text:
\section{Introduction}
\setcounter{equation}{0}

The conceptual backbone for the modeling of
the kinematics of quantum
systems is the Heisenberg commutation relations which have found
their mathematical expression in various guises. There is an
extensive literature analyzing their properties, starting
with the seminal paper of Born, Jordan and Heisenberg on the
physical foundations and reaching a first mathematical satisfactory
formulation in the works of von Neumann and of Weyl.

These canonical systems of operators may all be presented in
the following general form: there is a real
(finite or infinite dimensional) vector space $X$
equipped with a non--degenerate symplectic form
$\sigma : X \times X \rightarrow {\mathbb R}$ and a linear map $\phi$  from $X$
onto the generators of a polynomial *--algebra $\PA$ of
operators satisfying the canonical commutation relations
$$
\big[\phi(f),\,\phi(g)\big]=i\sigma(f,\, g) \, {{\bf 1}}, \quad
\phi(f)^* =\phi(f) \, .
$$
In the case that $X$ is finite dimensional, one can reinterpret
this relation in terms of the familiar quantum mechanical
position and momentum operators, and if $X$ consists
of Schwartz functions on some
manifold one may consider $\phi$ to be a bosonic quantum field.
As is well--known, the operators $\phi(f)$ cannot all be bounded.
Moreover, the algebra  $\PA$  does not admit
much interesting dynamics acting on it by automorphisms; in
fact there are in general only transformations
induced by polynomial Hamiltonians
which leave it invariant \cite{Di}. Thus
$\PA$ is not a convenient kinematical algebra in either
respect.

The inconveniences of unbounded operators
can be evaded by expressing the basic commutation relations
in terms of bounded functions of the generators $\phi(f)$.
In the approach introduced by Weyl, this is done by
considering the C*--algebra generated by the set of
unitaries  $W(f) \, \hat{=} \, \exp(i\phi(f))$, $f \in X$
(the Weyl operators) satisfying the Weyl relations
$$
W(f) W(g) =
e^{-i \sigma(f,g)/2} \, W(f+g) \, , \quad
W(f)^* = W(-f) \, .
$$
This is the familiar Weyl (or CCR) algebra $\WA$. Yet
this  algebra still suffers from the fact that its automorphism
group does not contain physically significant dynamics \cite{FaVB}.
This deficiency can be traced back to the
fact that the Weyl algebra is simple, whereas any unital
C*--algebra admitting an expedient variety of dynamics must
have ideals \mbox{\cite[Sec.~10]{BuGr1}}, cf.\ also the conclusions.

For finite systems this problem can be solved by
proceeding to the twisted group algebra~\cite{Ka}
derived from the unitaries $W(f)$, $f \in X$.
By the Stone--von Neumann theorem this
algebra is isomorphic to ${\cal K}({\cal H})$, the compact operators
on a separable Hilbert space, for any finite dimensional $X$.
This step solves the problem of dynamics for finite systems, but
it cannot be applied as such to infinite systems
since there $X$ is not locally compact. Moreover, one pays
the price that the original operators, having continuous spectrum,
are not affiliated with ${\cal K}({\cal H})$. So
one forgets the specific properties of the underlying quantum
system.

This unsatisfactory situation motivated the formulation of
an alternative version of the C*--algebra of canonical commutation
relations, given in \cite{BuGr1}. Here one considers
the C*--algebra generated by the resolvents of the
basic canonical operators which are formally given by
${R(\lambda,f)} \, \hat{=} \, (i \lambda {\bf 1} - \phi(f))^{-1}$
for $\lambda \in {\mathbb R} \backslash \{0\}$, $f \in X$. All algebraic
properties of the operators~$\phi(f)$ can be expressed in terms of
polynomial relations amongst these resolvents. Hence,
in analogy to the Weyl algebra generated by the exponentials,
one can abstractly define a unital \mbox{C*--algebra}
$\RA$ generated by the resolvents, called the resolvent
algebra.

In accordance with the requirement of admitting sufficient dynamics
the resolvent algebras have ideals. Their ideal
structure was recently clarified in \cite{BuGr2}, where it was shown
that it depends sensitively on the size of the underlying
quantum system. More precisely, the specific nesting of the
primitive ideals encodes information about the dimension of
the underlying space $X$. This dimension,
if it is finite, is an algebraic invariant which labels the
isomorphism classes of the resolvent algebras.
Moreover, the primitive ideals are in one--to--one
correspondence to the spectrum (dual) of the respective
algebra, akin to the case of commutative algebras.
The resolvent algebras are postliminal (type I) if the
dimension of $X$ is finite and they are still nuclear if
$X$ is infinite dimensional. Thus these algebras not only encode
specific information about the underlying systems but
also have comfortable mathematical properties.

The resolvent algebras already have proved to be useful in several
applications to quantum physics such as the representation theory of abelian
Lie algebras of derivations \cite{BuGr3}, the study
of constraint systems and of the BRST method in a
C*--algebraic setting \cite{BuGr1,Co}, the
treatment of supersymmetric
models on non--compact spacetimes and the rigorous construction
of corresponding JLOK--cocycles \cite{BuGr4}. Their
virtues also came to light in the formulation and
analysis of the dynamics of finite
and infinite quantum systems \cite{BuGr1,BuGr5}.

In the present article we give a survey of the basic
properties of the resolvent algebras and an outline of
recent progress in the construction of dynamics,
shedding light on the role of the ideals.
The subsequent section contains the formal definition
of the resolvent algebras and some comments on their relation
to the standard Weyl formulation of the canonical
commutation relations. Section~3 provides a
synopsis of representations of the resolvent algebras
and some structural implications and Sect.~4 contains the
discussion of observables and of dynamics. The article
concludes with a brief summary and outlook.

\section{Definitions and basic facts}
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Let $(X, \sigma)$ be a real symplectic space;
in order to avoid pathologies we make the standing assumption
that $(X, \sigma)$ admits a unitary structure \cite{Ro}.
The pre--resolvent algebra $\RAO$ is the universal \mbox{*--algebra}
generated by the elements of the set
$\{{R(\lambda,f)} : \lambda \in {\mathbb R} \backslash \{0\}, \; f\in X \}$
satisfying the relations
\begin{eqnarray}
\label{Resolv}
{R(\lambda,f)} - R(\mu,f) & \! \! = \! \! & i (\mu-\lambda) {R(\lambda,f)} R(\mu,f)  \\[1mm]
\label{Rinvol}
{R(\lambda,f)}^*& \! \! = \! \! &R(-\lambda,f) \\[1mm]
\label{Rccr}
\big[{R(\lambda,f)}, \, {R(\mu,g)} \big] & \! \! = \! \! &
i\sigma(f,g)\,{R(\lambda,f)} \, {R(\mu,g)}^2 {R(\lambda,f)} \\[1mm]
\label{Rhomog}
\nu \,R(\nu \lambda,\, \nu f)   & \! \! = \! \! &  {R(\lambda,f)} \\[1mm]
\label{Rsum}
{R(\lambda,f)} {R(\mu,g)} & \! \! = \! \! & R(\lambda+\mu, f+g) \big( {R(\lambda,f)} + {R(\mu,g)}
+ i\sigma(f,g) {R(\lambda,f)}^2 {R(\mu,g)} \big) \\[1mm]
\label{Riden}
R(\lambda,0)& \! \! = \! \! & -{\textstyle \frac{i}{\lambda}} \, {\bf 1}
\end{eqnarray}
where $\lambda,\, \mu, \, \nu \in {\mathbb R} \backslash \{0\} $ and $f,\,g\in X$,
and for (\ref{Rsum}) we require $\lambda+\mu\not=0$.
That is, start with the free unital *-algebra generated by
$\{{R(\lambda,f)} : \lambda \in {\mathbb R} \backslash \{0\}, \; f\in X \}$
 and factor out by the ideal generated by the relations
(\ref{Resolv}) to (\ref{Riden}) to obtain the *-algebra $\RAO$.

\medskip
\noindent \textbf{Remarks:}
(a) Relations (\ref{Resolv}), (\ref{Rinvol}) encode the
algebraic properties of the resolvent of some self--adjoint operator,
(\ref{Rccr}) amounts to the canonical commutation relations and relations
(\ref{Rhomog}) to (\ref{Riden}) correspond to the linearity of the
initial map $\phi$ on $X$. \\
(b) \ The *-algebra $\RAO$ is nontrivial, because it has nontrivial
representations. {}For instance, in a Fock representation
${(\pi, {\cal H})}$ one has self--adjoint operators $\phi_\pi (f)$, $f \in X$
satisfying the canonical commutation relations over $(X, \sigma)$
on a sufficiently big domain in the Hilbert space ${\cal H}$
so that one can define $\pi({R(\lambda,f)}) \doteq (i\lambda{\bf 1} - \phi_\pi (f))^{-1}$
to obtain a representation~$\pi$ of~$\RAO$.

