text:
\section{Introduction}
A wide variety of techniques have been proposed to model multiphase fluid systems with incompressible, immiscible fluids.
These include explicit front-tracking techniques \cite{UNVERDI1992, Tryggvason2001}, the volume-of-fluid method \cite{Weymouth2010, HIRT1981},
the phase-field method \cite{Jacqmin1999, Anderson1998, Aland2010},
and the level set method \cite{Shin2005, Chang1996}.
A constant challenge in each of these techniques is the conservation of mass during the course of the simulation.
Each of these methods handle this challenge differently. For example, the volume-of-fluid methods has excellent mass conservation
properties \cite{Baraldi2014} at the expense of requiring complex
heuristic interface reconstructions techniques to calculate geometric quantities such as curvature \cite{Cummins2005,Pilliod2004,Raessi2007}.

Unfortunately, there are many physical systems where high accuracy of geometric quantities are required. One example are models of liposome vesicles
where interfacial forces depend on high order derivatives of the interface's curvature \cite{Salac2011,Kolahdouz2015b,Yazdani2012}.
Unlike volume-of-fluid techniques, front tracking, phase-field, and level-set methods are able to provide higher geometric accuracy.
This accuracy is obtained at the expense of natural volume conservation and thus special care must be taken to ensure
that mass does not change over the course of a simulation.

There have been numerous attempts to improve the mass conservation of such methods. For example, level set methods have
been adjusted by Lagrangian particles
which are used to correct the level set function \cite{Enright2005,Enright2002} or level sets have been combined with
volume of fluid methods \cite{Sussman2000}. Other simply shift the interface function to match the volume constraint \cite{Yap2006}.

The major issue with these types of corrections is that the interface becomes decoupled from the underlying flow field. As an example consider
an interface on a uniform Cartesian grid, Fig. \ref{fig:velocitydecouple}. The interface begins on the left side and the underlying fluid flow-field
dictates that it should move to the right one grid spacing. After this time step it is determined that errors in the simulation
resulted in a mass gain. A simple and often-used correction is to simply move the interface to obtain mass conservation. In this case the effect is
the interface will not move the full amount that the underlying flow field dictates. The movement of the interface and the underlying flow field
have become decoupled. Any externally applied correction for front-tracking, phase-field or level set methods will demonstrate similar behavior.
\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=1.5in]{VelocityDecouple}
		\caption{Schematic of velocity-interface decoupling. The interface begins on the left (solid line). The underlying flow field dictates that the interface
		move to the right and should end as the dash-line after a single time step. Due to an externally applied correction the interface instead ends the time step
		at the dotted line.}
		\label{fig:velocitydecouple}
	\end{center}
\end{figure}

In this manuscript a different approach is taken. Instead of adjusting the interface to achieve mass conservation a novel
Navier-Stokes projection method is developed which ensures mass conservation. Unlike the previous method the interface is simply advected
due to the underlying flow-field, it is the flow-field itself which explicitly takes into account any possible mass-loss. As will
become apparent later in the manuscript it is useful to think of this method as modifying the pressure so that it can handle not only
local incompressibility but also global mass conservation. Note that while this manuscript will focus on a particular Navier-Stokes
numerical implementation, the concept presented here extends to any type of multiphase fluid simulation that uses a projection method.

The remainder of the manuscript is as follows. In Section \ref{sec:single-fluid} the single-fluid formulation of multiphase fluid flow
is briefly described. The novel mass-preserving Navier-Stokes projection method is described in Section \ref{sec:projection-method}.
The numerical implementation is given in Section \ref{sec:numerics}, which is followed by numerical experiments in Section \ref{sec:experiments}.
A short conclusion is presented in Section \ref{sec:conclusion}.

\section{Single-Fluid Navier-Stokes Equations}
\label{sec:single-fluid}
In this sections a brief introduction to the single-fluid formulation of multiphase fluid flow is presented. In a multiphase fluid system
the interface between two immiscible fluids evolves over time. In this work a level-set description of the interface is used. Let the
evolving interface be given as the set of points where the level-set function is zero, $\Gamma(t)=\{\vec{x}:\phi(\vec{x},t)=0\}$.
The evolution of the level set function $\phi(\vec{x},t)$ over time will implicitly determine the location of the interface.
Following convention the interior fluid, $\Omega^-$, is given by $\phi<0$ while the outer fluid, $\Omega^+$, is given by $\phi>0$, Fig. \ref{fig:levelset}.
The entire domain is given as $\Omega=\Omega^+\cup\Omega^-$. Using the level-set description it is possible to obtain geometric
information of the interface easily. For example, the outward facing normal is simply $\vec{n}=\nabla\phi/\|\nabla\phi\|$ while the total curvature can be calculated
as $\kappa=\nabla\cdot\vec{n}$.
\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=2in]{levelset}
		\caption{The level-set description of a multiphase fluid system. The outward facing normal is shown for clarity.}
		\label{fig:levelset}
	\end{center}
\end{figure}

In each domain the Navier-Stokes equations hold,
\begin{align}
	\label{eq:NS} \rho^\pm\dfrac{D\vec{u}^\pm}{Dt}&=-\nabla p^\pm+\nabla\cdot\left(\mu^\pm\left(\nabla\vec{u}^\pm+\nabla^T\vec{u}^\pm\right) \right)+\vec{g}^\pm, \\
	\label{eq:DIV} \nabla\cdot\vec{u}^\pm&=0,
\end{align}
where $\rho$ is the fluid density, $\mu$ is the fluid viscosity and $\vec{g}$ is any body force term, such as gravity.
The fluid equations are coupled by a jump in the stress at the interface,
\begin{equation}
	\left[-p\vec{n}+\mu\left(\nabla\vec{u}+\nabla^T\vec{u}\right)\right]\cdot\vec{n}=\vec{f},
	\label{eq:interfaceJump}
\end{equation}
where $[\;]$ indicates the jump of a quantity (outside minus inside) across the interface and $\vec{f}$ are any forces, such as tension, which act on the interface.

A difficulty in multiphase fluid simulations is the solution of this set of coupled but discontinuous differential equations. Some techniques, such as the
Immersed Interface Method \cite{Lee2003}, augment the discretization of the differential equations to take into account the jumps across the interface. Another technique,
which is explained here, is to model the domain as a ``single" fluid with spatially varying properties \cite{Chang1996, BRACKBILL1992}. The interface condition, Eq. (\ref{eq:interfaceJump}),
is accounted for by converting the singular force contribution at the interface into a body-force term localized around the interface.

Define the smooth Heaviside, $H_\varepsilon(\phi)$, function as
\begin{equation}
	H_{\varepsilon}(\phi)=
	\begin{cases}
		0 & \phi<-\varepsilon\\
		\tfrac{1}{2}\left[1+\tfrac{\phi}{\varepsilon}+\tfrac{1}{\pi}\sin\left(\tfrac{\pi\phi}{\varepsilon}\right)\right] & |\phi|\leq \varepsilon \\
		1 & \phi>\varepsilon
	\end{cases}
\end{equation}
where $\varepsilon$ is proportional to the grid spacing.
From the definition of the Heaviside function define the smoothed Dirac-Delta function as $\delta_\varepsilon(\phi)=\partial H_\varepsilon(\phi)/\partial \phi$,
\begin{equation}
	\delta_{\varepsilon}(\phi)=
	\begin{cases}
		0 & |\phi|>\varepsilon\\
		\tfrac{1}{2\varepsilon}\left[1+\cos\left(\tfrac{\pi\phi}{\varepsilon}\right)\right] & |\phi|\leq \varepsilon
	\end{cases}.
\end{equation}
The use of the Heaviside and Dirac functions allows for the calculation of the volume enclosed by a given level set as well as the surface area as a integrals over the domain,
\begin{align}
	V(\phi)=&\int_\Omega\left(1-H_\varepsilon(\phi)\right)\|\nabla\phi\|dV,\\
	A(\phi)=&\int_\Omega\delta_\varepsilon(\phi)\|\nabla\phi\|\; dV.
\end{align}

Using the Heaviside function the density and viscosity are given by
$\rho_\varepsilon(\phi)=\rho^-+\left(\rho^+-\rho^-\right)H_\varepsilon(\phi)$ and $\mu_\varepsilon(\phi)=\mu^-+\left(\mu^+-\mu^-\right)H_\varepsilon(\phi)$.
These functions also allow for the transformation of singular interface forces into localized body force terms. For example, let the only
singular interfacial force be from a uniform surface tension, $\vec{f}=\sigma \kappa\vec{n}$, where $\sigma$ is the coefficient of surface tension.
In conjunction with the smoothed density and viscosity
definitions this results in a single Navier-Stokes equation valid in the entire domain,
\begin{align}
	\rho_\varepsilon(\phi)\dfrac{D\vec{u}}{Dt}=&-\nabla p+\nabla\cdot\left(\mu_\varepsilon(\phi)\left(\nabla\vec{u}+\nabla^T\vec{u}\right)\right)\nonumber\\
	\label{eq:singleNS} 	&-\sigma \delta_\varepsilon(\phi) \kappa\nabla\phi+\vec{g}_\varepsilon(\phi),\\
	\label{eq:singleDiv} \nabla\cdot\vec{u}=&0,
\end{align}
where the body force term has been written as a smooth function of the level-set.
This type of single-fluid formulation for multiphase flow has been used to model bubbles and droplets \cite{Sussman1997,Sussman2012} and vesicles \cite{Kolahdouz2015b,Kolahdouz2015a}.

The results below focus on bubbles and droplets under the influence of surface tension and gravity. Therefore, the interface and body force are restricted
to this particular case. The dimensionless form of the single-fluid Navier-Stokes equations can be obtained by defining a characteristic length $l_0$,
and velocity $u_0$, from which a characteristic time can be obtained, $t_0=l_0/u_0$. The density and viscosity are normalized by the values of the
outer domain,
\begin{align}
	\rho(\phi)&=\lambda+(1-\lambda)H(\phi), \\
	\mu(\phi)&=\eta+(1-\eta)H(\phi),
\end{align}
where $\lambda=\rho^-/\rho^+$ and $\eta=\mu^-/\mu^+$ are the density and viscosity ratio, respectively, and the $\varepsilon$ has been dropped for clarity.

Define the the Reynolds number as
\begin{equation}
	\textrm{Re}=\dfrac{\rho^+ l_0 u_0}{\mu^+},
\end{equation}
the Weber number as
\begin{equation}
	\textrm{We}=\dfrac{\rho^+ l_0 u_0^2}{\sigma},
\end{equation}
and the Froude number as
\begin{equation}
	\textrm{Fr}=\dfrac{u_0^2}{g_0 l_0},
\end{equation}
where $g_0$ is the strength of gravity pointing in the $\bar{\vec{g}}$ direction. The Navier-Stokes equations now become
\begin{align}
	\rho(\phi)\dfrac{D\vec{u}}{Dt}=&-\nabla p+\dfrac{1}{\textrm{Re}}\nabla\cdot\left(\mu(\phi)\left(\nabla\vec{u}+\nabla^T\vec{u}\right)\right)\nonumber\\
			&-\dfrac{1}{\textrm{We}}\delta(\phi)\kappa\nabla\phi+\dfrac{1}{\textrm{Fr}}(\rho(\phi)-1)\bar{\vec{g}}.
\end{align}
Note that in this particular formulation gravity is written as $(\rho(\phi)-1)\bar{\vec{g}}$ so that uniform acceleration of the entire
domain does not occur.

