text:
\section{Introduction}
Problems of combinatorial optimization pervade all walks of science~\cite{nemhauser,papadimitriou}. This is particularly true in modern data-driven era, as analyzing the data becomes much harder than obtaining the data. Combinatorial optimization problems revolve around finding the optimal value of some criteria function on a discrete set. That set is usually very large, typical example being the set of all combinations or variations of some elements in a given order. Classical combinatorial optimization problems include linear and quadratic assignments, bin packing problem, vehicle routing problem, graph partitioning, etc.~\cite{schrijver}.

The difficulty of these problems is hidden in the size of the discrete set on which optimization is to be done. Most of them are NP-hard, which means that there is no algorithm to solve them that works in polynomial time (unless P=NP). In practice, we can optimally solve such problems only for small or medium size instances, while larger instances can usually be solved only approximately and without any approximation guaranty. Combinatorial optimization community developed over the years an array of exact and approximate methods. The winning strategy is usually a combination of general methods with problem-specific ones, like the Hungarian method for linear assignment problem~\cite{schrijver}. Another common approach is to formulate a combinatorial optimization problem as a (non)linear optimization problem with integer constraints and apply Branch and Bound method. But again, these methods quickly fail for larger instances of the problem. The standard alternative is to use various heuristic algorithms, which often involves utilisation of some general meta-heuristic algorithm, like genetic algorithm, simulated annealing, ant-colonies, etc.~\cite{Baghel2012,korte,woeginger2003}.

Luckily, many optimization problems can be represented in several different ways~\cite{sanz2016}. While this does not make them easier to solve, it helps to  conceptualize them from a different angle and enables different methodology approach them. Excellent example are quantum algorithms, whose promise for solving diverse optimization problems is nowadays clear~\cite{farhi2001}. Quantum simulations that solve optimization problems typically involve finding the ground state of a suitably designed quantum system~\cite{albash2018,johnson2011}. This refers to a variety of optimization problems~\cite{somma2008,wang2018}, including graph coloring~\cite{gaitan2012,gaitan2014,mcmahon2016,kudo2018}.

On a different scientific front, closer to statistical physics of complex systems, large efforts went into scrutinizing collective dynamics of oscillatory units on complex networks (graphs). Most often this involves studying synchronization and mechanisms of its emergence in relation to the coupling between the oscillators and the underlying graph topology~\cite{arkady,arenas2008,porter,levnajic2010,luciano2011,jacimovic2018,anna2020}. Frequent paradigm are simple phase-oscillators, such as Kuramoto oscillators, where synchronization emerges via positive coupling~\cite{acebron2005,rodrigues2016,crnkic2017,zankoc2019,pietras2019}. The evolving dynamics makes phases of the coupled oscillators `attract' and eventually synchronize. Much less researched is the opposite paradigm, namely, when the oscillators are coupled negatively, i.e., in the phase-repulsive way~\cite{levnajic2011,levnajic2012,goldstein2015,astakhov2016}. Here, the dynamics pushes the phases of the coupled oscillators away from each other.

In this paper we propose a new way to represent combinatorial optimization problems focusing on the well-known graph coloring problem (vertex coloring). Specifically, we design a family of interaction functions that couple phase-oscillators on a graph and show that natural evolution of such system -- the search for the stationary equilibrium state -- is equivalent to the search for the solution of the vertex coloring problem for the underlying graph. So, to test whether a given graph is colorable with a given number of colors we run the dynamics of phase-oscillators on that graph, coupled via particular interaction function corresponding to the selected number of colors, and examine the equilibrium stationary state. We demonstrate both analytically and numerically that if the graph is indeed colorable with that many colors, oscillators tend to arrange their phases as colors in a solution of the corresponding graph coloring problem.

Our work is a step forward in recent efforts to use the collective dynamics in various types of oscillators for solving graph (vertex) coloring  problems~\cite{wahwu,wujiao,novikov,parihar}. This includes both theoretical~\cite{wujiao,novikov} and experimental results relying on electric circuits~\cite{wahwu,parihar}. In particular, researchers in \cite{wujiao} and \cite{novikov} observed that by tuning the coupling strength in classic Kuramoto model one finds interesting clustering of equilibrium phases, which can serve as a heuristic for graph coloring. In opposition, our dynamical model involves not only Kuramoto interaction functions (with only the first harmonic), but an entire family of interaction functions, each function with a specific number of harmonics that depends on how many colors are we testing the graph for. As we show in what follows, this enables a better theoretical justification behind our method.

\section{The graph coloring problem}
We begin be precisely formulating the (vertex) graph coloring problem as the combinatorial optimization problem. Let $G=(V,E)$ be a non-directed graph with vertex set $V$ and edge set $E$. A $K$-coloring of vertices $V$ is a function $\varphi \colon V \rightarrow \{ 1,2,\ldots,K \}$, which maps adjacent vertices into different numbers, i.e., $\varphi(u) \neq \varphi(v)$ for all $uv\in E$. Graph is called $K$-colorable if and only if there exists a $K$-coloring of it vertices. The minimal $K$, such that there exists a $K$-coloring is called \textit{chromatic number} of graph $G$ and is denoted by $\chi(G)$~\cite{chartrand}:
\begin{equation}
    \chi(G):=\min \{K \mid \exists \,K\mbox{-coloring of vertices of G}\}.
\end{equation}
Finding the chromatic number of given graph is referred to as \textit{graph coloring problem}, and is a classical problem of combinatorial optimization. There is a vast amount of theoretical results for this problem on different families of graphs and on the problem's extensions~\cite{jensen11}. Graph coloring serves as a theoretical model for several practical problems, including timetable scheduling \cite{ganguli17} or frequency assignment problem \cite{aardal07}.

The problem can be formulated in the context of optimization theory in several different ways~\cite{Gvozdenovic2008,Govorcin2013,Jabrayilov2018}. Here we present the most basic version, so-called assignment-based ILP model~\cite{Jabrayilov2018}:
\[
\begin{array}{rrl}
    \chi(G)~=&\min & \sum_{i=1}^{H}w_i \\
     &\mbox{s.t.}&\sum_{i=1}^Hx_{vi}~=~1,~~\forall v\in V\\
     && x_{ui}+x_{vi}\le w_i,~~\forall uv\in E,~i=1,\ldots,H\\
     && x_{vi},w_i\in\{0,1\},~~\forall v\in V,~i=1,\ldots,H
\end{array}
\]
In this model $H$ is a fixed upper bound for $\chi(G)$ obtained by applying some of the many theoretical upper bounds or heuristic algorithms. Binary variables $x_{vi}$ represent the assignment of vertices to colors. Each vertex must be assigned exactly to one color (the first constraint) while two adjacent vertices can not have the same color, which is controlled by binary $w_i$ (the second constraint).

Graph coloring problem is simple for some special families of graphs. Trivial examples are $K$-partite graphs which are $K$-colorable. For the complete graph on $N$ vertices $K_N$ is trivially $\chi(K_N)=N$. For perfect graphs, the chromatic number equals to the size of the largest clique and this number can be efficiently computed by semi-definite programming. This follows from the definition of perfect graphs and by the famous Sandwich theorem~\cite{knuth1994}, while the Strong perfect graph theorem simply characterises these graphs~\cite{schrijver}.

As for solving the graph coloring problem, exact algorithms that solve it to optimality start with one of the mathematical programming formulations and try to feed it into appropriate nonlinear programming solver, if the problem is small enough and if the there is a solver available for that formulation~\cite{Eppstein2003}. For larger graph sizes, linear and semi-definite programming relaxations are used within Branch and Bound framework. However, for practical needs, a vast amount of heuristic algorithms is available~\cite{Malaguti2010,lewis}.