\medskip
It has been shown in  \cite[Prop.~3.3]{BuGr1} that the following
definition is meaningful.

\begin{definition} Let $(X, \sigma)$  be a symplectic space.
The supremum of operator norms with regard to all
cyclic *--representations ${(\pi, {\cal H})}$  of  $\, \RAO$
$$ \| R \| \doteq \sup_{(\pi, {\cal H})} \, \| \pi(R) \|_{\cal H} \, , \quad R \in \RAO  $$
exists and defines a C*--seminorm on $\RAO$.
The resolvent algebra $\RA$ is defined as the C*--completion
of the quotient algebra $\RAO / \ker \| \cdot \|$,
where here and in the following the symbol $\ker$ denotes
the kernel of the respective map.
\end{definition}

Of particular interest are representations of the resolvent
algebras, such as the Fock representations, where the abstract
resolvents characterized by conditions (\ref{Resolv}), (\ref{Rinvol})
(sometimes called pseudo--resolvents)
are represented by genuine resolvents of self--adjoint operators.

\begin{definition}
A representation ${(\pi, {\cal H})}$ of $\RA$ is said to be regular if
for each $f \in X$ there exists a self--adjoint operator
$\phi_\pi(f)$ such that $\pi({R(\lambda,f)}) \doteq (i\lambda{\bf 1} - \phi_\pi
(f))^{-1}$, $\lambda \in {\mathbb R} \backslash \{0\}$. (This is equivalent to the condition
that all operators $\pi({R(\lambda,f)})$ have trivial kernel.)
\end{definition}

The following result characterizing regular representations, cf.\
\cite[Thm.~4.10 and Prop.~4.5]{BuGr1}, is of importance, both in the
structural analysis of the resolvent algebras and in their
applications. It implies in particular that the resolvent algebras
have faithful irreducible representations ({\it e.g.\ } the
Fock representations), so their centers are trivial.

\begin{proposition} \label{faithfulrep}
Let ${(\pi, {\cal H})}$ be a representation of $\RA$.
\begin{itemize}
\vspace*{-2mm}
\item[(a)] If ${(\pi, {\cal H})}$ is regular it is also faithful,
{\it i.e.\ } $\| \pi(R) \|_{\cal H} = \| R \| $ for $R \in \RA$.
\vspace*{-2mm}
\item[(b)] If ${(\pi, {\cal H})}$ is faithful and
the weak closure of $\pi(\RA)$ is a factor, then ${(\pi, {\cal H})}$ is regular.
\end{itemize}
\end{proposition}

The regular representations of the resolvent algebras are in
one--to--one correspondence with the regular representations
of the Weyl--algebras, cf. \cite[Cor.~4.4]{BuGr1}.
(Recall that a representation ${(\pi, {\cal H})}$ of $\WA$
is regular if the maps $\nu \in {\mathbb R}\mapsto\pi(W(\nu f))$
are strong operator
continuous for all $f$.) In fact one has the following result.

\begin{proposition} \label{one-one}
Let $(X, \sigma)$ be a symplectic space and
\begin{itemize}
\vspace*{-2mm}
\item[(a)] let ${(\pi, {\cal H})}$ be a regular representation of
the resolvent algebra $\RA$ with associated self--adjoint operators
$\phi_\pi(f)$ defined above. The exponentials
$W_\pi(f) \doteq \exp(i \phi_\pi(f))$, $f \in X$ satisfy the Weyl relations
and thus define a regular representation of the Weyl algebra~$\WA$
on~${\cal H}$;
\vspace*{-2mm}
\item[(b)] let ${(\pi, {\cal H})}$ be a regular representation of
the Weyl algebra $\WA$ and let $\phi_\pi(f)$ be the
self--adjoint generators of the Weyl operators. The resolvents
\mbox{$R_\pi(\lambda,f) = (i\lambda{\bf 1} - \phi_\pi (f))^{-1}$}
with $\lambda \in {\mathbb R} \backslash \{0\}$,
$f \in X$ satisfy relations (\ref{Resolv}) to (\ref{Riden}) and
thus define a regular representation of the resolvent algebra
$\RA$ on ${\cal H}$.
\end{itemize}
\end{proposition}

Whilst this proposition establishes the
existence of a bijection between the regular
representations of $\RA$ and those of $\WA$, there is no such
map between the non--regular representations of the two
algebras. In order to substantiate this point
consider for fixed nonzero $f \in X$  the  two commutative subalgebras
$C^*\{ R(1,sf) : s \in {\mathbb R} \} \subset \RA$
and $C^*\{ W(sf) : s \in {\mathbb R} \} \subset \WA$.
These algebras are isomorphic
respectively to the continuous functions on
the one point compactification of ${\mathbb R}$, and the continuous functions on
the Bohr compactification of ${\mathbb R}$. Now the point measures
on the compactifications having support in the complement of ${\mathbb R}$
produce non-regular states (after extending to the full C*--algebras by
 Hahn--Banach) and there are many more of these for
the Bohr compactification than for the one point compactification of
${\mathbb R}$. Proceeding to the GNS--representations it is
apparent that the Weyl algebra has substantially more
non-regular representations than the resolvent~algebra.

\section{Ideals and dimension}
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Further insight into the algebraic properties of the resolvent
algebras is obtained by a study of its irreducible representations.
In case of finite dimensional symplectic spaces these representations
have been completely classified \cite[Prop.~4.7]{BuGr1}.

\begin{theorem}
Let $(X, \sigma)$ be a finite dimensional symplectic space and let
${(\pi, {\cal H})}$ be an irreducible representation of $\RA$. Depending
on the representation, the space $X$ decomposes as follows,
cf.\ Fig.\ 1.
\begin{itemize}
\vspace*{-2mm}
\item[(a)] There is a unique subspace $X_R \subset X$ such that
there are self--adjoint operators $\phi_\pi(f_R)$ satisfying
$\pi(R(\lambda,f_R)) =   (i\lambda{\bf 1} - \phi_\pi (f_R))^{-1}$ for
$\lambda \in {\mathbb R} \backslash \{0\}$, $f_R \in X_R$.
\vspace*{-2mm}
\item[(b)]
Let $X_T \doteq
\{f\in X_R : \sigma(f,g)=0 \ \mbox{for all} \ g\in X_R \}$.
Then $\phi_\pi$  restricts on $X_T$ to a linear functional
$\varphi : X_T \rightarrow {\mathbb R}$ such that
$\pi(R(\lambda,f_T)) =   (i\lambda - \varphi (f_T))^{-1} {\bf 1}$
for $f_T \in X_T$, $\lambda \in {\mathbb R} \backslash \{0\}$.
\vspace*{-2mm}
\item[(c)] For $f_S \in X_S \doteq X \backslash X_R$ and $\lambda \in
  {\mathbb R} \backslash \{0\}$ one has $\pi(R(\lambda,f_S)) = 0$.
\end{itemize}
\vspace*{-2mm}
Conversely, given subspaces $X_T \subset X_R \subset X$ and
a linear functional $\varphi : X_T \rightarrow {\mathbb R}$ there exists a
corresponding irreducible representation ${(\pi, {\cal H})}$ of $\RA$,
unique up to equivalence, with the preceding three properties.
\end{theorem}

\begin{figure}[h] \label{figure}
 \hspace*{60mm}
\epsfig{file=decomposition.eps,height=48mm}
\caption{Decomposition of $X$ fixed by an
irreducible representation}
 \label{fig1}
 \end{figure}

This result may be regarded as an extension of the
Stone--von Neumann uniqueness theorem for regular representations
of the CCR algebra. It shows that the only obstruction to
regularity is the possibility that some of the underlying
canonical operators are infinite and the corresponding
resolvents vanish. This happens in
particular if there are some canonically conjugate operators
having sharp (non--fluctuating) values in a representation, as is the case for
constraint systems \cite[Prop.~8.1]{BuGr1}.
But, in contrast to the Weyl algebra,
the non--regular representations of the resolvent algebra
only depend on the values of these canonical operators. So
the abundance of different singular representations of the Weyl
algebra shrinks to a manageable family on the resolvent algebra.

The preceding theorem is the key to the structural analysis
of the resolvent algebra for symplectic spaces of arbitrary
finite dimension. We recall in this context that the
primitive ideals of a C*--algebra are the (possibly zero)
kernels of irreducible representations and that the spectrum of the algebra
is the
set of unitary equivalence classes of irreducible representations.
The following
result has been established in \cite{BuGr2}.

\begin{theorem}
Let $(X, \sigma)$ be a finite dimensional symplectic space.
\begin{itemize}
\vspace*{-2mm}
\item[(a)] The mapping $\widehat{\pi} \mapsto \ker \widehat{\pi}$
from the elements $\widehat{\pi}$ of the
spectrum (dual) of the resolvent algebra $\RA$ to its primitive
ideals \ $\ker \widehat{\pi}$ is a bijection.
\vspace*{-2mm}
\item[(b)] Let $L \doteq
\sup \, \{l \in {\mathbb N} : \ker \widehat{\pi}_1 \subset
\ker \widehat{\pi}_2 \dots \subset \ker \widehat{\pi}_l \}$
be the maximal length of proper inclusions of primitive ideals
of $\RA$. Then $L  = \dim(X) / 2 + 1$.
\end{itemize}
\end{theorem}

\noindent \textbf{Remarks:}
Property (a) is a remarkable feature of the resolvent algebras,
shared with the abelian \mbox{C*--algebras}. It rarely holds for
non-commutative algebras and also fails if $X$ is infinite dimensional.
The quantity $L$ defined in (b) is an algebraic invariant,
so this result shows that the dimension $\dim(X)$ of the underlying systems
is algebraically encoded in the resolvent algebras. As a matter of fact,
$\dim(X)$ is a complete algebraic invariant of resolvent algebras
in the finite dimensional case.