\section{A Mass-Preserving Projection Method}
\label{sec:projection-method}
Projection methods are a common technique to solve the Navier-Stokes equations \cite{Brown2001}. In such schemes a tentative velocity field
is first determined and then the pressure is calculated to enforce the local volume conservation.
In addition to conservation of local volume, Eq. (\ref{eq:singleDiv}), it would be advantageous to explicitly enforce global volume conservation.
Even if local volume conservation is performed exactly errors in the advection of the interface can introduce unwanted
volume change over the course of a simulation. In this novel projection method both local and global volume conservation are enforced.

The first step is to determine a tentative velocity field at time $t^{n+1}$ in the absence of the pressure and forces.
In this work the semi-Lagrangian formulation is used:
\begin{equation}
	\rho\left(\phi^n\right)\dfrac{\vec{u}^*-\vec{u}^n_d}{\Delta t} = \dfrac{1}{\textrm{Re}}\nabla\cdot\left(\mu\left(\phi^n\right)\left(\nabla\vec{u}^*+\nabla^T\vec{u}^n \right)\right).
	\label{eq:SemiImplicitStep}
\end{equation}
The departure velocity, $\vec{u}^n_d$, for a grid point $\vec{x}$ is determined by tracing characteristics backward in time,
\begin{align}
	\vec{x}_d &= \vec{x} - \Delta t\vec{u}^n\left(\vec{x}\right), \\
	\vec{u}_d^n &= \vec{P}_u\left(\vec{x}_d,t^n\right),
\end{align}
where $\vec{P}_u$ is an interpolant of the velocity field at time $t^n$. Additional information regarding the semi-Lagrangian method for Navier-Stokes equations
is given in Ref \cite{Xiu2001}.

The next step of a standard projection method would be to determine the pressure to satisfy local volume conservation, Eq. (\ref{eq:singleDiv}).
In this work global conservation is also considered. Global volume conservation can be enforced by looking at the rate
at which the enclosed volume, $V$, will change over time \cite{Kolahdouz2015b,Laadhari2012}
\begin{equation}
	\int_\Gamma\vec{n}\cdot\vec{u}^{n+1} dA = \dfrac{dV}{dt}
	\label{eq:GlobalConservation}
\end{equation}
for an outward facing unit normal, $\vec{n}$, to the interface $\Gamma$.
Unfortunately, the pressure field alone can not satisfy both constraints. Therefore the pressure is split into a constant and spatially-varying component:
\begin{equation}
	p = \tilde{p} + (1-H(\phi))p_0.
\end{equation}
The constant pressure component, $p_0$, only has a contribution to the overall pressure field in the region given by $\phi<0$ and can be thought of enforcing
global volume conservation. The spatially-varying component, $\tilde{p}$, takes the place of the standard pressure and will be used to enforce
local volume conservation. Expanding the pressure component results in
\begin{equation}
	-\nabla p = -\nabla(\tilde{p}+(1-H(\phi^n))p_0) = -\nabla\tilde{p} + \delta(\phi^n)p_0\nabla\phi^n,
\end{equation}
which demonstrates that $p_0$ only has a contribution near the interface.

Using this definition of pressure the projection step is now
\begin{align}
	\dfrac{\vec{u}^{n+1}-\vec{u}^*}{\Delta t} =& -\dfrac{1}{\rho(\phi^n)}\nabla\tilde{p}+\dfrac{\delta(\phi^n)p_0\nabla\phi^n}{\langle\rho\rangle}\nonumber\\
				&-\dfrac{\delta(\phi^n)\kappa\nabla\phi}{\langle\rho\rangle\textrm{We}}+\dfrac{\rho(\phi^n)-1}{\rho(\phi^n)\textrm{Fr}}\bar{\vec{g}}.
	\label{eq:projection}
\end{align}
For quantities localized around the interface the average density is used, $\langle\rho\rangle=(\rho^+ + \rho^-)/2$.

To determine $\tilde{p}$ and $p_0$ both conservation conditions are applied to the projection step, Eq. (\ref{eq:projection}).
Applying local conservation, Eq. (\ref{eq:singleDiv}), results in
\begin{align}
	\nabla\cdot\left(\dfrac{1}{\rho(\phi^n)}\nabla\tilde{p}\right) - \dfrac{p_0}{\langle\rho\rangle}\nabla\cdot\left(\delta(\phi^n)\nabla\phi\right) = \nonumber\\
	\nabla\cdot\left(\dfrac{\vec{u}^*}{\Delta t}+\dfrac{\delta(\phi^n)\kappa\nabla\phi}{\langle\rho\rangle\textrm{We}}-\dfrac{\rho(\phi^n)-1}{\rho(\phi^n)\textrm{Fr}}\bar{\vec{g}}\right).
	\label{eq:localSolve}
\end{align}
Global conservation can be enforced by combining Eqs. (\ref{eq:GlobalConservation}) and (\ref{eq:projection}) and also
requiring that the time-evolution of the volume be such that any previous errors are corrected,
\begin{align}
	\int_\Gamma \dfrac{1}{\rho(\phi^n)}\vec{n}\cdot\nabla\tilde{p}\;dA-\dfrac{p_0}{\langle\rho\rangle}\int_\Gamma\delta(\phi^n)\|\nabla\phi\|\;dA =\nonumber\\
	\int_\Gamma\left(\dfrac{\vec{n}\cdot\vec{u}^*}{\Delta t} +\dfrac{\delta(\phi^n)\kappa\|\nabla\phi\|}{\langle\rho\rangle\textrm{We}}
		-\dfrac{\rho(\phi^n)-1}{\rho(\phi^n)\textrm{Fr}}\vec{n}\cdot\bar{\vec{g}}\right)dA\nonumber\\
		-\dfrac{1}{\Delta t}\dfrac{V^0-V^n}{\Delta t}
	\label{eq:globalSolve}
\end{align}
where $V^0$ is the initial enclosed volume.

Let $\tilde{\vec{p}}$ represent the vector holding the the values of $\tilde{p}$ over the entire discretized domain.
The surface integrals in Eq. (\ref{eq:globalSolve}) can be written as a summation over a discretized domain, $\int_\Gamma f dA\approx \sum\delta_{i,j,k}f_{i,j,k}dV$
where grid points are given by $\vec{x}_{i,j,k}$, $\delta_{i,j,k}=\delta(\phi(\vec{x}_{i,j,k}))$, $f_{i,j,k}$ is the function value at the grid points, and $dV$ is the
volume of each cell surrounding a grid point.
It is then possible to define the following
linear operators:
\begin{align}
	\label{eq:linear1} \nabla\cdot\left(\dfrac{1}{\rho(\phi^n)}\nabla\tilde{p}\right) \approx&\;\vec{L}\tilde{\vec{p}}, \\
	\label{eq:linear2} \dfrac{p_0}{\langle\rho\rangle}\nabla\cdot\left(\delta(\phi^n)\nabla\phi\right) \approx&\;p_0 \vec{l},\\
	\label{eq:linear3} \int_\Gamma \dfrac{1}{\rho(\phi^n)}\vec{n}\cdot\nabla\tilde{p}\;dA \approx&\;\vec{s}^T\tilde{\vec{p}}\\
	\label{eq:linear4} \dfrac{p_0}{\langle\rho\rangle}\int_\Gamma\delta(\phi^n)\|\nabla\phi\|\;dA \approx&\; p_0 a.
\end{align}
The two constraint equations, Eqs. (\ref{eq:localSolve}) and (\ref{eq:globalSolve}), can now be written in block matrix-vector form,
\begin{equation}
	\begin{bmatrix}
		\vec{L} & \vec{l} \\
		\vec{s}^T & a
	\end{bmatrix} \;
	\begin{bmatrix}
		\tilde{\vec{p}} \\
		p_0
	\end{bmatrix} =
	\begin{bmatrix}
		\vec{d} \\
		e
	\end{bmatrix},
\end{equation}
where $\vec{d}$ is the discretization of the right-hand-side of Eq. (\ref{eq:localSolve}) while $e$ is the evaluation of the integral on the right-hand-side of Eq. (\ref{eq:globalSolve}).
This particular block matrix form can be solved efficiently through the use of a Schur decomposition,
\begin{equation}
	\begin{bmatrix}
		\tilde{\vec{p}}\\
		p_0
	\end{bmatrix} =
	\begin{bmatrix}
		\vec{I} & 0 \\
		-a^{-1}\vec{s}^T & 1
	\end{bmatrix}
	\begin{bmatrix}
		\vec{S}^{-1} & 0 \\
		0 & a^{-1}
	\end{bmatrix}
	\begin{bmatrix}
		\vec{I} & -a^{-1}\vec{l} \\
		0 & 1
	\end{bmatrix}
	\begin{bmatrix}
		\vec{d}\\
		e
	\end{bmatrix}.
\end{equation}
The Schur complement is a rank-1 update on the matrix $\vec{L}$:
\begin{equation}
	\vec{S}=\vec{L}-\dfrac{1}{a}\vec{l}\vec{s}^T,
\end{equation}
which has an inverse given by the Sherman-Morrison formula,
\begin{equation}
	\vec{S}^{-1}=\vec{L}^{-1}+\dfrac{\vec{L}^{-1}\vec{l}\vec{s}^T\vec{L}^{-1}}{a-\vec{s}^T\vec{L}^{-1}\vec{l}}.
\end{equation}
So long as $a-\vec{s}^T\vec{L}^{-1}\vec{l}$ is non-zero the inverse is defined. This condition is not proven here, but a check is performed
during all simulations and has never been violated. Note that as the method as described does not depend on any particular spatial
discretization. The particular discretization used in the numerical experiments below will be discussed in Sec. \ref{sec:projectionDiscrete}.

\section{Numerical Implementation}
\label{sec:numerics}
In this section the particular numerical implementation used in the numerical experiments of Sec. \ref{sec:experiments} is briefly discussed.
In particular the description and advection of the interface and the discretization of the projection method is presented.
\subsection{Interface Description and Advection}

The interface separating the multiple fluid phases is described here using a gradient-augmented level set method \cite{Nave2010,Seibold2012}. In this extension
of the level set method the gradient field of the level set is explicitly tracked in addition to the level set function. This allows for the accurate
determination of interfacial quantities such as curvature \cite{Nave2010, Seibold2012}. In particular the results shown below use a recent semi-implicit
extension of the original level set Jet scheme \cite{Velmurugan2015}.

Consider the implicit tracking of an interface by way of a level set function: $\Gamma(t)=\{\vec{x}:\phi(\vec{x},t)=0\}$. In addition to the level set
also track the gradient field of the level set, $\vec{\psi}=\nabla\phi$. This allows for the determination of Hermite interpolants using only
information local to a cell on a Cartesian grid \cite{Nave2010}. Unfortunately, the original Jet scheme does not work well for stiff
advection equations, such as those involving surface tension, and the recent SemiJet method was developed in response \cite{Velmurugan2015}.

