text:
\section{Introduction }  \label{Introduction}

The study of adsorption of molecules on the surface of different bodies covers
an extremely wide class of problems of physics and chemistry and is one of the
most important problems both from the theoretical point of view and for
practical applications. The results of numerous investigations show that
adsorption of molecules on surfaces of bodies leads to changes in various
physical and chemical characteristics of these bodies. The detailed analysis of
changes in the properties of the surface due to adsorption is given in
\cite{ref.Mor,ref.RobMc,ref.Adam,ref.KiK,ref.Vol,ref.Zan,ref.Zhd,ref.LNP,ref.Nau,ref.VaS}.
The results of investigation of changes in properties of the surface due to
adsorption--desorption processes are also widely used in the design of various
sensors (physical, chemical, and biological) \cite{ref.MyaSKZ,ref.Cat} whose
action is based on the use of the change in a certain characteristic of a
sensitive element of the sensor due to adsorption of molecules on its surface.

The results of the theory of adsorption are extremely important for
investigation of heterogenous-catalytic reactions because processes of
adsorption and desorption are integral stages of these reactions.

The classical Langmuir theory that describes adsorption of a gas on solid
surfaces is based on several assumptions. Numerous theoretical investigations,
which, to a large extent, were stimulated by many experimental data that did
not agree with conclusions of the Langmuir theory, were aimed at the
construction of more general models free of one or several restrictions of the
Langmuir theory. An extensive material obtained on the basis of these models
and applications to various problems of adsorption and catalysis are widely
presented in the literature (see, e.g.,
\cite{ref.Zhd,ref.FrK,ref.Bar,ref.Rog,ref.Roz,ref.KrSh,ref.Tov}). In
particular, it is established that taking account of lateral interactions
between adsorbed molecules can lead to a qualitative change in adsorption
isotherms, namely, to a hysteresis of isotherms and to structural changes in
the surface of bodies (surveys of theoretical and experimental results are
given, e.g., in \cite{ref.Adam,ref.Zan,ref.Zhd,ref.LNP,ref.Nau,ref.Tov}).

At the same time, as early as in 1938, in \cite{ref.Zeld}, Zeldovich has
suggested an idea on a change in the surface in the course of adsorption and
desorption due to the presence of adsorbed molecules on it. Using this idea, he
has predicted a hysteresis of adsorption isotherms if the typical time of
adsorption and desorption is much less than the relaxation time of the surface.

Note that a change in adsorption isotherms due to lateral interactions between
adsorbed molecules can also be interpreted as a consequence of a certain change
in adsorption properties of the surface caused by adsorption. However, as far
as we know, the problem of variation in adsorption properties of the surface
itself in the course of adsorption and desorption of molecules of gas on it in
the absence of interactions between adsorbed molecules and the possibility of
hysteresis of adsorption isotherms in the this case remains open.

The present paper is devoted to investigation of specific features of the
behavior of adsorption isotherms and the kinetics of the surface coverage by
adsorbate molecules with regard for variations in adsorption properties of the
surface in the course of adsorption--desorption.

In Sec.~2, we propose a  self-consistent model taking into account variations
in adsorption properties of the surface in the process of
adsorption--desorption of molecules of gas on it. We introduce a dimensionless
coupling parameter that characterizes the interaction of an adsorbed molecule
with polarized medium. We obtain adsorption isotherms and establish that their
behavior essentially depends on the value of this parameter (Sec.~3). It is
shown that, within the framework of the proposed model, the Zeldovich
hysteresis is possible. In Sec.~4, we investigate specific features of the
kinetics of the surface coverage. It is established that variations in
adsorption properties of the surface in the course of adsorption--desorption
cardinally change the Langmuir kinetics.

\section{Model of the Surface with Varying Adsorption Properties }  \label{Modell}

We consider a problem of adsorption of molecules of a one-component gas on the
surface of a solid adsorbent. According to the classical Langmuir theory,
molecules of gas are adsorbed on adsorption centers located on the adsorbent
surface and the number of centers does not change with time. Furthermore, all
centers have equal adsorption activity (energy-uniform surface), do not
interact with each other, and each adsorption center can be bound only with one
adsorbate molecule. The Langmuir kinetics of the quantity of adsorbed substance
is described by the differential equation \cite{ref.FrK}

\begin{equation}
 \frac{d \theta}{dt} = k_a C \bigl( 1 - \theta \bigr) - k_d \, \theta,
\end{equation}

\noindent where $\theta(t) = N_b(t)/N$ is the surface coverage by adsorbate,
$N$ is the total number of adsorption centers, $N_b(t)$ and $N_0(t)$ are,
respectively, the numbers of occupied and free ($N_0(t) = N - N_b(t)$)
adsorption centers at the time  $t$, $k_a$ and $k_d$ are the adsorption and
desorption rate constants, respectively, and $C$ is the concentration of
molecules in the gas phase that is kept constant.

The solution of this equation with zero initial condition $\theta(0) = 0$ has
the form \cite{ref.Roz}

\begin{equation}
 \theta(t) = \theta^{st}_L \biggl[ 1 - \exp\Bigl(-\frac{t}{\tau_{ad}} \Bigr) \biggr],
\end{equation}

\noindent where

\begin{equation}
 \theta^{st}_L = \frac{l}{1 + l}
\end{equation}

\noindent is the stationary surface coverage (Langmuir isotherm), which is
defined by the single dimensionless quantity (dimensionless concentration) $l =
C K$, $K = k_a/k_d$ is the adsorption--desorption equilibrium constant for the
given concentration of gas,

\begin{equation}
 \tau_{ad} = \frac{1}{k_a C + k_d} = \frac{\tau_d}{1 + l}
\end{equation}

\noindent is the time taken for the surface coverage to reach the stationary
value $\theta^{st}_L$, and $\tau_d = 1/k_d$ is the typical lifetime of a
complex (adsorption center + adsorbed molecule).

According to (2) and (3), there is a single-valued correspondence between the
gas concentration and the surface coverage. At the same time, in
\cite{ref.Zeld}, Zeldovich has suggested an idea on a change in adsorption
properties of the surface in the process of adsorption and desorption and,
using this idea, predicted a hysteresis of adsorption isotherms if the typical
time of adsorption and desorption is much less than the relaxation time of the
surface.

In the present paper, within the framework of the Langmuir model, we take into
account variations in properties of the surface of a homogenous adsorbent with
plane boundary in the process of adsorption and desorption of molecules of a
one-component gas on it. We introduce the Cartesian coordinate system centered
at the surface of the adsorbent with $0X$-axis directed into the adsorbent
perpendicularly to its surface so that the adsorbent and the gas occupy the
regions $x \ge 0$ and $x < 0$, respectively. Each adsorption center is
simulated by a one-dimensional linear oscillator of mass $m_0$ that oscillates
perpendicularly to the surface about its equilibrium position $x = 0$.  In the
absence of adsorbate, the motion of an absorption center is described by the
well-known equation of motion of a free linear oscillator

\begin{equation}
 m_0 \frac{d^2x}{dt^2} + \alpha \frac{dx}{dt} + \kappa x = 0,
\end{equation}

\noindent where $\kappa$ is the restoring force constant, $\alpha$ is the
friction coefficient, and $x$ is the coordinate of the oscillator.

In the general case, due to occupation of the adsorption center with a molecule
of adsorbate, the electron structure of the center changes, which results in a
change in the interaction of the center with neighboring atoms of the
adsorbent, i.e., to a local polarization of the adsorbent. As a result, the
medium acts on the complex with a certain force $\vec{F}_p(\vec{r},t)$, where
$\vec{r}$ is the running coordinate of the complex, which is the reaction of
the medium on the electron reconstruction of the adsorption center. Under the
action of this force, the complex tends to a new equilibrium position different
from the equilibrium position  $x = 0$ of the free center. If the polarization
of adsorbent caused by the formation of the complex is axially symmetric about
the axis passing through the complex and parallel to the  $0X$-axis, then this
force has only the component normal to the boundary, $\vec{F}_p(\vec{r},t)=
\vec{e}_x\, F_p(x,t)$, where $\vec{e}_x$ is the unit vector along the
$0X$-axis; for convenience, the center has the coordinate  $\vec{r} = (x, 0,
0)$.

The force  $\vec{F}_p(\vec{r},t)$ either acts on each oscillator of the system
if the oscillator is occupied with a molecule of gas or does not act if it is
free, i.e., the oscillator interacts with the polarized medium only for
discrete time intervals. Instead, we consider an approximation where the
oscillator--medium interaction is continuous in time and the oscillator is
permanently bound with molecule with the time-dependent probability equal to
the surface coverage $\theta$. In this case, $F_p(x,t) = F_p(x)\, \theta$. This
approximation is analogous to the mean-field approximation used in problems of
adsorption with regard for lateral interactions between adsorbed particles
\cite{ref.Zhd}.

We represent the force  $F_p(x)$ in the form $F_p(x) = -
\frac{dU_{int}(x)}{dx}$, where $U_{int}(x)$ is the potential energy of
interaction of the polarized medium with the complex. Expanding the quantity
$U_{int}(x)$ in the Taylor series and keeping only the linear term, we obtain

\begin{equation}
 U_{int}(x) \approx -\chi \, x,
\end{equation}

\noindent where the parameter of complex--polarized medium interaction $\left.
\chi = - \frac{dU_{int}(x)}{dx} \right|_{x = 0}$ is the force acting on the
complex by the medium polarized by this complex.

Ignoring the internal motion in the bound molecule--center system, i.e.,
considering the motion of the complex as a whole, and taking into account a
change in the mass of the oscillator in the process of adsorption--desorption
within the framework of this approximation, we obtain the following equation of
motion for the oscillator:

\begin{equation}
 \frac{d}{dt} \biggl( m_{eff}(\theta) \frac{dx}{dt} \biggr) + \alpha
  \frac{dx}{dt} + \kappa x = \chi \, \theta,
\end{equation}

\noindent where $ m_{eff}(\theta) = m_0 + m \, \theta$ is the effective mass of
the complex that varies in the process of adsorption--desorption and  $m$ is
the mass of an adsorbate molecule. Since $\theta \leq 1$, the effective mass of
the complex does not exceed its total mass $ M = m_0 + m \equiv m_{eff}(1)$.

It follows from Eq.~(7) that bonding of an adsorbate molecule with center leads
to a shift of the equilibrium position of the oscillator by  $(\chi/ \kappa)\,
\theta$ and to a change in the potential energy of the free oscillator equal to
$\kappa x^2/2$ by  $U_{int}(x) \, \theta$. In the limiting case where all
centers are bound, i.e,  $\theta = 1$, the equilibrium position $x_{max} =
\chi/ \kappa$ of the bound oscillator is maximally distant from the surface and
the potential energy of the oscillator $U(x)$ at this equilibrium position is
minimal and equal to a half of the energy of interaction of the bound
oscillator with the polarized medium, $U_{min} \equiv U(x_{max}) = U_{int}/2$,
$\ U_{int} \equiv U_{int}(x_{max})= - \chi^2/\kappa$.