\vspace*{2mm}
As indicated above, there is an algebraic difference between the resolvent
algebras for finite dimensional $X$ and those where $X$ has infinite dimension.
A further difference is seen through the minimal (nonzero) ideals \cite{BuGr2}.

\begin{proposition} \label{compactideal}
Let $(X, \sigma)$ be a symplectic space of arbitrary dimension and
let ${\cal I} \subset \RA$ be the intersection of all nonzero ideals of $\RA$.
\begin{itemize}
\vspace*{-2mm}
\item[(a)] If \ $\dim(X) < \infty$ then ${\cal I}$ is isomorphic to the
C*--algebra ${\cal K}({\cal H})$ of compact operators.
Moreover, in any irreducible regular representation ${(\pi, {\cal H})}$ one has
$\pi({\cal I})={\cal K}({\cal H})$.
\vspace*{-2mm}
\item[(b)] If  \ $\dim(X) = \infty$ then ${\cal I} = \{0\}$. In fact,
there exists no nonzero minimal ideal of $\RA$ in this case.
\end{itemize}
\end{proposition}

If $(X, \sigma)$ is infinite dimensional the
resolvent algebra $\RA$ is the \mbox{C*--inductive} limit of
the net of its subalgebras ${\cal R}(Y,\sigma)$ where
$Y \subset X$ ranges over all finite dimensional
non--degenerate subspaces of $X$, cf.~\cite[Thm.~4.9]{BuGr1}.
This fact in combination with the first part of the preceding result
is a key ingredient in the construction of
dynamics, see below. It also enters in the proof
of the following statement \cite{BuGr2}.

\begin{proposition}
Let $(X,\sigma)$ be a symplectic space of arbitrary
dimension.
\begin{itemize}
\vspace*{-2mm}
\item[(a)] $\RA$ is a nuclear C*-algebra,
\vspace*{-2mm}
\item[(ii)] $\RA$ is a postliminal (type~I) C*--algebra
if and only if  $\dim(X)<\infty$.
\end{itemize}
\end{proposition}

Recall that a C*--algebra is said to be postliminal (type~I) if
all of its irreducible representations contain the
compact operators and that postliminal C*--algebras
as well as their C*--inductive limits are nuclear,
{\it i.e.\ } their tensor product with any other C*--algebra
is unique. It should be noted, however, that the
resolvent algebras are not separable \cite[Thm.~5.3]{BuGr1}.
With this remark we conclude our outline of pertinent
algebraic properties of the resolvent algebras.

\section{Observables and dynamics}
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The main virtue of the resolvent algebra consists of the fact
that it includes many observables of physical  interest and
admits non--trivial dynamics. In order to illustrate this
important feature we discuss in detail a familiar example of a
finite quantum system and comment on infinite systems at the
end of this section.

Let $(X,\sigma)$ be a finite dimensional symplectic space, {\it i.e.\ }
$\dim(X) = 2N$ for some \mbox{$N \in {\mathbb N}$}. Since
regular representations of the resolvent algebras
are faithful, cf.\ Proposition~\ref{faithfulrep},
it suffices to consider a regular irreducible representation
$(\pi_0,{\cal H}_0)$ of ${\cal R}(X,\sigma)$ which is unique up to
equivalence.
Choosing some symplectic basis $f_k, g_k \in X$ and putting
\mbox{$P_k \doteq \phi_{\pi_0}(f_k)$},  $Q_k \doteq  \phi_{\pi_0}(g_k)$,
\mbox{$k = 1, \dots N$} we identify the self--adjoint operators
fixed by the corresponding resolvents with the momentum and
position operators of~$N$ particles in one spatial dimension.

The (self--adjoint) quadratic Hamiltonian
$$
H_{0} \doteq \sum_{k=1}^N \, ({\textstyle \frac{1}{2m_k}} P_k^2 +
{\textstyle \frac{m_k \omega_k^2}{2}} Q_k^2)
$$
describes the free, respectively oscillatory motion of these particles,
where $m_k$ are the particle masses and $\omega_k \geq 0$ the
frequencies of oscillation, $k = 1, \dots N$. The interaction of the
particles is described by the operator
$$
V \doteq \sum_{1 \leq k < l \leq N} \, V_{k l}(Q_k - Q_l)
$$
where we assume for simplicity that the potentials  $V_{k l}$
are real and continuous, vanish at infinity, but are
arbitrary otherwise. Since $V$ is bounded, the
Hamiltonian $H \doteq H_{0} + V$
is self--adjoint on the domain of $H_{0}$ and its resolvents
are well defined.

\begin{proposition} \label{observables}
Let $H$ be the Hamiltonian defined above. Then
$$ (i \mu {\bf 1} - H)^{-1} \in \pi_0({\cal R}(X,\sigma)) \, ,
\quad \mu \in {\mathbb R} \backslash \{0\} \, . $$
\end{proposition}

\vspace*{2mm}
\noindent \textbf{Remark:} Since $\pi_0$ is faithful
its inverse $\pi_0^{-1} : \pi_0(\RA) \rightarrow
\RA$ exists,
so this result shows that $H$ is affiliated with the
resolvent algebra. Note that this is neither true
for the Weyl algebra $\WA$ nor for the corresponding twisted
group algebra ${\cal K}({\cal H})$ if one of the frequencies~$\omega_k$
vanishes. Thus $\RA$ contains many more observables
of physical interest than these conventional  algebras.

\vspace*{2mm}
\noindent \textit{Proof:}
Let $X_{k} \subset X$ be the two--dimensional subspaces
spanned by the symplectic pairs $(f_k,g_k)$, let
$\sigma_{k} \doteq \sigma \upharpoonright X_{k} \times  X_{k}$
and let $(\pi_k, {\cal H}_k)$ be regular irreducible representations
of ${\cal R}(X_k,\sigma_k)$, $k = 1, \dots N$.
Then $\pi_0 \doteq \pi_1 \otimes \cdots \otimes \pi_N$
defines an irreducible representation of the C*--tensor product
${\cal R}(X_1,\sigma_1) \otimes \cdots \otimes {\cal R}(X_N,\sigma_N)$
on the Hilbert space
${\cal H}_0 \doteq {\cal H}_1 \otimes \cdots \otimes {\cal H}_N$.
It extends by regularity to the Weyl algebra
$ \WA \simeq
{\cal W}(X_1,\sigma_1) \otimes \cdots \otimes {\cal W}(X_N,\sigma_N)$
and hence to a regular representation of $\RA$, cf.\
Proposition \ref{one-one}.

Disregarding tensor factors of ${\bf 1}$ one has
$H_{0 k} \doteq
(i \mu {\bf 1} - \frac{1}{2m_k} P_k^2 - \frac{m_k \omega_k^2}{2} \, Q_k^2)^{-1}
\in \pi_k({\cal R}(X_k,\sigma_k))$, $k = 1, \dots N$. If~$\omega_k > 0$
this follows from the fact that the resolvent of the
harmonic oscillator Hamiltonian is a compact operator
and hence belongs to the compact ideal of
$\pi_k({\cal R}(X_k,\sigma_k))$, cf.\ Proposition~\ref{compactideal}.
If $\omega_k = 0$ one resorts to the fact that the
abelian C*--algebra generated by the resolvents
\mbox{$(i\lambda {\bf 1}  - P_k)^{-1}$}, $\lambda \in {\mathbb R} \backslash \{0\}$
coincides with $C_0(P_k)$, the algebra of all
continuous functions of~$P_k$ vanishing at infinity.
Hence $C_0(P_k) \subset \pi_k({\cal R}(X_k,\sigma_k))$
and since $(i \mu {\bf 1} - \frac{1}{2m_k} P_k^2)^{-1} \in
C_0(P_k)$ the preceding statement holds also for
$\omega_k = 0$.

Now $C_0({{\mathbb R}}_+{}^{\! \! \! N}) = C_0({{\mathbb R}}_+)
\overbrace{\otimes \cdots \otimes}^N  C_0({{\mathbb R}}_+)$ and
$u_1, \dots, u_N \mapsto (i \mu - u_1
\dots - u_N)^{-1}$ is an element of $C_0({{\mathbb R}}_+{}^{\! \! \! N})$.
Since the resolvents of the
positive self--adjoint  operators $H_{0 k}$ generate
the abelian C*--algebras $C_0(H_{0 k})$, $k=1, \dots , N$,
it follows from continuous functional calculus that
$(i \mu {\bf 1} - H_0)^{-1}=\big( i \mu {\bf 1} - H_{0 1} \dots - H_{0 N} \big)^{-1}
\in C_0(H_{0 1}) \otimes \cdots \otimes C_0(H_{0 N}) \subset
\pi_0(\RA)$.

Similarly,
for the interaction potentials one uses the
fact that the abelian C*--algebras generated by the resolvents
\mbox{$(i \lambda {\bf 1} - (Q_k - Q_l))^{-1}$}, $\lambda \in {\mathbb R} \backslash \{0\}$ coincide
with $C_0(Q_k - Q_l)$. So as $V_{k l}\in C_0({\mathbb R})$, one also has that
$$
V = \sum_{1 \leq k < l \leq N} \, V_{k l}(Q_k - Q_l)
\in \pi_0(\RA).
$$
In summary one gets
$(1 - (i \mu {\bf 1} - H_0)^{-1}V) \in \pi_0(\RA)$
and the inverse of this operator exists if
$|\mu| > \|V\|$. Hence
$(i \mu {\bf 1} - H)^{-1} = (1 - (i \mu {\bf 1} - H_0)^{-1}V)^{-1}
(i \mu {\bf 1} - H_0)^{-1} \in \pi_0(\RA)$
for such $\mu$. The statement for arbitrary $\mu \in {\mathbb R} \backslash \{0\}$
then follows from the resolvent equation for $H$,
completing the proof.

As a matter of fact, the preceding proposition holds for a much
larger class of interaction potentials, including discontinuous
ones. It does not hold, however, for certain physically
inappropriate Hamiltonians such as that of the
anti--harmonic oscillator \cite[Prop.~6.3]{BuGr1}. The
characterization of all Hamiltonians which are affiliated
with resolvent algebras is an interesting open problem.

