text:
\section{Problem setting}\label{s-ps}
We consider the matrix differential equation of the form
\begin{equation}\label{mde}
 x'(t) = Qx(t),\qquad x(0) = x_0.
\end{equation}
where $Q$ is replaced with a compact convex set of matrices, called an \emph{interval matrix}, usually we denote it by ${\mathcal Q}$; and the intial vector $x_0$ is replaced by an interval vector. Interval matrices have been proven to efficiently model the uncertainty in parameter estimates, such as in modelling discrete time Markov chains with unknown parameters (see for instance \citet{hart:98,decooman-2008-a,skulj:09}). One of our main motivations is the modelling of continuous time Markov chains with uncertain parameters. We build on the work of \citet{hart:ctmc}, who studied the related problem requiring that $Q(t)$ is a piecewise continuous matrix function whose values lie between given matrices $\underline Q$ and $\overline Q$. Moreover, he requires zero column sums of all matrices $Q(t)$ and that the elements of $x_0$ sum to one. He then shows that the sets of all possible solutions are compact and convex, and studies their properties when time $t$ approaches infinity.

In this paper we relax most of the \citeauthor{hart:ctmc}'s assumptions. Thus we allow arbitrary compact convex sets of matrices and in addition we omit any continuity assumptions. With regard to solutions we provide a directly applicable methods for finding approximate solutions, based on linear programming techniques, together with error estimates.

There are two most common interpretations of the equation \eqref{mde} if instead of a fixed matrix $Q$, a set of possible matrices ${\mathcal Q}$ is given. The first one is the sensitivity analysis interpretation, where it is assumed that the matrix $Q$ is constant in time although all that is known is that it is contained in ${\mathcal Q}$ (see for  instance \citet{oppen:88,goldsztejn:2009}).

In this paper however we adopt another interpretation, which is similar to the one adopted by \citet{hart:ctmc}. That is, we allow that the matrix $Q$ depends on time $t$ in some very general unspecified way. Unlike \citeauthor{hart:ctmc} we do not even make any continuity assumptions. The only requirement being that $Q(t)$ is an element of ${\mathcal Q}$ at every time $t$.

Another related problem is when the matrix $Q(t)$ is a known function of time $t$ (see e.g. \citet{bohner:2001,dacunha:2005} and references therein). Our approach would rougly correspond to finding the bounds for all possible such solutions when only the bounds are given for $Q(t)$.

When the general equation \eqref{mde} is considered the closed form solution is obtained using matrix exponentials:
\begin{equation*}\label{general-solution}
 x(t) = e^{tQ}x_0.
\end{equation*}
The approach to the solution of the generalised problem that we propose is based on the generalisation of the matrix exponential for the case of interval matrices. We show that this is indeed possible and provide methods to find approximations, with arbitrary precision that can be estimated in advance.

The paper has the following structure. In Section~\ref{s-imde} we formalize the generalisation of matrix differential equations. The solution of the genaralised equations is based on the defintion of an exponential of interval matrix defined and analysed in Section~\ref{s-eim}, where a method for practical approximations is presented along with the estimation of errors. In Section~\ref{s-simde} we prove that the exponentials of interval matrices indeed provide the solution of the interval matrix differential equation. Finally, in Section~\ref{s-rcimc} we explain how the method can be applied to continuous time imprecise Markov chains, and give an example. Some of the technical auxiliary results are listed in the appendix.

\section{Interval matrix differential equations}\label{s-imde}
\subsection{Intervals of vectors and matrices}\label{ss-ivm}
All vectors and matrices used here are finite-dimensional and all vectors are assumed to be column vectors. We will write $x\le y$ when $x_i\le y_i$ holds for all their components. Moreover, we will involve sets of vectors and sets of matrices. All operations between sets of matrices or sets of vectors are meant to be elementwise. Thus, for instance, given a set of matrices ${\mathcal Q}$ and a set of vectors $\mathcal X$, we denote
\begin{equation*}
 {\mathcal Q} \mathcal X = \{ Qx\colon Q\in {\mathcal Q}, x\in \mathcal X \}.
\end{equation*}
\begin{defn}
Let $\underline x$ and $\overline x$ be vectors such that $\underline x\le \overline x$. Then the set of  vectors $[\underline x, \overline x]:=\{ x\colon \underline x\le x \le \overline x\}$ is called an {\em interval vector}.
\end{defn}
\begin{defn}
 A set of matrices ${\mathcal Q}$ has {\em separately specified rows} if for every $Q, Q'\in {\mathcal Q}$ the matrix $\tilde Q$, such that $\tilde Q_{ij} = Q_{ij}$ for all $j$ and $i\neq i_0$ and $\tilde Q_{i_0j} = Q'_{i_0j}$ for all $j$, also belongs to ${\mathcal Q}$.
\end{defn}
This means that rows of matrices in ${\mathcal Q}$ can be chosen independently from some sets of row vectors. In the case, such as \citeauthor{hart:ctmc}'s, where the set of matrices consists of all matrices $Q$ that satisfy the equation $\underline Q\le Q\le \overline Q$, this property is already implicity assumed by the defintion.
\begin{prop}\label{prop_interval}
Let ${\mathcal Q}$ be a compact convex set of matrices with separately specified rows and $x$ a vector. Then there exist matrices $Q_0, Q_1\in{\mathcal Q}$ such that $Q_0x \le Qx$  and $Q_1x\ge Q x$ for every $Q\in{\mathcal Q}$.
\end{prop}
\begin{proof}
 Let $\underline y_i = \min_{Q\in {\mathcal Q}} (Qx)_i$ and let $Q_i$ be a matrix that minimizes $(Qx)_i$. Then let $\tilde Q$ be the matrix whose $i$th row is equal to the $i$th row of $Q_i$ for every $i$. Then, clearly, $(\tilde Qx)_i = (Q_ix)_i$ which implies that $(\tilde Qx)_i\le (Qx)_i$ for every $Q\in {\mathcal Q}$ and every component $i$, and this completes the proof.
\end{proof}
If ${\mathcal Q}$ is a compact convex set of matrices with separately specified rows and $x$ a vector then their product is the interval
\begin{equation}\label{Q-interval}
 {\mathcal Q} x = [\underline Qx, \overline Qx],
\end{equation}
where $\underline Qx := \min_{Q\in {\mathcal Q}} Qx$ and $\overline Qx := \max_{Q\in {\mathcal Q}} Qx$, which exist by the above proposition. Moreover, when an interval matrix is multiplied with an interval vector, the result is again an interval:
\begin{equation*}
{\mathcal Q} [\underline x, \overline x]= [\underline Q\,\underline x, \overline Q\,\overline x].
\end{equation*}
Thus every compact convex set of matrices ${\mathcal Q}$ with separately specified rows defines a (non-linear) transformation of the set of interval vectors to itself.
\begin{defn}
An {\em interval matrix} is a compact and convex set of matrices ${\mathcal Q}$ with separately specified rows.
\end{defn}
\subsection{The formulation of the interval matrix differential equation}\label{ss-fimde}
We have shown that an interval matrix ${\mathcal Q}$ maps an interval vector $\luint{}{x}{}$ to the interval vector $[\underline Q\,\underline x, \overline Q\,\overline x]$. Consider all vector valued maps $x(t)$ satisfying the following inequalities:
\begin{align}
 & x(0) \in \luint{}{x}{_0} \label{initial_conds} \\
 & \limsup_{h\to 0} \frac{x(t+h)-x(t)}{h} \le \overline{Q}x(t) \label{diff_ineq_sup}\\
 & \liminf_{h\to 0} \frac{x(t+h)-x(t)}{h} \ge \underline{Q}x(t), \label{diff_ineq_inf}
\end{align}
for every $t\ge 0$.

Let $\mathcal X$ denote the set of all possible solutions of the above system of differential inequalities, i.e. the set of all vector functions $x(t)$ satisfying the inequalities \eqref{initial_conds}, \eqref{diff_ineq_sup} and \eqref{diff_ineq_inf}. We are concerned with the sets of vectors $\mathcal X(t) := \{ x(t) \colon x\in \mathcal X\}$ that the possible solutions can take at time $t$. We will show that every set $\mathcal X(t)$ is an interval vector of the form $[\underline x(t), \overline x(t)]$. Moreover, we will provide the method to efficiently calculate its bounds.

The solution of the classical matrix differential equation \eqref{mde} is obtained by multiplying the initial vector $x_0$ with the exponential of the matrix $Q$ to obtain $x(t) = e^{tQ} x_0$. In the next section we define a generalised exponential of an interval matrix which will provide the solution of the generalised equation.
\subsection{Computational considerations}\label{ss-cc}
Two crucial considerations when doing computation with interval matrices are needed. One is the representation of interval matrices and the other is the computation of the interval  bounds \eqref{Q-interval}.

Convex sets, and in particular iterval matrices, are represented via sets of constraints that in practical cases are usually finite and allow the computation of the bounds of the intervals of the form \eqref{Q-interval} using linear programming techniques. One such example of constraints is the most commonly used interval of matrices
\begin{equation*}\label{matrix-interval}
 [\underline Q, \overline Q] = \{ Q\colon \underline Q \le Q\le \overline Q\}.
\end{equation*}
Problems usually occur when the set of matrices is transformed in a non linear way. In our case the transformation is the exponential of an interval matrix defined in the next section. When such transformations are applied, not only it is not easy to find the constraints representing the transformed set, but also the cardinality of the set of such constraints may exponentially increase or even become infinite. For this reason we strive to besides giving the adequate theoretical definitions we also develop methods for efficient calculations, which are usually approximate methods.