Thus, due to the interaction of adsorbate molecules with adsorption centers,
centers shift relative to the nonperturbed surface of the adsorbent, i.e., this
interaction leads to the formation (for adsorption) and healing  (for
desorption) of local defects of the surface. For $U_{int}(x) <0$, these defects
are ``pits'' (for $\chi > 0$) or ``hills'' (for  $\chi < 0$) whose depth and
height depend on the properties of both the adsorbate and the adsorbent. In the
special case where all atoms of the surface are adsorption centers, this
interaction leads to a shift of the surface of the adsorbent either inwards
(for $\chi > 0$) or outwards (for $\chi < 0$), i.e., to the relaxation of the
surface \cite{ref.Nau}. In other words, the processes of adsorption and
desorption result in a deformation of the surface of the adsorbent, which
leads, in the general case, to changes in the adsorption and desorption rates
and, as a consequence, the surface coverage. Within the framework of the
Langmuir theory of kinetics on the nondeformable surface ($\chi = 0$), the
adsorption and desorption rate constants $k_a$ and $k_d$ do not depend on the
concentration of gas and are defined by the Arrhenius relations

\begin{equation}
 k_a = k_+ \exp{\biggl(-\frac{E_a}{k_B T} \biggr)}, \qquad \qquad
 k_d = k_- \exp{\biggl(-\frac{E_d}{k_B T} \biggr)},
\end{equation}

\noindent where $E_a$ and $E_d$ are the activation energies for adsorption and
desorption, respectively, $k_+$ and  $k_-$ are the preexponential factors, $T$
is the absolute temperature, and $k_B$ is the Boltzmann constant.

A molecule bound with center, due to its interaction with polarized medium, is
in a deeper potential well than in the case of the nondeformable surface.
Therefore, for its desorption, the molecule requires an energy greater than
$E_d$ by the value  $\chi \, x$, where $\chi \, x = |U_{int}(x)|$ is an
additional energy that the bound molecule must acquire to break the bond with
polarized medium.

Generally speaking, the polarization of the medium can also affect the number
of free molecules of gas that can overcome the adsorption barrier $E_a$, i.e.,
a peculiar activation of free molecules of gas occurs and varies in the process
of adsorption--desorption. Here, we do not take into account a change in the
activation energy for adsorption (some results obtained with regard for a
decrease in the activation energy for adsorption in the process of
adsorption--desorption are presented in Appendix A). Supposing that the
preexponential factor $k_-$ is not changed, we obtain the following expression
for  $k_d$:

\begin{equation}
 k_d(x) = k_d \exp{\biggl(-\frac{\chi \, x}{k_B T} \biggr)}.
\end{equation}

\noindent It is worth noting that this quantity already depends on the
concentration of gas because it is defined by the current state of the surface
(the quantity $x$) that depends on the concentration of gas. Therefore,
adsorption and desorption of molecules proceed on the surface whose adsorption
characteristics vary with time.

Introducing the dimensionless coordinate of oscillator  $\xi = x/x_{max}$, we
obtain the following autonomous system of nonlinear differential equations,
which describes the kinetics of the quantity of adsorbed substance with regard
for variations in adsorption properties of the surface in the process of
adsorption--desorption:

\begin{eqnarray}
 \frac{d \theta}{dt} = k_a C \bigl( 1 - \theta \bigr)
  - k_d \, \theta \exp{\left(-g\,\xi \right)}, \\
 \frac{d}{dt} \biggl( m_{eff}(\theta) \frac{d\xi}{dt} \biggr) + \alpha \frac{d\xi}{dt}  =
  \kappa \, \bigl( \theta - \xi \bigr), \
\end{eqnarray}

\noindent where the dimensionless parameter  $g = |U_{int}|/k_B T$, which
characterizes the interaction of an adsorbed molecule with polarized medium,
can be called a coupling parameter. In the absence of interaction (the linear
case) where  $\chi = 0$ ($U_{int} = 0$), the parameter  $g = 0$.

Note that system (10)--(11), in many respects, is analogous to the system of
equations given in \cite{ref.GaKh}, which describes a transport of electrons in
a system of molecules of biological nature with regard for
electron-conformation interaction.

\section{Stationary Case}  \label{Stationary}

In the stationary case, it follows from Eq.~(11) that  $\xi =\theta$.
Therefore, the equilibrium state of system (10)--(11) is defined not by the
pair of quantities  ($\theta^{st}, \xi^{st}$), as is typical of dynamical
systems of two equations \cite{ref.AVKh,ref.BaL}, but only by one quantity
$\theta^{st}$, which is a solution of the equation

\begin{equation}
 l = F(\theta), \qquad \qquad \mbox{where} \qquad\qquad
 F(\theta) = \frac{\theta}{1 - \theta} \, \exp{\Bigl( - g\, \theta \Bigr)}.
\end{equation}

In the general case, it is hardly possible to solve the transcendental equation
(12) in the explicit form. Nevertheless, based on this equation, we can make a
qualitative conclusion on the influence of a change in adsorption properties of
the surface on the surface coverage. To this end, note that the ratio
$\theta/(1 - \theta)$ is equal to  $N_b/N_0$ and the quantity  $l$ is also the
ratio $N_b/N_0$ but in the linear case. Rewriting relation (12) in the form

\begin{equation}
 \frac{N_b}{N_0} = l \, \exp{\Bigl(g\, \theta \Bigr)},
\end{equation}

\noindent we see that a change in properties of the surface caused by
adsorption and desorption leads to an increase in the surface coverage for any
concentration of gas.  The difference between the numbers of bound centers in
the nonlinear ($g \neq 0$) and linear cases increases with the coupling
parameter $g$.

This conclusion can also be made by taking into account that, in the stationary
case, the desorption rate characteristic (9) has the form

\begin{equation}
 k_d(\theta) = k_d \, \exp \bigl(-g\, \theta \bigr),
\end{equation}

\noindent  where the surface coverage $\theta$ is a solution of Eq.~(12). For a
system whose adsorption properties vary in the process of
adsorption--desorption, the adsorption--desorption equilibrium constant

\begin{equation}
 K(\theta) = \frac{k_a}{k_d(\theta)} = K \, \exp \bigl( g\, \theta \bigr)
\end{equation}

\noindent  is greater than the classical adsorption--desorption equilibrium
constant $K $. Therefore, the equilibrium of the system shifts towards an
increase in the number of adsorbed molecules.

It follows from (14) that the interaction of adsorbed molecules with polarized
medium results in an increase in the typical lifetime of complex

\begin{equation}
 \tau_d(\theta) = \frac{1}{k_d(\theta)} = \tau_d \, \exp \bigl(g\, \theta \bigr)
\end{equation}

\noindent and, hence, an increase in the surface coverage.

To analyze solutions of Eq.~(12), we use a standard procedure \cite{ref.BaL},
namely: we consider the plane  $(\theta, l)$ and take into account that  $l
\geq 0$ and $0 \leq \theta \leq 1$ (Fig.~1). The required solutions of Eq.~(12)
are the abscissas of the points of intersection of a horizontal line
corresponding to the given concentration  $l$ (the left-hand side of Eq.~(12))
with the curve $F(\theta)$ (the right-hand side of Eq.~(12)), which is shown in
the figure for different values of the parameter  $g$. Since the behavior of
the function $F(\theta)$ is essentially different for  $g < g_c$ and $g > g_c$,
where $g_c = 4$, it is convenient to represent the parameter $g$ in the form $g
= a_g\, g_c$, where $a_g \geq 0$. For $g < g_c$ (Fig.~1a), the function
$F(\theta)$ monotonically increases and lies to the right of curve 1 for the
linear case ($g = 0$). Thus, as in the linear case, for any given
concentration, the surface coverage has the unique value $\theta^{st}_1 >
\theta^{st}_L$, which agrees with conclusion made above on the basis of
relations (13) and (15). With increase in $a_g$, the curve $F(\theta)$ becomes
more deformed and its deviation from curve 1 increases, which leads to an
increase in the difference $\theta^{st}_1 - \theta^{st}_L$ between the values
of the surface coverage in nonlinear and linear cases. For $a_g = 1$, the
function $F(\theta)$ (curve 1 in Fig.~1b) has an inflection point for  $\theta
= \theta_c = 1/2$ for the concentration $l = l_c = \exp{(-2)} \approx 0.135$.

\begin{figure}
 \centering{
 \epsfig{figure=pfig_1a.eps, width=12cm, height=9.5cm} \\
  \vspace{0.8cm}
 \epsfig{figure=pfig_1b.eps, width=12cm, height=9.5cm} \\
  \bigskip
 \caption {Graphical solution of Eq.~(12) for different values of the parameter $a_g$:
  (a) $a_g$ = 0 (1), 0.1 (2), 0.5 (3), 0.9 (4);
  (b) $a_g$ = 1 (1), 1.5 (2), 2 (3), 3 (4).
  Horizontal dashed straight lines correspond to constant values of the dimensionless
  concentration $l$. }
 }
 \label{fig._1}
\end{figure}

For $a_g > 1$, the behavior of the function $F(\theta)$  essentially changes:
for the concentrations $l^b_1$ and $l^b_2$ ($l^b_1< l^b_2 < l_c$) depending on
the value of the parameter $a_g$, the function $F(\theta)$ has a minimum and a
maximum at the points $\theta = \theta^b_1 > \theta_c$ and $\theta = \theta^b_2
< \theta_c$, respectively, which are roots of the quadratic equation

\begin{equation}
  \theta^2 - \theta + \frac{1}{g} = 0
\end{equation}

\noindent and are equal to

$$
 \theta^b_1 = \frac{1}{2} \Biggl( 1 + \sqrt{1 - \frac{4}{g}}\,\, \Biggr),
  \qquad \qquad
 \theta^b_2 = \frac{1}{2} \Biggl( 1 - \sqrt{1 - \frac{4}{g}}\,\, \Biggr).
$$
\noindent The concentrations  $l^b_1$ and $l^b_2$ corresponding to these
surface coverages are defined as follows:

$$
 \l^b_n = \bigl( g \, \theta^b_n - 1 \bigr)\, \exp{(-g \, \theta^b_n)},
  \qquad \qquad  n = 1,2.
$$

In Fig.~1b, the concentrations  $l^b_1$ and $l^b_2$ and the surface coverages
$\theta^b_1$ and $\theta^b_2$ for them are shown with the use of dashed
straight lines for the special case $a_g = 3/2$.  Thus, for concentrations
$l^b_1 < l < l^b_2$, Eq.~(12) has three solutions $\theta^{st}_1 <
\theta^{st}_2 < \theta^{st}_3$, furthermore, only the first solution
$\theta^{st}_1$ lies near the linear $\theta^{st}_L$. With increase in  $a_g$,
the concentrations $l^b_1$ and $l^b_2$ decrease and the difference between
maximum and minimum solutions $\theta^{st}_3 - \theta^{st}_1$ increases.

Analysis of the system of equations (10)--(11) shows that its stationary
solutions $\theta^{st}_1$ and $\theta^{st}_3$ are asymptotically stable and the
solution $\theta^{st}_2$ is unstable.