We turn now to the analysis of the dynamics induced by the
Hamiltonians given above. The exponentials of the quadratic
Hamiltonians $H_0$ induce symplectic transformations, so one has
$\Ad{e^{itH_0}} (\pi_0(\RA)) = \pi_0(\RA)$ for $t \in {\mathbb R}$.
For the proof that the resolvent algebra is also stable
under the adjoint action of the interacting dynamics the
crucial step consists of showing that the cocycles
$\Gamma(t) = e^{itH} e^{-itH_0}$ are elements of
$\pi_0(\RA)$. Putting \mbox{$V(t) = \Ad{e^{itH_0}}(V)$} one
can present the cocycles in the familiar form of a Dyson series
$$
\Gamma(t) = 1 + \sum_{n=1}^\infty \, i^n
\int_0^t \! dt_1 \int_0^{t_1} \! dt_2 \dots \int_0^{t_{n-1}} \! dt_n \,
V(t_n) \cdots V(t_1)
$$
and this series converges absolutely in norm since the
operators $V(t)$ are uniformly bounded. Moreover, the functions
$t \mapsto V(t)$ have values in the algebra $\pi_0(\RA)$;
but since they are only continuous in the strong operator
topology it is not clear from the outset that their
integrals, defined in this topology, are still contained
in this algebra. Here
again the specific structure of the resolvent algebra
matters. It allows to establish the desired result.

\begin{proposition} \label{dynamics}
Let $H$ be the Hamiltonian defined above. Then
$$
\Ad{e^{itH}} (\pi_0(\RA)) = \pi_0(\RA) \, , \quad t \in {\mathbb R} \, .
$$
\end{proposition}

\vspace*{2mm}
\noindent \textbf{Remark:} Since $\pi_0$ is faithful
it follows from this result that
$\alpha_t \doteq \pi_0^{-1} \Ad{e^{itH}}  \pi_0$, $\, t \in {\mathbb R}$
defines a one--parameter group of automorphisms of $\RA$.
It should be noted, however, that its action is not
continuous in the strong (pointwise norm) topology of $\RA$.

\vspace*{2mm}
\noindent \textit{Proof:}
Let $k,l \in 1, \dots , N$ be different numbers,
let $(f_k,g_k)$ and $(f_l, g_l)$ be symplectic pairs
as in the proof of the preceding proposition and let
$X_{k l} \subset X$ be the space spanned by
$h_{k l}(t) \doteq ((\cos{\omega_kt}) \, g_k - (\cos{\omega_lt}) \, g_l
+ (\sin{\omega_kt})/{m_k \omega_k} \, f_k -
(\sin{\omega_lt})/{m_l \omega_l} \, f_l)
$, $t \in {\mathbb R}$, where we stipulate
$(\sin{\omega t})/\omega = t$ if $\omega = 0$.
This space is non--degenerate and, depending
on the masses and frequencies, either two or four dimensional.
We  put $\sigma_{k l} \doteq \sigma \upharpoonright X_{k l} \times X_{k l}$.
Let \mbox{$V_{k l}(t) \doteq \Ad{e^{itH_0}} (V_{k l}(Q_k - Q_l)) $}, where
$V_{k l}(Q_k-Q_l)$ is any one of the
two--body potentials contributing to $V$. Then, for any $t \in {\mathbb R}$,
$$ V_{k l}(t)
=  V_{k l}((\cos{\omega_kt}) \, Q_k - (\cos{\omega_lt}) \, Q_l
+ (\sin{\omega_kt})/{m_k \omega_k} \, P_k -
(\sin{\omega_lt})/m_l \omega_l \, P_l)
\in \pi_0({\cal R}(X_{k l},\sigma_{k l})) \, .
$$
Now the function $s_1, \dots s_d \mapsto V_{k l}(s_1) \cdots V_{k l}(s_d)$
is continuous in the strong operator
topology and, for almost all $s_1, \dots s_d$,
an element of the compact ideal of $\pi_0({\cal R}(X_{k l},\sigma_{k l}))$,
provided $d \geq \dim(X_{k l})$.
The latter assertion follows from the
fact that $V_{k l}(s)$ is, for given $s$, an element of the abelian C*--algebra
generated by the resolvents $\pi_0(R(\lambda,h_{k l}(s)))$,
$\lambda \in {\mathbb R} \backslash \{0\}$ and that the compact ideal coincides with
the principal ideal of  $\pi_0({\cal R}(X_{k l},\sigma_{k l}))$
generated by $\pi_0(R(\lambda_1,h_1) \cdots R(\lambda_d,h_d))$
for any choice of $\lambda_1, \dots \lambda_d \in {\mathbb R} \backslash \{0\}$ and of
elements $h_1, \dots h_d \in X_{k l}$ which span $X_{k l}$ \cite{BuGr5}.
It is then clear that
$\big( \int_0^t \! ds \, V_{k l }(s) \big)^d =
\int_0^t \! ds_1 \cdots \int_0^t \! ds_d
\, V_{k l}(s_1) \cdots V_{k l}(s_d)$
is contained in the compact ideal of $\pi_0({\cal R}(X_{k l},\sigma_{k l}))$
and this is also true for the operator $\int_0^t \! ds \, V_{k l }(s)$
since it is self--adjoint. As $k,l$ were arbitrary this implies
$\int_0^t \! dt_1  V(t_1) \in \pi_0(\RA)$.

The proof that all other terms in the Dyson
series are likewise elements of $\pi_0(\RA)$ is given by induction.
Let $I_n(t) \doteq
\int_0^t \! dt_1 \int_0^{t_1} \! dt_2 \dots \int_0^{t_{n-1}} \! dt_n \,
V(t_n) \cdots V(t_1) \in \pi_0(\RA)$, $t \in {\mathbb R}$; then
$I_{n + 1}(t) = \int _0^t \! dt_1 I_n(t_1) V(t_1)$, where
the integrals are defined in the strong operator topology.
Now $t \mapsto I_n(t)$ is continuous in norm, hence
$I_{n + 1}(t)$ can be approximated according to
$$
I_{n + 1}(t) = \lim_{J \rightarrow \infty} \, \sum_{j = 1}^J
 I_n(jt/J)  \int _{(j-1)t/J}^{jt/J} \! dt_1 V(t_1)  \, ,
$$
where the limit exists in the norm topology. Since
each term in this sum is an element of $\pi_0(\RA)$
according to the induction hypothesis
it follows that $I_{n + 1}(t) \in \pi_0(\RA)$.
Because of the convergence of the Dyson series this
implies $\Gamma(t) \in \pi_0(\RA)$, $t \in {\mathbb R}$,
completing the proof of the statement.

\vspace*{2mm}
Having illustrated the virtues of the resolvent algebras
for finite systems we discuss now the situation for
infinite systems. There the results are far from being complete,
though promising. For the sake of concreteness we consider
an infinite dimensional symplectic space $(X, \sigma)$ with a
countable symplectic basis $f_k, g_k \in X$, $k \in {\mathbb Z}$.
Similarly to the case of finite systems one can analyze
the observables and dynamics  associated with $\RA$ in any
convenient faithful representation
$(\pi_0, {\cal H}_0)$, such as the Fock representation.

As before, we identify the self--adjoint operators fixed
by the resolvents with the momentum and position operators of
particles,
$P_k \doteq \phi_{\pi_0}(f_k)$, $Q_k \doteq \phi_{\pi_0}(g_k)$, $k \in
{\mathbb Z}$.
In view of Haag's Theorem \cite{Em} it
does not come as a surprise that global observables,
such as Hamiltonians having a unique ground state
or the particle number operator are no longer
affiliated with the resolvent algebra of such infinite
systems. In fact, one has the
following general result \cite{BuGr5}.

\begin{lemma} Let $(X, \sigma)$ be an infinite dimensional symplectic
space, let $(\pi_0,{\cal H}_0)$ be a faithful irreducible
representation of $\RA$ and let $N$ be a (possibly unbounded)
self--adjoint operator on ${\cal H}_0$ with an isolated
eigenvalue of finite multiplicity.
Then $(i \mu {\bf 1} - N)^{-1} \notin \pi_0(\RA)$ for
$\mu \in {\mathbb R} \backslash \{0\}$, {\it i.e.\ } $N$ is not affiliated with $\RA$.
\end{lemma}

Observables corresponding to finite subsystems of
the infinite system are still affiliated with $\RA$.
Relevant examples are the partial Hamiltonians of the
form  given above,
$$
H_\Lambda \doteq
 \sum_{k \in \Lambda} \, ({\textstyle \frac{1}{2m_k}} P_k^2 +
{\textstyle \frac{m_k \omega_k^2}{2}} Q_k^2) \ + \!
\sum_{k,l \, \in \, \Lambda} \, V_{k l}(Q_k - Q_l) \, ,
$$
where $\Lambda \subset {\mathbb Z}$ is any finite set.
By exactly the same arguments as in the proof of
Proposition~\ref{observables} one can show that
any such $H_\Lambda$ is affiliated with $\RA$. Clearly,
these Hamiltonians may have isolated eigenvalues,
but these have infinite multiplicity. By the preceding
arguments one can also show that the resolvent algebra
is stable under the time evolution induced by the
partial Hamiltonians. Moreover, for suitable
potentials the evolution converges to some global
dynamics in the limit $\Lambda \nearrow {\mathbb Z}$. The precise
results are as follows.

\begin{proposition}
Let $H_\Lambda$, $\Lambda \subset {\mathbb Z}$
be the partial Hamiltonians introduced above,
where $V_{k l}$ are continuous functions tending
to $0$ at infinity, $k,l \in {\mathbb Z}$.
\begin{enumerate}
\vspace*{-2mm}
\item[(a)] Then $\Ad{e^{itH_\Lambda}} \, (\pi_0(\RA)) =  \pi_0(\RA)$,
$t \in {\mathbb R}$.
\vspace*{-2mm}
\item[(b)] Let $C, D$ be positive constants such that
$\|V_{k l} \| \leq C$ and $V_{k l} = 0$ for $|k - l| \geq D$, $k,l \in
{\mathbb Z}$. Then $\lim_{\Lambda \nearrow {\mathbb Z}} \, \Ad{e^{itH_\Lambda}} $, $t \in
{\mathbb R}$ exists  pointwise on $\pi_0(\RA)$ in the norm topology.
\end{enumerate}
\end{proposition}

\noindent The proof of this statement is given in \cite{BuGr5}.
It generalizes the results on a class of models describing
particles which are confined to
the points of a one--dimensional lattice by a harmonic pinning
potential and interact with their nearest neighbors \cite{BuGr1}.
In the present more general form it also has applications to other models of
physical interest. These results provide evidence
to the effect that the resolvent algebras are an expedient
framework also for the discussion of the dynamics of infinite
systems. Yet a full assessment of their power for the treatment
of such systems requires further analysis.