\section{The dynamical model that colors the graph}
Coming back to phase-oscillators, we begin explaining the contribution of this paper via illustrative toy-example. Consider a graph with $N$ nodes defined by the adjacency matrix $A$. Phase-oscillators specified by phases $\varphi_i$ are assigned to its nodes, each with degree $k_i$. Oscillators' frequencies are identical so we set them all to $\omega_i=0$ without loss of generality. The coupling strength is $\varepsilon=-1$. The equation for this phase-repulsive graph of oscillators reads:
\begin{equation}
\dot \varphi_i = - (1/k_i) \sum_j A_{ji} \sin (\varphi_j - \varphi_i) \; .
\label{eq-1}
\end{equation}
If our graph had only $N=2$ oscillators (nodes) connected by a link, dynamics would push their phases away from each other to the opposite values. Eventually, the phase difference between them would reach $\pi$, since this is the stable equilibrium for this system~\cite{levnajic2011}. Assume now that our graph is \textit{2-partite}. Dynamics pulls the phases of the connected oscillators apart, i.e., `stretches' the phase differences along each link, trying to reach the maximum stretch value $\pi$. Since the graph is 2-partite, this equilibrium is easily attained. Eventually, oscillators in two groups will have the opposite phase values and the graph will be perfectly anti-synchronized. An illustration of this situation is shown in Fig.~\ref{fig-1}a.
\begin{figure*}[!htb]
\centering
    \includegraphics[width=0.4\linewidth]{fig-1.eps}
    \caption{(a) Illustration of a simple 2-partite graph (network): link connect only nodes in one group with the nodes in the other group. Groups are shown in grey. (b) Illustration of a simple graph that is not 2-partite, but it is 3-partite.}
    \label{fig-1}
\end{figure*}

But now, inverting this problem, Eq.~\ref{eq-1} can be used to check whether a given graph is 2-partite. Once the dynamics of Eq.~\ref{eq-1} reaches equilibrium, one looks at the phase differences along links: if they are all $\pi$ the graph is 2-partite. Actually, consider a graph that is not 2-partite. For simplicity, let it have 3 nodes connected in a triangle, like in Fig.~\ref{fig-1}b. It is a 3-partite graph, since nodes can be grouped in three groups with links going only between the groups (but not in two groups). The dynamics of Eq.~\ref{eq-1} will no longer be able to 'stretch' the phase differences along links to $\pi$. Instead, at least some links will be ``frustrated" or ``squeezed" to a phase difference less than $\pi$. This will happen for any choice of initial phases, indicating that the underlying graph is not 2-partite.

To articulate this dynamical situation we introduce frustration $f_{ij} = f_{ij}(\varphi_j - \varphi_i)$ along the link $i-j$ ($A_{ij} = 1$) as:
\begin{equation}
 f_{ij}(\varphi_j - \varphi_i) = 1 + cos(\varphi_j - \varphi_i) \; .
 \label{eq-2}
\end{equation}
Frustration captures how ``squeezed" is a link, $f_{ij}=0$ means that link $i-j$ is not frustrated (phase difference is $\pi$). If we picture the oscillators' interactions as springs, frustration can be thought of as elastic potential energy due to spring being squeezed~\cite{levnajic2011}. In fact, gradient of $f_{ij}$ yields the sinusoidal interaction in Eq.~\ref{eq-1}, so $f$ plays the role of dynamical non-equilibrium potential~\footnote{Frustration can be equivalently defined for phase-attractive case, but it is always zero since for identical oscillators full synchronization is the only equilibrium state. In contrast, for the phase-repulsive case graph topology is intimately related to frustration~\cite{levnajic2011}}. We define the total frustration $F$ of a graph as the sum of frustrations along the individual links:
\begin{equation}
F = \sum_{i,j} A_{ij} f_{ij}(\varphi_j - \varphi_i) \; ,
\label{eq-total}
\end{equation}
so that $F=0$ guarantees that $f_{ij}$ is zero along all links. The graph in Fig.~\ref{fig-1}a will always have $F=0$, whereas the one in Fig.~\ref{fig-1}b will always have at least some frustrated links and hence $F>0$. This shows that former graph is 2-partite, while the latter is not. Eq.~\ref{eq-1} can be used to confirm 2-partitness, but not to confirm 3-partitness.

So, can we develop a dynamical model inspired by Eq.~\ref{eq-1} that could confirm $K$-colorability of a graph for any $K$ (not just for $K=2$)? Such model would offer a dynamical solution to the combinatorial optimization problem~\cite{sanz2016}, finding the chromatic number \textit{dynamically}, just by naturally relaxing to its equilibrium. Note the analogy with searching for the ground state in quantum systems~\cite{kudo2018} (quantum algorithms here serve \textit{only} as inspiration and have no relevance otherwise).

Equipped with above preliminaries, we now present exactly such a model. We are given a non-weighted and non-directed graph and set out to find a dynamical model that checks its $K$-colorability for any $K$. We need the specific coupling functions and frustration function for each $K$. For $K=2$ we have Eq.~\ref{eq-1} and Eq.~\ref{eq-2}: a graph is 2-colorable if for at least one choice of initial phases we obtain $F=0$.

We define the $K$-frustration $f^K_{ij}$ along the link $i-j$ for $K \geq 2$ as:
\begin{equation}
f^K_{ij} (\varphi_j - \varphi_i) = 1 + c_K \Big[ (K-1) \cos (\varphi_j - \varphi_i) + (K-2) \cos 2(\varphi_j - \varphi_i) + \cdots + \cos (K-1)(\varphi_j - \varphi_i) \Big] \; .
\label{eq-Kfrustation}
\end{equation}
Constants $c_K$ are fixed to have $f^K_{ij} (\frac{2\pi}{K})=0$ for every $K$. Each $f^K$ is a trigonometric polynomial of the degree $K-1$. For $K=2$ we have $f^2_{ij} = 1 + c_2 \cos (\varphi_j - \varphi_i)$. Requesting $f^2_{ij}(\pi)=0$ we get $c_2=1$ and recover Eq.~\ref{eq-2}.

We define the total $K$-frustration $F^K$ similarly to Eq.~\ref{eq-total} as the sum of $K$-frustrations along the individual links
\begin{equation}
F^K  = \sum_{i,j} A_{ij} f^K_{ij} (\varphi_j - \varphi_i) \; .
\end{equation}
$F^K$ is the non-equilibrium potential for the dynamical model behind Eq.~\ref{eq-Kfrustation}. We have:
\begin{equation}
\dot \varphi_i = - \frac{1}{k_i} \frac{\partial F^K}{\partial \varphi_i} = - \frac{1}{k_i} \sum_j A_{ij} \frac{\partial}{\partial \varphi_i} f^K_{ij}  (\varphi_j - \varphi_i) =
- \frac{1}{k_i} \sum_j A_{ij} C^K (\varphi_j - \varphi_i) \;  .
\label{eq-Kdynamics}
\end{equation}
For coupling functions $C^K$ we get:
\[ C^K (\varphi_j - \varphi_i) = c_K \Big[ (K-1) \sin (\varphi_j - \varphi_i) + 2 (K-2) \sin 2(\varphi_j - \varphi_i) + \cdots + (K-1) \sin (K-1)(\varphi_j - \varphi_i) \Big] \; , \]
where the constants $c_K$ are fixed as before. Since $c_2=1$, Eq.~\ref{eq-Kdynamics} for $K=2$ reduces to the phase-repulsive graph of Kuramoto oscillators Eq.~\ref{eq-1}.

For higher values of $K$, in addition to the first harmonic like in Eq.~\ref{eq-1}, oscillators are also coupled via higher harmonics. Consider the example of $K=3$. We have:
\begin{equation}
\begin{aligned}
 f^3_{ij} (\varphi_j - \varphi_i) = 1 + \frac{4}{3} \cos (\varphi_j - \varphi_i) + \frac{2}{3} \cos 2(\varphi_j - \varphi_i)  \; , \\
 \dot \varphi_i = - \frac{4/3}{k_i} \sum_j A_{ji} \Big[ \sin (\varphi_j - \varphi_i) + \sin 2 (\varphi_j - \varphi_i) \Big] \; .
\end{aligned}
\end{equation}
For phase differences $\varphi_j - \varphi_i = \frac{2\pi}{3}$ or $\varphi_j - \varphi_i = \frac{4\pi}{3}$ we get $f^3_{ij}=0$ and hence $\dot \varphi_i = 0$. In contrast, for all other phase differences we have $\dot \varphi_i \neq 0$. Hence, above dynamical model looks for equilibrium states where the phase differences along links are integer multiples of $\frac{2\pi}{3}$. Back to the graph from Fig.~\ref{fig-1}b, under dynamics governed by $F^3$ its three nodes will ultimately have three different phase values, equidistant on the circle, and separated by $\frac{2\pi}{3}$. In this situation we have $F^3=0$, indicating that this graph is 3-colorable. Meanwhile, $F=F^2$ will stay positive, since graph is not 2-colorable (every $K$-colorable graph is also $(K+1)$-colorable, but not vice versa).

In general, \textit{any graph is $K$-colorable if and only if the global minimum of $F^K$ is zero} and the  smallest such $K \geq 2$ is the graph's chromatic number $\chi(G)$. In fact, if for some arrangement of equilibrium phase values around the nodes we find $F^K=0$, then there are exactly $K$ different equilibrium phase values. This means some nodes will have common phase values, but never those that are connected. Instead, phases of the connected nodes will always be integer multiples of $\frac{2\pi}{K}$ apart (if they were not, this would not be an equilibrium state, and $F^K$ would be positive). Identifying $K$ colors as $K$ equilibrium phase values, we have that such graph is $K$-colorable. This is the core result of our paper and it is rigorously proven in the Appendix.