\section{The exponential of an interval matrix}\label{s-eim}
\subsection{Definition}\label{ss-d}
Our aim is defining a set of matrices $\exp({\mathcal Q})$ that will provide the solution to the set of inequalities \eqref{initial_conds}, \eqref{diff_ineq_sup} and \eqref{diff_ineq_inf} in the following sense:
\begin{equation*}
 \mathcal X(t) = \exp(t{\mathcal Q})[\underline x_0, \overline x_0] = \{ Px \colon P\in \exp(t{\mathcal Q}), x \in [\underline x_0, \overline x_0] \}.
\end{equation*}
We have explained in Section~\ref{s-ps} that there are different interpretations of interval matrix differential equations. When sensitivity analysis interpretation is used, i.e. when it is assumed that the differential equation in question is a classical matrix differential equation with an unknown matrix $Q$, that is known to belong to the set ${\mathcal Q}$, the adequate way to calculate its exponential is to take the set of all exponentials of matrices in ${\mathcal Q}$:
\begin{equation}\label{pseudo_exponential}
 \exp'({\mathcal Q}) =  \{ e^{Q}\colon Q\in{\mathcal Q} \}.
\end{equation}
This type of exponential and the methods for its efficient calculations have been provided by \citet{goldsztejn:2009,oppen:88}.

The above exponential, however, is not suitable when our interpretation is adopted. Let us explain the main reason, why the exponential \eqref{pseudo_exponential} does not solve the set of inequalities \eqref{initial_conds}, \eqref{diff_ineq_sup} and \eqref{diff_ineq_inf}. Suppose we are trying to find the upper bound $\overline x(\Delta t_1)$ for the set of solutions at time $\Delta t_1$. We may have, for instance:
\begin{equation*}\label{lin_approx}
 \overline x(\Delta t_1) = e^{\Delta t_1Q_1}\overline x_0.
\end{equation*}
where $Q_1$ is the matrix in ${\mathcal Q}$ that maximizes the value $e^{\Delta t_1Q}\overline x_0$. At time $\Delta t_1+\Delta t_2$ let $Q_2$ be the matrix that maximizes $e^{\Delta t_2Q}\overline x(\Delta t_1)$. But the resulting upper vector $\overline x(\Delta t_1+\Delta t_2)$ is then in general greater than $e^{(\Delta t_1+\Delta t_2)Q}\overline x_0$ for every $Q\in {\mathcal Q}$.

The exponential serving our purposes is instead defined as follows.
\begin{defn}\label{interval_exponential}
 Let ${\mathcal Q}$ be an interval matrix with separately specified rows. Then we define
 \begin{equation}\label{interval_exponential_formula}
  e^{{\mathcal Q}} = \mathrm{cl}\left\{ \prod_{i=1}^n e^{d_iQ_i}\colon n\in\mathbb{N}, \sum_{i=1}^nd_i = 1,  Q_i\in {\mathcal Q}, \text{ for } 1\le i \le n\right\},
 \end{equation}
 where $\mathrm{cl}$ denotes the closure of the set.
\end{defn}
Since interval matrices are compact by definition, this implies that the norms of their elements are uniformly bounded by a constant, say $M$. By Lemma~\ref{norm_exp} it easily follows that the norms of elements in $e^{{\mathcal Q}}$ are uniformly bounded as well with the constant $e^M$. Hence, $e^{{\mathcal Q}}$ is a closed set of matrices with uniformly bounded norms, which implies that it also compact. Every element of $P\in e^{{\mathcal Q}}$ is therefore the limit of a sequence $\{ P_n \}$ of the form
\begin{equation}\label{sequence-exponential}
 P_n = \prod_{i=1}^{k_n} e^{d^n_iQ^n_i},
\end{equation}
where $\sum_{i = 1}^{k_n}d^n_i = 1, d^n_i\ge 0, Q^n_i\in {\mathcal Q}$ for every $n\in \mathbb{N}$ and $1\le i\le k_n$.
\begin{lemma}\label{additivity-exponential}
  For every $\alpha$ and $\beta$ we have that
  \begin{equation*}
   e^{(\alpha+\beta) {\mathcal Q} } = e^{\alpha {\mathcal Q} }e^{\beta {\mathcal Q} }.
  \end{equation*}
\end{lemma}
\begin{proof}
 Clearly $e^{\alpha {\mathcal Q} }e^{\beta {\mathcal Q} }\subseteq e^{(\alpha+\beta) {\mathcal Q} }$. To see the converse inclusion take an element of the form $\prod_{i=1}^n e^{d_iQ_i}$ where $\sum_{i=1}^n d_i = \alpha + \beta$ and let $k$ be the largest index such that $\sum_{i=1}^{k-1} d_i \le \alpha$. Let $d_k = d'_k+d''_k$ such that $\sum_{i=1}^{k-1} d_i + d'_k = \alpha$. Then we have that
 \begin{equation*}
  \prod_{i=1}^n e^{d_iQ_i} = \left(\prod_{i=1}^{k-1} e^{d_iQ_i} e^{d'_kQ_k}\right) \left(e^{d''_kQ_k} \prod_{i=k+1}^{n} e^{d_iQ_i}\right),
 \end{equation*}
which is an element of $e^{\alpha {\mathcal Q} }e^{\beta {\mathcal Q} }$.

Let $P$ be an arbitrary element of $e^{(\alpha+\beta) {\mathcal Q}}$. Then there is a sequence $\{ P_n\}$ of the form \eqref{sequence-exponential} such that $P = \lim_{n\to\infty} P_n$. Now, by the above considerations, every $P_n$ is a product of $P'_n\in e^{\alpha {\mathcal Q} }$ and $P''_n\in e^{\beta {\mathcal Q} }$, and since both $e^{\alpha {\mathcal Q} }$ and $e^{\beta {\mathcal Q} }$ are compact sets, both sequences $\{ P'_n\}$ and $\{ P''_n\}$ have subsequences that converge to some $P'$ and $P''$ respectively, whence $P = P'P''$ immediately follows and completes the proof.
\end{proof}
For simplicity of notation we will denote an element of $e^{{\mathcal Q}}$ by $\tilde e^Q$, although, as follows from the definition there might be no $Q\in {\mathcal Q}$ such that $\tilde e^Q = e^Q$.
\subsection{Partitions}\label{ss-p}
Let $\mathcal T = \{ t_0, t_1, \ldots, t_n \colon t_{i-1}< t_i, \text{for every } 1\le i \le n\}$ be a {\em partition} of the interval $[t_0, t_n]$. Suppose that $\mathcal T'= \{ t'_0, t'_1, \ldots, t'_m\}$ is another partition such that $t'_0 = t_0$ and $t'_m = t_n$ and $\mathcal T'\subseteq \mathcal T$. Then we will say that $\mathcal T'$ is a {\em refinement} of $\mathcal T$.

For every partition $\mathcal T$ define
\begin{equation}\label{exp-partition}
 e_\mathcal T^{\mathcal Q} = \left\{ \prod_{i=1}^n e^{(t_i-t_{i-1})Q_i}\colon Q_i\in {\mathcal Q}, \text{ for } 1\le i \le n\right\},
\end{equation}
which is a closed set if ${\mathcal Q}$ is closed.
Clearly we have that $e_\mathcal T^{\mathcal Q} \subseteq e_{\mathcal T'}^{\mathcal Q}$ if $\mathcal T'$ is a refinement of $\mathcal T$. The definition \eqref{interval_exponential_formula} can now be equivalently rewritten as
\begin{equation*}
 e^{{\mathcal Q}} = \mathrm{cl}\bigcup_{\mathcal T} e_\mathcal T^{\mathcal Q},
\end{equation*}
where $\mathcal T$ runs over all possible partitions of the unit interval $[0, 1]$.

Further, for every partition $\mathcal T= \{ t_0, t_1, \ldots, t_n\}$, we denote
\begin{align*}
 \delta (\mathcal T) & = \min_{1\le i\le n} t_i - t_{i-1}, \\
 \Delta (\mathcal T) & = \max_{1\le i\le n} t_i - t_{i-1} \text{ and }\\
 | \mathcal{T}| & = n.
\end{align*}
We will call a refinement $\mathcal T'$ of $\mathcal T$ an {\em elementary refinement} if there is at most one $t'_k\in \mathcal T'$ in every interval $[t_{i-1}, t_i]$ for $1\le i \le n$. Further, let $\mathcal T$ be a partition. Then we recursively define the following sequence of refinements:
\begin{equation*}
 \mathcal T^{(1)} = \mathcal T \cup \left\{ \frac{t_{i-1}+t_i}{2}\colon t_i \in \mathcal T, 1\le i \le n \right\}
\end{equation*}
and
\begin{equation*}
 \mathcal T^{(n)} = (\mathcal T^{(n-1)})^{(1)}.
\end{equation*}
Every partition in this sequence is therefore an elementary refinement of the previous one.
\begin{lemma}\label{lemma-subdivs}
 Let $\mathcal T$ and $\mathcal T'$ be partitions such that $\Delta (\mathcal T')< \delta (\mathcal T)$. Then $\mathcal T\cup \mathcal T'$ is an elementary refinement of $\mathcal T'$.
\end{lemma}
\begin{proof}
 Let $t_k\in [t'_{i-1}, t'_i]$ where $t'_{i-1}$ and $t'_i$ are two consecutive elements of $\mathcal T'$. Then $\Delta(\mathcal T') < \delta(\mathcal T)\le t_{k+1}-t_k$ and therefore $t_{k+1}$ cannot lie in the same interval.
\end{proof}
\subsection{Approximations}\label{ss-a}
The calculations involving the interval matrix exponentials cannot be performed directly, as is the case with the exponential in the case of a single matrices, but rather using approximations defined in the previous subsection. In this subsection we will derive an approximation method based on linear programming techniques along with error estimation. The accuracy of the approximations of the form \eqref{exp-partition}, which also are sets of matrices, will be measured using the \emph{Hausdorff metric} induced by the operator norm for matrices. Let $x$ be an element in a normed space and $\mathcal Y$ a compact set. Then the Hausdorff distance between them is
\begin{equation*}\label{hausdorff_1}
 d_H\left( x,  \mathcal Y\right) = \min_{y\in \mathcal Y} \left\| x-y \right\|
\end{equation*}
and the distance between two compact sets is
\begin{equation*}\label{hausdorff}
 d_H\left( \mathcal X, \mathcal Y\right) = \max \left\{ \max_{x\in \mathcal X} d_H\left(x,  \mathcal Y\right), \max_{y\in \mathcal Y}d_H\left( y, \mathcal X \right) \right\}.
\end{equation*}
It follows from the above definition that if $\mathcal X\subseteq \mathcal Y$ then
\begin{equation*}
d_H(\mathcal X,  \mathcal Y) = \max_{y\in \mathcal Y}d_H\left( y, \mathcal X \right).
\end{equation*}
For every set $\mathcal X$ let $\| \mathcal X\| = \sup_{x\in \mathcal X}\| x\|$.
\begin{defn}
Given a partition $\mathcal T$ with $|\mathcal T| = N$ and $\Delta(\mathcal T)=D$, and an interval matrix ${\mathcal Q}$ with $\|\mathcal  Q\| = M$, let
\begin{equation*}
 A({\mathcal Q}, \mathcal T) = N(DM)^2e^{(1+D)M} (1.5+3e^{DM}).
\end{equation*}
\end{defn}
\begin{prop}\label{A-Tn}
\begin{equation*}
 A({\mathcal Q}, \mathcal T^{(n)})\le 2^{-n} A({\mathcal Q}, \mathcal T).
\end{equation*}
\end{prop}
\begin{proof}
 We have that $|\mathcal T^{(n)}| = 2^nN, \Delta(\mathcal T^{(n)})=2^{-n}D$ and $e^{(1+\frac{D}{2^n})M} (1.5+3e^{\frac{D}{2^n}M})\le e^{(1+D)M} (1.5+3e^{DM})$, whence the above inequality easily follows.
\end{proof}
\begin{lemma}\label{approx-lemma}
 Let $\mathcal T'$ be an elementary refinement of $\mathcal T$. Then
 \begin{equation}\label{approx-lemma-equation}
  d_H\left(e_{\mathcal T}^{\mathcal Q}, e_{\mathcal T'}^{\mathcal Q}\right) \le A({\mathcal Q}, \mathcal T).
 \end{equation}
\end{lemma}
\begin{proof}
 We will assume that every interval $[t_{i-1}, t_i]$ contains one $t'_k\in\mathcal T'$, since otherwise, we can form a refinement of $\mathcal T'$ with the centers added of those intervals $[t_{i-1}, t_i]$ that do not contain elements from $\mathcal T'$. The result would be a finer partition $\tilde{\mathcal T}'$, and such that $d_H\left(e_{\mathcal T}^{\mathcal Q}, e_{\mathcal T'}^{\mathcal Q}\right)\le d_H\left(e_{\mathcal T}^{\mathcal Q}, e_{\tilde{\mathcal T}'}^{\mathcal Q}\right)$.