\section{Conclusions}
\setcounter{equation}{0}

In the present survey we have outlined some recent
structural results and instructive applications of
the theory of resolvent algebras. These algebras
are built from the resolvents of the canonical
operators in quantum theory and their algebraic relations
encode the basic kinematical features of quantum
systems just as well as the Weyl algebras. But, as we have shown,
the novel approach cures several shortcomings of this
traditional algebraic setting.

The resolvent algebras comply with the condition that
kinematical algebras of quantum systems must have
ideals if they are to carry various dynamics of
physical interest. This requirement can easily be inferred
from the preceding arguments in case of a single
particle: there the cocycles $\Gamma(t) = e^{itH} e^{-itH_0}$
appearing in the interaction picture have the
property that the differences $(\Gamma(t) - 1)$ are
compact operators for generic interaction potentials. Hence
$(e^{itH} W e^{-itH}  - e^{itH_0} W e^{-itH_0})$ is a compact
operator for any choice of bounded operator $W$.
It is then clear that any unital \mbox{C*--algebra} which is stable under
the action of these dynamics must contain compact operators
and consequently have ideals.

The resolvent algebras, respectively their subalgebras
corresponding to finite subsystems, contain these ideals
from the outset. As we have demonstrated by several physically significant
examples, the ideals play a substantial role in the
construction of dynamics of finite and infinite
quantum systems. For they accommodate the terms in
the Dyson expansion of the cocycles resulting from the
interaction picture and
thereby entail the stability of the resolvent algebras
under the action of the perturbed dynamics.
In order to cover a wider class of models it would, however,
be desirable to invent some more direct
argument, avoiding this expansion and the ensuing
questions of convergence.

The ideals of the resolvent algebras also play a prominent
role in their classification. The nesting of primitive ideals
encodes precise information about the size of the
underlying quantum system, {\it i.e.\ } its dimension. It
is a complete algebraic invariant in the finite dimensional case.
There is also a sharp algebraic distinction between finite
and infinite quantum systems in terms of their minimal ideals.
In either case the resolvent algebras have comfortable
algebraic properties: they are nuclear, thereby
allowing to form unambiguously tensor products with other
algebras which plays a role in the discussion of coupled
systems.

In company with the resolvents of the canonical
operators all their continuous functions
vanishing at infinity are contained in the resolvent
algebras. This feature ensures, as we have shown,
that many operators
of physical interest are affiliated with the resolvent algebras.
It also implies that these algebras contain
multiplicative mollifiers for unbounded operators which
appear in the algebraic treatment of supersymmetric models
\cite{BuGr4} or of constraint systems~\mbox{\cite{BuGr1,Co}}.
Thus the resolvent algebras provide in many respects a
natural and convenient mathematical setting for the discussion
of finite and infinite quantum systems.

redpajama_set_name: RedPajamaArXiv


So it all began on the 28.02.16 when Snuggy Sooty and Janey joined for their their debut outing, they have never looked back! In Janey's first year competing within the first ridden classes the two superstars WON the HOYS FR at NPS and went on to win Mini Champion.
There is NOTHING that a Sweet Itch pony can not achieve and Sooty is the perfect example. Sooty is based with Team Holder who have an extremely strict daily routine in place for Sooty to ensure that his Sweet Itch is controlled at all times, this is a perfectly manageable condition once you know how, do not let Sweet Itch ruin your ponies life.
Make it your time to stand out from the crowd like Snuggy Sooty.
For further information and advice please take a look at Sooty's story in the videos below.


The Office of Corporate Relations at the University of Illinois Fire Service Institute is the point of contact with business and industry. While providing corporations with access to the Institute's greatest resources, our staff, facilities and programs, we coordinate training programs to reduce the risk of loss and enhancing corporate profits.
The Institute also works together with the corporate community to develop specialized training programs to meet specific company needs.
Our goal is to develop long-term corporate partnerships to strengthen the bottom line while continuing to build fire training and safety programs to insure a productive work environment.


The Dolgeville Volunteer Fire Department call report for April 26.
•The Dolgeville Volunteer Fire Department was requested by Fulton County Fire Control to stand-by at their station at 9:22 a.m. due to a structure fire at 482 County Highway 151 in Fulton County. At 9:26 a.m., Dolgeville firefighters were requested to respond with a tanker to 482 County Highway 151 and to continue to stand-by at the fire station. At 9:52 a.m., the tanker returned to the station.
•The Dolgeville Volunteer Fire Department was dispatched to a commercial fire alarm at North Hudson Woodcraft Corp., 152 N. Helmer Ave., Dolgeville, at 9:56 a.m. Upon arrival, and after an investigation, firefighters determined the alarm was caused by a system malfunction. The scene was turned over to maintenance personnel and firefighters returned to service at 10:30 a.m.


The conservation of ecosystems and species, a daunting task for humans, has been thwarted in part by insufficient and inappropriate use of science, particularly in the way that scientific uncertainty is reported and incorporated into decisionmaking. Ecological systems are complex, and our understanding of them will always include scientific uncertainty. However, we assert that such uncertainty must not be used to avoid responsible ecological decisionmaking. Failing to act today to conserve ecosystems and prevent species extinctions will have significant social and ecological costs tomorrow.
include full disclosure of value judgements, and assumptions that underlie the interpretation of data and information.
Herein, we propose standards for disclosing uncertainty (making uncertainty explicit), reducing uncertainty, and managing in the face of uncertainty.
Disclosing Uncertainty The levels and kinds of uncertainty relevant to a decision should be fully and clearly identified and described in ways that are understandable to informed participants.
According to sound scientific practice, the description of uncertainty should include: sample sizes, duration and geographic extent of studies, and estimates of precision associated with point estimates; the limitations and dangers of extrapolating results to other geographical areas, time periods; and assumptions used to deal with uncertainty in the analysis and modeling.
All data that are used to calculate point estimates and confidence intervals and are used in other ways to arrive at a decision should be disclosed. Those data should also be made available to all participants in a decision and, open debate and discussion of alternative interpretations encouraged as a further means of ensuring that all uncertainties are identified.
Because assumptions are also sources of uncertainty, the likely social, economic and biological consequences of their being invalid should be disclosed.
publication should include communication to all relevant audiences, not just peer-reviewed scientific journals.
Managing in the Face of Uncertainty Decisions must be made despite uncertainty. In such cases, decision makers should use the precautionary principle to discharge their duties.
shift the burden of proof to those who advocate a potentially harmful action .
A precautionary approach to conservation problems for which there is limited scientific information should entail the review of all relevant scientific information and its interpretation in the context of ecological theory, making specific the link between data, theory and interpretation. This process produces qualified insights, which should clarify irreducible uncertainties in the scientific information.


text:
Apple unveils upgraded MacBook Pro laptops
Santa Rosa processors, higher-end graphics cards, memory space, and energy efficiency features are all part of Tuesday's upgrade announcement.
June 5, 2007 8:03 a.m. PT
Apple announced Tuesday that it has updated its high-end MacBook Pro laptops with faster processors, better graphics, and more memory.
The laptops are still available in three models: two 15.4-inch versions that retail for $1,999 and $2,499 and a larger 17-inch edition that costs $2,799. But with Tuesday's update, the lower-end 15-inch model is now equipped with 2.2GHz Intel Core 2 Duo processors (previously, it had a 2.16GHz chip); the higher-end 15-incher and the 17-incher now come with 2.4GHz processors, up from 2.33GHz. These new processors are indeed Intel's latest Centrino (or "Santa Rosa") chips, which were introduced in early May.
All three MacBook Pros additionally now can be upgraded from 2GB to 4GB of RAM at an extra cost, and have faster NVIDIA GeForce 8600M GT graphics cards designed for media-intensive animation and gaming.
The 15-inch models also have an energy efficiency boost, with LED-backlit displays; Apple has touted these not only as power-saving but also mercury-free.
The announcement follows another Apple laptop upgrade in May, when the company released new versions of its lower-priced MacBook machines. Those laptops received processor and memory upgrades in addition to a hard drive space boost (which was not part of Tuesday's MacBook Pro boost), but did not come with "Santa Rosa" processors or LED-backlit screens as had been rumored.
The last major update to the MacBook Pro laptops was in October, when they were first manufactured with Intel Core 2 Duo processors.
It's shaping up to be a hot summer for Apple. The company is expected to make more product announcements--though no one's really sure what they will be--at its Worldwide Developers Conference next week. And then the iPhone, will hit stores on June 29.
Discuss: Apple unveils upgraded MacBook Pro laptops

