\n\n1. H. Akaike (1973). Information theory and an extension of the maximum likelihood principle. Proc. Second Intern. Symp. Information Theory, Budapest\n2. D.B. Rubin (1976). Inference and missing data. Biometrika\n3. G. Wahba (1978). Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. Royal Statistical Society\n4. G.W. Imbens and J.D. Angrist (1994). Identification and estimation of local average treatment effects. Econometrica.\n5. Box, G.E.P. and Lucas, H.L (1959) Design of experiments in nonlinear situations. Biometrika\n6. S. Fienberg (1972) The multiple recapture census for closed populations and incomplete 2k contingency tables Biometrika\n\nOf course, there are others that come close to the above, like Besag\u2019s 1975 Series B paper. Or Fisher\u2019s 1922 foundational paper. But the list is already quite long. (In case you wonder, I would not include Bayes\u2019 1763 paper in the list, as it is just too remote from statistics.)\n\nAnd this year some of his students are reading the following papers:\n\n1. W.K.Hastings (1970) Monte Carlo sampling methods using Markov chains and their applications, Biometrika\n2. G. Casella & W. Strawderman (1981) Estimation of a bounded mean Annals of Statistics\n3. A.P. Dawid, M. Stone & J. Zidek (1973) Marginalisation paradoxes in Bayesian and structural inference J. Royal Statistical Society\n4. C. Stein (1981) Estimation of the mean of a multivariate normal distribution Annals of Statistics\n5. D.V. Lindley & A.F.M. Smith (1972) Bayes Estimates for the Linear Model\u00a0 J. Royal Statistical Society\n6. A. Birnbaum (1962) On the Foundations of Statistical Inference J. American Statistical Assoc.\n\nI think it is also a good list for my own reading.","date":"2017-12-17 23:24:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 5, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.48186394572257996, \"perplexity\": 1649.4033314366563}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948599156.77\/warc\/CC-MAIN-20171217230057-20171218012057-00290.warc.gz\"}"}
| null | null |
News , Trending
Issa Rae Gives Candid Response On Kenya Barris' Reported ABC Censoring
Trey Mangum
In an in-depth, revealing feature with GQ, Issa Rae covered a number of topics.
She was asked about the reported disagreement between Kenya Barris and ABC on a black-ish episode which would have tackled NFL players protesting the state of race in America by kneeling for the national anthem.
"That would infuriate me. You know? Like, I'm out here telling the truth, and I'm telling my authentic experience, and you pride yourself on having this show that exposes the plight of a black family in the United States, and then you're censoring: No, not that. We don't want to see that part. The world isn't ready for that. America's not ready. That's crazy to me…. Kenya tries to couch so much in a family show, and get so much across, in a way that I really respect and admire. But a lot of the time it is just mired in the Disney, ABC of it all."
After news surfaced that Barris was contemplating leaving his deal at ABC Studios due to the recent disagreements, ABC president Channing Dungey said, "We have loved working with Kenya and would love to continue."
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,816
|
Q: How to prevent $ interpretation in JSP to make jQuery templates work I have a page generated with data from both backend and front. The front end is using a jquery template, and page itself is written in JSP, and there is a conflict of using $ sign:
<script type="text/javascript">
${title}
</script>
For example, I want the above code to be interpreted by front end, but JSP is translating to something else. How do I prevent this from happening?
thanks
Oliver
A: Add this to the top of the page:
<%@ page isELIgnored="true" %>
This should only be in the page which defines the template. Include that page from the main page, if you want to use EL in it.
A: You could try using the following for any problematic lines
out.print("${title}")
A: Put backslash before dollar sign and it won't be interpreted as JSP EL.
\${title}
So, ${1+1} prints 2 and \${1+1} prints ${1+1}.
Otherwise, you can do what Jared says.
A: Why not enclose them between jsp:text or move all your template code in a .js file if possible.