If the concentration  $l$ tends to the end point of the interval $[l^b_1, \,
l^b_2]$ (to the value $l^b_1$ or $l^b_2$), then the stable  $\theta^{st}_3$ (or
$\theta^{st}_1$) and unstable  $\theta^{st}_2$ solutions approach each other
and, in the limit  $l = l^b_1$ (or $l = l^b_2$), coalesce into one solution
$\theta^b_1$ (or $\theta^b_2$) (in Fig.~1b, for $a_g = 3/2$, these cases are
shown for curve 2.) Therefore, $l = l^b_1$ and $l = l^b_2$ are the bifurcation
concentrations for which the dynamical system (10)--(11) is structurally
unstable \cite{ref.AVKh,ref.BaL} and has the compound (double) equilibrium
states $\theta^b_1$ and $\theta^b_2$. These special cases should be
investigated in their own rights.

\begin{figure}
 \centering{
 \epsfig{figure=pfig_2.eps, width=12cm, height=9.5cm} \\
  \bigskip
 \caption {Bifurcation curve: branches 1 and 2 of the curve correspond to
  the bifurcation concentrations  $l^b_1$  and  $l^b_2$, respectively. }
 }
 \label{fig._2}
\end{figure}

Using relations (12) and (17), we plot a bifurcation curve in the plane of
parameters ($a_g, \, l$). This curve defined in the parametric form

\begin{equation}
 a_g  =  \frac{1}{4 \, \theta \, \left( 1 - \theta \right) } \, , \qquad\qquad
 l = \frac{\theta}{ 1 - \theta} \, \exp \biggl(- \frac{1}{1 - \theta} \biggr )
\end{equation}

\noindent is shown in Fig.~2.  For any point of this plane lying between the
branches of the bifurcation curve, the system of equations (10)--(11) has three
structurally stable equilibrium states: two states are stable and one is
unstable. If a point lies outside these branches, then the system has one
structurally stable equilibrium state. At any point of the bifurcation curve,
except for the cusp ($a_g = 1$, $l = l_c$), the system has two equilibrium
states: one is structurally stable and another is double \cite{ref.BaL}. At the
cusp, the system of equations (10)--(11) has one triple equilibrium state
\cite{ref.BaL}.

The $S$-shaped adsorption isotherm depicted in Fig.~3 for $a_g > 1$ (curve 1)
essentially differs from the Langmuir isotherm (curve 2) and, on the
qualitative level, reproduces the Zeldovich hysteresis predicted in
\cite{ref.Zeld}.

With quasistatic increase in the concentration from zero, the surface coverage,
at the initial section of the lower stable branch  $0A$ of the isotherm,
coincides with Langmuir one. For these concentrations, a released adsorption
center manages to relax to the nonperturbed state before it binds with other
molecule, furthermore, $\tau_d(\theta) \approx \tau_d$. With increase in the
concentration up to the bifurcation value $l^b_2$, the difference between the
typical lifetimes of complex  $\tau_d(\theta)$ and $\tau_d$ increases. In this
case, a free adsorption center can bind with a subsequent molecule before it
relaxes to the nonperturbed state. In this section of the lower stable branch
of the isotherm, occupation of the surface by adsorbate molecules is determined
by two factors: an increase in the concentration of gas and a change in
adsorption properties of the surface. Due to the last factor, the isotherm
deviates from the Langmuir isotherm, and this deviation increases with
concentration. The pattern cardinally changes as soon as the concentration
negligibly exceeds $l^b_2$. In this case, the lower stable branch of the
isotherm disappears and a new (unique) equilibrium state of the complex is
considerably more distant from the surface than the previous one for $l \le
l^b_2$. Furthermore, the passage to this state is performed for a constant
concentration, i.e., solely due to a change in adsorption properties of the
surface of adsorbent (according to the terminology used in \cite{ref.Zeld}, a
slow adsorption occurs). This passage can require many molecules that
successively take part in the process of adsorption--desorption on the same
adsorption center. Thus, in this stage, a certain interaction between the
molecule leaving the adsorption center and the molecule binding with it occurs.
In Fig.~3, this stage of a sharp increase in the surface coverage for a
constant concentration is shown by the dashed straight line $AB$.

In passing to a stable equilibrium state lying on the upper stable branch of
the isotherm (the point $B$ in Fig.~3), the majority of adsorption centers is
bound. As a result, a subsequent increase in the concentration of gas slightly
affects an increase in the surface coverage. Such a ``saturation'' of the
surface with adsorbate, which rapidly increases with parameter  $a_g$
(Fig.~3b), occurs for concentrations considerably less than those in the linear
case.

\begin{figure}
 \centering{
 \epsfig{figure=pfig_3a.eps, width=12cm, height=9.5cm} \\
  \vspace{1cm}
 \epsfig{figure=pfig_3b.eps, width=12cm, height=9.5cm} \\
  \bigskip
 \caption {Adsorption isotherms for  $a_g$ = 1.5 (a), 2 (b);
  curve 2 stands for the Langmuir isotherm. }
 }
 \label{fig._3}
\end{figure}

In passing through the bifurcation concentration  $l^b_2$, the conditions of
desorption for adsorbed molecules become essentially worse due to a
considerable increase in the depth of the potential well and the displacement
of the equilibrium state of bound adsorption centers from the surface. As a
result, for returning the system to the lower branch of the isotherm, the
concentration should be considerably less than  $l^b_2$. With quasistatic
decrease in the concentration, the surface coverage decreases slightly and
only, in approaching the bifurcation value $l^b_1$, a variation in $\theta$
becomes noticeable. In passing through the bifurcation concentration $l^b_1$,
the upper stable branch of the isotherm disappears and an equilibrium state of
the complex lies considerably closer to the surface than the previous state for
$l \ge l^b_1$. As a result, the surface coverage sharply decreases for the
fixed concentration due to a change (restoration) in properties of the
adsorbent surface. The transition of the system from the upper branch of the
isotherm to its lower branch is shown by the dashed straight line $CD$ in
Fig.~3. Note that this stage of drop of the quantity  $\theta$ is absent in
\cite{ref.Zeld}. A subsequent decrease in the concentration is accompanied by a
decrease in the surface coverage along the lower branch of the isotherm, which,
in fact, coincides with Langmuir isotherm.

This behavior of the adsorption isotherm corresponds to the principle of
perfect delay \cite{ref.Gil} according to which a system, which is in a stable
state at the initial time, with variation in a parameter (concentration in the
case at hand), remains in this state until the state exists.

As the parameter  $a_g$  increases, the bifurcation concentration $l^b_1$
rapidly vanishes (see Fig.~2). Using the results of calculation, we can say
that, for $a_g \geq 3$, a change in adsorption properties of the adsorbent in
the process of adsorption--desorption leads to a peculiar adaptation of the
system to a state in which the majority of adsorption centers are bound up to
very low concentrations.

Note that the isotherms obtained above for the surface whose adsorption
properties vary in the course of adsorption--desorption (Fig.~3) are similar to
the isotherms obtained with regard for lateral interactions between adsorbed
molecules on a nondeformable surface \cite{ref.Adam,ref.Zhd,ref.Tov} and to the
Hill--de~Boer isotherms derived on the basis of the Hill--de~Boer equation of
state for adsorbed molecules (a two-dimensional analog of the Van der Waals
equation) \cite{ref.JaP}.

In analysis of adsorption isotherms with regard for lateral interactions
between adsorbed molecules (see, e.g., \cite{ref.BrMed}), for investigation of
possible surface phase transitions, a critical temperature $T_c$ is introduced
\cite{ref.Zhd,ref.LNP,ref.Nau,ref.JaP}. For the model considered in the present
paper, using the expression for the coupling parameter $g$, the critical value
$g_c$, and the analysis of adsorption isotherms performed above, the critical
temperature is defined as follows: $k_B T_c = |U_{int}|/4$. For a system of
adsorbed molecules, one stable state occurs for  $T > T_c$, whereas, for  $T <
T_c$, two stable states are possible. The corresponding phase diagram for the
adsorbed layer in the ``surface coverage--critical temperature'' coordinates is
determined by the relation

$$
 \frac{T}{T_c}  =  4\, \theta \, \left( 1 - \theta \right) \,
$$

\noindent and, as in the case taking into account lateral interactions between
adsorbed molecules on a nondeformable surface within the framework of the
mean-field approximation  \cite{ref.Zhd}, is symmetric about $\theta = 1/2$.

\section{Nonstationary Case}  \label{Nonstationary}

\subsection{Overdamped Approximation}  \label{Massless}

First, we investigate the kinetics of system (10)--(11) within the framework of
overdamped approximation where the masses of adsorption center and molecule are
low and the friction coefficient is so large that the first term on the
left-hand side of Eq.~(11) can be neglected as compared with the second term.
Using the well-known results for a linear free oscillator of constant mass
\cite{ref.AVKh}, this approximation is correct if

\begin{equation}
 \tau^2_M  \ll \tau^2_r,
\end{equation}

\noindent where  $\tau_M = 1/\omega_M$, $\omega_M = \sqrt{\kappa/M}$ is the
oscillation frequency of an oscillator of mass $M$, and $\tau_r =
\alpha/\kappa$ is the typical relaxation time of a massless oscillator. Since
$M$ is the maximally possible effective mass, condition (19) is even somewhat
high. In this approximation, the system of equations (10)--(11) is simplified
to the form

\begin{eqnarray}
 \frac{d \theta}{dt'} = l \, \bigl( 1 - \theta \bigr)
  - \theta \exp{\left(-g\,\xi \right)}, \\
 \frac{d\xi}{dt'}  = \frac{\theta - \xi}{\beta}, \qquad \qquad \qquad \qquad
  \end{eqnarray}

\noindent where $t' = t/\tau_d$ is the dimensionless time and  $\beta =
\tau_r/\tau_d\,$.

Analysis of system (20)--(21) performed on the basis of the qualitative theory
of differential equations \cite{ref.BaL,ref.ArP} shows that the stable
equilibrium states of the system $\theta^{st}_1$ and $\theta^{st}_3$ are stable
nodes and its unstable equilibrium state $\theta^{st}_2$ is a saddle. For the
bifurcation concentration $l = l^b_1$ (or $l = l^b_2$), the system is
structurally unstable and has a compound equilibrium state $\theta^b_1$ (or
$\theta^b_2$), namely, a saddle-node with two saddle sectors and one stable
nodal sector. The system is also structurally unstable for the critical
concentration $l = l_c$ and $a_g = 1$. In this case, the system has one
equilibrium state $\theta_c$, which is stable triple node.

The numerical analysis of the system of equations with initial conditions for
$t' = 0$

\begin{equation}
 \theta(0) = 0   , \qquad\qquad  \xi(0) = 0
\end{equation}

\noindent shows that, for any values of the parameters  $a_g$, $l$, and
$\beta$, the system monotonically evolutes to the nearest stable equilibrium
state. Therefore, for the bistable system ($a_g > 1$ and $l^b_1 \leq l \leq
l^b_2$), the stable equilibrium state $\theta^{st}_3$ is inaccessible. The time
taken for attaining the equilibrium state  $\theta^{st}_1$ considerably depends
on parameters, first of all, on the concentration.

Let us investigate the kinetics of the surface coverage for a system that can
be bistable for  $a_g = 1.5$. In this case,  $l^b_1 \approx 0.0329$ and $l^b_2
\approx 0.0754$; $\theta^b_1 \approx 0.789$ and $\theta^b_2 \approx 0.211$.