redpajama_set_name: RedPajamaCommonCrawl


text:
News|Russia-Ukraine war
Ukrainian court finds two Russian soldiers guilty of war crimes
Captured soldiers sentenced to more than 11 years in jail each for shelling a civilian area in Ukraine's northeastern Kharkiv region.
Russian soldiers Alexander Ivanov and Alexander Bobykin, left, attend their trial hearing in Kotelva, northeastern Ukraine [Bernat Armangue/AP Photo]
Published On 31 May 202231 May 2022
A Ukrainian court has sentenced two captured Russian soldiers to more than 11 years in jail each for shelling a civilian area in the country's east amid Moscow's offensive.
Tuesday's sentencing of Alexander Bobikin and Alexander Ivanov is the second verdict handed down in war crimes trials held by Ukraine since the start of Russia's invasion in late February.
Bobikin and Ivanov, who watched proceedings at the Kotelevska district court in central Ukraine from within a reinforced glass box, both pleaded guilty last week to "violating the laws and customs of war".
"The guilt of Bobikin and Ivanov has been proven in full," Judge Evhen Bolybok said. There was no immediate response to the judgement from Moscow.
The pair had acknowledged being part of an artillery unit that fired at targets in Ukraine's northeastern Kharkiv region from the neighbouring Belgorod region in Russia.
The shelling destroyed a school in the town of Derhachi but caused no casualties, prosecutors said. Bobikin and Ivanov, described as an artillery driver and a gunner, were allegedly captured after crossing the border and continuing the shelling.
Prosecutors had asked the court to jail the Russian service members for 12 years, while their defence asked for leniency, saying the two soldiers had been following orders and repented.
Thousands of civilians killed
Tuesday's developments came after a Kyiv court last week sentenced 21-year-old Russian soldier Vadim Shishimarin to life in prison for killing an unarmed Ukrainian civilian.
Shishimarin was accused of shooting 62-year-old Oleksandr Shelipov in the head in Ukraine's northeastern Sumy region in the early days of the war. His lawyer has said an appeal will be lodged against the verdict.
Many more such trials are expected in both Ukraine and Russia, which has routinely denied that its forces have targeted civilians or carried out war crimes and is thought to be preparing for its own hearings for captured Ukrainian soldiers.
Ukraine's Prosecutor General's Office has said it is investigating more than 14,000 cases of alleged war crimes carried out by Moscow's troops.
More than 4,000 civilians, including dozens of children, have been killed in Ukraine since President Vladimir Putin launched Russia's offensive, according to the United Nations.
The toll includes numerous suspected victims of war crimes in areas currently or previously occupied by Russian forces, such as Mariupol and Bucha.
The UN has said it believes that the true figure for civilian casualties is likely "considerably higher", with ongoing hostilities making efforts to count the dead difficult and many reports of killings still pending corroboration.
Source: Al Jazeera and news agencies


The Radisson Blu es. Hotel, Rome sits in the very heart of the city's center, only 10 minutes away from Roma Termini Railway Station. This ensures you easy access to the entire Italian capital through the Rome Metro.
Leonardo Da Vinci Fiumicino Airport is located 32 km west of the hotel When arriving by plane you can take the Leonardo Express non-stop train, departing every 15 or 30 minutes – depending on the time of day, from the airport and into the city center. The journey takes a mere 32 minutes, and all trains are equipped with dedicated seating and restrooms for passengers with reduced mobility. For updated ticket prices and departure times, please see the Leonardo Express website. When you arrive at Rome Termini Station, follow the indication along the rails for platform 25 to 27. In front of it, head down the stairs on the left, cross the street and you will arrive at the hotel.
Rome Termini Station, which is the central station for Rome and the base for all train transport, connects Rome with all of Italy and Europe. When you arrive at Rome Termini Station, follow the indication along the rails for platform 25 to 27. In front of it, head down the stairs on the left, cross the street and you will arrive at the hotel.
Metro lines A and B go around rather than through the historic city. The Vittorio Emanuele metro station is located two blocks away, which is serviced by metro line A. Termini metro station is a 10-minute walk away, and is serviced by both metro lines A and B.
Leonardo express is the non-stop service that connects Roma Termini station * and Roma Fiumicino airport in just 32 minutes, with programmed departures every 15 minutes (every 30 minutes during certain time periods). Price for a single ride is 14 euro.
*Roma Termini is the train station next to the conference hotel (5 minutes walk).


The International Journal of Community Currency Research (IJCCR) is a small, online journal that is run from the School of Environmental Science at the University of East Anglia. We have just edited a special edition which details some of the recent developments in the field of complementary currencies. Experiments in community and complementary currencies have been ongoing since the early 1980s. Over this period there has been increasing awareness, diffusion and experimentation in the field. The joint stimuli of economic and ecological crises now appears to be underpinning a new wave of experimentation.
Many of these currency experiments could be considered as forms of grassroots innovation in that they originate from within civil society. Increasingly systems are also being developed that are linked explicitly to sustainability objectives.


For entrepreneurs who want to start their own businesses, the vending machine business is a viable option. It's profitable (the industry generates $33 billion annually), and you need only one vending machine and stocked supplies to get started, according to Forbes magazine. Though a vending machine business is among the easiest to start, regular maintenance and stocking is required. Besides the initial legwork required to establish a vending business, vending-machine owners are often busy traveling from machine to machine refilling stock, collecting money, repairing machines and scouting new locations.
Research your options before purchasing a vending machine. Pay attention to the types and styles of vending machines around town and at places you frequent. Conduct additional research by going online and canvassing the variety of vending machines available and comparing prices. Business.com has a list of suppliers and tips on matching products to your market.
Consider the benefits that go with the vending machine you will buy, and see if it will help your business. Sites such as gumballs.com offer a wide variety of machines plus a variety of bulk supplies. Some vending machine sales companies also give free consultations and ongoing professional advice.
Size your machine or machines according to the space available in the location you'd like to place your machine.
Check out used machines. If you are thinking about buying a used machine, see how much the newer version of the same machine costs. Make sure the used machine you are considering is fully operational. Inspect the coin receptor and validator to ensure that they are updated and functional. If possible, choose a machine with a dollar-bill changer, which could raise sales by 30 percent.
Get a factory warranty for your vending machine. If you are buying a used machine, see if the previous owner can transfer his warranty to you.
If you don't have enough to pay for a new or used machine up-front, inquire about a payment plan.
Base the items you sell in your machine on the market profile of the traffic around your vending machine.
According to Vending Machines NC, beverages and ice cream usually sell well. Other suggestions for items to sell include stickers, toy capsules and tattoos for children; bouncy balls and candy for teens; and snacks and mints for adults. If your vending machine is in a company break room, survey a segment of the staff to find out what they like.
Shop around for wholesalers who have the types of products available for your customer demographic. Compare prices and buy in bulk when possible. Some vending machine sellers will assist you with finding local wholesalers.
Choose high-traffic locations for better sales. Forbes.com says schools and hotels are best. Other options are new businesses, sites with outdated equipment, billiard parlors, restaurants, warehouses, factories, car dealerships, industrial parks, malls, real estate offices, banks, tech schools, hospitals, nursing homes, skating rinks, golf courses, offices and service centers.
Make sure there is an electric outlet where you want to place your machine.
Ask the manager of the establishment where you desire to install your vending machine if you can place your machine on his site. If the manager gives you the go ahead, offer a commission and have him sign a written agreement outlining the terms of your partnership.
Observe your site for an hour or two to determine how many customers you may have. If your machine is in an office building, find out how many employees the company has.
Add your overhead costs such as transportation (car and fuel), vending machine supplies, and commissions. Then use this to determine what you will charge for items on sale in your machine.
Your expenses will vary based on your supplies. If one item doesn't sell well, you can switcch it to something else. The fact that you can tweak products along the way can increase your profits.
Fifty percent of making this business work lies in maintenance. Make sure your machines are always functioning well.
Many charities will give you stickers to put on your vending machines, usually for $1 per month per machine. Some store owners will allow you space in their stores for free if you are associated with a charitable organization.
Usually vending machines are safe from vandalism because they are in high-traffic areas. If the location is questionable, chain your machine to a sturdy fixture inside the store.


School of Law Prof. Thomas Geraghty.
When School of Law Prof. Steven Lubet received an 18-month grant that brought him to Northwestern, he never thought he would stay for more than 40 years –– but then he worked with School of Law Prof. Thomas Geraghty.
After serving as the Bluhm Legal Clinic's first director for more than 40 years, Geraghty (School of Law '69) retired on Aug. 31. The clinic, which opened in 1969, allows law students to gain experience trying cases while providing legal help to those in need, Geraghty said.
Geraghty said he is especially proud of his cases involving juvenile rights and the abolition of the death penalty in Illinois.
School of Law Prof. Juliet Sorensen succeeded Geraghty as the clinic's director on Sept. 1. She said about 140 law students are enrolled in clinical classes during the year, and 91 percent of Pritzker School of Law students graduate having participated in some clinical program.
Sorensen said even though Geraghty's shoes are "impossible" to fill, it's exciting to serve as the new director and continue to grow its programs.
Geraghty said he's proud of how much the clinic has grown since its inception. Original grant funding paid for the clinic to have two lawyers, he said. Now, the clinic has expanded to about 35.
School of Law Prof. Bob Burns has tried cases with Geraghty and said working with him has been a "complete pleasure" because of his technical proficiency as a trial lawyer, his diplomacy and his dedication to social justice.
Although he has retired from the clinic, Geraghty said he is still teaching at NU law school. Sorensen, the new director, said she's glad Geraghty will be nearby in case she needs his guidance.
Overall, Geraghty said he's proud of the opportunities the clinic has provided to students.
"(I hope) … that we continue to provide a combination of experiences to students involving work on individual cases, as well as on projects that are designed to achieve systemic reform of our justice system," Geraghty said.


text:
Q: file permission: one user with multiple permission types The situation I thought of is, that one user get's two links with different types of permission for the same file (didn't find a satisfying answer yet):

*

*Link A: read-only

*Link B: full access

Now if the user opens the file with Link A, what is he able to do?