 Now denote, for every $1\le i\le n, d_i=t_i-t_{i-1}, d'_i = t_i - t'_k$ and $d''_i = t'_k - t_{i-1}$, where $t'_k$ is the unique element of $\mathcal T'$ contained in the interval $[t_{i-1}, t_i]$.  Further, let $D=\Delta(\mathcal T)$, and then we have that $d'_i$ and $d''_i\le D$ for every $1\le i\le n$.

 Because of $e_{\mathcal T}^{\mathcal Q}\subseteq  e_{\mathcal T'}^{\mathcal Q}$ we have that
 \begin{equation*}
  d_H\left(e_{\mathcal T}^{\mathcal Q}, e_{\mathcal T'}^{\mathcal Q}\right) = \max_{\tilde e^{Q'}\in e_{\mathcal T'}^{\mathcal Q}}\min_{\tilde e^{Q}\in e_{\mathcal T}^{\mathcal Q}}\left\|\tilde e^{Q'} - \tilde e^{Q} \right\|.
 \end{equation*}
 Every element of $e_{\mathcal T'}^{\mathcal Q}$ is of the form
 \begin{equation}\label{elt_of_refined}
  \tilde e^{Q'} = \prod_{i=1}^n e^{d'_iQ'_i}e^{d''_iQ''_i}.
 \end{equation}
 Convexity of the set ${\mathcal Q}$ implies that the convex combinations $Q_i = \frac{d'_i}{d_i}Q'_i +\frac{d''_i}{d_i}Q''_i$ of the elements of ${\mathcal Q}$, for every $1\le i\le n$, belong to ${\mathcal Q}$ as well. To every element of the form \eqref{elt_of_refined} corresponds the element
 \begin{equation*}\label{elt_of_partit}
  \tilde e^{Q} = \prod_{i=1}^n e^{d_iQ_i}.
 \end{equation*}
 Using Lemma~\ref{refining}, we obtain that
 \begin{align*}
 \left\|\tilde e^{Q'}  -\tilde e^Q \right\| =
 \left\| \prod_{i=1}^n e^{d'_iQ'_i}e^{d''_iQ''_i} - \prod_{i=1}^n e^{d_iQ_i}\right\|  \le n(DM)^2e^{(D+1)M}(1.5+3e^{DM}),
 \end{align*}
 which implies \eqref{approx-lemma-equation} and thus completes the proof.
\end{proof}
\begin{cor}\label{dist-t-tn}
 Let a partition $\mathcal T$ and an interval matrix ${\mathcal Q}$ be given. Then
 \begin{equation*}
  d_H(e^{\mathcal Q}_{\mathcal T}, e^{\mathcal Q}_{\mathcal T^{(n)}}) \le 2A({\mathcal Q}, \mathcal T)
 \end{equation*}
 for every $n>0$.
\end{cor}
\begin{proof}
 By Proposition~\ref{A-Tn} and Lemma~\ref{approx-lemma}, we obtain
 \begin{multline*}
  d_H(e^{\mathcal Q}_{\mathcal T}, e^{\mathcal Q}_{\mathcal T^{(n)}})  \le \sum_{k=0}^{n-1} d_H(e^{\mathcal Q}_{\mathcal T^{(k)}}, e^{\mathcal Q}_{\mathcal T^{(k+1)}}) \\ \le \sum_{k=0}^{n-1} A({\mathcal Q}, \mathcal T^{(k)})
   \le \sum_{k=0}^{n-1} \frac{A({\mathcal Q}, \mathcal T)}{2^k}
   \le 2A({\mathcal Q}, \mathcal T),
 \end{multline*}
 where $\mathcal T^{(0)}:=\mathcal T$.
\end{proof}
\begin{thm}\label{thm_approx}
 Let a partition $\mathcal T$ and an interval matrix ${\mathcal Q}$ be given. Then we have that
 \begin{equation}\label{approx-final}
  d_H(e^{\mathcal Q}_{\mathcal T}, e^{\mathcal Q}) \le 2A({\mathcal Q}, \mathcal T).
 \end{equation}
\end{thm}
\begin{proof}
 Denote $N=|\mathcal T|, M=\|\mathcal  Q\|$ and $D=\Delta(\mathcal T)$. Let $\varepsilon>0$ be arbitrary and $\mathcal T_1$ a partition such that $d_H(e^{\mathcal Q}_{\mathcal T_1}, e^{\mathcal Q})\le \varepsilon$. Further let $n$ be large enough that $A({\mathcal Q}, \mathcal T^{(n)})\le \varepsilon$ and that $\Delta(\mathcal T^{(n)})= \frac{D}{2^n}< \delta(\mathcal T_1)$. Then let $\tilde{\mathcal T}=\mathcal T^{(n)}\cup \mathcal T_1$. We have that $\Delta(\tilde{\mathcal T}) < \frac{D}{2^n}$ and, by Lemma~\ref{lemma-subdivs}, $\tilde{\mathcal T}$ is an elementary refinement of $\mathcal T^{(n)}$ and also a refinement of $\mathcal T_1$. Therefore, $d_H(e^{\mathcal Q}_{\tilde{\mathcal T}}, e^{\mathcal Q}) \le d_H(e^{\mathcal Q}_{\mathcal T_1}, e^{\mathcal Q})\le \varepsilon$ and $d_H(e^{\mathcal Q}_{\tilde{\mathcal T}}, e^{\mathcal Q}_{\mathcal T^{(n)}}) \le A({\mathcal Q}, \mathcal T^{(n)}) \le \varepsilon$, by Lemma~\ref{approx-lemma}, which implies that $d_H(e^{\mathcal Q}, e^{\mathcal Q}_{\mathcal T^{(n)}})\le 2\varepsilon$ and then, by Corollary~\ref{dist-t-tn}, $d_H(e^{\mathcal Q}, e^{\mathcal Q}_{\mathcal T})\le 2(A({\mathcal Q}, \mathcal T)+\varepsilon)$, and since $\varepsilon$ can be made arbitrarily small, this implies \eqref{approx-final}.
\end{proof}
\begin{cor}
 Let $\mathcal T$ be a partition with $\Delta(\mathcal T) = D, \delta (\mathcal T)=d$ and $\|{\mathcal Q} \| = M$. Then
 \begin{equation*}
 d_H(e^{\mathcal Q}_{\mathcal T}, e^{\mathcal Q}) \le \frac{2D^2}{d}M^2e^{(1+D)M} (1.5+3e^{DM}).
 \end{equation*}
\end{cor}
\begin{proof}
 The inequality is a direct consequence of Theorem~\ref{thm_approx} and the fact that $|\mathcal T| \le \frac 1d$.
\end{proof}