A: also you can use this
<%="${name}"%>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,822
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/196579\/what-is-the-distribution-of-the-difference-between-two-ar1-processes","text":"# What is the distribution of the difference between two AR(1) processes?\n\nI am reading a paper published in a good economics journal. An econometric model is presented in the paper. A part of the described model is not very clear to me. Please let me state a couple of equations and ask my question.\n\n$\\upsilon_{s}$ and $\\omega_{s}$ are both first-order autoregressive processes and take the form\n\n$\\upsilon_{s} = \\rho \\upsilon_{s-1} + \\phi_{s}$, $\\mathbb{E}_{s-1}(\\phi_{s}) = 0$\n\nand\n\n$\\omega_{s} = \\rho \\omega_{s-1} + \\psi_{s}$, $\\mathbb{E}_{s-1}(\\psi_{s}) = 0$\n\nfor $s = t+1, \\dotsc, S$.\n\nThen $\\xi_{s}$ is defined as\n\n$\\xi_{s} = \\upsilon_{s} - \\omega_{s}$.\n\nIt is then \"assumed\" that $\\xi_{s}$ is normally distributed with variance $\\sigma_{\\xi}^{2}$.\n\nHow strong is the assumption that $\\xi_{s}$ is normally distributed? Under which conditions can one claim that the difference between two AR(1) processes is normal? I know that the difference between two I(1) processes can be I(0) (cointegration), but I do not know the distribution of the difference between two AR(1) processes. The paper does not provide any argument on the assumption. I am wondering if this is a strong assumption, or if it is somewhat easy to argue\/expect that $\\xi_{s}$ is normally distributed.\n\nThanks.\n\nUsing your notation; $$\\xi_{s}= \\upsilon_{s}-\\omega_{s} = \\rho( \\upsilon_{s-1}-\\omega_{s-1}) + \\phi_{s} - \\psi_s$$ $$=\\rho \\xi_{s-1} + \\epsilon_s$$ where $\\epsilon_s = \\phi_{s} - \\psi_s$, so $E[\\epsilon_s]=0$ and $$var(\\epsilon_s)=var(\\phi_s)+var(\\psi_s)-2cov(\\phi_s,\\psi_s)$$\nSo it does seem that $\\xi_s$ is probably a stationary AR(1). There are some caveats though.\n1. $\\epsilon_s$ may not be normally distributed if $(\\phi_s,\\psi_s)$ are not multivariate normal ($\\xi_s$ would still be an AR(1) though, but with non-normal disturbance terms).\n2. $\\epsilon_s$ may not be mutually independent white noise if there is lagged cross-correlation between $\\phi_s$ and $\\psi_s$. This will violate the assumptions of the AR(1) if it is estimated via maximum likelihood.\nIn sum, you are right to suspect that $\\xi_s$ may not be normally distributed. In econometrics we sometimes assume things like multivariate normality without stating it (it is wrong, but it happens). Look to see if the author did state an assumption about independence or multivariate normality for the AR(1) disturbance terms.","date":"2020-01-28 19:36:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7438159584999084, \"perplexity\": 265.4555274992591}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251783000.84\/warc\/CC-MAIN-20200128184745-20200128214745-00445.warc.gz\"}"}
| null | null |
$: << File.dirname(__FILE__)
require 'rubygems'
require 'fsevents'
require 'stringio'
require 'ext/kernel'
class Benchmarker
class << self
def go(path)
@path = path
check_path_exists
add_sigint_handler
stream = FSEvents::Stream.watch(path) do |events|
output = capture_stdout do
yield
end
times << grab_times(output.string).flatten
display_output
end
stream.run
end
def times
@times ||= []
end
def add_sigint_handler
trap 'INT' do
puts "Bye!"
raise Interrupt, nil # let the run loop catch it
end
end
private
def display_output
output = ""
last_time = @times[-2]
now = @times[-1]
output << "Last Time: #{last_time.inspect}\n" if last_time
output << "Now : #{now.inspect}\n"
output << calculate_difference(last_time, now) if last_time
output << "-" * 50
puts output
end
def calculate_difference(last_time, now)
changes = []
last_time.each_with_index do |time, index|
a = "#{(((time.to_f - now[index].to_f) / time.to_f) * 100).to_i}%"
changes << a
end
"Change : #{changes.inspect}\n"
end
def grab_times(output)
output.scan(/(\d+\.?\d+)\)/)
end
def check_path_exists
unless File.directory?(@path)
raise "Error: Wheres your libs?"
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,834
|
For resources and guidance on the coronavirus outbreak, click here. Granite Staters with questions can dial the NH Emergency Operations Center at 2-1-1.
For Granite Staters
Students and Kids
Academy Nominations
Tour Requests
Coffee with Jeanne
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Jeanne's Priorities
About Jeanne
Meet Jeanne
Contact Jeanne
SHAHEEN ANNOUNCES $1.8 MILLION FOR NEW HAMPSHIRE POLICE FORCES
(Washington, DC) – U.S. Senator Jeanne Shaheen (D-NH) announced today that New Hampshire has been awarded $1,827,737 in Department of Justice (DOJ) grants to support police departments and operations across the state. The awards are administered through DOJ's Edward Byrne Memorial Justice Assistance Grant Program (JAG), which allows states and units of local government to support a broad range of crime-prevention activities based on local needs.
"I'm pleased New Hampshire will receive these funds to bolster its police forces and help keep our citizens safe" Shaheen said. "With state and local budgets strained under current economic pressures, these awards will help give the brave policemen and women around our state the resources they need to perform their jobs effectively and better serve the communities they protect."
A summary of the awards received by state and local governments is included in the table below. Some jurisdictions applied jointly as noted.
Award Recipient
N.H. Department of Justice
$1,377971
Drug interdiction efforts, cyber crime prevention, completion of statewide integrated criminal justice information system, youth crime prevention.
Equipment, community policing initiatives, crime prevention programs.
Purchase and outfit of a patrol vehicle, firearms, ammunition, non-lethal weapons, supplies.
Training, equipment, supplies, criminal justice information systems.
New equipment and training.
Purchase of a mobile data system to increase effectiveness of police communications.
Purchase video monitors, radio equipment and a mobile data terminal to increase law enforcement services.
Hire anti-drug personnel.
Purchase a police cruiser and equipment to use for roadway safety and traffic enforcement.
Purchase a police cruiser and surveillance cameras.
Press Office, (202) 224-5553
Pursuant to Senate Policy, petitions, opinion polls and unsolicited mass electronic communications cannot be initiated by this office for the 60-day period immediately before the date of a primary or general election. Subscribers currently receiving electronic communications from this office who wish to unsubscribe may do so here.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,335
|
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