*

*just view the file
or

*edit it because the 2nd permission is stronger?

A: Based on my research, when sharing links are created, permissions along with other information are stored in a hidden SharePoint List called Sharing Links.
For example, if link is shared with read permission, RoleDefinitionId 1073741826, and with editing, RoleDefinitionId 1073741827, are stored in the list. So, the permission of document defined in the shared link is retrieved from this list when the link is accessed.
Assuming the user receiving the link does not have any other permissions given directly to the site/library, the user will have the permission as defined during creation of the respective link, that is, in this case, Link A will have Read Only and Link B will have full access regardless the order of accessing the links.
You can read more on Sharing Links here
For RoleDefinitionId, here

redpajama_set_name: RedPajamaStackExchange


Eat An Ounce Of This Each Day, Live 20% Longer!
Is boosting your lifespan as easy as eating a handful of these a day? According to a Harvard Medical School study, yes! And this is no wimpy, small scale study either. It looked at 75,000 women and 40,000 men over a 30 year period.
The participants completed food surveys every few years over several decades. And while many of the foods they ate showed positive benefits, one stood out to the researchers.
"What we find is that regular nut consumers have about a 20% reduction in all-cause mortality" over the course of the study, said Charles S. Fuchs, professor of medicine at Harvard Medical School and author of the study. The regular nut-eaters were also thinner than those who didn't eat nuts.
"The most obvious benefit was a reduction of 29 percent in deaths from heart disease — the major killer of people in America," Fuchs went on to say.
All nuts showed benefits from almonds, walnuts, pecans, macadamia, pistachio, Brazil, and even peanuts, which are technically a legume.
Many earlier studies found an association between higher nut consumption and a lower risk of heart disease, type 2 diabetes, colon cancer, gallstones, and diverticulitis. Regular nut consumption also shows reduced cholesterol, inflammation, oxidative stress and insulin resistance.
Bottom line: EAT AN OUNCE OF NUTS EACH DAY FOR A LONGER, HEALTHIER LIFE!


Named after Buena Vista Park and located in the area immediately surrounding it, this charming neighborhood is known for its panoramic views and palm-tree-lined streets. The park itself, which is dotted with hiking trails, pathways, tennis courts and staircases, is the oldest official park in San Francisco, built in 1867. The homes here are primarily single-family Edwardians and Victorians, but there are also some more modern houses and multi-unit properties.
Grattan Elementary is one of the highest-rated public elementary schools in San Francisco.
UCSF Parnassus Medical Center is regarded as one of the world's leading universities in health sciences.
The Mount Sutro Open Space Reserve is a largely undeveloped, 61-acre area just south of the UCSF Parnassus Heights campus.
Lined with stores reminiscent of its counter-culture past, Haight Street is also home to trendy book, record and clothing shops, as well as lively bars and restaurants.


text:
Q: Setting up GLEW and GLFW in VS Code with Visual C++ I have a basic Visual C++ environment in Visual Studio Code. I've recently started to read about OpenGL and have been trying to set up GLEW and GLFW in VS Code. Almost all of the tutorials I find fall into one of the 2 categories:

*

*GLEW and GLFW setup with MSVC, but in Visual Studio.

*GLEW and GLFW setup in VS Code, but with minGW.

I've read the documentation for MSVC command line and its options a few times and can't figure it out. I'm not familiar with Cmake and such, so I've been trying to do it via VS Code tasks.json file.

My tasks.json file is as follows:
{
    "version": "2.0.0",
    "tasks": [

        // BUILD TASK
        {
            "type": "shell",
            "label": "C/C++: build project",
            "command": "cl.exe",
            "args": [
                // exception handling args
                "/EHsc",
                // including additional headers
                "/I${workspaceFolder}\\include\\glew\\include",
                "/I${workspaceFolder}\\include\\glfw\\include",
                // obj files path
                "/Fo${workspaceFolder}\\bin\\",
                // exe file path
                "/Fe:", "${workspaceFolder}\\bin\\program.exe",
                // source files
                "${workspaceFolder}\\src\\*.cpp",
                // library files
                "opengl32.lib",
                "${workspaceFolder}\\include\\glew\\lib\\Release\\x64\\glew32s.lib",
                "${workspaceFolder}\\include\\glfw\\lib-vc2019\\glfw3.lib"
            ],
            "options": {
                "cwd": "${workspaceFolder}"
            },
            "problemMatcher": [
                "$msCompile"
            ],
            "group": {
                "kind": "build",
                "isDefault": true
            }
        },

        // CLEAN TASK
        {
            "type": "shell",
            "label": "C/C++: clean project",
            "command": "del",
            "args": [
                "${workspaceFolder}\\bin\\*",
            ],
            "options": {
                "cwd": "${workspaceFolder}"
            }
        }

    ]
}

And my main.cpp file has a few lines of code to open a window.
#include <iostream>
#include <GL/glew.h>
#include <GLFW/glfw3.h>

using namespace std;

int main()
{
    if (!glfwInit())
    {
        cout << "Failed to initialize GLFW" << endl;
        return -1;
    }

    glfwWindowHint(GLFW_SAMPLES, 4);
    glfwWindowHint(GLFW_CONTEXT_VERSION_MAJOR, 3);
    glfwWindowHint(GLFW_CONTEXT_VERSION_MINOR, 3);
    glfwWindowHint(GLFW_OPENGL_PROFILE, GLFW_OPENGL_CORE_PROFILE);

    GLFWwindow* window;
    window = glfwCreateWindow(1000, 700, "Test window 01", NULL, NULL);
    if (window == NULL)
    {
        cout << "Failed to open GLFW window" << endl;
        return -1;
    }
    glfwMakeContextCurrent(window);

    glewExperimental = true;
    if(glewInit() != GLEW_OK)
    {
        cout << "Failed to initialize GLEW" << endl;
        return -1;
    }
}

Full terminal output while running build task at this pastebin

A: I had a similar issue recently and solved it by using the dlls, instead of static linking
For this, you'd add glfw3dll.lib instead of glfw3.lib (which you should have in the same zip you've downloaded from glfw website, alongside the dll) and place the dll nearby the executable.
You should also define GLFW_DLL before including any other glfw headers, as the documentation states

A: Right I am using MinGW as the compiler and after many days trying to fix this problem my self I found that this works.
Image
For the task file you need to put (in the args in the task json for vs code):
"-g",
"-std=c++17",
"${file}",
"-I",
"./include",
"-L",
"./lib",
"-lopengl32",
"-lglew32",
"-lglfw3dll",
"-o",
"${fileDirname}\\${fileBasenameNoExtension}.exe"

Then it should work. Side note you also need glfw3.dll with glew32.dll


Margaret M. Zeigler, Ph.D., is the Deputy Director of the Congressional Hunger Center. Her responsibilities include founding and directing the Mickey Leland International Fellows Program, establishing educational initiatives about hunger and food security, overseeing programs and financial operations, and helping establish a permanent endowment to fund the congressional fellowship programs. She holds a Ph.D. in Geography and International Development, and she teaches a course examining humanitarian emergencies in the 20th and 21st centuries entitled "Complex Humanitarian Emergencies" at George Washington University's Elliott School of International Development.
Margaret also serves as the co-chair of the Committee on Development Policy and Practice at InterAction, a consortium of over 160 non-governmental organizations working in humanitarian relief and international development. She was selected by the Massachusetts Institute of Technology to participate in a year-long national security fellowship program, Seminar XXI, in 1999-2000. In 1997-1998, she obtained funding from the US Institute of Peace to conduct research on coordination of humanitarian NGO's and the military in Bosnia, and she provides regular humanitarian training for the Southern Command of the US military and other Latin American military units during peacekeeping exercises. She provides international food security updates regularly for Bread for the World Institute's annual Hunger Report, publishes articles in journals such as Humanitarian Affairs Review and is a founding member of the Council on Faith and International Affairs (CFIA).