\begin{cor}
 Let $\mathcal T$ be a partition with $\Delta(\mathcal T) = \delta (\mathcal T)=D$ and $\|{\mathcal Q} \| = M$. Then
 \begin{equation*}
 d_H(e^{\mathcal Q}_{\mathcal T}, e^{\mathcal Q}) \le 2DM^2e^{(1+D)M} (1.5+3e^{DM}).
 \end{equation*}
\end{cor}
For actual calculations, however, calculating exponential at every time slice is not very practical. We now show that the matrix exponential can be approximated with linear factors.
\begin{thm}\label{thm_linear}
 Let $\mathcal T$ be a partition of the unit interval and ${\mathcal Q}$ an interval matrix. Further let $\Delta(\mathcal T)=D, |\mathcal T| = N$ and $\| {\mathcal Q}\| = M$. Denote $d_i = t_i-t_{i-1}$ for every $1\le i \le N$. Then for every $\tilde e^Q\in e^{{\mathcal Q}}$ there exist matrices $Q_1, \ldots, Q_N$ so that
 \begin{equation}\label{thm_linear_eq}
  \left\| \tilde e^{Q} - \prod_{i=1}^N (I+d_iQ_i) \right\| \le N(DM)^2(2e^{(1+D)M}(1.5+3e^{DM})+0.5e^{M}).
 \end{equation}
\end{thm}
\begin{proof}
 By Theorem~\ref{thm_approx} there exists $\tilde e^{Q'} = \prod_{i=1}^N e^{d_iQ_i}\in e^{\mathcal Q}_{\mathcal T}$ such that $d\left( \tilde e^{Q'}, \tilde e^{Q} \right) \le 2A({\mathcal Q}, \mathcal T).$ By Corollary~\ref{approx-linear-alt} we have that
 \begin{equation*}
  \left\| \prod_{i=1}^N e^{d_iQ_i}-\prod_{i=1}^N (I+d_iQ_i) \right\| \le \frac{N}{2}(DM)^2e^M.
 \end{equation*}
 Combining the above inequalities gives \eqref{thm_linear_eq}.
\end{proof}
\begin{cor}
 Let ${\mathcal Q}$ be an interval matrix with $\|{\mathcal Q} \| = M$. Then for every $\tilde e^Q\in e^{{\mathcal Q}}$ and every $N\in \mathbb{N}$ there exist matrices $Q_1, \ldots, Q_N$ so that
 \begin{equation*}
  \left\| \tilde e^{Q} - \prod_{i=1}^N \left(I+\frac 1N Q_i\right) \right\| \le 2\frac{M^2}{N}e^{\left(1+\frac 1N\right)M} (1.5+3e^{\frac MN}+0.5e^M).
 \end{equation*}
\end{cor}

\section{Solutions of the interval matrix differential equation}\label{s-simde}
In this section we prove that the exponential of an interval matrix ${\mathcal Q}$ indeed provides the solution to the inequalities \eqref{initial_conds}, \eqref{diff_ineq_sup} and \eqref{diff_ineq_inf}, and that the solutions are interval vectors.
\begin{thm}\label{interval-separately}
  Let ${\mathcal Q}$ be an interval matrix. Then, for every $x$, the set $e^{\mathcal Q} x$ is an interval, which we denote by $[\underline{e^{\mathcal Q}}x, \overline{e^{\mathcal Q}}x]$.
\end{thm}
\begin{proof}
 We will use the same notation as in the proof of Theorem~\ref{thm_linear}.

 Let $\mathcal T$ be a partition of the unit interval such that the right hand side in \eqref{thm_linear_eq} is smaller than $\varepsilon$, where $\varepsilon>0$ is arbitrary. Because the rows of ${\mathcal Q}$ are separately specified, so are the rows of $I+d_i{\mathcal Q}$ for every $1\le i\le N$ and therefore, by Proposition~\ref{prop_interval} and the discussion following it, the set $\{\prod_{i=1}^N (I+d_iQ_i)x\colon Q_i\in{\mathcal Q} \text{ for every } 1\le i \le N \}$ is an interval $[\underline x_\mathcal T, \overline x_\mathcal T]$. Moreover, for every $\tilde e^Q\in e^{\mathcal Q}$ we have that $\overline x_\mathcal T \ge \tilde e^Qx - \varepsilon \| x\|$ and similarly $\underline x_\mathcal T \le \tilde e^Qx + \varepsilon \| x\|$.

 Although $\underline x_\mathcal T$ and $\overline x_\mathcal T$ may not belong to $e^{\mathcal Q} x$, there are, by Theorem~\ref{thm_linear}, some $\underline y_\mathcal T$ and $\overline y_\mathcal T\in e^{\mathcal Q} x$ such that $\| \underline x_\mathcal T- \underline y_\mathcal T\|$ and $ \| \overline x_\mathcal T- \overline y_\mathcal T\|\le \varepsilon$. This together with closedness of $e^{\mathcal Q} x$ clearly implies that $\underline x = \inf_{\mathcal T} \underline x_\mathcal T$ and $\overline x = \sup_{\mathcal T} \overline x_\mathcal T$ belong to $e^{\mathcal Q} x$ which is then equal to the interval $[\underline x, \overline x]$.
\end{proof}
\begin{cor}
 Let ${\mathcal Q}$ be an interval matrix. Then, for every interval vector $[\underline x, \overline x]$, the set $e^{\mathcal Q} [\underline x, \overline x]$ is an interval, which we denote by $[\underline{e^{\mathcal Q}}\, \underline x, \overline{e^{\mathcal Q}}\, \overline x]$.
\end{cor}
Every $\tilde e^Qx_0$, where $\tilde e^Q\in e^{\mathcal Q}$ and $x_0\in [\underline x_0,  \overline x_0]$, satisfies the inequalities \eqref{initial_conds}, \eqref{diff_ineq_sup} and \eqref{diff_ineq_inf} by construction. The following theorem shows that the converse also holds.
\begin{thm}
 Let $x(t)$ satisfy \eqref{initial_conds},  \eqref{diff_ineq_sup} and \eqref{diff_ineq_inf}, where ${\mathcal Q}$ is an interval matrix. Then $x(1)\in [\underline{e^{\mathcal Q}}x_0, \overline{e^{\mathcal Q}}x_0]$.
\end{thm}
\begin{proof}
 Because of symmetry we only prove that $x(1)\le \overline{e^{\mathcal Q}}x_0$. Suppose that $\| {\mathcal Q}\| \le M$ and let $\varepsilon>0$ be given. Compactness of the interval $[0, 1]$ implies the existence of $n$ large enough that
 \begin{equation*}
  x\left(t+\frac 1n\right) - x(t) \le \frac 1n \overline{{\mathcal Q}}x(t) + \varepsilon \frac 1n
 \end{equation*}
for every $t\in [0, 1-1/n]$. Further, let $n$ be large enough so that
\begin{equation}\label{dist_lin_exp}
 \left\|\prod_{i=1}^n \left(I + \frac 1n Q_i\right) - \prod_{i=1}^n \left(e^{\frac 1n Q_i}\right) \right\| \le \varepsilon,
\end{equation}
for every sequence of matrices $Q_1, \ldots, Q_n\in {\mathcal Q}$, which is possible by Corollary~\ref{approx-linear-alt}.

Now let $Q_k$ be the matrices such that
\begin{equation*}
 \overline{{\mathcal Q}}x\left(\frac {k-1}n\right) = Q_k x\left(\frac {k-1}n\right).
\end{equation*}
and
\begin{equation*}
 \tilde x\left(\frac {k}n\right) = \prod_{i=1}^k \left(I+\frac 1n Q_i\right)x_0.
\end{equation*}
By \eqref{dist_lin_exp},
\begin{equation}\label{dist_to_exp}
d_H(\tilde x(1),  e^{\mathcal Q} x_0)\le \varepsilon.
\end{equation}
 Now let
\begin{equation*}
 d_k = \max_{i}\left(x_i\left(\frac {k}n\right) - \tilde x_i\left(\frac {k}n\right) \right)
\end{equation*}
where $x_i$ denotes the $i$th component of the corresponding vector.

We have
\begin{align*}
 x\left(\frac {k}n\right) - \tilde x\left(\frac {k}n\right) & \le x\left(\frac {k-1}n\right) + \frac 1n Q_k x\left(\frac {k-1}n\right) + \frac 1n \varepsilon \\
 & \qquad - \tilde x\left(\frac {k-1}n\right) - \frac 1n Q_k \tilde x\left(\frac {k-1}n\right) \\
 & \le d_{k-1} + \frac 1n \| Q_k \| d_{k-1} + \frac 1n \varepsilon \\
 & \le d_{k-1}\left(1+\frac Mn\right) + \frac 1n \varepsilon.
\end{align*}
In the above equations all constants are used as constant vectors. The values of $d_k$ are therefore bounded from above by the solutions of the recurrence equations $d_{k} = d_{k-1}\left(1+\frac Mn\right) + \frac 1n \varepsilon$, whose solution is $d_k = \frac{\varepsilon}{M}\left(\left(1+\frac{M}{n}\right)^k -1 \right)$ and therefore $d_n\le \frac{\varepsilon}{M}(e^M-1)$. By \eqref{dist_to_exp}, the distance between $x(1)$ and some element in $e^{\mathcal Q} x_0$ is smaller than $d_n+\varepsilon$, which can be made arbitrarily small by taking $\varepsilon$ small enough. Therefore we conclude that $x(1)-\overline {e^{\mathcal Q}}x_0\le 0$ which is equivalent to saying that $x(1)\le \overline{e^{\mathcal Q}}x_0$. This concludes the proof.
\end{proof}

\section{The relation with continuous time imprecise Markov chains}\label{s-rcimc}
The method described in previous sections has a direct applicability in the theory of continuous time Markov chains. Consider Kolmogorov's backward equation
\begin{equation*}
	\frac{d}{dt} P(t) = QP(t)
\end{equation*}
and let $x_0$ be an arbitrary column vector. Then we denote $x(t) = P(t) x(0)$, where $x(0)=x_0$. The Kolmogorov's equation then translates into
 \begin{equation*}
	\frac{d}{dt} x(t) = Qx(t),
\end{equation*}
which is exactly the same equation as \eqref{mde}. When imprecision is involved it can be represented through a convex set of generator matrices $\mathcal Q$, which then allows the generalisation of the above equation as in \eqref{initial_conds}-\eqref{diff_ineq_inf}. The solutions of the generalised Kolmogorov equations are then of the form of function $t\mapsto \mathcal X(t)$, where all $\mathcal X(t)$ are interval vectors. Hence we can write $\mathcal X(t) = [\underline x(t), \overline x(t)]$. This induces the operators
\begin{align*}
\underline T(t) & \colon x \mapsto \underline x(t)
\intertext{and}
\overline T(t) & \colon x \mapsto \overline x(t),
\end{align*}
which are known from the theory of discrete time imprecise Markov chains as the lower and upper transition operator, now dependent on time. Therefore we have a natural correspondence between the solutions of the generalised Kolmogorov's equation and transition operators, similarly as in the precise case of continuous time Markov chains.