The SemiJet method begins by performing a semi-implicit, semi-Lagrangian update on the level set field,
\begin{equation}
	\dfrac{\phi^{n+1}-\phi^n_d}{\Delta t}=\beta\nabla^2 \phi^{n+1}-\beta\nabla^2 \phi^n,
	\label{eq:semijet-phi}
\end{equation}
where $\phi^n_d$ is the level set value at the departure location and $\beta=0.5$ is a constant. \

To update the gradient field note that the smoothing operation can be captured using a smoothing operator,
\begin{equation}
	S_\phi=\beta\nabla^2 \phi^{n+1}-\beta\nabla^2 \phi^n,
	\label{eq:semijet-source}
\end{equation}
where $\phi^{n+1}$ and $\phi^n$ are both known at grid points after solving Eq. (\ref{eq:semijet-phi}). At each grid point define a sub-grid
of spacing $\epsilon\ll h$, where $h$ is the grid spacing:
\begin{equation}
	\vec{x}^{\vec{q}}=\vec{x}+\vec{q}\epsilon \textrm{ for } \vec{q}\in\{-1,1\}^p,
\end{equation}
with $p=2$ in two-dimensions and $p=3$ in three-dimensions. This results in a sub-grid of either 4 or 8 points, depending on the dimension of the simulation.
On each of these sub-grid points an updated level-set value is obtained using an explicit semi-Lagrangian update with the smoothing source term
given in Eq. (\ref{eq:semijet-source}),
\begin{equation}
	\dfrac{\phi^{\vec{q}}-\phi^{\vec{q}}_d}{\Delta t}=S_{\phi}\left(\vec{x}^{\vec{q}}\right).
	\label{eq:semijet-subgrid}
\end{equation}
Once the level set values on the subgrid are calculated updated gradient values at grid points can be obtained using finite difference approximations. For example,
in two-dimensions the gradient in the $x$-direction at a grid is given by $\left(\phi^{(1,1)}-\phi^{(-1,1)}+\phi^{(1,-1)}-\phi^{(-1,-1)}\right)/4\epsilon$.

Curvature values are calculated at grid points using
\begin{equation}
	\kappa = \frac{\phi_{xx}\phi_y^2+\phi_{yy}\phi_x^2-2\phi_{xy}\phi_x\phi_y}{\left(\phi_x^2+\phi_y^2\right)^{3/2}},
\end{equation}
in two-dimensions with a similar expression for three-dimensions.
The values used to calculate the curvature were calculated by averaging derivatives. For example, when computing $\phi_x$, $\phi_{xx}$ and $\phi_{xy}$
for use in the curvature calculation the following averaging would be used:
\begin{align}
	\phi_x &= \tfrac{1}{2}\left(\psi^x + D_x\phi\right),\\
	\phi_{xx} &= \tfrac{1}{2}\left(D_x\psi^x + D_{xx}\phi\right),\\
	\phi_{xy} &= \tfrac{1}{3}\left(D_y\psi^x + D_x\psi^y + D_{xy}\phi\right),
\end{align}
where $D_x$, $D_y$, $D_{xx}$ and $D_{xy}$ are the finite difference approximations for the derivatives and $\psi^x$ and $\psi^y$ are the components of the
gradient field tracked by the Jet. The curvature was then extended away from the interface in the normal direction.
Complete information about the SemiJet method and curvature calculation can be found in Ref. \cite{Velmurugan2015}.

\subsection{Discretization of the Projection Method}
\label{sec:projectionDiscrete}
The Navier-Stokes equations and projection method are discretized on a collocated Cartesian mesh using standard finite difference approximations. The variable coefficient
Poisson equation, Eq. (\ref{eq:linear1}), is discretized using compact second-order finite difference approximations. In one-dimension this is written as
\begin{equation}
	\vec{L}\tilde{p}_{i}=\dfrac{\rho^{-1}_{i+1/2}\left(\tilde{p}_{i+1}-\tilde{p}_{i}\right)-\rho^{-1}_{i-1/2}\left(\tilde{p}_{i}-\tilde{p}_{i-1}\right)}{h^2}
\end{equation}
where the inverse density at half-grid points is obtained using harmonic averaging,
\begin{equation}
	\rho^{-1}_{i+1/2}=\dfrac{\rho_{i}+\rho_{i+1}}{2\rho_{i}\rho_{i+1}}.
\end{equation}
It was found that the harmonic averaging improved the stability for large density variations. Similar expressions hold in other dimensions.
The remaining discretizations in Eqs. (\ref{eq:linear2})-(\ref{eq:linear4}) were also performed using centered second-order finite difference approximations.

Note that the method described here is an approximate pressure discretization. It is well known that this particular discretization does not suffer from
the pressure decoupling effect which can occur from a discrete discretization on a collocated mesh \cite{PERIC1988,Rauwoens2007}. On the other hand the fact that
the divergence-free velocity field condition is not discretely enforced means that small mass errors will accumulate over time. An approximate discretization was
purposefully chosen to demonstrate the ability of the proposed method to conserve mass even if the underlying method is not discretely mass conserving.

\section{Numerical Experiments}
\label{sec:experiments}
The mass-preserving projection method described above is tested using four different multiphase fluid systems. These are: 1) a static bubble, 2) a rising bubble, 3) a
fluid thread undergoing a Plateau-Rayleigh instability, and 4) a droplet in shear flow. In all cases the mass-preserving method, denoted as the conserving scheme, is compared
to a simulation without the method described above. This non-conserving scheme is computed by using the same discretization as the conserving scheme with
the pressure calculated as the solution to Eq. (\ref{eq:localSolve}) with $p_0=0$.
All of the results will be three-dimensional results using uniform grid spacing, although some figures will show two-dimensional slices instead of three-dimensional figures.

\subsection{A Static Bubble}
The first numerical experiment is the evolution of a droplet in the absence of gravity, Fig. \ref{fig:StaticBubbleEvolution}.
The viscosity ratio is set to $\eta=0.01$ while the density ratio
is $\lambda=0.001$. The Reynolds number is 10 and the Weber number is 1. Gravity is ignored, which is equivalent to setting $\textrm{Fr}=\infty$.
The initial shape is an ellipsoidal interface with an axis of 1 in the $x-$ and $z-$directions and 2 in the $y-$direction.
The computational domain is a cube given by $[-4,4]^3$ and a $65^3$ grid, resulting in a grid spacing of $h=0.125$. The time step is $\Delta t=0.01$.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[t=0]{
			\includegraphics[width=1.0in]{StaticBubble_3D_00000}
		}
		\subfigure[t=0.5]{
			\includegraphics[width=1.0in]{StaticBubble_3D_00005}
		}
		\subfigure[t=1]{
			\includegraphics[width=1.0in]{StaticBubble_3D_00010}
		}\\
		\subfigure[t=2]{
			\includegraphics[width=1.0in]{StaticBubble_3D_00020}
		}
		\subfigure[t=10]{
			\includegraphics[width=1.0in]{StaticBubble_3D_00100}
		}
		\subfigure[t=20]{
			\includegraphics[width=1.0in]{StaticBubble_3D_00200}
		}
		\caption{Evolution of a bubble under surface tension with $\textrm{Re}=10$, $\textrm{We}=1$, and $\textrm{Fr}=\infty$. The density ratio is $\lambda=0.001$ while
		the viscosity ratio is $\eta=0.01$. The final sphere radius is 1.261.}
		\label{fig:StaticBubbleEvolution}
	\end{center}
\end{figure}

Under the influence of surface tension the bubble evolves into a sphere with the same volume. It is therefore expected that
the final shape will be a sphere with a radius of 1.26. The simulated final radius is 1.261, very close to the theoretical value.

The distribution of the spatially-varying pressure component, $\tilde{p}$,
and the total pressure $p$ along the $x-$axis for $y=0$ and $z=0$ at a time of $t=20$
are shown in Fig. \ref{fig:StaticBubblePressure}. Results are similar in the other directions.
It is clear that at steady-state the majority of the pressure is due
to $p_0$, which has a value of $p_0=1.584$.
According to the Laplace-Young condition it is also expected that the difference between the inner and outer
pressure should be $p^- -p^+=2\kappa/\textrm{We}$. Using an analytic final radius of 1.26 and the simulation
parameters this results in an expected pressure jump of 1.587, which matches well with the simulated final pressure.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[Spatially Varying Pressure Component: $\tilde{p}$]{
			\includegraphics[width=2.75in]{StaticBubble_PTilde_00200}
		}
		\subfigure[Total Pressure: $p=\tilde{p}+(1-H(\phi))p_0$]{
			\includegraphics[width=2.75in]{StaticBubble_P_00200}
		}
		\caption{The pressure along the $x$-axis with $y=0$ and $z=0$ at a time of $t=20$ for the static bubble shown in Fig. \ref{fig:StaticBubbleEvolution}.
		The spatially varying pressure component, $\tilde{p}$, is nearly zero while the pressure constant is $p_0=1.584$. This is very near to the predicted static pressure of 1.587.}
		\label{fig:StaticBubblePressure}
	\end{center}
\end{figure}

A comparison between the mass conserving and non-mass conserving simulations is given in Fig. \ref{fig:StaticBubbleVolume}. Both the volume and volume-error over the
course of the simulation are presented. The volume error is computed as $(V(t)-V(0))/V(0)$ where $V(t)$ is the instantaneous volume and $V(0)$ is the initial volume.
The use of an approximate projection method, combined with the non-conservation properties of the level set method, results in excessive mass loss in the non-conserving
simulation. This mass loss results in the complete disappearance of the bubble in the non-conserving simulation, while the conserving scheme has a maximum volume
error of approximately $2\times 10^{-5}$.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[Volume]{
			\includegraphics[width=2.75in]{StaticBubble_Volume}
		}
		\subfigure[Volume Error]{
			\includegraphics[width=2.75in]{StaticBubble_VolumeError}
		}
		\caption{Comparison between volume preservation in the conserving and non-conserving schemes for a static bubble with the initial condition shown in Fig. \ref{fig:StaticBubbleEvolution}.
			After $t=15$ the bubble has completely disappeared in the non-conserving simulation.}
		\label{fig:StaticBubbleVolume}
	\end{center}
\end{figure}

\subsection{A Rising Bubble}
The next numerical experiment is that of a rising bubble.
The computational domain is a cube given by $[-5,5]^3$ using a $101^3$ grid, giving a grid spacing of 0.1. The time step is chosen to be $0.01$.
The initial bubble is a sphere centered at the origin with a radius of 1.