To confirm that a graph is $K$-colorable one needs to run the dynamical model Eq.~\ref{eq-Kdynamics} from many initial phases and examine the final equilibrium states, searching for the one with $F^K=0$. Simplest way to do this is to track $F^K(t)$ (in addition to several `tricks' that we discuss later). Note that $F^K$ \textit{always has} a global minimum: the question is, if it is zero. Thus, checking the $K$-colorability of a graph is translated into evaluating the global minimum of a real function $F^K$, which is done via natural evolution of the collective dynamics Eq.~\ref{eq-Kdynamics}. We have thus translated a combinatorial into a continuous optimization problem: rather than permuting arrangements of node colorings (combinatorial problem), we run the Eq.~\ref{eq-Kdynamics} by integrating ordinary differential equations (continuous problem).

However, if $F^K=0$ is not found, this still does not prove that graph is not $K$-colorable. $F^K$ is a trigonometric polynomial whose functional properties depend on (and in fact, capture) the topology of the underlying graph. Therefore, in addition to the global minimum, such function will in general have many local minima where $F^K>0$. Dynamics could easily get ``stuck" in one of them, and confuse it for the global minimum, leading to (potentially false) conclusion that graph is not $K$-colorable. Dynamics can miss the true global minimum for several reasons, e.g. because its basin of attraction is too small. This is a notorious problem in functional optimization.

Furthermore, there are situations where the minimization converges not to a point, but to a critical set of bigger dimensionality, for example a limit cycle. However, this does not change the principle of our method: we look for $F^K=0$, regardless of the dimensionality of the set on which this occurs. In fact, larger the set with $F^K=0$ is, easier is for convergence process to find it. On the other hand, convergence can become stuck more easily in a local minimum of bigger dimensionality (than in a point). Partial remedy for this is in `tricks' that we discuss later.

\paragraph{An important remark.} Once the non-negative function $F^K$ is defined, its local and global minima are what they are (zero or otherwise), regardless of whether our dynamics Eq.~\ref{eq-Kdynamics} finds them or not. So, running the dynamics Eq.~\ref{eq-Kdynamics} \textit{is not the only way} to evaluate the global minimum of $F^K$. There are numerous approaches in functional optimization for evaluating the minima of real functions, such as gradient descent, differential evolution, etc~\cite{sundaram}. In fact, some of them could be more efficient than the natural evolution of Eq.~\ref{eq-Kdynamics}. In other words, once the combinatorial problem of $K$-colorability is converted into the optimization (evaluation) problem, it can be approached either via Eq.~\ref{eq-Kdynamics} or via any of the standard methods in functional optimization.

\section{Simulations}
To illustrate our result, we construct a simple graph with 8 nodes and chromatic number 4. We run the dynamics of Eq.~\ref{eq-Kdynamics} with coupling function $C^4$, whose corresponding frustration is $F^4$. Initial phases are set at random, but instead of picking them from uniform distribution on the circle, we use von Mises distribution~\cite{mardia} (it can roughly be thought of as a Gaussian on the circle, where $\kappa \geq 0$ plays the role of standard deviation, so that uniform distribution is obtained for $\kappa = 0$). The effect of this is that initially all phases are close to each other, so the system is very frustrated, which gives it a bit of ``push" and makes phases diverge more rapidly.

In Fig.~\ref{fig-2} we show two realizations of dynamical evolution on the graph via four snapshots (last snapshot for $t=15$ is the equilibrium state). In top panel the dynamics reaches the state with $F^4=0$, which confirms that graph is 4-colorable. There are indeed only 4 different final phase values and no connected pair of nodes is colored the same. In bottom panel the dynamics settles into a state with $F^4>0$, since some links remain frustrated and there are more than 4 different equilibrium phase values. This is a local minimum, in the sense that small perturbations will not ``kick" it out of it.
\begin{figure*}[!hbt]
\centering
    \includegraphics[width=\textwidth]{fig-2.eps}
    \caption{Two realizations of evolution for illustrative graph with 8 nodes, shown in four time snapshots as indicated above. Both evolutions start from initial phases randomly chosen from von Mises distribution on the circle with $\kappa=10$~\cite{mardia}. The phases of the nodes are shown by colors and frustration of links in gray-scale (see two colorbars on the right). Top panel is an example of dynamics that reaches $F^4=0$, while bottom panel is an example of dynamics that settles in a local minimum with $F^4>0$.}
    \label{fig-2}
\end{figure*}
This shows how not all runs finish in the global minimum, regardless of it being zero or not. Still, we need only one realization as in top panel to confirm that graph is 4-colorable.

But what if the basin of attraction for the state $F^4=0$ was much narrower? In that case it could happen that no runs finish in the global minimum. This would falsely lead us to believe that graph is not 4-colorable. Establishing whether some minimum is local or global is a notorious problem in functional optimization. To investigate its repercussions here, we trace $F^K (t)$ starting from 50 initial phases for above graph and report the results in Fig.~\ref{fig-3}. Situation in earlier Fig.~\ref{fig-2} corresponds to  Fig.~\ref{fig-3}b ($K=4$). Only a fraction of runs finish in $F^4=0$ (red curves), and the rest finishes in $F^4>0$ (blue curves). Examples of the former and the latter correspond to the top and the bottom panels in Fig.~\ref{fig-2}, respectively. One needs to run the dynamics from many different initial phases to find one state with $F=0$, assuming there are any.
\begin{figure*}[!hbt]
\centering
    \includegraphics[width=\textwidth]{fig-3.eps}
    \caption{(a) Plots of $F^3 (t)$ for 50 runs of Eq.~\ref{eq-Kdynamics} with $C^3$ for the same graph as in Fig.~\ref{fig-2}. (b) The same for $C^4$ and $F^4$, two examples on this dynamics are shown in Fig.~\ref{fig-2}. (c) The same for $C^5$ and $F^5$. Plots are shown until $t=15$ for consistency with Fig.~\ref{fig-2}. Red curves are runs that finish in $F^K=0$ and blue curve are runs that finish in $F^K>0$.}
    \label{fig-3}
\end{figure*}
To put this in context, we re-do the same runs with $K=3$ are show the plots in Fig.~\ref{fig-3}a. Clearly, none of the runs finish with $F^3=0$, since graph is not 3-colorable. However, the same situation could happen for $K=4$ unless we run enough realizations. In contrast, in Fig.~\ref{fig-3}c we show the same for $K=5$: almost all runs lead to $F^5=0$, since graph is easily colorable with 5 colors. Confirming the colorability for higher $K$ is easier, since $F^K$ touches zero in more points, despite $F^K$ being a higher-order trigonometric polynomial for higher $K$. As we get closer to the minimal $K$ for which the graph is $K$-colorable, less and less initial phases lead to $F^K=0$, since the basin of attraction for states with $F^K=0$ shrinks. That is why pinpointing the chromatic number of a graph is challenging, as illustrated in Fig.~\ref{fig-3}.

We ran many more simulations for graphs of varying size and chromatic numbers. Overall, we found that our simulations correctly identify the chromatic numbers obtained via traditional combinatorial algorithms. However, as graph size grows beyond $N=50$, the number of necessary runs increases rapidly. In fact, the dimensionality of space on which $F^K$ is defined is $N$. As $F^K$ becomes higher and higher dimensional, pinpointing the basin of attraction for $F^K=0$ becomes harder and harder. But still, if the chromatic number is (relatively) low, the corresponding $F^K$ is a low-order trigonometric polynomial, so our dynamical graph coloring works.

On the other hand, except taking the initial phases from specific distributions, there are other enhancements (or `tricks') one could use. One of them is to introduce kicking in the oscillator dynamics, i.e., make Eq.~\ref{eq-Kdynamics} a stochastic differential equation by adding a noise term on its RHS. The effect of this is that evolution can get kicked out of a local minimum, possibly finding the global minimum later. This is similar to the effect that temperature has in simulated annealing. We confirmed that this enhancement indeed improves the overall performance of our method, but it still does not solve the challenge of large graphs and high chromatic numbers.

\section{Improper coloring of weighted graphs}
Our approach is actually useful for a harder problem known as improper graph coloring~\cite{riste1999,mohar2003}. Here, one looks into colorings with number of colors smaller than graph's chromatic number. Therefore, one allows for some connected nodes to be colored the same, but looks for the arrangement of colors that minimizes the number of such pairs. In this improper sense, any graph is colorable with any $K$ number of colors, but the question is, how to minimize  the number of connected nodes having the same color. If this number is zero for some $K$, the graph is $K$- colorable in the proper sense, and we have $\chi(G)\le K$. Several variations of this problem have been studied in the literature, although not very frequently, with diverse motivations, for example related to scheduling and timetabling problems~\cite{mishra2005}.