Let us conclude with an example.
\begin{ex}
Let an imprecise generator matrix be given in terms of a matrix interval $\mathcal Q = [\underline Q, \overline Q]$, where
\[ \underline Q = \begin{pmatrix}
-7 & 4 & 0 \\
2 & -4 & 1 \\
0 & 3 & -6 \\
\end{pmatrix} \qquad \text{and} \qquad \overline Q = \begin{pmatrix}
-5 & 5 & 2 \\
3 & -3 & 2 \\
1 & 4 & -4 \\
\end{pmatrix}, \]
and all $Q\in \mathcal Q$ are assumed to have zero row sums.

Let us approximate $e^{0.2\mathcal Q}$ with $\left(I+\frac {0.2}n \mathcal Q\right)^n$. When $n=80$, the maximal theoretical error is 0.12 and the approximate lower and upper bounds are
 \[ \underline P(0.2) = \begin{pmatrix}
 0.3164 & 0.3839   & 0.0421       \\
0.1545   & 0.5826   & 0.0927       \\
0.0635   & 0.3340   & 0.4019       \\
\end{pmatrix} \quad \overline P(0.2) = \begin{pmatrix}
0.4945   & 0.4984   & 0.2338   \\
0.2864   & 0.6921   & 0.2338   \\
0.1853   & 0.4432   & 0.5323
\end{pmatrix} \]
With $n=200$, the maximal theoretical error drops to 0.02 and the lower and upper bound are approximated with
 \[ \underline P(0.2) = \begin{pmatrix}
0.3181   & 0.3830   & 0.0420       \\
0.1541   & 0.5836   & 0.0924       \\
0.0633   & 0.3332   & 0.4033       \\
\end{pmatrix} \quad \overline P(0.2) = \begin{pmatrix}
0.4957   & 0.4972   & 0.2333   \\
0.2858   & 0.6928   & 0.2333   \\
0.1849   & 0.4421   & 0.5334  \\
\end{pmatrix} \]
\end{ex}
The presented method is apparently not very efficient in terms of convergence. Therefore, it remains a challenge for further work to find more efficient methods for estimating imprecise continuous time Markov chains.

redpajama_set_name: RedPajamaArXiv


The lifeblood of a consulting business consists of the relationships that you build with your clients.You must care for them as if they were plants. Water them liberally, prune them as needed, and treat them as best you can.
Treating a client well in no way guarantees that they will reciprocate. Most clients do, but there are a few who have an alternate agenda, which isn't typically seen until it's too late. So what can happen you ask?
I present you with this cautionary tale about how to not lose $40,000.


Dimensions Concrete 12X24 Glazed Porcelain tile is Versatile and Elegant. With its shade of Grey's It stands up to today's demanding applications both indoors and outdoors. Easy to clean and low maintenance, this product is ideally suited for both residential and commercial applications.


Glamping sounds great, I am glad the travel gods have been on your side and you found this little gem of a place. A well deserved rest and fun after your housesitting. I am looking forward to the big reveal of your future plans.


text:
Left Book Club
Independent Radical Publishing
Brett Scott
Brett Scott is a journalist, campaigner and the author of The Heretic's Guide to Global Finance: Hacking the Future of Money (Pluto, 2013). He writes for publications such as the Guardian, New Scientist, Wired Magazine and CNN. He is a Senior Fellow of the Finance Innovation Lab, he helps facilitate a course on power and design at the University of the Arts London, and facilitates workshops on alternative finance with The London School of Financial Arts.
View Author Website
The Heretic's Guide to Global Finance
Hacking the Future of Money
Shows how activists can tap into the internal dynamics of the sector to disrupt it and showcases the growing alternative finance movement.
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Like most websites, we use cookies to improve our service and make your user experience better. See our updated Privacy Policy to find out more about cookies and how we use your data.

redpajama_set_name: RedPajamaCommonCrawl


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redpajama_set_name: RedPajamaGithub


Such a beautifully written piece of art. Kelly Minter writes about her eye and heart opening trip to the Amazon jungle with such detail that you can picture everything as if you are there.
I have to review this book as I read along so I don't forget anything! This paragraph really stood out to me. The Holy Spirit brought to her attention a young man and a word for him. The word for him was that God saw him and he was going to a great leader in his village. That brought a smile to the face of this sad child. He responded with, "No one has ever seen me before. God sent you to see me." HOW POWERFUL IS THAT?!
"We may desire deep friendships, ironclad marriages, thriving children, a secure economy, healthy bodies, a trip to Europe- but what we really want is an encounter with the living God. To be loved, known, and seen by Him. We pursue all the other stuff because it's the stuff that tastes, but Jesus is the One who nourishes."
Kelly's story is inspiring! We don't have to take a trip to the Amazon jungle to experience these changes in our lives. We are surrounded by our own jungle daily. I make it a point to help out someone in need if I am able. My children also notice those in need a lot more than they use to. I have so many testimonies of times that God has provided us with the opportunity to be a blessing to others in their time of need.
As we pack up our house I realize how MUCH we have and so much of it we don't really need. We are happy to fill up bags to donate. In the Amazon and so many other countries (including our own) kids don't have beds to sleep in, shoes on their feet, or any idea where their next meal will come from. We can help with that!
My son's class took a trip this week to a place where they helped pack 200,000 meals for children and their families. That night at bible study he lifted up those children and their families in prayer. That trip impacted him in a good way. He has always been my helper, but this was on a different level.
This can be a convicting story to some. It made me want to pack my bags so I can go help the people of the Amazon. Which takes me back to what I stated earlier. I can help in my own neighborhood, city, state. I don't have to go far to see the same types of difficulties.
Thank you Jesus for blessing my family so that we may bless others.
Thank you Christian Family for providing this book to review. Opinions are my own.


This colorful spherical ice bucket will keep your ice for a long time thanks to its double wall. Coordinating colors makes it a unique and valuable, useful during your relaxing moments. His practice also serves as a handle closing the bucket. Comes with ice tongs, it is dishwasher-safe.
Note that STELLA was created in the 70s: he became one of the most iconic brand GUZZINI.


In addition to IF Stjernen, 93 other teams from 16 different countries played in Boys 16. They were divided into 15 different groups, whereof IF Stjernen could be found in Group 4 together with TUS Aumühle-Wohltorf, Ski IL 1, Tollarps IF, Enköpings HF and IK Sävehof 1.
IF Stjernen continued to A-Play-off after reaching 2:nd place in Group 4. In the playoff they made it to 1/32 Final, but lost it against Eskilstuna Guif with 10-17. In the Final, Skanderborg HB won over Norrköpings HK and became the winner of A-Play-off in Boys 16.
IF Stjernen also participated in Boys 15 during Partille Cup 2017. They reached the 1/32 Final in B15 A-Play-off, but lost it against Hebraica with 12-16.


Should I be worried about my baby's weight?
My babyis 10months old and weighs 7.3kgs should i be worried?
That does seem rather small for 10 mos, but it really depends as to whether your pediatrician is concerned. If you are going to your pediatrician regularly then voice your question to him or her. Or even call them and ask, it will make you feel better to find out.


We are international sales agents and distributor, focused on docs, fiction and tv formats.
Started in 2016, we are located in two different countries and we operate all over the world.
Our mission to connect East with West thanks to the films.
We have our own marketing and communication agency that helps us to promote and communicate our catalogue.
Our experienced team can support you since the beginning of your project. We can advise you in benchmarking, branding, positioning, target audience, communication, promotion and media plan.
There are no differences between big and small project; there's a difference between good and bad. The penguin can manage locations, do product placement, help in shooting and editing, support in DCP and IMF export and subtitling.
Thank to our own agency, we help you promoting your products by providing strategic advices and creating powerful communication plans.
We know that it must have been very hard to get your film; we'll put in the same effort to share it with the world.
BPF deals with distribution focus on festival, theatrical, broadcaster, SVOD, VOD, Educational. As you may have guessed, the blue penguin will follow you anywhere you want to go, crossing borders!
Follow the Penguin, find festivals!


The effect of Customer Capital on Loyalty with Satisfaction and Commitment as Intervening Variables: An Empirical Investigating in Credit Division at PT Bank Sumut, Branch Coordinator of Medan, Indonesia.
Loyalty is essential to maintaining good relationships with customers. The banking activity is trust by the client, so the bank is a financial institution that is the place for companies, government agencies, and private parties, as well as the public in saving funds. Customer capital is one of the elements of intellectual capital, a harmonious and good relationship that the company has with its partners. It comes from a loyal and satisfied customer of the company's service. Customer satisfaction as a fulfillment response, from comparing customer experience with expectations of service. Customer commitment is a constant desire to maintain a good relationship with the company. The purpose of the study is to examine the influence of customer capital, consumer loyalty and customer satisfaction towards loyalty. This study uses quantitative methods with a sample of 100 customers of the credit division of PT Bank Sumut branch coordinator of Medan. Using Partial Least Square Structural data analysis (PLS) by testing external models and internal models. The results of this study show that customer capital is changing and customer satisfaction has a positive and significant effect on loyalty with a 5% interest level, but consumer commitment has an insignificant and positive impact on customer capital, customer capital has an insignificant effect on consumer loyalty through consumer commitment. Customer capital has a significant effect on customer loyalty through customer satisfaction. It is concluded that changing customer capital is greater in influencing consumer loyalty. The implications in this study are to conduct ongoing follow-ups, become financial consultants, to increase financing activities and to facilitate credit procedures.
Universitas Sumatera Utara, Medan, Indonesia.
Endang Sulistya Rini, Faculty of Economics & Business, Universitas Sumatera Utara, Medan, Indonesia.
Khaira Amalia Fachruddin, Faculty of Economics & Business, Universitas Sumatera Utara, Medan, Indonesia.