In Fig.~4, the kinetics of the surface coverage $\theta(t')$ is shown for
concentrations less ($l = 0.05$, Fig.~4a) and greater ($l = 0.1$, Fig.~4b) than
the bifurcation concentration $l^b_2$.  For comparison, the Langmuir kinetics
is shown in this figure by curve 2. For $l < l^b_2$, the behavior of
$\theta(t')$ is analogous to that in the Langmuir case: the quantity
$\theta(t')$ monotonically increases from zero to the nearest stationary value
$\theta^{st}_1$ that lies near the linear value  $\theta^{st}_L$ (Fig.~4a).
With increase in the concentration, this behavior remains true up to the
bifurcation value $l^b_2$ (moreover, both the stationary value $\theta^{st}_1$
and the time taken for its attaining increase).

\begin{figure}
 \centering{
 \epsfig{figure=pfig_4a.eps, width=12cm, height=9.5cm} \\
  \vspace{1cm}
 \epsfig{figure=pfig_4b.eps, width=12cm, height=9.5cm} \\
  \bigskip
 \caption {Kinetics of the surface coverage  $\theta(t')$ on deformable
  (curve 1) and nondeformable (the Langmuir kinetics, curve 2) surfaces for
  $a_g = 1.5$, $\beta = 1$; $l = 0.05$ (a), $0.1$ (b). }
 }
 \label{fig._4}
\end{figure}

For $l > l^b_2$, the system has only one stable equilibrium state, furthermore,
in this state, the surface coverage, which is close to the maximum possible
value, is essentially greater than that in the linear case.  Moreover, both the
shape of the kinetic curve $\theta(t')$ and the time taken for attaining the
stationary value considerably differ from the Langmuir ones (Fig.~4b). With
increase in the concentration, the time taken for attaining the stationary
value decreases.

\begin{figure}
 \centering{
 \epsfig{figure=pfig_5a.eps, width=12cm, height=9.5cm} \\
  \vspace{1cm}
 \epsfig{figure=pfig_5b.eps, width=12cm, height=9.5cm} \\
  \bigskip
 \caption {Kinetics of the surface coverage  $\theta(t')$ for concentrations
  close to the bifurcation concentration $l^b_2$:
  $a_g = 1.5$, $\beta = 1$;
  $\delta = 0$ (1), $10^{-4}$ (2), $10^{-3}$ (3), $10^{-2}$ (4), $10^{-1}$ (5). }
 }
 \label{fig._5}
\end{figure}

The value of the parameter  $\beta$ affects only the time taken for the system
to attain the stationary value but does not qualitatively change the kinetics
of   $\theta(t')$ both for  $l < l^b_2$ and for $l > l^b_2$. This time
decreases if $\beta$ decreases and increases if $\beta$ increases, which is
quite natural because a variation in $\beta$ is equivalent to a variation in
the friction coefficient $\alpha$.

For concentrations  $l > l^b_2$ near the bifurcation concentration $l^b_2$, the
behavior of  $\theta(t')$ qualitatively changes. The behavior of the quantity
$\theta(t')$  for $l = l^b_2 \, \left(1 + \delta \right)$, for low values of
the relative concentration  $\delta = \left(l - l^b_2\right)/l^b_2 \geq 0$, is
shown in Fig.~5. If the concentration slightly exceeds the bifurcation
concentration $l^b_2$ (curves 2 and 3), then the evolution of  $\theta(t')$ can
be conditionally divided into three stages: (i) from the initial zero value to
a value of  $\sim\theta^b_2$ corresponding to the bifurcation concentration
$l^b_2$; (ii) a very slow (in the limit $\lim\limits_{\delta \rightarrow 0}\,$,
infinitely slow) variation in the neighborhood of $\theta^b_2$; (iii) from
$\sim\theta^b_2$ to the stationary value $\theta^{st}_1$. For low values of
$\delta$, the time taken for attaining the stationary level $\theta^{st}_1$ is
determined mainly by the second (``quasistationary'') stage in which the
system, in fact, does not change (curve 2 in Fig.~5b). This behavior of the
surface coverage $\theta(t')$ is caused by the well-known effect of slowing
down of a system near a singular point for the bifurcation value of a parameter
\cite{ref.Gil,ref.Hak,ref.Sug,ref.Byk} in the case where a phase trajectory of
the system moves near this point.

This behavior of system (20)--(21) can be clearly explained on the basis of
analysis of its phase trajectories in the phase plane  $(\theta, \, \xi)$. The
phase trajectories of the system with zero initial condition (22) are shown for
concentrations less (Fig.~6a), equal (Fig.~6b), and slightly greater (Fig.~6c)
than the bifurcation concentration  $l^b_2$. The dashed lines in these figures
stand for the main isoclines of the system: the isocline of horizontal slopes
$\xi = \theta$ and the isocline of vertical slopes $\xi = (1/g)\, \ln(\theta/l
\, (1 - \theta))$. The points of intersection of these isoclines are singular
points of the system.

\begin{figure}
 \centering{
 \epsfig{figure=pfig_6a.eps, width=7.5cm, height=7.5cm} \hspace{0.5cm}
 \epsfig{figure=pfig_6b.eps, width=7.5cm, height=7.5cm} \\
  \vspace{0.6cm}
 \epsfig{figure=pfig_6c.eps, width=7.5cm, height=7.5cm} \\
  \bigskip
 \caption {Phase trajectories of system (20)--(22) for concentrations less ($l = 0.065$) (a),
  equal($l = l^b_2$) (b), and slightly greater
  ($l = l^b_2 \,\left(1 + \delta \right)$,  $\delta = 0.1$) (c)
  than the bifurcation concentration $l^b_2$;
  $a_g = 1.5$, $\beta = 1$.
  Dashed lines stand for the main isoclines of the system.}
 }
 \label{fig._6}
\end{figure}

For $l < l^b_2$ (Fig.~6a), the singular points  $A$ and  $C$ are stable (stable
nodes) and the singular point  $B$ is unstable (saddle). The phase trajectory
in Fig.~6a starting from the origin of coordinates and going to the nearest
singular point $A$ completely lies between the main isoclines. Moreover, the
immediate analysis of system (20)--(21) shows that all phase trajectories of
system (20)--(21) with initial values belonging to the domain bounded by the
sections of the main isoclines before their intersection at the point  $A$ also
completely lie between the main isoclines. A change in the parameter  $\beta$
does not qualitatively change the behavior of the phase trajectories and only
shifts them to one of the main isoclines: for  $\beta << 1$ and $\beta >> 1$,
the phase trajectories are closely pressed to the isoclines of horizontal and
vertical slopes, respectively.

For $l = l^b_2$, the singular points  $A$  and $B$ coalesce into one (compound)
singular point  $D \equiv D(\theta^b_2, \, \xi^b_2)$, which is a point of
tangency of the main isoclines (Fig.~6b). In this case, the phase trajectory is
analogous to that in the previous case. For $l > l^b_2$, the system has only
one (stable) singular point  $C$ (Fig.~6c). If the relative concentration is
low, then a gap between the main isoclines in the neighborhood of their point
of tangency  $D$ for  $l = l^b_2$ is also small. Since a phase trajectory does
not leave the domain bounded by the main isoclines, it goes through the gap
and, in a neighborhood of the point  $D$,  its motion becomes slower.
Furthermore, the less the relative concentration, the narrower the gap between
the main isoclines and the closer the phase trajectory approaches the point $D$
and, hence, the more its slowing down near the point. This behavior of the
system corresponds to the effect of critical slowing down near a degenerate
critical point \cite{ref.Gil,ref.Hak,ref.Sug}.

As a result, for low values of  $\delta$, the function  $\theta(t')$ in Fig.~5b
has the form of a double step (curves 2 and 3). The first plateau of the step
corresponds to the quasistationary state  $\theta^b_2$ and the second
corresponds to the stable state $\theta^{st}_1$.

For $\beta << 1$, the kinetics of the surface coverage, which is shown in
Figs.~4 and 5 for the intermediate case  $\beta = 1$, can be analyzed in a
standard way with the use of a potential  \cite{ref.Gil,ref.Hak}. It follows
from Eq.~(21) that $\xi =\theta$. Substituting this relation into (20), we
obtain the following equation for  $\theta$:

\begin{equation}
 \frac{d \theta}{dt'} = - \frac{dV(\theta; l, g)}{d \theta},
\end{equation}

\noindent where the potential $V(\theta; l, g)$ can be represented as the sum
\begin{equation}
 V(\theta; l, g) = V_L(\theta; l) + V_{ind}(\theta; g),
\end{equation}

\noindent  where

\begin{equation}
 V_L(\theta; l) = \frac{l + 1}{2}\, \theta \left(\theta - 2 \theta_L^{st}\right)
\end{equation}

\noindent is the parabolic potential for the Langmuir kinetics (1) and

\begin{equation}
 V_{ind}(\theta; g) = \frac{1}{g^2}\, \Bigl\{1 - (1 + g\,\theta)\,
  \exp{\left(-g\, \theta \right)} \Bigr\} - \frac {\theta^2}{2}\, .
\end{equation}

\noindent is the potential caused by the action of the polarized medium on the
complex.

Thus, the behavior of the quantity   $\theta$ is completely defined by the form
of the potential  $V(\theta; l, g)$ as a function of  $\theta$. Analysis of the
potential  $V(\theta; l, g)$ shows that its form essentially depends on values
of the parameters  $l$ and $g$. For $g < g_c$, the function $V(\theta; l, g)$,
like  $V_L(\theta; l)$ in the linear case, has one minimum for a certain
$\theta_1^{st}$. With increase in  $l$, the value $\theta_1^{st}$ increases and
the minimum of  $V(\theta_1^{st}; l, g)$ decreases. The behavior of the
potential  $V(\theta; l, g)$ essentially changes for  $g > g_c$, $l^b_1 < l <
l^b_2$. In this case, the function $V(\theta; l, g)$ has the form of a double
well with local minima at  $\theta = \theta_1^{st}$ and $\theta =
\theta_3^{st}$ separated by a maximum at $\theta = \theta_2^{st}$. As the
concentration $l$ varies from $l^b_1$ to $l^b_2$, the positions of the extrema,
the depths of the wells, and the barrier between them $\delta V_{2,1}(l, g) =
V(\theta_2^{st}; l, g) - V(\theta_1^{st}; l, g)$ vary. For concentrations near
$l^b_1$, the second well with minimum at  $\theta = \theta_3^{st}$ is rather
flat, essentially shallower as compared with the first well with minimum at
$\theta = \theta_1^{st}$, $V(\theta_3^{st}; l, g) > V(\theta_1^{st}, l, g)$,
and corresponds to a possible metastable state of the system. As the
concentration increases, the second well becomes deeper and the barrier between
the wells decreases. At a certain concentration, the depth of the second well
becomes equal to the depth of the first one. For higher concentrations, the
second well is deeper than the first, $V(\theta_3^{st}; l, g) <
V(\theta_1^{st}; l, g)$, i.e., the system is in a metastable state in the first
well and in a stable state in the second well. With a further increase in the
concentration, the modulus of the difference between the minima of the wells
$|V(\theta_3^{st}; l, g) - V(\theta_1^{st}; l, g)|$ increases and the barrier
$\delta V_{2,1}(l, g)$ decreases (the slope of the first well between
$\theta_1^{st}$ and  $\theta_2^{st}$ is close to zero) and disappears for $ l =
l^b_2$. As soon as the concentration becomes greater than the bifurcation value
$l^b_2$, the first minimum disappears, furthermore, for concentrations near
$l^b_2$ ($l
> l^b_2$), the potential $V(\theta; l, g)$, in a neighborhood of  $\theta_2^b$,
has almost a zero slope, which leads to the well-known critical slowing down of
the system \cite{ref.Gil,ref.Hak,ref.Sug}. The behavior of this gradient
dynamical system corresponds to the well-known principle of perfect delay
\cite{ref.Gil} according to which a transition of the bistable system between
two stable states of equilibrium is absent.