To make our analysis even more general, in this section we face the problem of improper coloring of weighted graphs. In a weighted graph, links (edges) are not all of equal weight (``thickness"), but can be stronger or weaker, modeling the strength of interactions between the nodes. Consider a non-directed weighted graph with $N$ nodes, described by the symmetric weighted adjacency matrix $W_{ij}$. The entry $W_{ij}$ in this matrix denotes the (non-negative) weight of the link $i-j$. Given an integer $K \leq N$, the problem is to color the nodes of this graph in a way to minimize the sum of weights of links that connect nodes that are colored the same. In other words, we want to minimize the sum of $W_{ij}$ such that nodes $i$ and $j$ are colored the same (if $i$ and $j$ are not connected then of course $W_{ij}=0$). Traditionally, improper graph coloring is treated via similar combinatorial algorithms, which in weighted case have an additional layer of complexity.

As before, we assign phase-oscillators of identical frequencies $\omega=0$ to graph nodes and consider a tentative coloring number $K$. $K$-frustration $f^K_{ij}$ along $i-j$ remains defined by Eq.~\ref{eq-Kfrustation}, but the total $K$-frustration $F^K_W$ is now a weighted sum of individual link frustrations:
\begin{equation}
    F^K_W  = \sum_{i,j} W_{ij} f^K_{ij} (\varphi_j - \varphi_i) \; .
\end{equation}
The dynamical equations are obtained the same way:
\begin{equation}
  \dot \varphi_i = - \frac{1}{k_i} \frac{\partial F^K_W}{\partial \varphi_i} = - \frac{1}{k_i} \sum_j W_{ji} \frac{\partial}{\partial \varphi_i} f^K_{ij}  (\varphi_j - \varphi_i) =
- \frac{1}{k_i} \sum_j W_{ji} C^K (\varphi_j - \varphi_i) \;  ,   \label{eq-weightdynamics}
\end{equation}
where the coupling functions $C^K$ for each $K$ are the same as before. Our main result fully generalizes: if the system reaches the global minimum of $F^K_W$, then in that state there are exactly $K$ different equilibrium phase values on the nodes, separated by integer multiples of $\frac{2\pi}{K}$. The key difference is that now even in the global minimum we have $F^K_W>0$, since some links will be frustrated, as their nodes will have the same phase value (i.e. be colored the same). However, the sum of weights $W_{ij}$ of such links will be minimal, as we prove in the Appendix. Hence, the minimization of $F^K_W$ yields the improper coloring of a weighted graph using exactly $K$ colors. As in the non-weighted case, we can use either the dynamics itself or a range of other functional minimization methods.

There are several differences with the non-weighted case. First, a graph is now $K$-colorable for any $K$, the only question is, how frustrated it will be. Our results says that the dynamics evolves to (\textit{i}) minimize the total frustration $F^K_W$, and (\textit{ii}) squeeze the remaining frustration preferentially into weaker links. We have again translated a combinatorial into a continuous optimization problem, it is just that now we cannot recognize the global minimum via $F^K_W=0$, but via $K$ different final phase values. Second, the fact that coloring is improper does not specify how many connected node pairs are colored the same (at least one), but only that the joint frustration of their links is minimal. So, we are not minimizing the number of frustrated links, but the total frustration $F^K_W$ itself. Third, in weighted case dynamics can also get stuck in the local minima. For any $K$ there is a guaranteed improper coloring corresponding to minimal total frustration, but one might again resort to ``tricks" such as special choice of initial phases or stochastic evolution to find that minimum more efficiently. On the other hand, local minima can now be interpreted as Nash equilibria in the system: let $e^K_i$ be the $K$-frustration of the node $i$ defined as the sum of $K$-frustrations of all its adjacent links $ e^K_i = \sum_j W_{ij} f^K_{ij}$. Collective dynamics can now be seen as each node $i$ seeking to minimize its $e^K_i$. Pairs of nodes connected by weak links will be more successful in this.

Next we show the performance of our dynamical method for improper coloring. For simplicity we use a fully connected graph (clique) with $N=6$ nodes (this is a non-trivial problem, a variation of the {\it channel assignment problem} for wireless graphs known from telecommunications). We randomly assign weights to its 15 links and run the dynamics of Eq.~\ref{eq-weightdynamics} with coupling functions $C^6$, $C^5$, $C^4$, $C^3$, computing the corresponding frustrations $F^6_W$, $F^5_W$, $F^4_W$, $F^3_W$. The results are shown in Fig.~\ref{fig-4}, three snapshots for each example.
\begin{figure*}[!hbt]
\centering
    \includegraphics[width=\textwidth]{fig-4.eps}
    \caption{Four realizations of evolution for weighted fully connected graph with $N=6$ nodes, shown in three snapshots each, as indicated. Link weights are chosen randomly between 1 to 5. Initial phases are randomly selected from von Mises distribution with $\kappa=2$. The phases of the nodes are shown by colors and frustration of links in gray-scale, as in Fig.\ref{fig-2}. One realization of evolution for each coupling function $C^6$, $C^5$, $C^4$, $C^3$ and the corresponding frustration $F^6_W$, $F^5_W$, $F^4_W$, $F^3_W$ is shown in each panel, as indicated.}
    \label{fig-4}
\end{figure*}
For $C^6$, as expected, the dynamics settles into a state with zero total frustration and with graph (properly) colored with $K=6$ colors. For $C^5$, as the graph tries to get colored with 5 colors, at least one pair of nodes must be colored the same. The dynamics reaches a state with just one link frustrated, the weakest one. For $C^4$, the dynamics settles into a state where two weakest links are frustrated. For $C^3$, graphs is eventually colored with 3 colors. Four links remain frustrated and we verified that their joint weight is the smallest possible. So, as expected, in all cases our dynamical coloring reached the equilibrium with minimal total frustration squeezed into weakest links. In case of a complex topology (not clique), the minimization into weakest links would interplay with topology, making it more difficult to see that algorithm works as expected.

To examine the impact of the local minima, we repeat in Fig.~\ref{fig-5} the analysis from Fig.~\ref{fig-3}. We track $F^K_W (t)$ between $t=0$ and $t=15$ for 50 choices of initial phases for above graph for $K=6, 5, 4, 3$, in correspondence to four panels in Fig.~\ref{fig-4}. For $K=6$ most initial phases lead to $F^6_W=0$, but not all: since the graph is weighted, despite having 6 colors available, the graph sometimes fails to ``stretch" to zero total frustration. In such equilibria, the graph is not colored by 6 colors.
\begin{figure*}[!hbt]
\centering
    \includegraphics[width=\textwidth]{fig-5.eps}
    \caption{(a) 50 realizations of $F^6_W (t)$ for 50 runs of Eq.~\ref{eq-weightdynamics} with $C^6$ for the weighted graph from Fig.~\ref{fig-4}. (b) The same for $C^5$ and $F^5_W$. (c) The same for $C^4$ and $F^4_W$. (d) The same for $C^3$ and $F^3_W$ (all in correspondence with four panels in Fig.~\ref{fig-4}). Plots are shown until $t=15$ for consistency. Red curves are runs that finish in zero total frustration and blue curves otherwise.}
    \label{fig-5}
\end{figure*}
For $K=5, 4$ and 3, the global minimum that corresponds to improper coloring with $K$ colors decreases with $K$. For each of these values of $K$ there are many local minima, so the global minimum is somewhat less distinguishable, and has to be identified by actually checking if the graph is colored (although improperly) with $K$ distinct colors. We run additional simulations for larger weighted fully connected graphs and found the same scaling properties as in the non-weighted case. Global minima are increasingly harder to find as the graph size increases. Enhancements such as kicking (stochastic evolution) have the same effect as before -- they improve the search, but does not solve the problem of local minima.

\section{Discussion}
Using the analogy with repulsively coupled phase-oscillators we found a way to convert a combinatorial optimization problem (proper and improper graph coloring of non-weighted and weighted graphs) into the optimization problem of evaluating the global minimum of a real function. This function is the non-equilibrium potential for the collective dynamics of oscillators coupled through the edges of the graph (network). This dynamics can be used as a ``natural" way to search for the minimum, or alternatively, one can resort to the rich ensemble of methods from functional optimization. This conversion does not offer an immediate solution to the problem, but it allows us to approach the problem using completely different methodology. Examining in what situations this might lead to a more efficient or precise solution of the original problem remains the matter of future work, potentially extending to other discrete problems.