text:
The Leela Palace – New DelhiAgent2021-04-15T17:05:48+11:00
Home — Destinations — India — Rajasthan & North India — The Leela Palace – New Delhi
The Leela Palace New Delhi is an ultra-luxurious hotel inspired by maharajah's palaces located in New Delhi's upscale Diplomatic Enclave.
Located in Delhi's upscale Diplomatic Enclave, The Leela Palace New Delhi is a luxurious oasis in a city that never sleeps. Inspired by maharajahs' palaces of days gone by, this opulent property is built out of earthy sandstone and its interiors feature plenty of gleaming polished marble, plush Persian rugs and glittering chandeliers.
Accommodation at The Leela Palace New Delhi is fit for royalty, with even the smallest room larger than its counterparts across the city. There are 254 guestrooms and suites in total, beautifully appointed and filled with state-of-the-art amenities. The extravagant suites come with separate living areas, dedicated butler service and access to an exclusive lounge.
On the rooftop of The Leela Palace is the only terrace pool in Delhi – heated and with an infinity edge. The hotel also has a well-equipped gym and fabulous day spa, with a long list of therapies available for both women and men. Onsite dining options are equally impressive, with Indian, Japanese, French and Italian cuisine on offer, along with round-the-clock room service.
How to get to The Leela Palace New Delhi
The Leela Palace New Delhi is a convenient 12km from Indira Gandhi International Airport and 6km from the centre of the Indian capital.
The Leela Palace New Delhi Highlights
A luxury hotel inspired by maharajahhs' palaces
Boasting enormous guestrooms and opulent suites
Exquisite public spaces of marble, gold and glass
Home to Delhi's only rooftop infinity pool
World-class recreational facilities and outstanding dining options
Want to find out more about our India & Sri Lanka tours?
Get in touch one of our destination specialists.
Below are a selection of our preferred hotels in Rajasthan and North India. Please contact us to tailor accommodation options to suit the style of your trip and your budget.
Crystal Sarovar Premiere
Crystal Sarovar Premiere in downtown Agra offers stylish, modern accommodation and a peaceful atmosphere, while being just 3km from the Taj Mahal.
Fateh Garh is an eco-friendly hilltop hotel in Udaipur offering sweeping views of the City of Lakes, its ancient sites and the Aravalli mountain range.
ITC Mughal
The stately ITC Mughal offers sumptuous accommodation sprawled across 35 perfectly manicured acres in close proximity to the Taj Mahal in Agra.
Jamtara Wilderness Camp
Jamtara Wilderness Camp is a lovely tented camp at the edge of Pench National Park, the evocative setting for Rudyard Kipling's classic novel, The Jungle Book.
Jawai Bandh
Jawai Bandh is a splendid Relais & Châteaux luxury tented camp set in untrammelled wilderness where leopards still roam free.
The Le Meridien New Delhi is conveniently close to many famous monuments, from India Gate and Jantar Mantar to Connaught Place.
The extravagant 19th century Rambagh Palace offers spectacularly restored guestrooms that were once the private chambers of the Maharajah of Jaipur.
Samode Haveli
Samode Haveli is a unique boutique hotel tucked inside a nobleman's family home. The property is located within the ancient walls of the 'Pink City' of Jaipur.
Shahpura Bagh
Shahpura Bagh is one of Rajasthan's finest luxury homestays, located on a picturesque nobleman's estate between Udaipur and Jaipur.
Taj Jai Mahal Palace
The exquisite Taj Jai Mahal Palace provides a taste of regal Rajasthani heritage, occupying a painstakingly restored royal residence in the heart of Jaipur.
Taj Lake Palace
The Taj Lake Palace is a unique luxury hotel set in an 18th century palace of white marble that appears to float on Udaipur's Lake Pichola.
The Birder's Inn
The Birder's Inn is the perfect base for bird-watching in India's wonderful Keoladeo Ghana National Park which boasts over 350 avian species.
More of India


The shortest of love stories can sometimes have the longest consequences. Under false names, young Sandrine comes from the darkness of war into a sun-drenched retardation in Sweden. The year is 1945. Her only baggage is a child she's planning to get rid of, and a secret she's prepared to do anything to keep.
Seven months later she is getting married on an island far out in the archipelago. Female Counselor Ivan Ceder has applied for an exemption for his bride-to-be, the not yet legal Sandrine. Background: Unknown. Seventeen years old, with death behind her and life in front of her?
By saying yes to Ivan, Sandrine marries into a strange world: the great upper-class Ceder family. During the wedding night, a dangerous snowstorm enters the archipelago and the whole party is trapped on the island. It is as if this night foretells how the rest of the marriage will develop – an ice cold mesallians built on secrets, lies and fears.
Vera is Anne Swärd's fourth novel, about flight and survival, about mothers and daughters; about guilt and love that carry on for generations.
First published by Albert Bonniers Förlag, Sweden 2017.
"Vera" is reminiscent of Ingmar Bergman's austere existential dramas set in high society and in barren coastal landscapes /… / Anne Swärd is one of Sweden's most compelling contemporary authors.
Anne Swärd is a divinely gifted storyteller and a novelist of great European significance. Without a hint of hesitation she writes a beautiful and grand prose that evokes one image after the other and that unfolds like a Günter Grass novel towards a world that is at once childishly accurate and ingeniously wicked.
Her style is grand and epic, luminous and plot-driven and she unabashedly uses great emotional expressions /… / a bewitching fairy-tale.
Anne Swärd's web – full of secrets, lies and mysteries – is so intricate and astonishing, a genuine page-turner with just as much poetic finesse as terrifying events /… / Swärd's book is everything but black and white, it's rather blue like the sea. The sea as beautiful, like ice, as cold, like enigmatic, alluring and frightening /… / With "Vera", Anne Swärd has written a both menacing and poetically perfect novel.
Anne Swärd's novels always have a magnificent tone, written with a strong sense of atmosphere and an eloquent narrative – it's almost like a beautiful painting by Danish artist Vilhelm Hammershöi, foreboding secrets on the other side of the door, and when I describe "Vera" it resembles an evil fairy tale, illusively characterised as a Captain Bluebeard with an incarcerated woman.
Anne Swärd evokes the dark story with a remarkably powerful and convincing prose that is occasionally painful to read.
A majestic and astonishing novel.
An extraordinarily good novel /… / She is an epic storyteller.
The main character is an encapsulated princess and the people around her symbols and shadows, either good or bad. So in a near future I will probably believe this was something I read a very long time ago. It may well be what we call timeless.
The scenes are dramatic. A wedding party in the depths of midwinter, which is interrupted by a blizzard, a village at the top of a mountain that no roads lead to, a larch forest, an aristocratic apartment with sombre velvet curtains, silk noil nightgowns and an inertia-like slumber induced by powder white opiates /… / "Vera" is a powerful story that does not leave me unaffected.
… a multi-faceted, vivid novel /… / Anne Swärd's prose awakens strong emotions inside of me.
Anne Swärd has a peculiar flair for making stories come alive and for delving deep under the surface in a way that is not always comfortable, but very poignant and fascinating / … / Circumstances are shrouded in mystery. Everything is illusive and implicit. Still there is a sense of fervor and presence in the prose, which keeps me eagerly hooked.
"Vera" is an expertly written story; poetic, dramatic and surprising. Sometimes it reminds me of a fairy-tale.
This is a novel with ingredients such as escape, betrayal, abuse, love and lies. Swärd has a gift for describing events with small nuances that don't mean much, but at the same mean everything. As a reader you are pulled into the subdued, paralyzing mood, unable to let go.
"Vera" is a grand, epic story of loneliness, flight and dissolution. And perhaps of survival. The prose is flawless – the best I have read in a long time – as is the plotting. There isn't a false note anywhere.
I am (yet again) impressed by Anne Swärd's luminous prose. It is pure, atmospheric and full of shrewd and beautiful metaphors.


01/25/2019 Complaint Download Complaint filed. Endangered Species Act Lawsuit Filed Challenging Federal Approvals for Montana Silver and Copper Mine Project. A "coalition of traditional cultural leaders from the Ksanka Band of the Ktunaxa Nation and local, regional, and national conservation organizations" filed a lawsuit in the federal district court for the District of Montana asserting that federal agencies failed to comply with the Endangered Species Act when they authorized the Rock Creek Mine project in the Cabinet Mountains in northwest Montana. The complaint alleged that the copper and silver mine project would tunnel under one of the region's last undeveloped habitats for two threatened species, grizzly bear and bull trout. (The complaint alleged that bull trout were threatened by a number of factors and were particularly vulnerable to climate change because they require "especially cold water to spawn and rear.") The plaintiffs contended that the U.S. Fish and Wildlife Service (FWS) had concluded that a 2006 no-jeopardy determination for the grizzly bear remained valid without considering new mortality data. The plaintiffs also challenged the FWS's biological opinion for the bull trout as well as U.S. Forest Service authorizations that relied on the FWS determinations.