\subsection{Influence of the Masses of Center and Molecule on the Kinetics
of the Surface Coverage }  \label{Mass}

Now we investigate the kinetics of the surface coverage taking into account the
masses of adsorption center and molecule. The estimating condition (19) for the
overdamped approximation can be represented in the form $Q^2_M \ll 1$, where
$Q_M = \omega_M/2\, \gamma_M = \tau_M/\tau_r$ and $\gamma_M = \alpha/2M$ are,
respectively, the $Q$-factor and the damping constant of a free oscillator of
mass $M$. We also introduce the dimensionless quantities $\mu = m/m_0$  and $d
= (\tau_0/\tau_d)^2$, where $\tau_0 = 1/\omega_0$ and $\omega_0 =
\sqrt{\kappa/m_0}$ is the oscillation frequency of a free oscillator of mass
$m_0$. The quantity $Q_M$ can be represented in the form $Q_M = Q_0 \, \sqrt{1
+ \mu}$, where $Q_0 = \omega_0/2\, \gamma_0 = \tau_0/\tau_r = \sqrt{d}/\beta$
and $\gamma_0 = \alpha/2m_0$  are, respectively, the $Q$-factor and the damping
constant of a free oscillator of mass $m_0$.

Analysis of the singular points  $\theta^{st}_1$, $\theta^{st}_2$, and
$\theta^{st}_3$ of the system of equations (10)--(11) carried out in the phase
space gives the following:  For very low values of the $Q$-factor $Q_M$,
$\theta^{st}_1$, $\theta^{st}_3$ and $\theta^{st}_2$ are, as in the overdamped
case, stable nodes and a saddle, respectively \cite{ref.Kak}. With increase in
the $Q$-factor $Q_M$, starting from certain values that depend on values of the
parameters $a_g, \l, \beta$, and $\mu$, $\theta^{st}_1$ and $\theta^{st}_3$
become stable node-focuses. In the interval $(\theta^b_2, \, \theta^b_1)$, near
$\theta^b_1$, there appears a domain such that if $\theta^{st}_2$ falls within
this domain, then the singular point $\theta^{st}_2$ is a saddle-focus,
otherwise, it is a saddle. With increase in the $Q$-factor, this domain rapidly
grows and covers the entire interval $(\theta^b_2, \, \theta^b_1)$ so that the
unstable singular point $\theta^{st}_2$ is always a saddle-focus.

Below, we give the results of numerical analysis obtained for  $a_g = 1.1$,
i.e., for a system with possible bistability, and  $\beta = 1$.

\begin{figure}
 \centering{
 \epsfig{figure=pfig_7a.eps, width=12cm, height=9.5cm} \\
 \vspace{1cm}
 \epsfig{figure=pfig_7b.eps, width=12cm, height=9.5cm} \\
 \bigskip
 \caption {Influence of the masses of adsorption center and molecule on the kinetics of
  the surface coverage  $\theta(t')$  for the concentration $l = 0.11$ lying in
  the middle of the bistability interval;
  $a_g = 1.1$, $\beta = 1$; $m_0 = 0, m = 0$ (curve 1),
  $\mu = 1$; (a) $d = $ 1 (2), 5 (3), 10 (4); (b) $d = $ 50 (2), 100 (3). }
 }
 \label{fig._7}
\end{figure}

The curves shown in Fig.~7 illustrate the influence of the masses of adsorption
center and molecule on the kinetics of the surface coverage  $\theta(t')$ for
the concentration  $l = 0.11$  lying in the middle of the bistability interval
$[l^b_1, \, l^b_2]$, where $l^b_1 \approx 0.1064$ and $l^b_2 \approx 0.1154$.
Here, for comparison, we present the quantity $\theta(t')$ without regard for
the masses of adsorption center and molecule (curve 1) and the stationary level
$\theta^{st}_1$ for this concentration (dashed line). For low values of masses
(curve 2 in Fig.~7a), as in the overdamped case, the number of molecules
adsorbed on the surface monotonically increases with time and reaches its
maximum value $\theta^{st}_1$. With increase in the coefficient $d$, which is
proportional to the mass of adsorption center, the behavior of the quantity
$\theta(t')$ changes. The surface coverage reaches its stationary value
$\theta^{st}_1$ only after several oscillations about it (Fig.~7b). The
amplitude and the number of oscillations as well as the time taken for
attaining the stationary value $\theta^{st}_1$ increase with $d$ (curves 2 and
3 in Fig.~7b). This behavior is caused by the inertia of the oscillator that
overshoots its equilibrium position and the deviation of the oscillator from
the equilibrium position increases with its mass. Therefore, taking account of
the masses of adsorption center and molecule changes only the character of
attainment of the nearest equilibrium state by the system.

\begin{figure}
 \centering{
 \epsfig{figure=pfig_8a.eps, width=7.5cm, height=6.5cm}  \hspace{0.5cm}
 \epsfig{figure=pfig_8b.eps, width=7.5cm, height=6.5cm} \\
  \vspace{0.6cm}
 \epsfig{figure=pfig_8c.eps, width=7.5cm, height=6.5cm} \\
  \bigskip
 \caption {Influence of the masses of adsorption center and molecule on the kinetics of
  the surface coverage $\theta(t')$  for the concentration $l = 0.115$ lying near
  the end point of the bistability interval;
  $a_g = 1.1$, $\beta = 1$; $m_0 = 0, m = 0$  (curve 1), $\mu = 1$;
  (a) $d = $ 10 (2), 20 (3), 30 (4);
  (b) $d = $ 33 (2), 50 (3), 100 (4);
  (c) $d = $ 4900 (2), 5200 (3), 5300 (4). }
 }
 \label{fig._8}
\end{figure}

The behavior of the system can essentially change if the concentration is near
the end point of the bistability interval. This case is shown in Fig.~8 where
the stationary states of the system (the stable states $\theta^{st}_1$ and
$\theta^{st}_3$ and the unstable state  $\theta^{st}_2$) are shown by dashed
lines. The curves  $\theta(t')$ in Fig.~8a show that, for low values of $d$,
the behavior of the system, to a large extent, is analogous to its behavior for
low masses considered above (Fig.~7a). Due to inertia, the system penetrates
into the domain $(\theta^{st}_1, \, \theta^{st}_2)$, which is a domain of
attraction of the attractor $\theta^{st}_1$ (domain I) \cite{ref.BaL}. As in
the overdamped case, with time, the system attains its stationary level
$\theta^{st}_1$. However, as soon as the mass of the complex reaches a value
for which the kinetic energy of the complex is sufficient to overcome the
``barrier'' $\theta^{st}_2$, the behavior of the system qualitatively changes
(cf. curves 2 and 3 in Fig.~8b). Having fallen into the domain $(\theta^{st}_2,
\, \theta^{st}_3)$, which is a domain of attraction of the attractor
$\theta^{st}_3$ (domain II), the system moves toward its second stable
stationary state  $\theta^{st}_3$. Having reached this state, the system
oscillates about it with decreasing amplitude. As $d$  increases, the time of
attainment of the stationary state  $\theta^{st}_3$  decreases because the
system, in fact, is not delayed in the neighborhood of the unstable state
$\theta^{st}_2$ (curve 4 in Fig.~8b). Thus, unlike the overdamped case for
which the stable equilibrium state $\theta^{st}_3$ is not attainable, due to
the masses of adsorption center and molecule, the system can be in this state
rather than in the state $\theta^{st}_1$.

A further increase in  $d$ is accompanied by an increase in the amplitude of
oscillations of the system about the stationary level $\theta^{st}_3$ and the
number of oscillations (high-$Q$ system). However, the system does not leave
domain II (curve 2 in Fig.~8c). If the amplitude of oscillations exceeds
$(\theta^{st}_3 - \theta^{st}_2)$, then the system falls into domain I and its
subsequent behavior can be different. The system can remain in this domain and,
after a time, it attains the stable stationary level  $\theta^{st}_1$ (curve 3
in Fig.~8c) as in the overdamped case (curve 1) and in the case of low values
of masses (curves 2--4 in Fig.~8a and curve 2 in Fig.~8b).

\begin{figure}
 \centering{
 \epsfig{figure=pfig_9.eps, width=12cm, height=9.5cm} \\
 \bigskip
 \caption {Influence of the ratio of the masses of molecule and adsorption center on
  the kinetics of the surface coverage  $\theta(t')$;  $a_g = 1.1$, $\beta = 1$, $l = 0.115$;
  $m_0 = 0, m = 0$  (curve 1); $d = $ 5300, $\mu = $ 0.5 (2), 1 (3). }
 }
 \label{fig._9}
\end{figure}

\begin{figure}
 \centering{
 \epsfig{figure=pfig_10a.eps, width=12cm, height=9.5cm} \\
 \vspace{1cm}
 \epsfig{figure=pfig_10b.eps, width=12cm, height=9.5cm} \\
 \bigskip
 \caption {Influence of the masses of adsorption center and molecule on the kinetics of the
  surface coverage  $\theta(t')$
  for the concentration close to the bifurcation concentration $l^b_2$;
  $a_g = 1.5$, $\beta = 1$, $\delta = 10^{-4}$; $m_0 = 0, m = 0$ (curve 1);
  $\mu = 1$, $d = $ 1 (2), 10 (3), 20 (4), 30 (5). }
 }
 \label{fig._10}
\end{figure}

For a somewhat greater value of  $d$, the kinetic energy of the complex can be
sufficient for the system to overcome the ``barrier'' $\theta^{st}_2$ for the
second time. As a result, the system again penetrates into domain II and
attains the stationary level $\theta^{st}_3$ (curve 4 in Fig.~8c). It is clear
that, with increase in mass (the value of $d$), the system can again return
into domain I, etc. Therefore, the finial stable state of the bistable system
($\theta^{st}_1$ or $\theta^{st}_3$) depends on the number of intersections of
the unstable state $\theta^{st}_2$ by the system in the process of its
evolution with time. The value of the parameter $\beta$, which is proportional
to the friction coefficient, also affects overcoming of the ``barrier''
$\theta^{st}_2$ by the system. Decreasing this parameter, i.e., increasing the
$Q$-factor of the system, it is possible, in principle, to realize a mode in
which the system visits each of two stable stationary states many times. This
behavior of the system under study qualitatively agrees with the well-known
behavior of a Newton gradient system whose potential energy has two minima
separated by a barrier in the case of a low value of the damping parameter
\cite{ref.Gil}.