There are two ways of comparing our result to the state of the art. First, one could compare our collective dynamics as a method of evaluating the global minima of real functions to the existing optimization methods. We did not find this of interest, since this means comparing functional optimization methods among them, and for that there already exists abundant literature. Second and more interesting way is to examine whether solving graph coloring problem this way would be better than solving it via standard combinatorial algorithms. We did look into this, and as already stated, for all examined graphs we found the same chromatic numbers. In no case we found the chromatic number bigger or smaller than indicated by traditional algorithms. We checked this for 30 graphs with sized up to $N=100$. Another independent way of making this comparison is to look at the computational cost involved in two cases. However, this comparison strongly depends on the computing architecture, since the algorithms are of quite different nature (permutations of color arrangements vs. integrating a differential equation using e.g. Runge-Kutta integrator). An interesting direction of future work might be to find a computationally cheaper integrator for Eq.~\ref{eq-Kdynamics}~\cite{linda}.

A different issue revolves around scaling up the problem: how about coloring a graph of size $N=1000$? This represents a challenge for both combinatorial and our method. While the former suffer from combinatorial explosion, our method comes down to finding a minimum of 1000-dimensional real function, which is far from trivial. However, while cases of small graphs can in practice be easily treated by any approach, the true challenge lies in coloring large graphs, which is where our approach might make a difference. This remains the core direction of future work. Still, the degree of trigonometric polynomial in Eq.~\ref{eq-Kfrustation} is $K-1$. Hence, checking the colorability for small $K$ is relatively easy, since low-level trigonometric polynomial is not very ``wavy" and the minima is easier to find, even for large graphs. This might be a competitive advantage of our methods, since checking for these minima could beat the existing combinatorial optimization approaches. In contrast, when checking the colorability for high values of $K$, the degree of trigonometric polynomial is higher, which makes the function more wavy and hence the basins of attraction for local and global minima are steeper and harder to find. On the other hand, given the purely trigonometric nature of any $F^K$, it would be interesting to do frequency analysis on it.

We close the paper with another potentially interesting idea. Consider a graph $G$ with unknown chromatic number $\chi(G)=K_0$ and some very large tentative $K$, much larger than $K_0$ (say $K=N$). $F^K$ is zero on a potentially very large domain within $\mathbb{R}^N$, since the graph is easily $K$-colorable for this $K \gg K_0$. We call this domain support of $F^K$. Consider now $K-1$, where the support for $F^{K-1}$ is a large domain, although presumably smaller than support of $F^K$. Continuing on, one expects that support of $F^{K-2}$ is even smaller, but larger than the support of $F^{K-3}$, and so on. Eventually, the support of $F^{K_0}$ is the smallest such non-empty domain, since the support of $F^{K_0-1}$ is empty. What is the geometric relationship between these domains? For example, could they be nested subsets of each other? If so, one could be able to extrapolate $K_0$ by examining the geometric process of how these domains (supports) shrink as $K$ deceases. The value of $K$ just before this domain becomes empty is the chromatic number $K_0$, obtained without optimization.

\paragraph{Acknowledgements.} Work supported by the Slovenian Research Agency (ARRS) via programs P1-0383 and P2-0256, projects J5-8236, J1-8155, N1-0057 and N1-0071, and by EU via Marie Sklodowska-Curie Grant project 642563 (COSMOS). Part of it has been completed during the STSM of the third author (VJ) supported by the COST action CA15140. Thanks to colleagues Alexander Yurievich Gornov, Peter Koro\v sec, Ljup\v co Todorovski, and Riste \v Skrekovski for very useful suggestions and feedback.

\bigskip
\bigskip

redpajama_set_name: RedPajamaArXiv


HEBRON (Ma'an) -- Israeli authorities raided and shut down a Palestinian radio station in Hebron city in the southern occupied West Bank during predawn hours on Thursday for allegedly "inciting terrorist attacks."
Israeli soldiers raided the al-Hurriya (Freedom) Media Company in Hebron -- which runs the Manbar al-Hurriya radio station and al-Nawras TV -- and destroyed, damaged, and confiscated various equipment, according to Ayman Qawasmeh, the head of the radio station.
"We were not surprised at the occupation's barbaric policy and they will not be able to silence the voice of al-Hurriya," Qawasmeh told Ma'an, adding that it was not the first time the radio had been shuttered by Israeli forces.
However, he added, the soldiers had only issued the closure order for al-Hurriya radio station.
Qawasmeh told Ma'an that 56 employees work at the radio station, all of whom have families to support.
In Nov. 2015 al-Hurriya radio station was also closed for six months during an increase in violence that erupted across the occupied Palestinian territory and Israel. At the time, the radio station was also shut down for "incitement to violence," according Israeli daily Haaretz.
Israeli authorities have led a crackdown on alleged incitement since the increased violence, with Palestinian media outlets often being raided for allegedly inciting violence and hundreds of Palestinians detained over social media posts that Israeli forces believe amount to incitement against the Israeli state.
However, Israel has been accused of labeling any media that is critical of Israel and its policies in Palestinian communities as "incitement" in order to stifle criticisms of Israel's discriminatory policies in Israel, its half-century occupation of the West Bank, and its decade-long siege of the Gaza Strip that has collapsed the territory into an interminable humanitarian crisis.
In a high profile case, Israeli Prime Minister Benjamin Netanyahu has targeted the award-winning news network Al Jazeera with allegations of incitement, most recently during the two-week long civil disobedience campaign in occupied East Jerusalem held in protest of Israel's security measures at the holy site last month.
Netanyahu has since worked to shutter Al Jazeera's office in Israel and revoke the official credentials of Al Jazeera journalists.
Two weeks ago, Israel had announced that it would revoke the press card of senior Al Jazeera journalist Elias Karam. However, on Wednesday it was reported that Israeli authorities had postponed the decision and would "monitor" Karam's work for six months.
Karam, a Palestinian citizen of Israel, was threatened with the revocation of his credentials owing to an interview he had done with a Muslim Brotherhood television channel last year in which he said that he considered his work as a journalist to be a part of Palestinian "resistance." However, he recently denied this conflation during a hearing with Israeli authorities.


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Ramesses II
This article is currently undergoing copyediting process.
Following article heavily relies on religious and/or local-folk traditions and does not include verifiable archaeological or historical information. It should be rewritten to reflect the correct nature of the subject matter. Once done, this tag should be removed.
Pharaoh Ramesses II is the most widely regarded Pharaoh of the Exodus depending of the time frame of his reign era. Born c. 1303 BCE; died July or August 1213 BCE; reigned 1279-1213 BCE), also known as Ramesses the Great, was the third pharaoh of the Nineteenth Dynasty of Egypt. He is often regarded as the greatest, most celebrated, and most powerful pharaoh of the Egyptian Empire.
KV7: Tomb of Ramesses II
Ramesses II. Madainproject.com. (2022). Editors of the Madain Project. Retrieved on January 31, 2023, from https://madainproject.com/ramesses_ii
Intext citation: ("Ramesses II - Madain Project (en)", 2022)
Ramesses II. Madainproject.com, 2022, https://madainproject.com/ramesses_ii. Accessed 31 January 2023.
Intext citation: ("Ramesses II - Madain Project (en)")
"Ramesses II." 2022. Madain Project. https://madainproject.com/ramesses_ii.
View Subject Home > Rameses II
Contrary to the popular belief of death by drowning as mentioned in Quran and Bible, while in pursuit of Moses, Ramesses II died a natural death at the age of 90. At the time of the death he was suffering from severe dental problems and was plagued by arthritis and hardening of the arteries.
circa 100 BCE
The Mummy of Ramesses II at Cairo museum, it is believed that Ramesses II was essentially crippled with arthritis and walked with a hunched back for the last decades of his life. The pharaoh's mummy reveals an aquiline nose and strong jaw. It stands at about 1.7 metres (5 ft 7 in). Ramesses II originally was buried in the tomb KV7 in the Valley of the Kings, but because of looting, priests later transferred the body to a holding area, re-wrapped it, and placed it inside the tomb of queen Ahmose Inhapy.
Tomb of Rameses II, the entrance to the fourth corridor which descends further into the tomb towards a vestibule and the burial chamber.
Mummy of Ramesses II
Sarcophagus of Ramesses II
Ramesseum: Mortuary Temple of Ramesses II

redpajama_set_name: RedPajamaCommonCrawl


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DOI:10.7598/cst2019.1549
Effect of Synthesis Parameters on Photoactivity of CdS Nanoparticles
MAHENDRASINGH J. PAWAR*, ARCHANA D. INGLE and RAHUL K. TAYAWADE
Laboratory of Materials Synthesis, Department of Chemistry, ACS College, Kiran Nagar, Amravati- 444606, M.S., India
In the present work, CdS nanoparticles were synthesized by four different routes using ammonium sulphide ((NH4)2S) as source of (Sulphide) S2- ions. CdS nanoparticles were grown by simple chemical precipitation reactions in aqueous medium at room temperature. The effect of stabilizers on the stability and size of CdS nanoparticles was studied. The effect of different reaction parameters on the size of nanoparticles, absorbance and respective band gaps and crystalline structure were studied. The microstructure and morphology of the synthesized CdS nanoparticles have been characterized by x-ray diffraction (XRD), scanning electron microscopy (SEM) and transmission electron microscopy (TEM) analysis. XRD pattern reveals that as-synthesized CdS nanoparticles exhibit both hexagonal and cubic phases. The size of the particles calculated by Debye Scherrer formula according to the XRD spectra has been found to be about 3-30 nm. The optical properties of the sample have been studied by UV-Visible and photoluminescence spectroscopy. The existence of blue shift in UV-Visible spectroscopy reveals the quantum confinement effect of CdS nanoparticles. Finally, the synthesized CdS nanoparticles were exploited for the degradation of acid blue-29 (AB-29), under UV light. The photocatalytic efficiency of the synthesized nanoparticles, using different reaction routes were compared and optimized.
CdS Nanoparticles, Acid blue- 29 dye, Photoactivity, Photodegradation of dye
Received 24 May 2018
mjpawar09@outlook.com
This article has been viwed 171 times