18th June 2015. That was the date that Surrey's season reached a low. That evening, Surrey left the field at Guildford having been defeated by seven wickets by Glamorgan. This, after winning the toss, scoring 406 and having the Welsh county 106 for six in their first innings. They left that game 48 points adrift of leaders Lancashire at the halfway point of the County Championship.
The T20 Blast campaign was in disastrous shape, the London county picking up four points from their first six games, and rooted to the bottom of the South Group.
They had won four matches from their first 14. Things looked extremely bleak. After that Guildford defeat, however, things turned a corner.
Surrey won four of their remaining eight Blast games as they staged a late, albeit unsuccessful, charge for a place in the knockout stages. It was a disappointing end to the campaign, particularly considering defeat to rivals Middlesex all but confirmed their exit. This was a blessing in disguise, however, as they were able to focus on the Championship and the Royal London One-Day Cup.
Six wins from eight group stage games, including a no result, saw Surrey top Group A in the RLODC in relatively emphatic fashion – three of those victories were by at least 40 runs. Things became far more nerve-wracking after that, however, as knockout matches against Kent and Nottinghamshire almost ended in disaster thanks to centuries from Matt Coles and Greg Smith, who nearly clinched victory for their respective counties having stared defeat in the face.
Playing Gloucestershire in the final, a Jade Dernbach six-wicket haul, including a hat-trick, and half-centuries from Kumar Sangakkara and Rory Burns left Surrey with one hand on the trophy. Some poor batting and relentless good bowling from Gloucestershire saw them lose by six runs, but it had been a strong campaign, especially after winning just one game in this competition in 2014.
Before, and during, their success in the RLODC, Surrey needed to recover in the Championship. That's exactly what they did, winning their next four games, including an innings victory over Gloucestershire. Still second, the gap on Lancashire had closed to just ten points with four games remaining.
The antepenultimate match of the season saw them leapfrog Lancashire as a Gareth Batty hat-trick secured his side promotion. Surrey were left on the verge of the Division Two trophy – needing to avoid defeat to their contenders and pick up one win to guarantee the title. Despite their best efforts, they couldn't quite beat Lancashire or Northamptonshire, but Lancashire's draw at Essex saw Batty's men lift the trophy. They had won 17 from 27 after the Glamorgan defeat, and it had been quite the turnaround as desperation turned to joy in an ultimately successful season.
The focus this year has been on young players, and they're certainly in good supply at The Oval. The two standout stars were Tom and Sam Curran, but Matt Dunn had a solid season in the Championship, picking up 28 wickets from his nine games, James Burke played a huge role in Surrey's One-Day Cup campaign and another strong year for Zafar Ansari saw him called up to the England squad for the UAE.
In addition, the experience of Kumar Sangakkara and captain Gareth Batty has been invaluable this year, both on and off the field. Contributions from everyone, however, was the key to Surrey's success, and regardless of their age, it's clear to see an abundance of talent at The Oval.
Some have made comparisons to Surrey's almost identical 2011 season, where a mix of developing and experienced talent led them to promotion and a List A trophy. The death of Tom Maynard the following year was, understandably, a game-changer for that group of players, something that, one would hope, will not be an issue next year. They found form at the end of 2015, ending on a high, and with the developing players a year older, 2016 could be a really great year for Surrey.
Tom Curran had quite the season. He took 105 wickets at 24.9 in all formats in 2015, including five-wicket hauls on five occasions along with a ten-for in the final Championship game of the season against Northants. His good form this year has also led to him being picked in the England performance squad.
Tom Curran may have taken over 100 wickets across all formats this season, but it's his younger brother Sam who has been making all the headlines. At 17, he has 44 wickets from 23 matches since his debut in June, and consistently bowls at 85 miles per hour, along with 239 first-class runs at 47.8 in six games, and is a future England star in the making.
While it's difficult to pick any player who notably struggled during the year, Gary Wilson might have been slightly disappointed with his season. Unable to make a competitive century for the first time in three years, he also struggled with the gloves in the Nottinghamshire RLODC semi-final, and lost them to Ben Foakes in the following game.
On several occasions in the One-Day Cup, Surrey suffered a collapse caused by a top-heavy batting order. This was none more notable than in the final, where they lost their last eight wickets for just 71 runs and lost a game they had all but won. The middle order have batting talent – that's clear to see – but the inability to put it into practice as wickets fall cost them.
Defeat in the One-Day Cup final would have dampened the mood slightly at Surrey, but it's been an otherwise impressive season that has seen young talent emerge and experienced players at the top of their game as silverware returned to The Oval.


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100 First Animals; Little Tiger Parent Panel.
My Big Book of Lift And Learn. 100 First Animals.
Little Tiger Parent Panel Review.
We were happy to be accepted as 'Little Tiger Panellists' on there new Parent Panel Review Team. This gives us the opportunity to review a wide variety of books. Personally, I feel books are great at any age and encourage learning of the child. The first book we received for review was 100 First Animals. This book was perfect for us as Jack loves animals.
The book is a Lift the Flap style, so gets the parent and child to interact througout the book. There are twelve pages to the book and each features a different theme of animals including 'pets', 'farm', 'snow', 'desert' and more. Then included on each page are seven different animals that match the category.
The pictures on each of the pages are very bright so eye-catching to children, both young and older. Each page shows an image of the animals some even have the sounds in which they make. You then lift the flap to be shown a closer image of the animal on one side and the name of the animal on the open side of the flap.
I feel this is a great book for as children grow and learn as you could use it to remember animals names. Your child could try to guess the animal and then lift the flap to see if they are right.
Overall we found this a wonderful book with bright colours throughout which Jack found very amusing. We give this book a firm 5/5.
These books are great. Bud still loves his animal picture books even though he's nearly 3.


The laws in California permit for the impounding and elimination of illegally parked autos on both personal and public property. Throughout litigation, depositions become one of the useful tools for any legal professional or lawyer. In addition, individuals have time constraints that prohibit frequent visits to the lawyer's office.
We also can not refund any money paid by you straight to 3rd events, reminiscent of funds made by you directly to attorneys affiliated with our legal plans or attorney-assisted merchandise. States that haven't created simplified types will nonetheless help you file a formal title change case, but the kinds will have to be obtained from a legal forms enterprise.
You can organize to have legal research and writing done by a variety of providers. Instead of the usual that requires that trade or enterprise bills be incurred in the carrying on of a commerce or enterprise, the tax regulation imposes a requirement that the expenditure bear an affordable and proximate relation to the manufacturing or collection of revenue or to the administration, conservation, or maintenance of property held for the production of revenue.
Courts are more ceaselessly offering online dockets to the public, who can research the chronological summaries of any explicit case without having to visit the clerk and search by way of courtroom records. Moreover, many paralegals additionally provide their providers exterior the workplace of attorneys.
Land regulation varieties the basis for most kinds of property regulation, and is probably the most advanced. On this complex enterprise world individuals often want the service of a reliable business attorney. 27 This disruption benefits people who otherwise could not hire a lawyer by expanding their access to legal providers.
Family regulation covers marriage and divorce proceedings, the rights of youngsters and rights to property and money in the event of separation. Thus, generally, a corporation should incur legal and different skilled fees for its own benefit and can't deduct charges incurred only for the good thing about its stockholders.
In the midst of a person's life, many forms of legal companies are wanted. The main points: Provide expires March 31, 2019, at 11:fifty nine pm PT. Excludes shipping and filing fees, third-occasion merchandise and charges, legal kinds, lawyer-assisted merchandise, subscriptions that embody a free trial & renewals. The criticism is filed within the court docket clerks workplace and a replica must be despatched to the owners, to inform them of the lawsuit.
Moreover, peculiar and mandatory bills paid or incurred in the management, conservation, or maintenance of a constructing dedicated to rental purposes are deductible notwithstanding that there's truly no earnings there from within the taxable year, and regardless of the manner during which, or the aim for which, the property in query was acquired.
Rules on the usage of personal property fall underneath intellectual property, company legislation , trusts and commercial regulation An example of a basic case of most property regulation is Armory v Delamirie 1722. Just as one's personality may change over time, so can handwriting. We take a practical approach to teaching the meanings and functions of legal guidelines and legal systems with a focus on incomes in-demand certifications and building abilities you'll be able to take into the workplace.
The first class of bills which can be deductible contains those paid or incurred "for the production or collection of income." It isn't crucial that they relate to the production or assortment of income in the present taxable 12 months, so long as the legal expenses were presently paid or incurred.
About one million children are affected by divorces within the U.S. every single 12 months. Proof legislation involves which materials are admissible in courts for a case to be built. There is only one reason why our laws are so arduous, hid and sophisticated and it's as a result of it provides energy to legal professionals comparable to lawyers and legislators.
186 The law of the modern firm started with the Joint Inventory Companies Act 1856 , passed in the United Kingdom, which supplied investors with a simple registration procedure to realize limited liability below the separate legal personality of the corporation.
Breach of contract is the important claim made in the foreclosure lawsuit, when the bank complains to the court docket that the owners have didn't pay their mortgage as agreed.
Handwriting analysis, the science of understanding the character and character from one's handwriting, has loved elevated reputation and elevated use over the last decade. It should be famous, nonetheless, that deductible nonbusiness legal and other professional bills should nonetheless meet all the opposite necessities of deductible trade or business expenses. We acknowledge that in a time when there are soaring well being care prices, that hashish is really a matter of safety for people who find themselves affected by cancer and different ailments," Trufant mentioned.
Of, relating to, or attribute of the occupation of legislation or of lawyers: a legal mind. Rendering legal providers on a contingency foundation means that the consumer won't initially pay any amount of money as legal professional's charges to the lawyer. Civil law seeks to resolve non-legal disputes comparable to disagreements over the meaning of contracts, property ownership, divorce, little one custody, and damages for private and property damage.
If, nevertheless, the taxpayer vacates and converts his residence to rental property, legal expenses incurred in its administration as rental property are deductible. Such a lawyer handles any case that entails a person getting injured. Breach of contract is the essential declare made in the foreclosure lawsuit, when the bank complains to the courtroom that the homeowners have didn't pay their mortgage as agreed.
Some bills, that are paid or incurred in connection with revenue-producing activities (akin to investing) of a noncorporate taxpayer, will not be actually commerce or enterprise expenses and likewise can't be characterised as either personal bills or capital expenditures.
Watching the commercials on late-night time TELEVISION might be quite an training for the property planning lawyer. Though the property may not be at the moment productive and there is no probability that the property can be offered at a profit or in any other case will probably be productive of income, expenses paid or incurred in managing, conserving or sustaining it might be deductible.
A taxpayer cannot deduct legal and professional fees paid or incurred in a taxable 12 months (even when they're proven to be unusual, crucial and cheap and meet the opposite requirements herein discussed) except the taxpayer can show that the expense was paid or incurred in carrying on a commerce or enterprise.
The next are a few of the more common laws that some towing corporations violate. That is when a personal harm lawyer comes into play. Your personal damage lawyer may want additional photos or an knowledgeable to examine the car earlier than it is altered. Details about our clients is a crucial part of our enterprise, and EA would never sell your personally identifiable data to anybody.
Beneath this theory, the origin and character of the declare with respect to which the legal or other professional fee was paid or incurred is examined to determine whether or not it arose from a personal or capital transaction or whether it's related to a business or revenue-looking for exercise.
Handwriting evaluation, the science of understanding the character and persona from one's handwriting, has loved increased popularity and increased use over the last decade. Lis Pendens is a Latin time period that means "a swimsuit pending," and is most ceaselessly used to explain a foreclosures lawsuit that has been initiated in opposition to a specific property. Make sure that your Will conforms to the laws of your state. Each time a person is not satisfied by the decision in a decrease court they've the appropriate to appeal on the basis of the improper utility of the regulation or procedures.