To allow for the long-time evolution of the bubble to be modeled after every time step the level set Jet and complete fluid field are re-centered in the computational
domain using a cubic interpolation procedure. Using the values for the spherical cap Bubble A in Table 1 of Hnat and Buckmaster \cite{HNAT1976}
the density ratio is set to $\lambda=0.001142$ while the viscosity ratio is $\eta=0.008474$. Based on the dimension of the experimental bubble, the final
rise velocity, and the given surface tension the dimensionless parameters are $\textrm{Re}=9.694$, $\textrm{We}=7.6375$ and $\textrm{Fr}=0.7754$.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[t=0]{
			\includegraphics[width=1.0in]{BubbleRise_3D_00000}
		}
		\subfigure[t=1]{
			\includegraphics[width=1.0in]{BubbleRise_3D_00010}
		}
		\subfigure[t=2]{
			\includegraphics[width=1.0in]{BubbleRise_3D_00020}
		}\\
		\subfigure[t=4]{
			\includegraphics[width=1.0in]{BubbleRise_3D_00040}
		}
		\subfigure[t=10]{
			\includegraphics[width=1.0in]{BubbleRise_3D_00100}
		}
		\subfigure[t=20]{
			\includegraphics[width=1.0in]{BubbleRise_3D_00200}
		}
		\caption{Evolution of a three-dimensional bubble with density ratio of $\lambda=0.001142$ and viscosity ratio of $\eta=0.008474$ with $\textrm{Re}=9.694$,
		$\textrm{We}=7.6375$ and $\textrm{Fr}=0.7754$. These properties correspond to Bubble A in Table 1 in Hnat and Buckmaster \cite{HNAT1976}. Experimental results show the aspect
		ratio should be 2.7 while the simulation results in an aspect ratio of 2.8.}
		\label{fig:BubbleRiseEvolution3D}
	\end{center}
\end{figure}

\begin{figure}[!ht]
	\begin{center}
		\subfigure[t=0]{
			\includegraphics[width=1.0in]{BubbleRise_XY_00000}
		}
		\subfigure[t=1]{
			\includegraphics[width=1.0in]{BubbleRise_XY_00010}
		}
		\subfigure[t=2]{
			\includegraphics[width=1.0in]{BubbleRise_XY_00020}
		}\\
		\subfigure[t=4]{
			\includegraphics[width=1.0in]{BubbleRise_XY_00040}
		}
		\subfigure[t=10]{
			\includegraphics[width=1.0in]{BubbleRise_XY_00100}
		}
		\subfigure[t=20]{
			\includegraphics[width=1.0in]{BubbleRise_XY_00200}
		}
		\caption{Evolution of the $x-y$ plane of a three-dimensional bubble for the result shown in Fig. \ref{fig:BubbleRiseEvolution3D}.}
		\label{fig:BubbleRiseEvolution}
	\end{center}
\end{figure}

As the bubble rises it evolves into the expected spherical cap, see Fig. \ref{fig:BubbleRiseEvolution3D} for a three-dimensional figure and Fig. \ref{fig:BubbleRiseEvolution}
for the cross-section in the $x-y$ plane. The experimental results indicate that the aspect ratio of the bubble at steady-state should be 2.7, which compares to a value of 2.8 from
the simulation. With the given parameter set it is expected that the dimensionless rise velocity should be equal to one. The rise velocity over the course of the
simulation is given in Fig. \ref{fig:BubblRiseVelocity}, with a steady-state rise velocity of 0.9922.
\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=2.75in]{BubbleRise_RiseVelocity}
		\caption{The rise velocity of the center of mass for the three-dimensional bubble in Figs. \ref{fig:BubbleRiseEvolution3D} and \ref{fig:BubbleRiseEvolution}. The experimental results of Hnat and
		Buckmaster \cite{HNAT1976} indicate the final dimensionless rise velocity should be 1. The simulation results in a rise velocity of 0.9922.}
		\label{fig:BubblRiseVelocity}
	\end{center}
\end{figure}

The evolution of the volume over the course of the simulation is presented in Fig. \ref{fig:RisingBubbleVolume}. Note that due to discretization errors
the initial volume is not equal to $4\pi/3\approx 4.19$ but is calculated numerically as 4.252.
As with the static bubble example both the volume and the
volume error are given. Over the course of the simulation the volume for the non-conserving scheme varies from a minimum of 4.13 to a maximum of 4.39, while the
conserving scheme only varies slightly. It should also be obvious that if the simulation was allowed to continue volume errors in the non-conserving scheme
would continue to grow, which is not true for the conserving scheme.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[Volume]{
			\includegraphics[width=2.75in]{BubbleRise_Volume}
		}
		\subfigure[Volume Error]{
			\includegraphics[width=2.75in]{BubbleRise_VolumeError}
		}
		\caption{Comparison between volume preservation in the conserving and non-conserving schemes for a rising bubble with the initial condition shown
			in Figs. \ref{fig:BubbleRiseEvolution3D} and \ref{fig:BubbleRiseEvolution}.
			The volume gain will continue in the non-conserving scheme while the volume will remain constant in the conserving scheme.}
		\label{fig:RisingBubbleVolume}
	\end{center}
\end{figure}

\subsection{Plateau-Rayleigh Instability}
As a third test case consider the evolution of a fluid thread. It is well known that if the shape has an initial perturbation this thread will
undergo a Plateau-Rayleigh instability and split into separate droplets \cite{Rayleigh1878}. In this case the computational domain is a rectangle spanning
$[-5,5]\times[-10,10]\times[-5,5]$ while the number of grid points is $101\times 201\times 101$ and the time step is $\Delta t=0.05$. Periodic boundary conditions are given in all directions.

Initially a thread is aligned along the $y$-direction with an initial radius given by $1+0.2\cos(0.2\pi y)$ in the $x-z$ plane.
It is then allowed to evolve under the conditions of
$\textrm{Re}=10$, $\textrm{We}=1$, and $\textrm{Fr}=\infty$ with a density ratio of $\lambda=10$ and viscosity ratio of $\eta=10$.
The three-dimensional evolution of the interface is given in Fig. \ref{fig:PREvolution}. As time progresses the initially thin regions of the thread
become thinner. At a time of $t=28$ the thread is about to pinch off and extremely thin regions are observed between individual droplets, which are visible
at $t=28.5$. Over time the droplets become spherical to minimize the surface tension.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[t=0]{
			\includegraphics[width=1.5in]{PR_3D_00000}
		}
		\subfigure[t=20]{
			\includegraphics[width=1.5in]{PR_3D_00040}
		}\\
		\subfigure[t=28]{
			\includegraphics[width=1.5in]{PR_3D_00056}
		}
		\subfigure[t=28.5]{
			\includegraphics[width=1.5in]{PR_3D_00057}
		}\\
		\subfigure[t=30]{
			\includegraphics[width=1.5in]{PR_3D_00060}
		}
		\subfigure[t=100]{
			\includegraphics[width=1.5in]{PR_3D_00200}
		}
		\caption{Evolution of a fluid thread with a density ratio of $\lambda=10$, viscosity ratio of $\eta=10$, $\textrm{Re}=10$, $\textrm{We}=1$, and $\textrm{Fr}=\infty$.
		Periodicity is assumed in all directions. With the given initial shape the droplet undergoes a Plateau-Rayleigh instability and breaks into spherical individual droplets.}
		\label{fig:PREvolution}
	\end{center}
\end{figure}

A comparison between the conserving and non-conserving schemes for the Plateau-Rayleigh instability is presented in Fig. \ref{fig:PRVolume}. As with the static bubble
case the use of the non-conserving scheme results in complete volume loss. The conserving scheme has a maximum mass loss during the breakup observed
at approximately $t=28.5$, but quickly recovers to an error of less than $10^{-6}$.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[Volume]{
			\includegraphics[width=2.75in]{PR_Volume}
		}
		\subfigure[Volume Error]{
			\includegraphics[width=2.75in]{PR_VolumeError}
		}
		\caption{Comparison between volume preservation in the conserving and non-conserving schemes for a fluid thread undergoing a Plateau-Rayleigh instability
			with the initial condition shown in Fig. \ref{fig:PREvolution}.
			Due to volume loss the droplets in the non-conserving scheme disappear.}
		\label{fig:PRVolume}
	\end{center}
\end{figure}

\subsection{Droplet in Shear Flow}

As a final test consider the behavior of a droplet in the presence of shear flow. Under certain situations the droplet will extend and ``daughter" droplets may
form at the tip of the original droplet \cite{Li2000}. Here the computational domain spans $[-16,16]\times[-2,2]\times[-2,2]$ with a $513\times 65\times 65$ grid.
The time step is $\Delta t=0.025$. Periodic boundary conditions are taken in the $x-$ and $z-$ directions while wall-boundary conditions are taken in the $y-$direction.
The shear flow is imposed by specifying a velocity of $(y, 0, 0)$ on the $y=2$ and $y=-2$ walls.

The result for matched density and viscosity, $\lambda=1$ and $\eta=1$, with $\textrm{Re}=1$, $\textrm{We}=1$, and $\textrm{Fr}=\infty$ for an initially
spherical droplet of radius 1 is given in Fig. \ref{fig:ShearEvolution}. This figure shows the $x-y$ cross-section of the droplet up to the formation
of the first ``daughter" droplets from the tips.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[t=0]{
			\includegraphics[width=2.75in]{Shear_XY_00000}
		}\\
		\subfigure[t=5]{
			\includegraphics[width=2.75in]{Shear_XY_00020}
		}\\
		\subfigure[t=10]{
			\includegraphics[width=2.75in]{Shear_XY_00040}
		}\\
		\subfigure[t=15]{
			\includegraphics[width=2.75in]{Shear_XY_00060}
		}\\
		\subfigure[t=20.25]{
			\includegraphics[width=2.75in]{Shear_XY_00081}
		}	\\
		\subfigure[t=20.5]{
			\includegraphics[width=2.75in]{Shear_XY_00082}
		}
		\caption{Evolution of a droplet in the $x-y$ plane under shear flow. The density ratio is $\lambda=1$, the viscosity ratio is $\eta=1$, while $\textrm{Re}=1$,
		$\textrm{We}=1$, and $\textrm{Fr}=\infty$. The normalized shear rate is 1.}
		\label{fig:ShearEvolution}
	\end{center}
\end{figure}

Tracking the volume and volume error of the course of the simulation, Fig. \ref{fig:ShearVolume}, demonstrates results similar to the previous numerical tests. Over time the errors in the non-conserving
scheme accumulate and results in the complete disappearance of the droplet. The conserving scheme, on the other hand, is capable of ensuring that the mass errors do not grow
over time.
\begin{figure}[!ht]
	\begin{center}
		\subfigure[Volume]{
			\includegraphics[width=2.75in]{Shear_Volume}
		}
		\subfigure[Volume Error]{
			\includegraphics[width=2.75in]{Shear_VolumeError}
		}
		\caption{Comparison between volume preservation in the conserving and non-conserving schemes for the droplet in shear flow
			with the initial condition shown in Fig. \ref{fig:ShearEvolution}.
			Due to volume loss the droplet in the non-conserving scheme disappears.}
		\label{fig:ShearVolume}
	\end{center}
\end{figure}

\section{Conclusion}
\label{sec:conclusion}

In this work a new mass conserving projection method for multiphase Navier-Stokes systems has been presented. Various numerical tests have demonstrated that the
proposed scheme is capable of conserving the mass in a wide variety situations, including a static bubble, a rising bubble,
a fluid thread undergoing a Plateau-Rayleigh instability and a droplet in shear flow.
Unlike many other mass correction schemes the material interface advects with the underlying fluid field. The general idea is applicable to any
system being investigated using a projection-based Navier-Stokes solver and is not limited to the particular discretization used here.

\section*{Acknowledgments}
	This work has been supported by the National Science Foundation through the Division of Chemical, Bioengineering, Environmental, and Transport Systems Grant \#1253739.

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A Short History of Early Modern England presents thehistorical and cultural information necessary for a richerunderstanding of English Renaissance literature.
Peter C. Herman is Professor of English and Comparative Literature, San Diego State University, USA. His previous books include Royal Poetrie: Monarchic Verse and the Political Imaginary of Early Modern England (2010) and Destabilizing Milton: "Paradise Lost" and the Poetics of Incertitude(2008). His essays have appeared in Renaissance Quarterly, SEL, and Criticism.