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Event Coverage: 2017 Indianapolis 500 Qualifying
byDave Zipf
by David Zipf
Week one of "The Greatest Month in Racing" started with left and right turns, kicking off with the IndyCar Grand Prix last week. This week, the right turns are abandoned and the focus is on nailing down the fastest time at one of the fastest tracks on Earth. One week of practice leads up to two days of qualifying for the 101st running of the Indianapolis 500 Presented by PennGrade Oil.
Qualifying for Indianapolis is a unique affair in the IndyCar series. The procedure is now stretched over two days. On Saturday, all 33 cars make a qualifying attempt. A qualifying attempt at Indy happens over four flying laps. Cars are ranked by the average speed over all four laps. After all cars qualify, the top 9 cars are broken off from the rest of the pack. Times reset on Sunday, but the fastest 9 are guaranteed a starting position of no lower than 9th, while the rest of the field can start no better than 10th. After Sunday's qualifying, all times and positions are set, save for extenuating circumstances like James Hinchcliffe's horrific post-qualifying practice crash in 2015.
Qualifying begins with a final tech inspection. All cars are fueled up with 100% ethanol fuel provided by Sunoco, then are run through IndyCar's tech center to ensure alignment and weight are within IndyCar's rules. All cars line up here, awaiting their turn to be fueled, the first step in the tech inspection process.
Incredible effort goes into the 500. Teams have a fast oval-only car in storage for this event (as well as the followup super speedways of Pocono and Texas). These cars are still spec Dallara DW-12s, but hand modified to the limit of the rules. Control arms are shaped with airfoils and tape until they are perfectly smooth. Bodywork is fitted as tightly together as possible to minimize drag inducing gaps, which are then sealed over with tape. Suspension uprights are re-profiled to reduce any turbulence inside the wheel. Weeks of effort go into assembling these cars just for this race. While officially they are the same as their road course going counterparts, it is the tiny details that make the oval-only cars very different from the road-course spec DW-12s.
For some teams, just getting into the garage is where the effort goes. Juncos Racing is a new IndyCar team, but a veteran to the Mazda Road to Indy. Riccardo Juncos is a former Formula Renault driver who did a Roger Penske and became a team owner and engineer. He started his life and career in Brazil, but moved to the United States in 2009. He has been building ever since and now has a team in USF2000, Pro Mazda, Indy Lights, and now IndyCar. The demise of KV Racing allowed Juncos the chance to get into the 500 for the first time in 2017. This is what we were greeted to during the GP last week, but since then Juncos has set up shop and put two cars on the grid with Sebastian Saavedra and Spencer Pigot.
Teams always keep a large stock of pre-set wings ready to go in case of damage. This particular cluster of wings belongs to Andretti Autosport. Andretti is fielding six cars at Indy this year: his usual four full-timers (Ryan Hunter-Reay, Marco Andretti, Takuma Sato, and Alexander Rossi), along with rookie and former Indy Lights Champ, Jack Harvey, in a partnership with Michael Shank Racing. The sixth car you can plainly tell from this shot: Fernando Alonso's McLaren-Honda partnered car. Poor Jack Harvey has been, in his own words, "The forgotten rookie" next to his 2-time world champion teammate. Well here on MotoIQ, you get first billing today, Jack!
NX GTi-R: Hill Climb Edition
AntiGravity's XA 4-1 Multi-Function Flashlight!
Posted on May 25, 2017 July 2, 2018


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The Difference Between Research Design & Research Method
A research method is a general framework guiding a research project. Different methods can be used to tackle different questions. Research design is a specific outline detailing how your chosen method will be applied to answer a particular research question.
1 Research Methods
Research methods are generalized and established ways of approaching research questions (e.g., qualitative vs. quantitative methods). Not all methods can be applied to all research questions, so the choice of method is limited by the area of research that you wish to explore.
2 Research Design
Research design involves determining how your chosen method will be applied to answer your research question. The design of your study can be thought of as a blueprint detailing what will be done and how this will be accomplished. Key aspects of research design include: research methodology; participant/sample collection and assignment (if different conditions are being explored); and data collection procedures and instruments.
Choice of research methods and design should be thought of as a reciprocal process extending well into your study. For example, it may arise over the course of your study that there is a flaw in the design. Changing the design of the study may lead to the choice (or addition) of a different method which, in turn, may lead to subsequent changes in the design to accommodate the new method(s).
1 Google Books: "Applied Social Research: Planning, Designing and Conducting Real-World Research", Ralph Hall
Matthew Lee has been writing professionally since 2007. Past and current research projects have explored the effect of a diagnosis of breast cancer on lifestyle and mental health and adherence to lifestyle-based (i.e. nutrition and exercise) and drug therapy treatment programs. He holds a Master of Arts in psychology from Carleton University and is working toward his doctorate in health psychology.
How to Evaluate a Research Design
How to Use Qualitative Research Methods in a Case Study...
What Is the Difference Between Internal & External...
What Is a Preliminary Research Design?
How to Determine Which Research Design to Use
The 5 Basic Steps in Inquiry Science
How to Form a Theoretical Study of a Dissertation
What Is the Meaning of the Descriptive Method in Research?
How to Design and Conduct Mixed Method Research
How to Tabulate Survey Results
How to Do a Thesis Proposal Presentation
What Is a Qualitative Method?
How to Write an Analysis/Discussion for a Science Project
How to Do a Policy Analysis Paper
How to Write a Rationale for Your Dissertation
How to Put Together an Ethnographic Research Paper
Descriptive Research Techniques
Define Primary & Secondary Data
What Is the Chemical Formula for Magnesium Nitrate...


This was the first time I have entrusted the organisation of one of my golf holidays to a golf travel company and I have to say I was particularly impressed by the quality of service and especially by your personal efforts to help me identify viable options and pricings to then allow me to run this by my group for their selection.
I was confident that all the booking, rooms etc have been safely put to bed well before the trip, leaving me to focus on the specific events I arrange for the group over the break.
The spa area was utilised by the group and lockers/towels were freely available and the facilities there were clean and in proper working order. It served as a great place to recover from any aches and pains picked up during a long day on the fairways!
The catering was good for most tastes of the group, but Saturday didn't have a buffet style breakfast, which did add a little confusion on the day.
The catering team and bar staff were very friendly and helpful which adds to the quality the overall experience.
The two courses 'The Lakes' and 'The Garden' course were genuinely tough, particularly for the higher handicappers in our group, but they were very enjoyable to play with a mixture of holes that featured a lot of encroaching water hazards. The length of both courses also provided most of our group with a significant challenge.
I cannot praise highly enough the starter (Keith), as he went to great lengths to ensure we were made welcome and where able catered for our requests. E.g. I asked if we could play off the white tees, which are usually reserved for competitions, and as a result of his work, the club were happy to allow us this extra bonus for Thursday and Friday. As it turned out, the variation of tees we played from over the weekend added to the challenge, and allowed some of the higher handicappers more scope to gain better scores.
Having had trouble with buggy hire arrangements in the past, I was glad to find no such problems this time out. Everything was in order and as an added bonus they came at a relatively low cost compared with other venues.
Surprisingly I don't think we could have picked a better weekend. The weather was relatively windless, sunny and very hot. More sun tan, less woollies! In addition to the weather, the Masters was on the TV and so was the Grand National. A great way to top off a golf break!
The entire group reported having a great weekend away and all at a total cost that sits well in the pocket!
The bar prices were reasonable and being a hotel, the bar remained open until late in the evening.
Thanks again you for all your help in putting this golf trip together. I will certainly be using YGT in again and I look forward to testing your research skills for next year's trip!
Print article This entry was posted by Rory on May 6, 2011 at 9:04 am, and is filed under General. Follow any responses to this post through RSS 2.0. You can leave a response or trackback from your own site.