Yelm' first mayor, R. B. Patterson, local pharmacist, is the first to be interviewed in a series of sketches of Yelm's business men who will be separately biographed in this paper each week.
Born in Yamhill, Oregon, in 1890, Mr. Patterson attended college at Oregon State College in Corvallis, where he met and married Patsy Patterson in 1913. After a short residence in eastern Oregon, Mr. Patterson moved to Yelm where he opened his drugstore on the location now occupied by the office of Brown Bros. Garage, later buying the property where the tore is now located. He built a frame building which served as a residence as well as a place of business. Following the fire in 1924 he erected the building he now occupies.
In 1924 Mr. Patterson was elected as Yelm' first mayor and served three terms. His two daughters, Shirley and Jean, attended high school and later the University of Washington and are now married. In fact, the druggist became a grandfather a short time ago.


CHEMICALS USED IN PULP &PAPER MANUFACTURING AND COATING. While there are about 3000 or so different types of chemicals which "can be used" in papermaking, in practice only about 200 individual chemicals are typically used, each satisfying a specific need.
Manufacturer of Food Grade Chemicals - Gypsum Food Grade, Sodium Carbonate, Psyllium Husk and Flour Whitener offered by Nikunj Chemicals, Vadodara, Gujarat. Manufacturer of Food Grade Chemicals - Gypsum Food Grade, Sodium Carbonate, Psyllium Husk and Flour Whitener offered by Nikunj Chemicals, Vadodara, Gujarat.
And whether gypsum chemical is chemical, beverage, or food. There are 4,664 gypsum chemical suppliers, mainly located in Asia. The top supplying countries are China (Mainland), Kenya, and India, which supply 96%, 1%, and 1% of gypsum chemical respectively. Gypsum chemical products are most popular in Southeast Asia, Africa, and South America.
Heat resistant plaster is a building material used for coating walls and chimney breasts. Its purpose is to replace conventional gypsum plasters in cases where the temperature can get too high for gypsum plaster to stay on the wall. Heat resistant plaster should be used in cases where the wall is likely to exceed temperatures of 50 °C.
The Toxic Substances Control Act (TSCA) addresses the manufacturing, processing, distribution, use, and disposal of commercial and industrial chemicals. The Pollution Prevention Act (PPA) establishes pollution prevention as the national policy for controlling industrial pollution at its source.
"In the manufacture of super-phosphate fertilizer, phosphate rock is acidulated with sulfuric acid, and the fluoride content of the rock evolves as volatile silicofluorides. In the past, much of this volatile material was vented to the atmosphere, contributing heavily to pollution of the air and land surrounding the manufacturing site.
When gypsum is used as a soil amendment, moisture in the soil causes these toxic chemicals to leach into the earth as well. Raising the standard for drywall Fortunately, the industry is beginning to address these issues.
And whether gypsum chemical is chemical, beverage, or food. There are 4,483 gypsum chemical suppliers, mainly located in Asia. The top supplying country is China (Mainland), which supply of gypsum chemical respectively. Gypsum chemical products are most popular in Southeast Asia, Africa, and South America.
Gypsum is a by-product available from Phosphoric Acid Plant and used in production of Ammonium Sulphate. GSFC has been first in the country to develop and implement successfully Phospho-Gypsum process for manufacture of Ammonium Sulphate.
Gypsum free from coloring matter and of high purity is used in manufacturing of white cement. It is used as fertilizer. Fertilizing Corporation of India use natural gypsum in manufacturing of ammonium sulphate. In the manufacturing of P.O.P., ceramics and distemper gypsum of transparent variety is used.
Manufacturing of Concrete with Retarders R.Greesan¹, P.Prathap², R.Vijayakumar³, ... maintain the w/c ratio the paper suggests to use the chemical admixture called Gypsum ... • Can be used in cement-gypsum blends.
The few sources of synthetic gypsum are power utilities and acid neutralization. Many electric power utilities use high-sulfur bituminous coal as their source of fuel and in case of acid neutralization the chemical gypsum produced is of high quality, but it also is fine and wet, making it difficult handle.
Gypsum is available in two forms, viz, mineral gypsum and by-product gypsum. The mineral gypsum is mined from land deposits. By-product gypsum is obtained as by-product from different chemical plants.
Synthetic gypsum is the name used to classify a variety of gypsum by-products. The most common gypsum by-product is FGD gypsum. When referencing synthetic gypsum, most people are likely referring to FGD gypsum. However, there are small quantities of synthetic gypsum created through various acid-neutralizing industrial processes.
Typically, Synthetic Gypsum has lower trace metal content then what is typically found in residential soil standards. All of our synthetic gypsum wallboard products are just as safe as natural gypsum and are certified as low volatile organic compound (VOC) products by the GREENGUARD Environmental Institute.
FGD gypsum is a by-product of desulfurization of flue gas from the stacks of fossil-fueled power plants, and the emissions captured from smoke stacks are purified into a hard substance and manufactured into gypsum. FGD gypsum has been used to manufacture gypsum board for more than 30 years.
10 · EZ OFFICE is the fastest and easiest way to add a modular in-plant office in any warehouse or manufacturing facility. We have standardized this product line to include our most popular options and to use our best selling LEEDwall In-Plant Office System, which includes superior quality materials and a completely modular design.
Synthetic gypsum with potentially harmful materials is not used to manufacture gypsum board. Members of the Gypsum Association do not use phosphogypsum to manufacture any gypsum-based product. By using synthetic gypsum in its manufacturing process, the gypsum industry contributes to a cleaner environment in at least two ways.
commercially feasible. The Merseburg Process for manufacturing ammonium sulfate from gypsum is based on the chemical reaction between gypsum and ammonium carbonate. Ammonium carbonate is formed by the reaction of ammonia and carbon dioxide in aqueous solution. The reaction produces insoluble calcium carbonate and an ammonium sulfate solution.
Gypsum is commonly associated with rock salt and sulfur deposits. It is processed and used as prefabricated wallboard or as industrial or building plaster, used in cement manufacture, agriculture and other uses. Relation to Mining. IMAR 7 th Edition. Most of the world's gypsum is produced by surface-mining operations.
Whether they use natural gypsum (calcium sulphate dihydrate) or FGD gypsum (a by-product from flue-gas cleaning at power stations), manufacturers inspect all material continually to ensure quality and performance of the product. In use, plasterboard with FGD or natural gypsum is inert.
where is the chemicals used in the manufacture of gypsum. ... Besides the newly mined material, up to 20% of the gypsum used to manufacture drywall can be recycled from waste generated at the manufacturing plant or at construction sites. chemicals used in mining gypsum - Mine Equipments.
We are sourcing one of best Natural Gypsum Ore available in world, where the purity goes upto 96% which is best for Manufacturing of Plaster of Perris & Cement etc. Gypsum can also be used as a fertilizer by adding it directly to soil.


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Brian Finney
Follow @brianfinneywri1
Brian Finney is a prize-winning writer and professor emeritus in English literature. Born in London, he obtained a BA (hons) in English and Philosophy at Reading University and a PhD on D. H. Lawrence's shorter fiction at University of London, where he taught and arranged extra-mural courses in the arts from 1964 to 1987. After immigrating to Southern California he taught at UC Riverside, UCLA, the University of Southern California ad California State University Long Beach.
He has published eight books. His second book, Christopher Isherwood: A Critical Biography, was awarded the Jams Tait Black Memorial Prize for non-fiction that year. It was also voted one of the best three books of the year by Philip Toynbee in the Observer. He went on to publish, among others, The Inner I, a pioneering study of British literary autobiography, and a widely cited book about British fiction in the eighties and nineties, English Fiction Since 1984. In 2011 he published on Amazon Terrorized: How the War on Terror Affected American Culture and Society. He has also edited three editions of D. H. Lawrence's work, and published a range of essays and reviews in various journals and newspapers.
In 2019 he is publishing his first novel, Money Matters, in which an inexperienced young woman is persuaded to search for a woman who has disappeared and comes up against the powerful forces of big money, politics and a drug cartel. By the end of her search she has become a different person.
Will be launching his new novel, Money Matters, in May on KDP as an e-book and paperback. It combines psychological suspense with coming-of-age and social commentary.
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This bird was banded here at the house back on 8 December by Kelly Bryan.
That day there were two Selasphorus types. One was already banded, but on the left leg. Kelly bands all his hummingbird new-captures on the right leg. We really wanted that recap, because it wasn't local. Really, any recap is filled with goodness, but a bird not banded here...., oh well.
We got this kid. It is a hatch-year (HY) male Allen's. Late December and there is still this territorial sprite waiting for the sugar-water to thaw this morning. He'll also guard the drip and other portions of the back-40.
Well, in a week and a half from now we would have to celebrate his birthday. He'll have graduated to after-hatch-yr (AHY), we'll even know him to be a second-year Allen's. All that, if he sticks. Check the gorget, the sides are starting to show some shading..
Those who are interested in an Allen's Hummingbird for their life/year/state/county/town/block list are generally welcome to stop by. Just give us a shout or email to let us know you're coming. We can, in turn, let you know if we would've seen him that day.