Objectives, Introduction, Programs, Programming Language, Interpretation and Compilation, Programming Methodology, Bottom Up Approach, Top Down Approach, Programming Paradigms, Structured Programming, Procedural Programming, Object Oriented Programming, Basic Concepts of OOP, Advantage of Object Oriented Programming Language (OOP), Features of Object Oriented Programming Language, Functional Programming, Logic Programming, Summary, Review Questions.
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Current progress through the dungeon.
The Enclave of the Fairy Host is a dungeon that Kickstarter backers explore by completing achievements. Progressing through rooms can unlock additional backer rewards and progresses the story further. Along the way, there may be interactive decisions that will modify where the party goes.
Outside the Dungeon: Lioslaith leads you into a clearing and points her spear at the burial mound in its center. The barrow's towering stone arch stands open, revealing a dark corridor. The door that sealed it lies shattered before it.
"These mounds were old when the Fichti were new," she says. "My ancestors blocked that door ten generations past. It was broken open two days ago."
A cold wind pierces your armor as you enter the barrow. The first room is empty but for cobwebs and mould, but footprints in the dust lead further in.
"Your enemies have got here before you, perhaps," said Lioslaith.
The remains of a campfire blacken the floor in the center of this room. Chicken bones have been tossed in a corner, and crude Einarr runes have been scratched on the north wall. Galli, your dwarf head-knocker, translates them.
"Olf, we could wait no more. We've gone ahead."
There is nothing in the third chamber but a long smear of blood that ends at closed door. Behind the door you hear guttural muttering.
Lioslaith readies her spear. "Prepare yourselves, friends."
As you enter, a pair of trolls look up from dining on a corpse in Einarr armor. Your bard, Fiona, winces at the sight. "Looks like Olf should have stayed with the group."
The trolls stand, roaring, and toss aside tattooed arms and legs, then take up their clubs and charge you. A treasure chest glints in the darkness behind them.
With axe and sword and fire, you finally defeat the trolls, but cannot find the key to the chest. Then Shanko, your Trow thief, pipes up. "The daft bastards probably ate it!"
It's not the most pleasant task, but you gut the trolls and find the key in their intestines, still clenched in the fingers of a severed hand.
Shanko snatches it from you and opens the chest. "Loot!"
But as you survey your glittering prize, a faint melody reaches your ears. It sounds like someone singing in the depths of the barrow.
[Unlocked] - This room unlocks a physical code wheel to *all* physical tiers, including $65. Digital tiers will receive a digital representation of this code wheel.
It takes an hour and many blows from Galli's warhammer to break down the door to this room. It was barricaded from behind, perhaps against the trolls? A few Einarr helms and burnt out torches litter the floor.
You find a rolled up parchment near the door to the next room. Drawn upon it is what looks like a star field with lines between the stars. Your mage, Anders the Old, says they are elven constellations.
One of those constellations is carved into the door to the next room.
The floor of this room continues the star theme. It is patterned with the remaining constellations. It is also patterned with dead Einarr, all seemingly impaled by hidden traps in the floor.
By trial and error and constant checking with the parchment star map, you evade the traps and make it to the far side.
The singing comes again as you dodge the last line of stakes. Lioslaith hears it this time. She shudders. "Fairy song. Hear it three times and you are never seen again."
This room is bare and cold - so cold your breath comes out in freezing clouds. Frost starts to rime Galli's beard and moustache.
"Why is it so cold?" asks Shanko.
"Ghosts," said Siolaith. "It chills the bones to pass through their forms."
"Then there must be scores of them," says Fiona. "I'm chilled to the bloody marrow!" She cuts off as a strange clattering sound comes from the next room.
This room is knee deep in rusted weapons and human bones, and wading though them is a man in Einarr armor, picking up skulls and jeweled finger bones, then casting them aside. "Which of you is Mangar?" he mutters. "Which of you knows..." Suddenly he cries out and digs deep into the bones, then holds high a silver crown. "Is this the Crown of Mangar? Have I–"
He spots you and gasps, then drops the crown and bolts for the far door. "Hugin! Graef! They found us!"
Impossibly, the crown does not fall when the Einarr lets it go. It hangs in the air and an ethereal form coalesces beneath it, while all around it the bones begin to rattle and rise.
Before you can react, the room is full of skeletons, all hefting ancient swords and spears. Behind them, the wraith hisses and points at you, and they all clatter forward.
It is a desperate battle, but eventually, choking on bone dust and bleeding from rust flecked cuts, you defeat the army of the dead while Anders vanquishes the wraith with a spell of banishment.
There is no treasure chest in this room, but the wraith's silver crown and the rings on the fingers of the skeletons are ample reward.
There is nothing in this room but a door on the far wall, and a hole in the floor the size of a gold coin. Thinking the hole might be the catch for a secret door, Shanko probes it with his dagger. There's a click and greenish liquid starts to well from the hole.
"Oops." Shanko pulls his dagger out. The blade dissolves before his eyes. "Uh-oh."
You flee to the door. There appears no way to move it.
"Open it, Trow!" growls Galli. "You started this. Now get us out of it."
As Shanko starts poking and prodding the door, looking for a lever, the rest of you look uneasily at the ever spreading pool of acid.
Just as the acid is lapping at your boot heels, Shanko touches five points on the door one after the other and it slides open. You stumble into the next room and it slams closed behind you. You are in a treasure room beyond the dreams of avarice. Chests full of gold, gems, cups of crystal, and swords with jeweled pommels glitter in every corner of the room.
"Stop!" shouts Anders. "All is illusion! Look beyond it."
His words make you look again, and you see through the shimmer to discover that where the chests stand, there is actually no floor. You look down into the traps. They go down forever.
"Don't suppose we're lucky enough that the Einarr fell down those holes," says Fiona.
Lioslaith points to foot prints on the floor the go through the far door. "They are still ahead of us."
This room is half the height of the others, and has an iron grate for a floor. You hurry across it, afraid that at any moment you will be crushed between them.
That fear becomes panic as the ceiling begins to creak.
Panic drives you into this room in a blind rush, which is unfortunate, because you find yourself facing ninety-nine Einarr berserkers. Well, okay, only nine, but that's plenty. They're all well over six feet tall and frothing with rage.
Behind them, the Einarr you saw in the room of bones stands on a wheeled cart loaded with looted treasure. "Get them, brothers!" he shouts. "I'll guard the spoils!"
You turn to run back into the previous room and find that its ceiling is now waist high and dropping rapidly.
"We shouldn't be running anyway," says Shanko. "Running's for elves. Attack!" And with that he charges straight at the berserkers as they thunder forward.
"Scatter!" cries Anders, and casts a spell of darkness and a spell of cat's eyes in quick succession. You dodge the berserkers as they blunder blindly past you, then attack them from behind, your enchanted eyes easily piercing the sudden night. It's totally unfair.
In moments, the treasure is yours. But where did the cowardly Einarr go?
[Unlocked] - This room unlocks the Red Boots DLC, free for all backers $20 or above.
You pile the treasure you already won onto the cart, and push it into the next room, which is blissfully empty of berserkers. The cart barely moves.
"It's too heavy," says Anders. "We'll have to come back for it."
"Never!" cries Shanko. "I'll die before I leave the loot behind!"
"That's a distinct possibility," says Fiona.
"You want to bring it along," says Galli. "You push it."
The rest of you cross to door in the far wall as Shanko struggles to move the cart. He gives up and rejoins you just as you figure out how to open it. "Fine," he says, "But if all our loot is gone when we get back, I'm blaming you."
As the door slides aside, you hear the singing again, far in the distance.
You don't tell her you've already heard it three times.
Stepping into the room you see another party of adventurers stepping into it from the far side, and you go on guard. So do they. They're a rough looking bunch - a shady looking trow, a gaunt mage, a dwarf with an oft-broken nose, a female bard in ragged armor, and a Fichti spear daughter.
"It's a mirror!" laughs Fiona.
"Aye," says Galli. "But how do we get through it."
"Easy!" says Shanko, and hurls a dagger at it.
The dagger bounces off, and the mirror remains unscratched. Everybody else gives it a try too. Not one of you can damage it in the slightest.
"Wait a minute," says Fiona. "This reminds me of an old song." She sings it. "'If you see yourself acomin', ago the other way.'"
"But we've been that way," says Shanko.
"Humor me," says Fiona, and leads you back to the door you just came through. There are runes on it, but they're reversed.
"You may be on to something," say Anders. "These runes are a translation of the line you just sang."
On a hunch, Fiona sings the next line of the song, "For if you let you touch you, you'll not live another day."
The room is dark and empty, but Fiona's singing bounces back from the walls in an echo that doesn't fade away.
You hurry to the far door.
In the center of this room sits a stone sarcophagus surrounded by large funerary urns. Shanko rubs his hands with glee at the sight. "Smash 'em! Smash 'em! There's always loot in urns!"
He runs to the closest and cracks it with the pommel of his dagger. It shatters, but does not spill earth, bones and treasure. Instead an urn-sized chitinous form unfolds and rises to its eight legs.
"Giant spider!" shouts Fiona. "Kill it!"
You move to surround the spider, but strange scraping sounds distract you. The tops of all the other urns are rising up and falling aside, and saber-like spider legs are sprouting from the tops. The spiders rock forward and stand, their abdomens wearing the urns like armor. Venom drips from their fangs.
Anders snarls at Shanko. "From now on, no more smashing urns!"
Galli wades in to the spiders with his warhammer, smashing urns and shattering chitin, as Shanko and Lioslaith follow, guarding his sides and parrying slashing spider legs.
Fiona spurs them on with a ancient walking song while Anders casts a spell of entropy to make the spiders shells brittle and stiff.
Even with this help it's tough fight, and the party is sick with poison and weak from blood loss by the time the last spider falls.
Anders is on the floor, clutching a belly wound. "I... I'm not going to make it."
Fiona clutches her throat. "I don't think any of us is going to make it."
"Let's see what we died for," says Galli, and with a tremendous push, shoves the stone cover from the sarcophagus.
The ghostly form of an elven woman rises above the stone coffin. She looks around at you all. "Your death nears, interlopers, but it is not inevitable. Agree to repent your larceny and rid these halls of the vile Einarr who desecrate them, and you shall be saved. Refuse and this room will be your grave.
"Yes!" cries Anders from the floor. "Save us and we'll save you!"
"Hang on," says Fiona. "Let's talk terms first. Just how much repenting–"
But the ghostly elf is already spreading her hands. "Then live again, but know that your lives are forfeit if you break this compact."
A glow like sunlight and first love spreads through the room and your wounds heal as if they were never there. The fire of the spiders' venom fades from your veins.
"Further in, friends," echoes the elf woman's voice. "Further in."
[Unlocked] - This room unlocks 2 Julie Fowlis songs: the song from the in-engine video, and "Smeòrach Chlann Dòmhnaill (The Mavis of Clan Donald)" from her latest album, Gach Sgeul for all backers $20 or above.
You enter the next room arguing so fiercely you hardly notice your surroundings.
"I was dying!" says Anders. "Of course I took her offer."
"We were all dying," counters Fiona. "We still could have made a better deal."
"What's done is done," say Galli, then turns a hard eye on Shanko. "Just don't steal anything!"
"Okay, okay!" snaps the trow. "I don't know why you're looking at me. We're all thieves here."
With the argument over - for now - you finally take in the crumbling elven mosaics that cover the walls of this room in dusty green and gold hunting scenes. Elves on red stags charge through a forest with white-eyed hounds baying before them. It is slightly disconcerting to see that their quarry is... a man.
Voices reach you from deeper within the catacombs. It isn't the mournful female singing this time. It sounds like chanting.
The elven mosaics in this room depict a feast in an underground banquet hall with tree roots for pillars and witch-lights for lanterns. Elves in beautiful clothes eat and drink and listen to an elven bard. All the servants are humans. An elf maiden has a man for a pet. He curls on a pillow at her feet.
The chanting is louder here. You can distinguish the words now. You recognize them as ancient Einarr.
Unlock the all-new Exclusive Brian Fargo Bard Skin.
This page was last edited on 9 July 2015, at 13:35.