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'No one can answer whether he can do it' – Besiktas chief baffled by Dele Alli's poor form and hopes he gets 'better'
DELE ALLI'S time in Turkey is showing no signs of improving as he is performing 'below expectations' at Besiktas.
The former Tottenham midfielder is on loan at the Super Lig side from Everton but he has failed to refind his best form.
Dele Alli ios struggling to impress while on loan in TurkeyGetty
GettyBesiktas boss Senol Gunes had been disappointed with Dele Alli[/caption]
The 26-year-old has managed to score twice in his eight games but there is little enthusiasm surrounding his performances.
Besiktas sporting director Ceyhun Kazanci told TRT Spor: "No one can answer whether Dele Alli can do it right now.
"He is working very hard. He is a player who reached €100million (£87m) three years ago. He had a fall.
"Dele Alli is working intensively. We had intense talks with him before we signed him.
"Senol [Gunes] is famous for returning many players whose careers have been on the decline.
"We hope that he will be better in the second half of the season."
Club manager Senol Gunes has also admitted that Alli has been "disappointing" and that he is struggling with him, as he said earlier in the season: "I couldn't affect the players.
"Now I can easily touch them but we haven't touched Dele Alli yet.
HOW TO GET FREE BETS ON THE WORLD CUP
"Let's not discuss Dele Alli as a player but he is below expectations in terms of efficiency."
Not so long ago it appeared Alli had hit rock bottom after he saw his spot-kick against Belgian side Charleroi saved in a friendly match.
While it was a decently struck penalty, keeper Herve Koffi was still able to make the stop.
Former Everton defender Alan Stubbs doesn't see a way back for Alli at Goodison Park.
He recently told Football Betting: "Dele doesn't have a future at Everton.
"It's a strange situation because I don't think I can ever remember a player falling off a cliff as quickly as Alli has.
"I don't know what's gone on in his private life but to go from being an England international to not being anywhere near that level is really bizarre."
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December 11, 2012 | Customer News
Historic Boston plays host to the fifth Tecan Symposium
Männedorf, Switzerland, December 11, 2012 – Tecan recently hosted the fifth annual Tecan Symposium in the picturesque surroundings of Boston, Massachusetts, visiting the US for the first time. Held at the beautiful Taj Boston hotel, this year's symposium explored the topic of 'Mass spectrometry – the expanding role in life sciences and diagnostics' and gave delegates an insight into the exciting opportunities and new developments in this fast-moving field.
With mass spectrometry (MS) becoming increasingly popular throughout the life sciences sector, from fundamental research to medical diagnostics, the 2012 Tecan Symposium covered a diverse range of MS applications with four main themes: the use of MS in diagnostics and applied markets; the role of MS in basic research; the collection, preparation and storage of samples for MS; and emerging trends and technologies in MS. Prominent speakers were invited from academia and industry, including Imperial College London and Quest Diagnostics, and spoke on subjects as diverse as medical diagnostics, the value of MS in space science and identifying traces of chocolate on ceramics recovered from archeological sites. With a wealth of expertise in so many different fields, they provided practical insights into current applications of MS, and discussed the technique's potential for future applications. As in previous years, the main scientific program was complemented by a series of social events, including a tour of the Freedom Trail and an evening dinner in Boston Public Library, that allowed delegates to network in an informal atmosphere, contemplating the vast array of MS applications while hailing the success of the fifth Tecan Symposium.
To find out more about the Tecan Symposium, visit www.tecan.com/symposium
Cornelia Kegele, Antonietta Allocca
Tecan (www.tecan.com) is a leading global provider of laboratory instruments and solutions in biopharmaceuticals, forensics and clinical diagnostics. The company specializes in the development, production and distribution of automated workflow solutions for laboratories in the life sciences sector. Its clients include pharmaceutical and biotechnology companies, university research departments, and forensic and diagnostic laboratories. As an original equipment manufacturer (OEM), Tecan is also a leader in developing and manufacturing OEM instruments and components that are then distributed by partner companies. Founded in Switzerland in 1980, the company has manufacturing, research and development sites in both Europe and North America and maintains a sales and service network in 52 countries. In 2011, Tecan generated sales of CHF 377 million (USD 424 million; EUR 306 million). Registered shares of Tecan Group are traded on the SIX Swiss Exchange (TK: TECN/Reuters: TECZn.S/Swiss security number: 1210019).


Our subsea technologies include the most extensive range of remotely operated vehicles in the world. These rugged vehicles are used by the subsea industry for trenching, electronic observation, hydraulic work and submarine rescue. We also provide a full range of tethering, tooling, robotic components and launch and recovery systems. Our thrusters, tether management systems and buoyancy foam are used by other ROV manufacturers. Customers turn to Forum for analyses, pipeline routing, marine archaeology, arctic engineering and scenario simulation software.


During the 3rd Bodypainting Festival on Tenerife, Maike Karrer also offered an Airbrush Tattoo Workshop on the island.
As in the first two years of the festival, this year again great professionals were invited to share their knowledge with newcomers and inquisitive advanced students.
Shopping centres, bathing resorts etc.
Ideal for airbrush artists and artists who wanted to expand and offer their knowledge.
Erasmo Daaz of Senjo Color Spain guided the artists through the world of fake tattoos with Senjo Color INKS and stencils.


This house directly on the sea is surrounded by a big, beautifeul garden with fence. It has its own access to the sea, a bedroom with double bed, two singlebed sofas, kitchen, bathroom and porch, barbecue, outdoor shower and parking on the ground. It has sun from the morning to the evening, because it is directed towards the south. The village with shops, bars and restaurants is 1 km away, as well as the sandbeach of Seccheto.


Philadelphia, PA (Law Firm Newswire) November 23, 2018 - Recently on Court Radio, Dean Weitzman, managing partner of MyPhillyLawyer, discussed how Pennsylvania determines negligence in a car accidents and the state Lemon Law relating to vehicles.
When a driver does not drive safely, with due care and attention to the rules of the road, and there is an accident, that driver is negligent. Pennsylvania negligence laws require that damages in a car accident be reduced in proportion to the claimant's own level of fault, also referred to as contributory negligence.
Another caller wanted to talk about a car that should not have passed inspection and cost her out-of-pocket money a mere 45 days after buying it. She asserted she had bought a "lemon" and wanted to know what legal recourse she had to get money back. As it turned out, the Pennsylvania Automobile Lemon Law only applies to a new vehicle defined as being less than 12 months old or having less than 12,000 miles when bought and legally registered in Pennsylvania for personal use. "In general, the lemon law is a law designed to protect buyers from defective vehicles — but that only applies to newer vehicles," said Weitzman.
Weitzman went on to add that anyone contemplating buying a used vehicle really needs to have it inspected by a mechanic and make themselves aware of all existing and potential issues that may affect the vehicle. "If there is no inspection and no warranty or a very limited warranty, then its buyer beware," Weitzman added.