Since the $Q$-factor of the system is determined both by the mass of adsorption
center and by the mass of adsorbed molecule, these two characteristics affect
(but different in rights) the possibility of the system to overcome the
``barrier'' $\theta^{st}_2$. The curves  $\theta(t')$ depicted in Fig.~9, which
describe the kinetics of the surface coverage for systems with equal masses of
adsorption centers but different masses of adsorbed molecules, visually
illustrate this conclusion. The behaviors of systems that returned from domain
II into domain I are different. The kinetic energy of the complex with more
light-weight molecule (curve 2) is insufficient for the complex to overcome the
``barrier'' $\theta^{st}_2$ for the second time, and the system is stabilized
at the stable level $\theta^{st}_1$. At the same time, the complex with heavier
molecule is able to overcome this ``barrier'' and the system returns into
domain II and attains the stable level $\theta^{st}_3$ (curve 3).

Note that the behavior of the kinetic curves shown in Figs.~7--9, on the
qualitative level, agrees with conclusion on the kinetics of the surface
coverage made in Appendix~B with the use of the effective potential in the
special case where the relaxation time of the quantity   $\theta(t)$ and the
characteristic times of the quantity $\xi(t)$ are essentially different.

For concentrations $l = l^b_2 \, \left(1 + \delta \right)$ that slightly exceed
the bifurcation value $l^b_2$, taking account of the masses of adsorption
center and molecule can also qualitatively change the kinetics of the surface
coverage (Fig.~10). For equal masses of adsorption center and molecule ($\mu =
1$) and for values  $d \leq 1$ (i.e., for  $Q_0 \leq 1$), the influence of
masses on the behavior of $\theta(t')$ is insignificant (curves 1 and 2, in
fact, coincide). With increase in the mass, the plot of the function
$\theta(t')$ in the form of a double step typical of the overdamped case
remains true (curve 3 in Fig.~10b). However, the residence time of the system
in the ``quasistationary'' state somewhat decreases, i.e., the delay of the
system in a neighborhood of the singular point  $\theta^b_2$ corresponding to
the bifurcation concentration $l^b_2$ is less than in the overdamped case. As
the mass  increases, the residence time of the system in the
``quasistationary'' state rapidly decreases (curve 4 in Fig.~10b) and, starting
from certain values of  $d$,  the system moves to the stable stationary state
$\theta^{st}_1$ without delay near the ``quasistationary '' state $\theta^b_2$
(curve 5). Thus, the possibility of an intermediate ``quasistationary'' state
for the system essentially depends on relations between the inertial and
dissipative characteristics of the system.

\section{Conclusions }  \label{Conclusions}

In the present paper, we have proposed the self-consistent model taking into
account variations in adsorption properties of the adsorbent surface in the
process of adsorption--desorption of molecules of gas on it. Within the
framework of this model, we have introduced the dimensionless coupling
parameter that characterizes the interaction of an adsorbed molecule with
polarized medium. We have established that the system can be bistable if this
parameter is greater than critical and the concentration of molecules in the
gas phase belongs to a certain interval. We have investigated bifurcation
concentrations for which stable states of the system appear and disappear. We
have obtained adsorption isotherms that essentially differ from the classical
Langmuir isotherms. It is established the possibility of the Zeldovich
hysteresis within the framework of the proposed model. It is shown that
variations in adsorption properties of the surface in the course of
adsorption--desorption can lead to a peculiar adaptation of the system to the
state in which the majority of adsorption centers is bound up to very low
concentrations.

The detailed analysis of the kinetics of the surface coverage established that
taking account of variations in adsorption properties of the surface in the
course of adsorption--desorption leads to new phenomena: a ``quasistationary''
state for the overdamped approximation and damped self-oscillations of the
system in the general case.

\begin{acknowledgments}
The author expresses the deep gratitude to Prof. Yu.\,B.~Gaididei for the
statement of the problem, valuable remarks, and useful discussions of results.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Jack loves The Thompson House so much he wrote a song about us. We love you too Jack. We hope to see more of your great work for years to come.
I love The Thompson House it is the best.
I love The Thompson House.
You love The Thompson House.
If you want to have your project featuring The Thompson House posted on our website just send it to us and let us know. If you have a project, but don't want it posted on the website please send it too us anyway. We'd love to see it.
WHO'S READY FOR NICE HOT SUMMER DAYS AT THE THOMPSON HOUSE!!
Sure – we love the snow and the winter – but really!!
hope this letter finds you well and anxiously waiting for your "yellow" envelope to arrive meaning SPRING is here! It has been an interesting winter to say the least. Like everyone in the northeast we are all waiting for the warm weather to get here to thaw out our bones and get ready for a great summer.
Our families are all doing well – from our youngest Maddox to our oldest, John, we are all enjoying the great amount of snow that has fallen this season. John comes over every morning to help shovel off the skating rink and Maddox has already hit the slopes with his dad, Kurt. Even Dasha has been seen shushing off the top of Windham Mt. with some great instruction from Evan. The Pines hill has also brought many smiles as we've introduced Maddox to a great sleigh riding place. It seems like just yesterday it was Kurt, Brent and Evan sledding at the Pines. Boy how time flies!
Last fall John and Mickey (Eric's parents) were given an award from the Community of Windham Foundation in recognition of their "altruistic dedication and benevolent service toward the betterment of the Windham community". All of their children and most of their grandchildren were at the ceremony to witness this great honor. This is the 3rd year the award was given in honor of former town Supervisor, Pat Meehan, and Brent of course questioned who could possibly have done more than his grandparents!
For our outdoor enthusiasts we will again be offering discounted golf to our local golf courses on the mountain top. There is also some great news about the addition of a bridge for the Windham Path. Now our guests' can access the path right at the Windham Pharmacy right on Rt. 296. You will no longer have to walk along the road to see the beautiful nature of Windham and the surrounding Catskill Mountains. Also, the NY Dept. of Conservation has announced the expansion of the biking and hiking trails along the Elm Ridge/Escarpment Trail system. Now we will have almost 20 miles of mountain biking and hiking trails just a few miles from our resort.
Our 2015 summer season will begin on May 22, Memorial Day weekend and continue through to October 12, Columbus Day weekend. We are anticipating the weather will be warmer than it has been the past few weeks and hope that you can join us for some fun in the mountains.
Please note that we will not be answering the phone from March 28th to April 12th, but you are always welcome to send us an e-mail at TheThompsonHouse@gmail.com.


Please be advised that the information found on this website does not constitute legal advice, and is posted for informational purposes only. Every city official should consult with local legal counsel to ensure that any information complies with current law and the particular facts of each situation.
Texas Municipal Court Education Center (TMCEC): TMCEC is the premiere source for information about all things municipal court.


text:
Сергей Степанович Жилин (20 сентября 1926, Илетский сельхозрайон, Мари-Турекский кантон, Марийская автономная область — 17 декабря 1997, Кузнецово, Марий Эл) — советский хозяйственный, государственный и политический деятель, Герой Социалистического Труда (1965). Кавалер ордена Ленина (1965).

Биография
Родился в 1926 году в деревне Петухи. Член КПСС с 1947 года.

В Великую Отечественную войну после окончания Моркинского педагогического училища Сергей работал учителем в Сабанурской школе Параньгинского района Марийской АО. Призван 20 августа 1942 года по мобилизации в Красную Армию.

Участник Великой Отечественной войны. В составе 284-го отдельного батальона правительственной связи 1-й гвардейской танковой армии участвовал в боях на Курской дуге, на территории Польши и Германии, был ранен.

После демобилизации в 1946 году вернулся на родину. В 1946—1997 годах — первый секретарь Параньгинского райкома ВЛКСМ, второй секретарь Параньгинского райкома ВКП(б), директор Кузнецовской средней школы, председатель колхозы «Заветы Ильича», директор совхоза «Семёновский» Медведевского района, первый секретарь Медведевского райкома КПСС, директор Марийской государственной сельхозопытной станции.

В 1953 году С. С. Жилин заочно окончил Марийский пединститут им. Н. К. Крупской, работал директором Кузнецовской средней школы, затем руководил колхозом «Заветы Ильича».

В 1960 году он был назначен директором первого в Марийской АССР вновь созданного из семи хозяйств совхоза «Семёновский» в Медведевском районе.

Под его руководством многоотраслевой совхоз стал одним из крупнейших передовых хозяйств республики. Здесь была внедрена научная система земледелия, животноводства и организации труда, отрабатывались интенсивные технологии производства. Самые высокие урожаи, рекордные надои, привесы на откорме скота. Труженики хозяйства стали получать высокие урожаи и хорошие надои молока на комплексе.

В 1965 году совхоз «Семёновский» на площади 5828 гектаров в среднем получил зерновых по 17,3 центнера с гектара.

Указом Президиума Верховного Совета СССР от 26 ноября 1965 года за особые заслуги в развитии народного хозяйства Марийской АССР Жилину Сергею Степановичу присвоено звание Героя Социалистического Труда с вручением ордена Ленина и золотой медали «Серп и Молот».

Избирался депутатом Верховного Совета СССР 7-го созыва.

Доцент Марийского государственного университета, заведующий отделом экономики МарНИИ сельского хозяйства (1991—1996). Кандидат экономических наук (1972).

Умер 17 декабря 1997 года в с. Кузнецово Медведевского района Марий Эл. Похоронен на Туруновском кладбище в Йошкар-Оле.

Звания и награды

 Золотая медаль «Серп и Молот» (26.11.1965)
 Орден Ленина (26.11.1965)
 Орден Октябрьской революции (08.04.1971)
 Орден Трудового Красного Знамени (11.12.1973)
 Орден Отечественной войны I степени (11.03.1985)
 Орден «Знак Почёта» (1948)

 Медаль «За боевые заслуги»  (12.06.1945)
 Медаль «В ознаменование 100-летия со дня рождения Владимира Ильича Ленина» (6 апреля 1970)

 Медаль «За победу над Германией в Великой Отечественной войне 1941—1945 гг.» (9 мая 1945)
 Медаль «За освобождение Варшавы»  (9.06.1945)
 Медаль «За взятие Берлина»  (9.06.1945)
 Юбилейная медаль «Двадцать лет Победы в Великой Отечественной войне 1941—1945 гг.» (7 мая 1965)
 Юбилейная медаль «Тридцать лет Победы в Великой Отечественной войне 1941—1945 гг.»
 Медаль «Сорок лет Победы в Великой Отечественной войне 1941-1945 гг.»
 Медаль Жукова (1994)
 Юбилейная медаль «50 лет Победы в Великой Отечественной войне 1941—1945 гг.»
 Медаль «Ветеран труда»
 Медаль «30 лет Советской Армии и Флота»
 Юбилейная медаль «40 лет Вооружённых Сил СССР»
 Юбилейная медаль «50 лет Вооружённых Сил СССР» (26 декабря 1967)
 Юбилейная медаль «60 лет Вооружённых Сил СССР» (28 января 1978)
 Юбилейная медаль «70 лет Вооружённых Сил СССР» (28 января 1988)
 Знак «Гвардия»
 Знак МО СССР «25 лет Победы в Великой Отечественной войне» (24 апреля 1970)
 Золотая и серебряная медали ВДНХ (1966)
 Почётная грамота Президиума Верховного Совета Марийской АССР (1976, 1986)

Память
 Именем С. С. Жилина названа улица в районном центре Медведево Республики Марий Эл.
На могиле установлен надгробный памятник.
 Его имя увековечено в Главном Храме Вооружённых сил России, и навсегда запечатлено на мемориале «Дорога памяти»

Примечания

Литература

Ссылки
 Жилин Сергей Степанович
 Государственное Собрание РМЭ. Листая страницы истории. Летопись марийского парламента