The vLink Computer System approach to skiing is new and rather novel. It's a data collection sensor set (pictured above) that clips to the front of a pair of skis, and in real-time, monitor in real-time forward speed and lateral displacement data of the skis as the skier proceeds down a mountain.
The system is aimed at professional skiers who desire real-time information on their progress. A wireless unit allows real time performance management playback, but the devices can happily save this data for later use.
"Timely and accurate feedback of athlete performance is a key contributing factor to improving athletic performance. Alpine skiers get their feedback primarily through coaches' comments and video analysis, which usually comes a few hours after the performance. With this new system, skiers are able to make immediate corrections to their technique and have instant feedback of their adjustment", said the creator, Richard Kirby from Advanced Racing Computers, in the US.
However, it is their ability to record the displacement of the skis, and the angling process that is of interest not to racers, but as data collection for actual skiing processes. Knowing what happens with actual skiers, and having verifiable data to prove it, it becomes possible for the first time, if the skier allows the data to be released, for those in their homes and training centers using wither fitness or training fitness VR an important advantage.
It allows them to see where the pros are putting their feet and their weight on a course. A sensor system for improving the performance of those who seek to emulate their heroes, with a data collection system that a VR can use.


text:
Damian Iglesias Vilela (ur. 28 lipca 1993) – hiszpański zapaśnik walczący w stylu wolnym. Zajął piąte miejsce na mistrzostwach Europy w 2022. Srebrny medalista mistrzostw śródziemnomorskich w 2016 i brązowy w 2015 roku.

Przypisy

Hiszpańscy zapaśnicy
Urodzeni w 1993

redpajama_set_name: RedPajamaWikipedia


Looking at the candidate profiles, what do the numbers next to the question answers mean?
This relates to our super-secret algorithm. But just between you and me, that number tells you how far away the candidate's answer is from your ideal one.
What data/analytics do I have access to while I'm using Weirdly?
How do I email candidates?
Why are there four lists of candidates?
Why have free text and url answers if they can't be ranked?
How does the ranking and scoring work?
How do the text analysis donuts work?


Who doesn't like long hair? Irrespective of gender, long hair is a fashion today. Even men are loving to don long hair look. Don't they ? Coming to women, they just need long hair which grows in quick time. So, here are the superfoods which promote long hair and faster hair growth.
* Presence of Vitamin is an essential for faster and longer hair growth. Eat Vitmain A rich foods, especially Carrot.
Vitmain A promotes the production of natural sebum oil which is important for healthy roots and growth of every cell on the scalp.
* What hair is made up of ? Isn't it protein which builds hair ? So, why don't you include lots of protein in your diet ? Eat eggs regularly, prefer chicken and also plant proteins. But remember, in any ways, protein intake is compulsory.
* Biotin promotes cell proliferation. This element is found in whole grain.
In addition, whole grains also contain good amount of Vitamin B, iron and zinc. All these together fasten your hair growth.
How to obtain vitamin C ? It's simple, eat citrus family fruits. You can have a delicious Orange or a healthy lemon. Both give you vitamin C.
* Salmon fish comes with a package of omega 3 fatty acids. It also has load of vitamin D and protein. All these three elements are required for long hair and also faster hair growth.
* You need vitamin E for improved blood flow around your scalp. A good rate of blood flow helps the follicles work effectively on hair growth.
In addition, vitamin C also controls PH levels. Wonder how to get such good amount of Vitamin E ? Eat Avocados.
ఐపీఎల్ సీజన్ 12 వచ్చేసింది , ఐపీఎల్ 12 పూర్తి షెడ్యూల్ ఇదే...మొదటి మ్యాచ్ వీరి మధ్యే..
జుట్టు పెద్దగా మరియు త్వరగా పెంచాలంటే ఏం తినాలి ? What To Eat To Grow Long Hair In Quick Time ? జుట్టు పెద్దగా మరియు త్వరగా పెంచాలంటే ఏం తినాలి ?


text:
Niafunké is a small town on the Niger River in central Mali.  The town is the administrative center (chef-lieu) of the commune  of Soboundou in  the Niafunké Cercle of the Tombouctou Region.

Niafunké was the hometown of musician and farmer Ali Farka Touré, who moved there as an infant and went on to become the town's mayor in 2004. He invested much of his music income in an irrigation scheme. Niafunké is also the title of an album he recorded there in 1999.

Toure's son Vieux Farka Toure was also born in Niafunke.

References

Populated places in Tombouctou Region


The Baptist Foundation of SC has formed a Search Committee to seek the next President/CEO to succeed Barry B. Edwards who will retire at the end of August 2017. Reporting to the Trustees of the Foundation, the President/CEO provides leadership to accomplish the Foundation's established goals and ministry ends.
Applications will be accepted during the month of May and must be postmarked by May 31, 2017. Interested candidates should review the Candidate Profile at the link below to learn more about the position and desired qualifications.
All interested applicants MUST complete the Candidate Questionnaire at the link below. The questionnaire is in writable format.
All applications MUST be postmarked by May 31, 2017.


Horizon 7.0 AE Elliptical Trainer Review by Industry Experts!
The Horizon Fitness 7.0 AE Elliptical Trainer is On Sale!
This model is The 7.0 AE Elliptical from Horizon is an exciting option for those looking for both technology and performance with a price tag under $1000. The 7.0 AE provides the quality and comfort we have come to expect from Horizon products in a solid, built to last package.
The technology features on the 7.0 AE may not be the most extensive on the market, but the 7" LCD touchscreen is nothing to shake a stick at. The Advanced Bluetooth allows the 7.0 AE to sync with multiple devices on the speakers and the integrated tablet rack puts your entertainment right where you want it. All in all, the Horizon 7.0 AE offers durability, performance and an impressive array of features at a price you can't beat.
MID-WEEK SALE ENDS THURSDAY: Horizon priced the 7.0 AE Elliptical at $999 and they are throwing in FREE Shipping.
Smooth Elliptical Stride – The 7.0 AE uses a combination large drive pulley ratio and a heavy 23 lb. flywheel to create a smooth and efficient elliptical motion.
Narrow Pedal Distance – One of the benefits of a front drive elliptical trainer is that you can place the pedals very close to each other, this reduces stress to your hips. Wide pedal separation can be particularly stressful to women. The pedals on the 7.0 AE are about as close as you can get.
Optimal Handlebar Spacing – Handlebars are at a 45 degree and are within easy reach. They are positioned where your hands would naturally be on your side.
Straight Body Posture – Many elliptical trainers tend to force you to lean forward because of the positioning of your body over the mechanism. On the Horizon 7.0 AE your body is in an upright posture, resulting in more comfort and reducing the feeling of exhaustion.
Natural Elliptical Stride – Rather than an oval, bouncy stride, you get a flat elliptical pattern that mimics running or walking on air. This is also less stressful on your joints and lower back.
23 lb. Flywheel – While we already mentioned this as part of the "SixStar Frame', this is a substantially heavier flywheel than other treadmills in this price range. This places the 7.0 AE a step above its competitors for smooth, steady use.
20" Stride Length – A 20" stride length is something that we expect to see in ellipticals over the $1000 mark, so it's a nice surprise to find this feature on the 7.0 AE. This stride length makes the 7.0 AE comfortable for most users, regardless of height.
20 incline levels – Another great upgrade on the 7.0 AE is the incline feature. This is something that we don't always see on other Horizon ellipticals. The incline feature allows users of all ability levels to get a wider variety of great workouts. This also allows you to focus on various muscle groups to tone and strengthen specific areas.
20 levels of magnetic resistance – This feature is a step-up from the Horizon entry-level model. The resistance feature is another way to ensure that your workouts are varied and allows you to work a greater range of muscle groups.
194 lb. Unit Weight – Weighing in at 194 lbs. the 7.0 AE isn't messing around, weighing more than some of the more expensive premium models. This means that not only are you guaranteed a wobble-free workout, the 7.0 AE can accommodate users up to 325 lbs.
7" LCD Touchscreen – This may not boast the 10" screens that we are seeing on a lot of newer models, but it's still a definite step up from the LED displays on many models in this price range.
Advanced Bluetooth – This technology allows to you connect to multiple devices and stream media to the integrated speakers on the 7.0 AE. Plus, the free no-subscription app lets you push workout data to your favorite fitness apps and control your workout from your device. You can also be sure that you'll never lose your tunes or entertainment during your workout with the fast-charge USB port.
6 Workout Programs – No matter what you are looking for from your workout, the 7.0 AE has a program suited to you.
Warranty – Lifetime frame & brakes, 3 year parts, 1 year generator & labor.
The 7.0 AE maintains the high levels of quality, ease of use and performance that we have come to expect from Horizon products. Looking at the specs and features on the 7.0 AE one would very much expect to find a much higher price tag. In fact, the 7.0 AE offers many of the features that we see on the Evolve series machines, but the 7.0 AE is going to cost several hundred dollars less. Horizon has really stepped up to the plate with the 7.0 AE in terms of performance and features.
The incline is an awesome addition that we don't often see on Horizon models. This allows you to better tailor your sessions and work a larger range of muscle groups. The 23 lb. flywheel ensures that you get smooth use every time you step onto your machine. In addition, the heavy weight of that machine itself means that you will always feel solid and steady on your machine.
The 7.0 AE is also a great option for larger users, as it can accommodate a weight of up to 325 lbs. Tall users need not fear either, the 20" stride length means the 7.0 AE is comfortable for you too.
If you're looking for a lower priced model, you may want to check out the Horizon EX-59.


BETHESDA, Md., May 8, 2013 /PRNewswire/ -- American Capital Agency Corp. (NASDAQ: AGNC) ("AGNC" or the "Company") announced today that Peter Federico, Senior Vice President and Chief Risk Officer, is scheduled to make a presentation at the 12th Annual JMP Securities Research Conference on May 13, 2013 in San Francisco, CA. The AGNC presentation is scheduled to begin at 11:00 am PT (2:00 pm ET). The presentation will be webcast live and archived on the AGNC website in the Investor Relations section at http://ir.agnc.com.


text:
Q: biblatex: Storing result of \citeauthor for later use (based on an answer from Heiko Oberdiek)
*

*Ten years ago, Heiko provided the following solution for storing biblatex outputs (\citefield) for later use in headings (hyperref bookmarks).