I was at the press conference of Cre Arts Asia yesterday for a sneak preview of Southeast Asia's largest modern circus performance and questions & answers session. Cre Arts Asia will be performing for the first time in Malaysia at Kuala Lumpur Performing Arts Centre (KLPAC) from 29 September to 1 October 2017. Are you excited about this? My kids and I are extremely excited because we love to watch circus performances!
Cre is an Irish word meaning 'earth' and promises spectacular displays of gravity-defying performances and extraordinary feats of the human body, all set within stunningly beautiful backdrops that are a mainstay of contemporary performing arts. Cre started from a group of seven people.
Cre Arts Asia is collaborating with Viva Circus, an award-winning physical arts theatre company and Psycusix, Malaysia's leading modern and contemporary circus performing group.
The organiser of Cre, Chris Lo who is also the Event Director of Creative Events Solution Sdn. Bhd said, "The talents that are on display at Cre are truly world-class. In fact, some of the performers featured have won several international accolades and have performed at renowned venues across the world."
After the introduction of Cre, the throwback videos and Cre promotion video, came the best part which is showtime! I really enjoyed the mesmerizing performances. Here is a video I recorded.
If your family especially children love to watch these type of performances which is entertaining and meaningful, you can get your tickets HERE. Sign up as member, choose your preferred showtime, show map to select seat no., add to chart and pay by credit card or paypal. Cre Arts Asia will be donating part of the proceeds from ticket sales to the International Medical University Malaysia (IMU) Chariofare, the largest fundraising event at IMU.
Tickets for Cre are available for RM 98.00 (Fire), RM 128.00 (Water) and RM 168 (Earth). Tickets for kids below 12 years of age are available for RM 68.00 (Water & Earth only). Fire seats are free-seating while Water & Earth seats are numbered.
There is good news for the early birds. From now until 14 September 2017, you can enjoy the Early Bird Promotion for only RM 108 (Water) and RM 148 (Earth). A Special Package Promotion is also available where if you purchase five (5) Earth tickets, you are entitled for one (1) additional FREE Earth ticket.
My blogger friends and I are very excited and looking forward to the breathtaking contemporary circus performance. This is an experience not to be missed. Get your tickets now.
Yes, I am so excited about this. Thanks.
Thanks for the infor. It has been a while I did not go for circus show, this one is interesting.
It is a great performance. Get your early bird promo tickets.
Seeing fire so close is scary.
We have never seen a circus performance before. This looks awesome!
It will be a good experience to see a live circus performance!
Wow! Nice show. Should get the ticket now!
Yes, get the early bird ticket price to save money.
Nice show and the show price also attractive. Will share with my friends.
Awesome ! Definitely will bring my friends to watch the show !
Great! Good to watch with friends.
Do check it out. It's an experience not to be missed.
Happy to know you will get a ticket.


585052 800 Sq. Ft. 2 Bedrooms 1 Bathroom 1953 Year Built 5280 Sq. Ft.
Wonderful 2BD/ 1.5BA home in Midtown Key West with large lot and room for a pool. This home is move-in ready with updated baths, tiled floors and an open floor plan. Kitchen offers granite counter tops, built-in cabinets, new appliances and a spacious island. The property is fully fence and bordered by mature landscaping with a detached, covered deck for entertaining. Easy to maintain and provides ample off-street parking and carport. An additional structure, not included in the square footage, is attached to the back of the home along with a storage shed and outdoor shower. Tenant occupied and rented for $3,500/ month.


The Nora Geometric A-line Dress – Vence & Co.
Fit runs big; size down one size. Model is wearing small.
S - Sleeves 25" Bust 35" Waist 28" Length 37.5"
M - Sleeves 25" Bust 36" Waist 30" Length 38"
L - Sleeves 25.5" Bust 37" Waist 32" Length 38.5"


We have thousands of properties for sale all over France. But we've no property for sale in Maisons Laffitte at the moment. And that's something we want to change.
If you have a property to sell in Maisons Laffitte (or nearby) then act now to get a 50% discount.
Just place your advert in the normal way (Click here to place your advert). Then send us an email quoting "Maisons Laffitte" and telling us that you you saw this offer.
Maisons Laffitte is one of the most stylish and elegant neighbourhoods around Paris. It sits partly on an island in the River Seine, just a little way to the north-west of the city centre, and is a highly sought after suburb with great access to the capital (twenty minutes by rail) and a chic and tranquil air that is extremely attractive. All the benefits of Parisian life are available to residents of Maisons Laffitte, yet without the noise, grime and hassle of city centre living.
There is no shortage of heritage and history in Maisons Laffitte, and its gorgeous chateau is living proof. The fabulous building was commissioned by René de Longeuil in 1642, who was an incredibly wealthy financier and officer of the royal treasury. He chose Francois Mansart to be the architect and what resulted is a fine example of French baroque architecture, and is considered by some to be among the finest in France, in terms of its symmetry and perfection of design.
Originally called Château de Maisons, it was sold in 1818 to a Parisian banker called Jacques Lafitte, who started developing the surrounding land into more like how the town appears today. Members of his family also had an influence on the development, being involved in the building of the railway and the installation of the water supply. The family name "Lafitte" was added to the chateau and town in the late 19th century.
If Parisian life means first rate shopping and dining, then Maisons Laffitte will not disappoint. Well equipped to satisfy on both counts, the community offers all the usual shops and amenities, from supermarkets to the smaller boulangeries and boucheries, as well as nearly 50 bars and excellent restaurants, serving all types of food. Of course, just twenty minutes up the road you may have even more choice...!
Horse racing is big business in Maisons Laffitte, and the race track, or Hippodrome, is very popular among wealthy Parisians looking to enjoy a day out with a little excitement. The site occupies no less than 92 hectares of prime Maisons Laffitte land, and for aficionados of "the sport of kings" there is the Museum of the Horse to visit within the Maisons Lafitte Château as well. Of course, horse racing is not the only sport to be found in Maisons Laffitte, there is access to almost anything you can name in or near the town, including a swimming pool, golf centre and tennis courts.
In a discussion of places to visit close to Maisons Laffitte, where do I begin? When you have all of a city like Paris stretched out at your feet, it seems churlish to suggest any place else... but dare I mention Versailles?
Paris is so well supplied with airports, TGV stations, metro lines and road networks that access from the UK... or anywhere else for that matter, is simple. Maisons Laffitte is situated just twenty minutes from the centre of Paris. The airports of Paris are served by numerous airlines, including Air France, bmi, bmibaby, flybe, British Airways, easyJet, Ryanair, Jet2 and Aer Lingus. If you are travelling by train, the Eurostar will take you to Paris, from where you can connect to an onward train to Maisons Laffitte. If travelling by road, the town is close to both the A13 and A15, with the D308 going through the centre of Maisons Laffitte itself.
Desirable areas of Paris aren't cheap. Ever. Now that that question has been cleared up, it is also true to say that some areas are cheaper than others... but a chic bourgeois suburb such as Maisons Laffitte is certainly not the cheapest option. Not everyone is on a tight budget, however, and anyone with a little money to invest in an elegant property close to the world's most romantic capital city could do far worse than consider Maisons Laffitte. With a location desirability factor of ten out of ten, prices are unlikely to do anything but rise.
Do you know Maisons Laffitte better than we do? Do you have photos of Maisons Laffitte?
Maisons Laffitte is located in the north of central France, just north-west of Paris. It lies in the north-east of the Yvelines department.
Access: By air: Fly to Paris. By rail: The Eurostar will take you to Paris, and Maisons Laffitte is just twenty minutes from Paris by rail. By road: The D308 passes through the town.
Economy: The economy is strong and includes tourism, commerce and the leisure industry, specifically horse racing.
Interesting fact: Maisons Laffitte is partly situated on an island in the River Seine, providing one of Paris's most desirable and picturesque neighbourhoods.


text:
Kadmos Press Releases
Get the most up-to-date news about what's going on at Kadmos through our press releases. On this page we share company announcements, new product features, partner updates, and more.
Kadmos co-founders speak about digitalization in shipping at SMM Hamburg
Kadmos recently participated in SMM, a leading international maritime trade fair, during its 30th edition held in Hamburg. The maritime fair, running from September 6-9, served as a showcase for around 2,000 exhibitors and more than 40,000 visitors from over 100 countries.
NSB Group announces partnership with Kadmos
NSB Group chooses Kadmos to digitalize crew salary payment processes and provide seafarers with more control of their salary payouts.
CSM partners with Kadmos to automate crews' salary payouts
Columbia Shipmanagement (CSM) has announced a partnership with Berlin-based Fintech Kadmos to automate and digitalize its seafarers' salary payouts.
Kadmos raises €29m in Series A funding round led by Blossom Capital to streamline payment processes for cross-border workers
Berlin-based startup uses cutting-edge financial technology to make the salary journey of millions of migrant workers more transparent, convenient, and cheaper.
Kadmos Joins the International Maritime Employers' Council (IMEC) as Affiliate Member
Kadmos GmbH is happy to announce that it is now an affiliate member of the International Maritime Employers' Council Ltd (IMEC).
Fintech Kadmos closes €8.3M Seed round to revolutionize the salary payments experience of migrant workers
Kadmos provides a new solution for international salary payments, making the salary journey of millions of migrant workers more transparent, convenient, and less expensive. With ambitious goals set for 2022, the start-up will use the new funds to accelerate product development, expand the team, and facilitate its entry into further industries.


Today at the Global Citizen Festival: Mandela 100, HP (NYSE: HPQ) announced a new commitment to reach 100,000 learners across Africa over the next three years through the HP Foundation's HP LIFE (https://bit.ly/2QuAzsD) program and kicked off the commitment by opening a new tech-enabled HP LIFE Center in South Africa. HP's pledge supports the United Nations Sustainable Development Goal 4 (https://bit.ly/2HUe9gd) and furthers the company's goals to enable better learning outcomes for 100 million people by 2025 and to enrol a million HP LIFE users between 2016 and 2025, as outlined in the HP 2017 Sustainable Impact Report (https://bit.ly/2KOnogZ).
By 2030, Africa will be home to 32% of the population under the age of 30, and the largest working-age population by 2035. Yet, today's youth unemployment in Africa is up to 3x higher than adult unemployment. HP LIFE offers free, online learning for users to gain the skills to start and grow their own business or improve their employment opportunities. Building on the success of this global program, HP is working with partners to open technology-enabled HP LIFE innovation centres to further support entrepreneurship and workforce development across the continent.