Председатели колхозов
Первые секретари районных комитетов КПСС
Депутаты Верховного Совета СССР 7-го созыва

redpajama_set_name: RedPajamaWikipedia


text:
Hjem Publikasjoner Migrant workers occupational health resea[...]
Migrant workers occupational health research: an OMEGA-NET working group position paper
Temaovergripende
Objective: The aims of the study were: (1) to clarify the definitions of "migrant" used in occupational health research; (2) to summarize migrant workers' industry sectors, occupations and employment conditions; (3) to identify the occupational health and safety services available to migrant workers; (4) to summarize work-related health problems found among migrant workers; (5) to identify the methodological challenges to research into occupational health of migrant workers; and (6) to recommend improvements in migrant occupational health research. Methods: This position paper was prepared by researchers from several European countries and Australia, working within the EU COST Action OMEGA-NET. The paper drew on two recent systematic reviews on the occupational health of international migrant workers and other literature, and also identified uncertainties and gaps in the research literature. Migrants may, for example, be temporary or permanent, moving for specific jobs migrants or other reasons. Their ethnicity and language capabilities will affect their work opportunities. Results: The occupational health literature seldom adequately identifies the heterogeneity or characteristics of the migrant group being studied. Migrants tend to work in more physically and mentally demanding environments with higher exposures than native workers. Migrants tend to have an increased risk of physical and mental ill...
Aktas, Emine; Bergbom, Barbara; Godderis, Lode; Kreshpaj, Bertina; Marinov, Mario; Mates, Dana; McElvenny, Damien M.; Mehlum, Ingrid Sivesind; Milenkova, Valentina; Nena, Evangelia; Glass, Deborah C.
International Archives of Occupational and Environmental Health

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Excel cell shall contain required characters I would appreciate it if you could help to solve following issue:
When external User fulfil the defined cell in Excel, he/she is allowed to put predefined values within the one cell. Like I want to populate "Country" field and options available for me are: "Moscow" or "Moscow;New-York".
So that acceptable symbols are Latin letters in any case and Semicolon.
If the result is false (for some reasons I used comma after Country name), then the Error message shall be popped up, like "This symbol is not acceptable".
Thank you for your time and help in advance!

A: Seems to me that OP just wants to prevent the input of certain characters.
These might be detected e.g. by a negative Like comparison via
    Dim SearchString As String
    SearchString = "Moscow;New-York"
    If SearchString Like "*[! ;A-Za-z-]*" Then
        MsgBox "Found not accepted symbol."
        '... do other stuff
    Else
        MsgBox "Correct entry."
    Endif

Hints to pattern search via "*[! ;A-Za-z-]*"
The Like operator has only some rudimentary ressemblance with pattern searches in Regular Expressions, but can lead to helpful efficient code, too.

Especially "a group of one or more characters ( charlist ) enclosed in brackets ([ ]) can be used to match any single character in string and can include almost any character code, including digits."

An exclamation point (!) at the beginning of such a charlist means that
a match is made if any character except the characters in charlist is found in string.

The trick
Therefore it suffices to find just one single character other than the wanted character sequences (i.e. a blank  , a semicolon ;, letters within A-Z or a-z or a hyphen -) by a negative search pattern prefixed by ! to find out any intruder (the surrounding wild cards * allow to identify the intruder at any position).
Note that I positioned the hyphen - as last element in the pattern search, as within the char list the hyphen is used to identify a range of characters. - (Contrary to MS Help positioning - immediately after ! doesn't seem to work)
C.f. Ms Help Like operator
The above code splitter doesn't pretend in any way to be a complete solution, but intends to give a starting idea. It's up to the asker to develop further steps in code of his own.

A: You could use the INSTR function to check the cell after editing for specific characters you don't want.
for example:
If INSTR(1,cellreference,",")<>"0" then 'a comma was found in the cell
    msgbox "This symbol is not acceptable"
End if

redpajama_set_name: RedPajamaStackExchange


text:
Q: Multiple ble advertisements in a UWP app I am trying to publish multiple ble advertisements in a UWP app in c#. I can add 2 manufactor data to one publisher and those will be transmitted. However if I want to publish more the data is not possible. Is this even possible?
Sample code so far:
var publisher = new BluetoothLEAdvertisementPublisher();
publisher.Advertisement.ManufacturerData.Add(CreateData("Test");
publisher.Advertisement.ManufacturerData.Add(CreateData("AnotherTest");
publisher.start();

BluetoothLEManufacturerData CreateData(string data)
{
    var dataWriter = new DataWriter();
    dataWriter.WriteInt32(data.Length);
    dataWriter.WriteString(data);

    return new BluetoothLEManufacturerData(0xFFFE, dataWriter.DetachBuffer());
}

Tried to add another manufactor data, but I get an exception
Tried multiple bluetooth adapters but windows don't seem to start the second adapter.
Also tried multiple Windows.Devices.Bluetooth.Advertisement.BluetoothLEAdvertisementPublisher but that also does not do the trick

A: We can put multiple manufacturer data in one BluetoothLE Advertisement. However, please note that the max length of an advertisement payload is 31 bytes.
Ref ADVERTISING AND SCAN RESPONSE DATA FORMAT (BLUETOOTH SPECIFICATION Version 4.0 [Vol 3] Page 375 )

The format of Advertising data and Scan Response data is shown in Figure
  11.1. The data consists of a significant part and a non-significant part. The
  significant part contains a sequence of AD structures. Each AD structure shall
  have a Length field of one octet, which contains the Length value, and a Data
  field of Length octets. The first octet of the Data field contains the AD type field. The content of the remaining Length - 1 octet in the Data field depends on the value of the AD type field and is called the AD data. The non-significant part extends the Advertising and Scan Response data when necessary and shall
  contain all-zero octets.

BluetoothLEManufacturerData is one of these AD structures which AD Type is 0xFF.

For your first manufacturer data CreateData("Test"), its length is 1 (Lenght) + 1 (Type) + 2 (Company Id) + 8 (Your specific data), which is 12 bytes. And for the second manufacturer data, its length is 19 (1+1+2+15) bytes. With these two manufacturer data, the advertisement payload has already been 31 bytes. That's why you get an exception, when you try to add another manufacturer data. Actually, if you add one character in your date, for example using CreateData("Test=") instead of CreateData("Test"), you will also get the Invalid advertisement payload detected error.
So please make sure that the buffer length can fit within an advertisement payload. You can put multiple manufacturer data as long as its length is less than 31 bytes.
var publisher = new BluetoothLEAdvertisementPublisher();
publisher.Advertisement.ManufacturerData.Add(CreateData("T"));
publisher.Advertisement.ManufacturerData.Add(CreateData("A"));
publisher.Advertisement.ManufacturerData.Add(CreateData("S"));
publisher.start();


text:
Dr. John F. Burns '56
Dr. John F. Burns '56 died December 23, 2009, in Augusta.
Dr. John F. Burns '56 died December 23, 2009, in Augusta. He was born in Gardiner on July 2, 1934, and graduated from Brunswick High School. He attended Bowdoin from 1952 to 1955, and graduated from the New York University School of Dentistry in 1959. He practiced general dentistry in Brunswick for four years before returning to New York for internships and residencies in oral surgery at New York University – Bellevue Medical Center. He then served as a clinical instructor of oral surgery at New York University School of Dentistry and worked in private practice Staten Island. He also was chief of dentistry and director of oral surgery at Beekman Downtown Hospital in Manhattan, and chief of oral and maxillofacial surgery and on the board of directors of World Health Administrative Services, Inc., which provided medical services to the World Trade Center. He returned to Maine in 1978 to practice oral and maxillofacial surgery in Augusta, retiring in 1996. In 1984, he was instrumental in establishing the Impaired Dentists Program for the Maine Dental Association, and continued to serve on the committee after his retirement. He is survived by his wife of 19 years, Sara Loomer, and their son Ian Ross; his first wife, Eileen Marie Donahue, whom he married in 1956, and their sons Dr. Sean Burns and Michael Burns and daughter Jane Burns; a sister, Liz Alden; six grandchildren, including Michelle Burns '12; and his stepfather Wilfred Lajoie.
← Thomas R. P. Gibb Jr. '36
Dr. Robert B. Barlow Jr. '61 →


text:
Hudson Bennett 1850 Property Tax Map 2
1850 Tax Map showing the boundaries for land owned by Hudson Bennett (2 of 2)
Henry J. O'Bannon 1860 Census Slave Schedules
Image taken from the 1860 Slave Schedules for Loudoun Co., Va. Shows that Henry J. O'Bannon owned one female slave, aged 12.
Road Map of Area Surrounding Hicks Cemetery
An airline map of streets in Arlington and Alexandria, including the street layout of where the Hicks cemetery would have been located. The cemetery would have been to the north of 22nd Street North, to the west of the intersection with Lee Highway.
Frying Pan Meeting House
A picture of the Frying Pan Meeting House adjacent to the cemetery.
Charlotte Hunt Robey
Headstone belonging to Charlotte Hunt Robey. (May 16, 1843- October 8, 1919)
Located at Frying Pan Baptist Church, Herndon, Va.
Lewis H. Robey
Headstone belonging to Lewis H. Robey (Dec 29, 1837 - January 2, 1911)
Susan Kitchen Hunt Headstone
Headstone belonging to Susan Kitchen Hunt ( July 31 1806- February 4, 1892).
Elbridge G. Hutchison Headstone
Headstone belonging to Elbridge G. Hutchison (1853-1935)
Inscription reads:
"He who dies believing dies safely through thy love."
Martha G. O'bannon Headstone
Headstone belonging to Martha G. O'bannon (June 6, 1839- Sept 12, 1902) wife of Henry J. O'bannon ( Aug 5 1831-Nov 12, 1920).
Located in Frying Pan Baptist Church, Herndon, VA.
Henry J O'bannon Headstone
Headstone belonging to Henry J. O'bannon (Aug 5 1831-Nov 12 1920) . Husband of Martha E. O'bannon (June 6 1839-Sept 12 1902)
Recipient of the Southern Cross of Honor.