\makeatletter
\newcommand*{\StoreCiteField}[3]{%
  \begingroup
    \global\let\StoreCiteField@Result\relax
    \citefield{#2}[StoreCiteField]{#3}%
  \endgroup
  \let#1\StoreCiteField@Result
}
\DeclareFieldFormat{StoreCiteField}{%
  \gdef\StoreCiteField@Result{#1}%
}
\makeatother

*

*I tried to adapt the code so that it works with \citeauthor and failed.

*Any ideas?

This is my (failing) attempt to use the same approach for \citeauthior:
\newcommand*{\StoreAuthor}[2]{%
  \begingroup
    \global\let\StoreAuthor@Result\relax
    \citeauthor{#2}%
  \endgroup
  \let#1\StoreAuthor@Result
}
\DeclareFieldFormat{StoreAuthor}{%
  \gdef\StoreAuthor@Result{#1}%
}

I punch way above my weight class here and just tried to find analogies in Heiko's code and adapted it accordingly. I do not understand the code. Yesterday, the previous regular user cfr commented on a related question (How can I make \citetitle work in a hyperref section bookmark?) because he/she has the same problem. This motivated me to try again.

\documentclass{article}

\usepackage{xcolor}
\usepackage{biblatex}

\begin{filecontents}{\jobname.bib}
@book{key,
  title = {Book Title},
  year = {2000},
  author = {Givenname FamilyName},
}
\end{filecontents}
\addbibresource{\jobname.bib}

\usepackage{hyperref}
\usepackage{bookmark}

% https://tex.stackexchange.com/questions/73678
\makeatletter
\newcommand*{\StoreCiteField}[3]{%
  \begingroup
    \global\let\StoreCiteField@Result\relax
    \citefield{#2}[StoreCiteField]{#3}%
  \endgroup
  \let#1\StoreCiteField@Result
}
\DeclareFieldFormat{StoreCiteField}{%
  \gdef\StoreCiteField@Result{#1}%
}
%
\newcommand*{\StoreAuthor}[2]{%
  \begingroup
    \global\let\StoreAuthor@Result\relax
    \citeauthor{#2}%
  \endgroup
  \let#1\StoreAuthor@Result
}
\DeclareFieldFormat{StoreAuthor}{%
  \gdef\StoreAuthor@Result{#1}%
}
\makeatother

\begin{document}

%% StoreCiteField (Working)
% Store Results
\StoreCiteField\myTitle{key}{title}
% Use Results In Headings
\section{Title (Working): \textcolor{red}{\myTitle}}
\myTitle

\noindent\rule{\textwidth}{1pt}

% Store Results
\StoreCiteField\myYear{key}{year}
% Use Results In Headings
\section{Year (Working): \textcolor{red}{\myYear}}
\myTitle

\noindent\rule{\textwidth}{1pt}

%% StoreAuthor (Nor Working)
% Store Results
\StoreAuthor\myAuthor{key}
% Use Results In Headings
\section{Author (Not Working): \textcolor{red}{\myAuthor}}
\myAuthor

\end{document}

A: The problem here is that a name list is much more complex than a simple field.
Simple fields are relatively straightforward to handle because in the end, the field contents are stored in a macro which is used pretty directly by biblatex for printing. We "just" need get hold of this macro's expansion. (This can be done in the clever StoreCiteField way shown here or with more naive methods.) Specifically, we'll have something like
  \field{year}{1998}

in the .bbl and want to grab the complete 1998 for what we are trying to do here.
Name lists, however, are much more complex beasts. They are stored in an "abstract" way in the .bbl file
  \name{author}{2}{}{%
    {{hash=484061f383fb3b729627e12ab42c1963}{%
       family={Sigfridsson},
       familyi={S\bibinitperiod},
       given={Emma},
       giveni={E\bibinitperiod}}}%
    {{hash=b7e299b632e5db12681c2decc8ce023f}{%
       family={Ryde},
       familyi={R\bibinitperiod},
       given={Ulf},
       giveni={U\bibinitperiod}}}%
  }

You can't just grab "all" of these contents to get nice output. We need to post-process them heavily so that something useful comes out. This post-processing is usually done by biblatex. But it is done in a manner that is fundamentally unexpandable, which means it's no good to us here.
We essentially have to replicate the whole name formatting code in an expandable manner. For the family name this is done in the StoreName name format below. We add the family name part of the current name to \StoreName@Result in \protected@xappto\StoreName@Result{\namepartfamily}. The remainder of the code takes care of commas, "and", "et al." in an expandable manner (so everything is hard-coded).
\documentclass{article}

\usepackage{xcolor}
\usepackage{biblatex}
\addbibresource{biblatex-examples.bib}

\usepackage{hyperref}
\usepackage{bookmark}

% https://tex.stackexchange.com/questions/73678
\makeatletter
\newcommand*{\StoreCiteField}[3]{%
  \begingroup
    \global\let\StoreCiteField@Result\relax
    \citefield{#2}[StoreCiteField]{#3}%
  \endgroup
  \let#1\StoreCiteField@Result
}
\DeclareFieldFormat{StoreCiteField}{%
  \gdef\StoreCiteField@Result{#1}%
}
%
\newcommand*{\StoreName}[3]{%
  \begingroup
    \global\let\StoreName@Result\relax
    \blx@indexnamesetup
    \citename{#2}[StoreName]{#3}%
  \endgroup
  \let#1\StoreName@Result
}
\DeclareNameFormat{StoreName}{%
  \ifnumgreater{\value{listcount}}{\value{liststart}}
    {\ifboolexpr{
       test {\ifnumless{\value{listcount}}{\value{liststop}}}
       or
       test \ifmorenames
     }
       {\gappto\StoreName@Result{, }}
       {\gappto\StoreName@Result{ and }}}
    {}%
  \protected@xappto\StoreName@Result{\namepartfamily}%
  \ifboolexpr{
    test {\ifnumequal{\value{listcount}}{\value{liststop}}}
    and
    test \ifmorenames
  }
    {\gappto\StoreName@Result{et~al.}}
    {}%
}
\newcommand*\StoreAuthor[2]{\StoreName{#1}{#2}{author}}
\makeatother

\begin{document}
\StoreCiteField\myTitle{sigfridsson}{title}
\section{Title: \textcolor{red}{\myTitle}}
\myTitle

\noindent\rule{\textwidth}{1pt}

\StoreCiteField\myYear{sigfridsson}{year}
\section{Year: \textcolor{red}{\myYear}}
\myYear

\noindent\rule{\textwidth}{1pt}

\StoreAuthor\myAuthor{sigfridsson}
\section{Author: \textcolor{red}{\myAuthor}}
\myAuthor

\end{document}

redpajama_set_name: RedPajamaStackExchange


text:
De Wheelchair Tennis Masters zijn twee tennistoernooien voor rolstoeltennissers aan het eind van het seizoen. In het enkelspel is dat het NEC Wheelchair Singles Masters-toernooi voor acht mannen, acht vrouwen en acht quad-spelers. In het dubbelspel is er het ITF Wheelchair Doubles Masters-toernooi voor acht paar mannen, zes paar vrouwen en vier paar quad-spelers.

De toernooien worden beschouwd als de officieuze wereldkampioenschappen van het rolstoeltennis.

Resultaten enkelspel

Mannen

Vrouwen

Quads

Dubbelspel

Mannen

Vrouwen

Quads

ITF-toernooi


Bugatti has revealed the first image of their Vision Gran Turismo project. This virtual concept is to be presented at the International Motor Show (IAA) in Frankfurt from September 17 to 27, 2015.
With the "Bugatti Vision Gran Turismo" virtual concept, Bugatti will be presenting the new form and design language developed by the brand to celebrate the next chapter in its history following the end of the Veyron era.
Stay tuned for more updates including the official unveiling coming soon!


text:
Home Environment & Climate Indiscriminate blasting of mountains causing landslides, floods: Report
Indiscriminate blasting of mountains causing landslides, floods: Report
Indiscriminate blasting of mountains has rendered Uttarakhand more susceptible to landslides and floods, according to a report of GSI
India: Indiscriminate blasting of mountains to build roads or other infrastructure has rendered the fragile areas of Uttarakhand more susceptible to landslides and floods, according to a report of the Geological Survey of India (GSI). The report was submitted by Director of GSI, regarding the treatment of vulnerable sites in the aftermath of the flash floods of 2013.
According to experts from the International Road Federation – an organisation based in Geneva, had also suggested structural measures such as storing of rainwater in large storage reservoir, detention basins and non-structural measures but these had hardly been implemented on the ground. "To avoid loss of human life and damage to infrastructure, floodplain mapping programs must be carried out with the help of remote sensing and ground level river data."
This can predict in advance which areas will be flooded with a rise in river water level," said IRF chairman K K Kapila. "Floodplain mapping will enable us to minimize the impact of floods by not allowing habitation and development in flood-prone areas. Optimum combination of structural measures such as large storage reservoirs, detention basins and embankments built in advance to channelise rain water can be adopted. Non-structural measures like flood forecasting, flood plain zoning and catchment area treatment can be undertaken," he said.
He added that while planning the designing and rebuilding of the affected roads, it needs to be remembered that the outer and lesser Himalayas contain unstable strata so cutting of slopes in such ranges via blasting should be avoided. Kapila said that the best way to develop roads in hills was through the use of tunnels and connecting viaducts.
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Eagle Point launches Autodesk Construction Cloud Content on award-winning platform

redpajama_set_name: RedPajamaCommonCrawl


You are the 1911th visitor to this USGennet site as of Dec. 02, 2004.
In the Matter of the Estate of Austin Aaron, Deceased. To filing petition full blood approval.
To filing recording order approving deed. To issuing papers. To certified copy.
The Estate of Frances Hopson then Billy enrolled as Frances Aaron, Deceased.
Choctaw by Blood, Roll No. #14653.
In the Matter of the Estate of Harriet Aaron nee Taylor, Roll No. #2729.
In the Matter of the Estate of Mary Aaron, Deceased. To filing petition.
In the Matter of the Estate of Nebus Aaron, Deceased. To filing petition.
Deed of Wilburn Adams, Full Blood Choctaw Indian, Roll Number #9134.
In the Matter of the Estate of Leson Ahayatubbe, Deceased. To filing petition full blood.
approval. To filing and recording order approving deed. To certified copy.
In the Matter of the Estate of William Alexander, Deceased. To filing petition.
In the Matter of the Estate of Adaline Allen, Deceased. To filing petition.
To filing recording order approving deed. To issuing deed. To certified copy.
In the Matter of the Approval of Full Blood Deed of George Allen.
In the Matter of the Estate of John Allen, Deceased. To filing petition.
To filing recording order approving deed. To issuing deed. To certified copy and deed.
In the Matter of the Estate of Kana Amos, Deceased. To filing petition.
Bohanan. Full Blood Choctaw Roll Number #1609.
In the Matter of Full Blood Approval of Full Blood Deed.
In the Matter of the Estate of Allen Anderson, Deceased. To filing petition.
Case #184, Page 408, February 25, 1952 to July 1964.
In the Matter of the Estate of Edmond Anderson, Deceased. To filing petition.
filing recording order approving deed. To issuing papers. To certified copy.
In the Matter of the Approval of Deed of Frank Anderson. Approval of Oil and Gas Mining Lease.
In the Matter of the Estate of Hickman Anderson, Deceased. To filing petition.
In the Matter of the Estate of Levicy Anderson, Deceased. To filing petition.
In the Matter of the Estate of Levina Anderson, Deceased. To filing petition.
In the Matter of the Estate of Lizzie Anderson, Deceased. To filing petition.
In the Matter of the Estate of Nancy Anderson, Deceased. To filing petition.
In the Matter of the Estate of Parker Anderson, Deceased. To filing petition.
Deceased. 7/8 Blood Choctaw Indian.
Choctaw by Blood, Roll Number #5370.
Blood Choctaw Indian, Roll Number #4289.
In the Matter of the Approval of Deed of Katherine Walker Austin.
In the Matter of the Estate of Legion Austin, Deceased. To filing petition.


The CCOS supports five religious emblem opportunities for Girl Scouts: God is Love, I Live My Faith, Mary the First Disciple, e Spirit Alive, and Missio, that offer age appropriate encounters for girls to discover and learn the Catholic faith.
In addition to supporting these religious emblem programs, the CCOS offers an I Live My Faith weekend that affords girls in fourth through sixth grade the chance to earn this religious emblem in community with other girls and Troops throughout the area.
The Boy Scout awards included Light of Christ, Tiger and Wolf Cubs; Parvuli Dei (Child of God), Cub and Webelo Scouts; Ad Altare Dei, Pope Pius XII for high school Scouts and Venturers. Additionally Girl Scout Troops were awarded the Bishop Donovan Quality Troop Award and Cub and Boy Scout Troops were awarded the Pope Paul VI Quality Unit Award.
Adult Leaders were also honored for their outstanding contribution to scouting.


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Marshall County, South Dakota
U.S. county in South Dakota
Marshall County Courthouse in Britton
Location within the U.S. state of South Dakota
South Dakota's location within the U.S.
Marshall Vincent
48 sq mi (120 km2) 5.4%
marshall.sdcounties.org
Marshall County is a county in the U.S. state of South Dakota. As of the 2010 census, the population was 4,656.[1] Its county seat is Britton.[2] The county was created on May 2, 1885,[3] and was named for Marshall Vincent, who homesteaded near Andover, South Dakota.[4]
1.2 Protected areas
1.3 Adjacent Counties
2.1 2000 census
3.3 Census-designated place
Native vegetation based on NRCS soils information
Marshall County lies on the north side of South Dakota. Its north boundary line abuts the south boundary line of the state of North Dakota. Its terrain consists of rolling hills, with numerous lakes and ponds in the SE portion.[5] Its terrain slopes to the northeast, and its highest point is near its SE corner, at 2,034' (620m) ASL.[6]
Marshall County has a total area of 886 square miles (2,290 km2), of which 838 square miles (2,170 km2) is land and 48 square miles (120 km2) (5.4%) is water.[7] The county is drained by the Crow Creek, a tributary of the James River, and the Wild Rice River, a tributary of the Red River of the North.[8] A portion of the Lake Traverse Indian Reservation is located in the eastern part of the county.
South Dakota Highway 10
Protected areas[edit]
Abraham State Game Production Area
Aspen Slough State Game Production Area
Black Slough State Game Production Area
Bonham State Game Production Area
Buffalo Lakes State Game Production Area
Bullhead State Lakeside Use Area
Cattail State Game Production Area
Church State Game Production Area
Clear Lake State Game Production Area
Clear Lake State Lakeside Use Area
Douglas State Game Production Area
Eden State Game Production Area
Flat Creek State Game Production Area
Fort Sisseton Historic State Park
Fort Sisseton State Game Production Area
Four Mile-Clubhouse-Barretts State Game Production Area
Four Mile State Lakeside Use Area
Hamilton Township State Game Production Area
Hickman Dam State Game Production Area
Horeshoe State Game Production Area
Ivory Tower State Game Production Area
Knebel State Game Production Area (part)
Little Cottonwood State Game Production Area
Lost Lake State Game Production Area
Nine Mile State Game Production Area
North Church State Game Production Area
North Ottertail State Game Production Area
North Red Iron State Game Production Area
Renziehausen State Game Production Area (part)
Rock Crandall State Game Production Area
Roy Lake State Game Production Area
Roy Lake State Park
Schlosser Memorial State Game Production Area
Sica Hollow State Park (part)
Sorbell State Game Production Area
South Red Iron State Game Production Area
Turtlefoot State Game Production Area
White Lake State Game Production Area
Adjacent Counties[edit]
Sargent County, North Dakota - north
Richland County, North Dakota - northeast
Roberts County, South Dakota - east
Day County, South Dakota - south
Brown County, South Dakota - west
1930 9,540 −0.6%
Est. 2018 5,112 [9] 9.8%
US Decennial Census[10]
2000 census[edit]
As of the 2000 United States Census,[14] there were 4,576 people, 1,844 households, and 1,252 families in the county. The population density was 6 people per square mile (2/km²). There were 2,562 housing units at an average density of 3 per square mile (1/km²). The racial makeup of the county was 92.59% White, 0.09% Black or African American, 6.32% Native American, 0.09% Asian, 0.22% from other races, and 0.70% from two or more races. 0.76% of the population were Hispanic or Latino of any race.
There were 1,844 households out of which 29.80% had children under the age of 18 living with them, 57.20% were married couples living together, 6.50% had a female householder with no husband present, and 32.10% were non-families. 30.10% of all households were made up of individuals and 16.10% had someone living alone who was 65 years of age or older. The average household size was 2.43 and the average family size was 3.04.
The county population contained 27.00% under the age of 18, 5.10% from 18 to 24, 22.80% from 25 to 44, 23.80% from 45 to 64, and 21.30% who were 65 years of age or older. The median age was 42 years. For every 100 females there were 100.00 males. For every 100 females age 18 and over, there were 96.80 males.
As of the 2010 United States Census, there were 4,656 people, 1,815 households, and 1,154 families in the county.[15] The population density was 5.6 inhabitants per square mile (2.2/km2). There were 2,534 housing units at an average density of 3.0 per square mile (1.2/km2).[16] The racial makeup of the county was 85.4% white, 7.6% American Indian, 0.2% black or African American, 0.2% Asian, 5.6% from other races, and 1.1% from two or more races. Those of Hispanic or Latino origin made up 6.8% of the population.[15] In terms of ancestry, 45.6% were German, 30.6% were Norwegian, 6.8% were Swedish, 5.8% were Irish, 5.2% were English, and 2.7% were American.[17]
Of the 1,815 households, 24.4% had children under the age of 18 living with them, 52.8% were married couples living together, 6.0% had a female householder with no husband present, 36.4% were non-families, and 31.4% of all households were made up of individuals. The average household size was 2.36 and the average family size was 2.83. The median age was 43.2 years.[15]
The median income for a household in the county was $41,023 and the median income for a family was $56,544. Males had a median income of $31,639 versus $26,737 for females. The per capita income for the county was $22,441. About 9.3% of families and 17.1% of the population were below the poverty line, including 27.6% of those under age 18 and 12.8% of those age 65 or over.[18]
Cities[edit]
Britton (county seat)
Census-designated place[edit]
Kidder
Hillhead
Marlow[5]
West Britton[5]
Dumarce
Red Iron Lake
Wismer
Marshall County voters have tended to vote Democratic for the past several decades. In 64% of the national elections since 1960, the county selected the Democratic Party candidate.
Presidential election results
2016 54.2% 1,056 38.7% 754 7.0% 137
2000 52.8% 1,097 45.2% 939 2.1% 43
1996 38.3% 861 52.8% 1,185 8.9% 200
1984 57.7% 1,529 41.9% 1,111 0.3% 9
1924 51.5% 1,271 7.7% 190 40.8% 1,005
1920 56.0% 1,557 9.6% 266 34.5% 959
1916 45.6% 808 49.9% 885 4.5% 80
1912 0.0% 0 43.1% 541 56.9% 715
1908 59.5% 874 30.8% 453 9.7% 143
1904 67.9% 996 19.9% 292 12.1% 178
National Register of Historic Places listings in Marshall County, South Dakota
Marshall County, South Dakota (Official site)
^ a b "State & County QuickFacts". United States Census Bureau. Archived from the original on June 7, 2011. Retrieved November 25, 2013.
^ "Find a County". National Association of Counties. Retrieved June 7, 2013.
^ Individual County Chronologies/Marshall County (accessed February 6, 2019)
^ Legislative Manual, South Dakota, 2005, p. 596
^ a b c Marshall County SD Google Maps (accessed 6 February 2019)
^ "Find an Altitude" Google Maps (accessed 6 February 2019)
^ "2010 Census Gazetteer Files". US Census Bureau. August 22, 2012. Retrieved March 28, 2015.
^ Koopman, F. C. (1957). "Ground water in the Crow Creek-Sand Lake area, Brown and Marshall Counties, South Dakota " United States Geological Survey Water Supply Paper Series number:1425. Washington, DC: Government Printing Office. USGS website Retrieved November 18, 2017
^ "Population and Housing Unit Estimates". Retrieved May 14, 2019.
^ "US Decennial Census". US Census Bureau. Retrieved March 28, 2015.
^ "Historical Census Browser". University of Virginia Library. Retrieved March 28, 2015.
^ Forstall, Richard L., ed. (March 27, 1995). "Population of Counties by Decennial Census: 1900 to 1990". US Census Bureau. Retrieved March 28, 2015.
^ "Census 2000 PHC-T-4. Ranking Tables for Counties: 1990 and 2000" (PDF). US Census Bureau. April 2, 2001. Retrieved March 28, 2015.
^ "American FactFinder". US Census Bureau. Retrieved May 14, 2011.
^ a b c "Profile of General Population and Housing Characteristics: 2010 Demographic Profile Data". US Census Bureau. Retrieved March 12, 2016.
^ "Population, Housing Units, Area, and Density: 2010 - County". US Census Bureau. Retrieved March 12, 2016.
^ "Selected Social Characteristics in the US – 2006-2010 American Community Survey 5-Year Estimates". US Census Bureau. Retrieved March 12, 2016.
^ "Selected Economic Characteristics – 2006-2010 American Community Survey 5-Year Estimates". US Census Bureau. Retrieved March 12, 2016.
^ Election Results
Places adjacent to Marshall County, South Dakota
Sargent County, North Dakota Richland County, North Dakota
Municipalities and communities of Marshall County, South Dakota, United States
County seat: Britton
Indian reservation
Lake Traverse Indian Reservation‡
Pierre (capital)
Famous South Dakotans
Seal of South Dakota
Coteau des Prairies
Coteau du Missouri
Dissected Till Plains
Drift Prairie
Siouxland
Larger cities
pop. over 5,000
pop. 1,000 - 5,000
Elk Point
Garretson
Parkston
Tyndall
Largest CDPs
Colonial Pine Hills
North Eagle Butte
North Spearfish
Rapid Valley
Largest towns
pop. over 500
Beadle
Bon Homme
Charles Mix
Corson
Deuel
Jerauld
Minnehaha
Oglala Lakota
Ziebach
Retrieved from "https://en.wikipedia.org/w/index.php?title=Marshall_County,_South_Dakota&oldid=923195082"
South Dakota counties
1885 establishments in Dakota Territory
Pages using infobox U.S. county with unknown parameters