Stanley Tee, Legal activities, listed under "Legal Activities" category, is located at Star House Braintree Essex, CM7 2QP, United Kingdom and can be reached by (01376) 552277 phone number. Stanley Tee has currently 0 reviews.
Browse all Legal Activities in Braintree Essex.


text:
Kazakhstan|economy|October 9, 2019 / 04:31 PM
Air Astana presents its first Airbus A321LR aircraft at ceremony in Nur-Sultan
AKIPRESS.COM - Air Astana of Kazakhstan presented its first Airbus A321 Long Range (LR) 'Super Arrow' aircraft at a ceremony held in the airline's technical center at Nursultan Nazarbayev International Airport today, the Aviation24.be reported.
The airline is the first operator of the aircraft in the CIS and is scheduled to take a further six Airbus A321LR aircraft on lease from AerCap of the United States. The new Super Arrows will gradually replace the fleet of Boeing 757s, which have been in operation since 2003.
Air Astana now has 19 Airbus aircraft in its fleet, including eight A320, four A321, three A320neo, three A321neo and one A321LR. The average age of Air Astana's fleet is now 7.1 years, one of the lowest in the industry.
"The arrival of the A321LR 'Super Arrow' aircraft represents another major step in the process of fleet modernization, with the new aircraft offering the latest technologies, low emissions & noise, operational efficiency and passenger comforts," said Peter Foster, President and CEO of Air Astana. "The specification of the Business Class flatbeds, Economy Class slim seats and other cabin amenities will ensure that passengers experience the very highest standards of comfort and service on routes to Western Europe, Southeast Asia, Moscow, Dubai, Istanbul and Beijing."
The lie-flat business class seats are equipped with individual 16-inch screens, with four of 16 seats offering additional personal space. In the economy class, the Recaro seats provide increased comfort for long flights and are equipped with individual 10-inch screens. The design of the interior trim, curtains and seat covers for both business and economy classes have been specially developed for Air Astana, with a combination of elegance and sophistication that embraces a clear national identity.
The A321LR cabin is equipped with a Zodiac (RAVE) in-flight entertainment system, which is in common with all other Airbus and Boeing aircraft in the fleet.
The aircraft is equipped with the latest generation Pratt & Whitney engines, which reduce fuel consumption by 20%, maintenance costs by 5%, carbon emissions by 20% and noise levels by 50% compared with the current generation of aircraft.
During the presentation of the new aircraft, Air Astana announced a donation of US$20,000 to the National Board of Paralympics, with funds to be allocated for the development and promotion of Paralympic sports in Kazakhstan and for the training of Kazakhstan Paralympic athletes participating in international sports events.


Fresh flowers and gifts delivered to Lower Hotham, Western Australia.
Local florist delivers fresh flowers, flower baskets, chocolates, fruit baskets and gifts to Lower Hotham.
Lower Hotham Florist deliver fresh flowers, flower baskets and gifts to Lower Hotham area.
Have your flowers and gifts delivered to Lower Hotham area churches, hospitals, retirement and nursing homes, Lower Hotham businesses, offices, private homes, anywhere in Australia by our local Lower Hotham florist.


Organic baked beans in a hearty tomato sauce!
Probably the healthiest thing ever to hit bread. Optimum nutrition. Low fat. Bursting with antioxidants. This is a plate-load of gymnastic superheroes. And they are ready for you!


Condolence: A wonderful wife, mother, grandmother and friend rest in peace Gerry!
Condolence: My deepest and sincere condolences to you Bill, Brian and Carol.. I will miss seeing her and her smile at our family functions. Rest in Peace Geri!! I'll miss you!
Condolence: My sincere condolences Bill, Brian and Carol. I will truly miss Gerry's beautiful smile and kind heart.


Masks is a wonderful game, capturing the essence and joy of running a drama filled, teenage superhero game. A staple of super hero stories is time travel, which is also one of the most complicated stories you can tackle.
In my Monday Masks game, we've had a time travel thread going for the last few weeks. Friction, the Janus kinetic controller, has been replaced by her future self, although none of her teammates know. This week, we finally follow the story of the present-day friction exploring the future.
When we switched future and present Friction, her player came up with a few facts about the future on her own.
1. The death of her younger brother, Jack, was the perceived turning point in the timeline for when it became a bleak future.
2. The Calvary, the international group of superheroes, split up into individual cells. Some are good, some are bad.
3. Firefly, the Outsider alien princess, was usurped by a sibling and is in hiding on Earth.
4. Jishaku, the Reformed electromagnetic controller, is in control of a Calvary cell, and it is unknown whether she is good or bad.
5. Friction is the leader of a vigilante group, fighting against both the good and bad.
6. Finally, the bleakest fact: Nexus, the Transformed nanite lifeform, has been cut into pieces to be incorporated into other people. Integrating Nexus body parts grants incredible power, so organizations hunt down body parts and steal them from each other.
Overall, the future Halcyon city has a very Sin City feel, but with a higher level of technology.
A little bit of this, a little bit of Batman Beyond.
The facts we have are very invocative, but leave a lot open. For our session, we decided to dive into this further, and did it completely improvised.
The players of Firefly, Jishaku, and Nexus were allowed complete freedom to decide facts and features of the future, and push the story as they liked. My job was to introduce Friction to the future, and set her on the path to meeting her teammates.
I introduced a few features to the future: Hoverbikes, futuristic police blimps, unregistered metahumans, and the Manhunters.
Probably not these Manhunters. Probably.
Friction met with her vigilante group, and went to meet Jason Bellamy, the best friend of Nexus.
Nexus' player was controlling Jason, as Nexus was unable to be played, being in many pieces and all. Jason fleshed out some information on what happened to Nexus, and put them on the path to meeting Jishaku.
The player of Jishaku fleshed out the role of the Calvary cells. They have a warlord feeling to them; they're officially a part of the government, but can do whatever they want. The group decides to try and find a way to send a message back in time, and change the future. They go to look for Firefly and reunite the group.
The player of Firefly decided that after being usurped by a sibling, she made herself completely undetectable. She cut her psychic powers off, hid all of her alien tech, and tried living as a completely normal, boring person.
The group restored Firefly's memories, and went about finding a way to either send a message back, or even better, send Friction back to her normal time. Eventually, they sent Friction back to her timeline, where she can save her friends from their current predicament.
Throughout the whole story, Friction acted as a beacon of hope in a dark, bleak future. The players of the other characters did an amazing job at portraying a dark future, and because they were the ones deciding facts about it, they added things that were personal that I would have missed. Friction found a group of people who had grown apart, and brought them back together again. She brought the future the one thing it needed most- hope.
Time travel stories are complicated, confusing, and full of paradoxes. For a tabletop game, you don't need to worry about keeping an internal consistency the entire time. You're creating the game as you go, so you can always add things later. Go crazy. Let your players come up with facts and features about the future. It's their characters, so it's a great opportunity to preview what their plans are for their character, or delve into the mindset of them.
Have you ever run a time traveling adventure or story? Let me know how it went!


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UK progress on Covid is now wasted, warns senior scientist | Coronavirus
Nancy Owens December 5, 2021	UK Economy Comments Off on UK progress on Covid is now wasted, warns senior scientist | Coronavirus 4 Views
The emergence of the Omicron variant shows that the world is "closer to the start of the pandemic than to the end," warned one of Britain's top scientific figures, lamenting a lack of political leadership on Covid.
Sir Jeremy Farrar, the director of the Wellcome Trust who stepped down as science adviser to the government last month, said progress in tackling Covid-19 since its emergence was "wasted".
Write in the Observer, he declared that the rich countries had taken "a very indiscriminate internal concentration, letting themselves go into thinking that the worst of the pandemic was behind us". He said that while he cautiously hoped that current vaccines would protect against serious illnesses from Omicron, that might not be true for future variants.
"The more this virus continues to spread in largely unvaccinated populations around the world, the more likely it is that a variant capable of overcoming our vaccines and treatments will emerge," he writes. "If that happens, we could be close to square one.
"This political drift and lack of leadership prolongs the pandemic for everyone, with governments unwilling to tackle inequitable access to vaccines, tests and treatments. There were wonderful speeches, warm words, but not the actions needed to ensure equitable access to what we know works and would end the pandemic. "
He said the urgent action needed had not changed – "wearing masks indoors, increasing testing, social distancing, isolation if positive (with support to do so). ) and vaccination will all help reduce transmission and protect against disease. "
Farrar's intervention came as a major charity raised concerns about the government's recall campaign, which it sees as the current best method of tackling the virus. Caroline Abrahams, director of charity at Age UK, said the program was "downright messy". She said: "It's nowhere near where it should be, and hearing this week that one in five nursing home residents haven't had their boosters yet was just alarming."
She spoke after the suspension of certain goals for general practitioners to allow them to focus on the administration of jabs. Routine check-ups for those over 75 and for new patients may be postponed under the new directive.
"In this context, we agree that the top priority now is to supercharge the booster program, to ensure that as many elderly and vulnerable people as possible get their injections, and quickly," Abrahams said. "In the end, this situation is yet another demonstration of our GP system being underpowered. Unless and until we invest more in it, we will continue to see these difficult choices arise every time a crisis strikes, and it is certainly not in the best interests of anyone, let alone that of the elderly.
Booster jabs administered at St John's Church in West London on December 4th 2021. Photograph: Daniel Leal / AFP / Getty Images
Health Secretary Sajid Javid called on all eligible people to receive a reminder before spending time with their families this Christmas. "It is absolutely crucial that all eligible people receive their booster shot to boost their immunity before spending time with their loved ones," he said. "As our brilliant scientists learn more about the new Omicron variant, we must do everything possible to strengthen our defenses, and vaccines are the best way to do that."
On Saturday, Saturday, Downing Street announced it was introducing pre-departure testing for people traveling to the UK. Starting at 4 a.m. on Tuesday, travelers will need to take a lateral flow test or PCR 48 hours before entering the country. Nigeria has been added to the red list, and from 4 a.m. on Monday, arrivals will need to be quarantined at hotels. Officials said the measures came as evidence emerged that the Omicron strain has been reported in more than 40 countries, with community transmission apparent in places like Norway. They also said there was evidence the incubation period was shorter than for the other variants.
Labor had pushed for the introduction of the tests. A source from Downing Street said new evidence on the spread of the variant and its characteristics motivated the decision. This will cause further frustration for the travel industry, which has been hit hard by the Covid restrictions.
The sudden spread of Omicron has dismayed many scientists. "What really surprised me is the rapidity of the increase in the number of reported cases in South Africa," said Professor François Balloux of University College London. "The cases are doubling at a rate that I have never seen before. It is very disturbing. "
Early estimates suggest that this increase is due to the fact that the Omicron variant can infect between three and six times as many people as Delta, during the same period. This is an alarming figure that was offset by early reports which suggested that the variant is, at present, less likely to lead to hospitalization.
However, scientists warned last week that one should be careful with such data. "It's already a tense situation. Even though Omicron is less likely to cause serious illness, it can still cause more hospitalizations and put more strain on health services, as many more people will be infected, "said Professor Rowland Kao, of the University of Edinburgh.
There is also evidence that more than one in four UK workers (27%) would show up at work with symptoms of the Omicron variant because they needed the money, including 23% of those whose work exposes them. at an increased risk of infection. Additionally, 16% of workers say they could work from home, but are not allowed to do so by their employer, according to a survey by the Royal Society for Arts, Manufacturers and Commerce (RSA). Only 31% say they would get support from their employer beyond statutory sick pay.
"To ensure economic recovery, workers must be economically secure," said Anthony Painter, RSA research director. 'In the short term we need to see a' self-leave 'system and more generous sick pay, but in the long term Tories and Labor need to explain how they are going to deal with the deep economic insecurity that Covid does 'did that expose. "
Saturday saw further protests in Europe against the measures of various governments against coronaviruses. In Austria, the first EU country to announce that it will make vaccination against Covid compulsory, more than 40,000 joined a demonstration in Vienna.
The Dutch city of Utrecht has seen several thousand people demonstrate against the restrictions introduced over the weekend. Protesters carried banners reading "Medical Freedom Now!" Many protesters denounced what they see as growing pressure from the Dutch government to get vaccinated if they want to participate in mainstream society.
In Frankfurt, Germany, police interrupted a protest of several hundred people for failing to wear masks or maintain social distancing, using batons and pepper spray after being attacked by a group of protesters. In Berlin, small groups gathered to protest after a large demonstration was banned.
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