text:
Tag Archives: Twitter UK
Oct 18, 2018 · 5:02 pm
Simon Jay's play is not about John Lewis or John Lewis but John Lewis
The things you miss when you do not go to the Edinburgh Fringe…
This year, I missed the extraordinary Simon Jay reprising his highly-praised Trumpageddon show and a new show @JohnLewis: Never Knowingly Undertweeted. The latter is about the American academic and "computer science educator" who owns the @johnlewis Twitter handle and gets mistakenly deluged by enquiries intended for the famed chain of British department stores.
The British store chain's slogan is Never Knowingly Undersold.
Simon Jay answering my questions
Last night, I went to see@JohnLewis: Never Knowingly Undertweeted – only on at the Drayton Arms Theatre in London until Saturday.
And, this morning, I Skyped Simon Jay for a pre-arranged chat. He overslept and was barely conscious.
I always find this is the best way to chat with people for blogs…
JOHN: So what did John Lewis, the American, say when you contacted him?
SIMON: He is very modest and self-deprecating. So he said: "Why would anyone write a play about me?"
JOHN: Modest and self-deprecating? Are you sure he's an American?
SIMON: Yes. He lives in Blacksburg, Virginia.
JOHN: He was shocked you wanted to write a play about him?
John Lewis, confusing but not confused
SIMON: (LAUGHS) Humbled. Originally, the play was going to be much more like a Jon Ronson investigation, where I tried to get into his life and how it all came about. He gave his blessing but then said he was incredibly busy. So it turned into a sort of Joyce Grenfell character piece.
JOHN: Did he have to approve everything?
SIMON: No. I mean, I wasn't going to take the piss. I wasn't going to betray him. He didn't even see the script. He said he didn't have time to read anything, so he just took it all on trust, really. I put updates from our rehearsal on Twitter. I don't know anything about his mother, but I portrayed her as some kind of religious zealot. I Tweeted a picture of me full-crazy and he said: "Oh! It's uncanny!"
JOHN: When you quote him saying things in the play, are those quotes he gave you?
SIMON: Well, based on things that he said, yeah. When I interviewed him, I was only able to get a couple of lines on each question I asked him. And he did give a few interviews early on to people like the Daily Telegraph.
JOHN: Has the quirky story been picked up in the US or only in the UK?
US Congressman John Lewis
SIMON: Well, John Lewis, the store, isn't known in America. In the US, he is confused with Congressman John Lewis.
JOHN: Who you also mention in the play.
SIMON: Yes, he gets a lot of racist abuse on Twitter intended for the Congressman, which I was going to explore. But then I thought it might be a dramatic turning point too far – from a light-hearted show about mistaken identity to the racist underbelly of the US.
JOHN: Why racist abuse?
SIMON: Because the Senator he shares a name with is a black Senator from Georgia, a civil rights campaigner who talks a lot about how racist America is. So lots of people take to Twitter to abuse him (LAUGHS) to prove how right he is!
JOHN: You talked to the John Lewis Partnership – the stores – about the play…
SIMON: Yes. They were a bit bewildered at first. But it's very much a warm, fluffy thing; John Lewis are hardly going to be offended and sue me. And the story is not about them; it's about mistaken identity. The fact it is about John Lewis is incidental, really.
JOHN: It's all good, light-hearted publicity for them.
SIMON: Well, some people genuinely think it's a conspiracy – that John Lewis have employed someone to pretend to be this man with their name!
JOHN: And the store's reaction?
John Lewis: amused and bemused
SIMON: They just think: Oh! It's just an amusing thing that has happened and we don't really get involved, but we are very grateful for the fact he forwards everything on.
JOHN: He does?
SIMON: He always puts the @JLPartners tag when he answers on Twitter. (They are also @jlandpcustserv)
JOHN: In the play, you have some characters who are John Lewis staff reacting to things, sending him presents and so on.
SIMON: Well, they really did send him a hamper at Christmas and even embroidered his @JohnLewis Twitter handle into a cushion. It's all fact. That's all true.
JOHN: The staff in the play are slightly bitchy to each other. Did John Lewis, the store, object?
SIMON: Well, their Regional Marketing Manager came to see a preview. I think they were checking to see it wasn't libellous. She liked it. I think she just accepted we all know characters like that.
JOHN: Any chance you might be in one of the famous traditional John Lewis Christmas ads on TV?
Never knowingly under-represented online
SIMON: Well, apparently the management did talk to the director of this year's ad and there was some talk about that and they were looking to get the 'real' John Lewis involved and the way to do that was to have lots of different John Lewises in the ad and I would be one of the John Lewises. Maybe they would have had archive footage of the Congressman. But, like all these things, it was probably just an idea on a piece of paper and they will have gone with something much more outlandish.
JOHN: So whither the show now? The West End and Vegas?
SIMON: There's a thought of taking it on tour, but I think I will have a break.
JOHN: Doing what?
SIMON: Sleeping.
(AND I ADDED THIS BIT ON 19th NOVEMBER 2018…)
Alas, Simon did not appear in the John Lewis Christmas ad on UK television but, today, The Guardian carried a story headlined MAN NAMED JOHN LEWIS STARS IN TWITTER UK's CHRISTMAS AD. Here is the Twitter ad…
…and here is John Lewis (the department store)'s actual 2018 Christmas ad – starring Elton John…
Filed under Humor, Humour, Marketing, Theatre
Tagged as @johnlewis, Blacksburg, Christmas ad, humor, humour, John Lewis, John Lewis Partnership, Simon Jay, Trumpageddon, Twitter UK, Virginia


text:
Education/Youth
Wednesday, 20/1/2021
Vietnamese Edition
Edu/Youth
Society Customs officer suspended after complaint, pending no penalty
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Customs officer suspended after complaint, pending no penalty
TN News
Thursday, December 29, 2011 18:30 Email Print
A senior customs officer has been suspended after a businessman complained about customs services in a letter to Vietnam's finance minister earlier this month.
But there's no sign that the suspension will be followed by a punishment as the first investigation by the ministry concluded the official innocent, news website VnExpress reported Thursday.
Tang Xuan Phat, an officer from the Smuggling Investigation Department at Vietnam Customs, has been temporarily removed from his office until issues mentioned in the letter can be clarified.
In mid-December, Ho Chi Minh City-based Tran Hoang Huy sent an eight-page letter to finance minister Vuong Dinh Hue saying that Phat's unit since late October has been keeping his cargo. He said the unit suspected the cargo was illegal, even after necessary documents were presented.
But the first investigation into the matter by the Finance Ministry on December 29 said the unit's officers were not corrupt.
"Their only shortcoming was to lengthen the check time for the cargo, which was due to their low capability," said Nguyen Phi Hung, head of the Customs' Smuggling Investigation Department, quoting the ministry's conclusion.
Meanwhile, it said there were signs that Huy's company evaded paying domestic taxes for the cargo, and the ministry will look deeper to determine whether the cargo was smuggled.
In the letter, the 31-year-old said that his company Huy Phat signed a contract worth VND1.6 billion (US$76,150) to buy waste tires from a company in the northern city Hai Phong. The cargo was stopped when it reached Ba Ria-Vung Tau Province, neighboring HCMC, and held by the Customs Inspection Team No.3 under Phat for many days.
Huy himself left his business in HCMC to stay at Vung Tau waiting for the customs' response, he said in the letter.
On Thursday he was notified that his cargo was released, after the cargo ship had been held for nearly two months.
It costs around VND500 million ($23,800) each month for the boat to stay idle, according to the company that rented the boat to Huy.
More Society News
Vietnamese judge rebuked for lenient ruling in underage sex case
A Vietnamese-American man was found guilty of having sex with a minor and almost escaped jail time
Hydropower dam tunnel collapses in central Vietnam flash floods, many missing
Uncooperative woman stays in car, towed to police station
Typhoon Rai weakens after hitting central Vietnam
2 get death sentence for smuggling 7kg heroin into Vietnam
Editor-in-Chief: Nguyen Quang Thong
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Thanh Nien is the tribune of Vietnam's Youth Association
Publication permit No. 14/GP-BC, granted by Press Department, Vietnam Ministry of Culture and Information.
Copyright © 2008 by Thanhniennews.com. All rights reserved. [ About us ]
Thank you readers, contributors and friends!Thanh Nien Newspaper conferred with 1st-class Labor OrderAdvertise with us


text:
While 23andMe Raises $11 Million, Mohr Davidow Sells Stake To Invest In Rival
Leena Rao	@LeenaRao / 11 years
23andme, a company that helps consumers understand and decipher their genomes, has raised a partial $11 million out of a $24.26 million B round of funding, according to regulatory filings. 23andme, which was co-founded by Google co-founder Sergey Brin's wife, Anne Wojcicki, raised $9 million in Series A funding from Google, Genentech, New Enterprise Associates, and Mohr Davidow Ventures in 2007.
Now PEHub is reporting that Mohr Davidow Ventures divested its stake in 23andme after investing in a direct competitor Navigenics. Losing one of your main investors to a competitor is not a good sign. 23andme maps customers' DNA and helps them find information about their ancestry and their risks of getting certain diseases (Michael tried it last year). Google ended up taking a $3.9 million stake in 23andMe in May 2007, after Brin had personally loaned the company $2.6 million. The debt was repaid after the Google investment. We're curious as to how much, if any, of this round's $11 million is Brin's or Google's money.
Also, we'd like to know why Mohr Davidow shifted its funds to a 23andme's rival. Navigenics' service is similar to 23andme's with slight differences. For example, 23andme offers one package, priced at $399, to test your saliva. Navigenics offers two packages of testing priced at at $499 and $2,499, with the higher priced test including genetic counseling and testing for more health conditions. Most recently, California's Department of Public Health issued cease and desist letters to 23andMe, Navigenics and 11 other genetic testing startups, mandating that the labs demonstrate that they have been certified by both the state and federal government, and, perhaps more importantly, that all genetic tests were ordered by a patient's doctor, which is required by state law.
CrunchBase Information
Navigenics
Information provided by CrunchBase


text:
Live on Sky Sports: Middlesbrough boss Aitor Karanka expecting 'good test' against QPR
Ryan Percival
The Football League weekend kicks off with Middlesbrough against Queens Park Rangers live on Sky Sports on Friday night, and Boro boss Aitor Karanka is anticipating a tough test.
Karanka's side are fourth in the table but endured the international break after losing to leaders Hull City 3-0. Rangers, meanwhile, are down in 13th having won only two of their last 11 games.
But the Teessiders' manager isn't taking anything for granted ahead of the 7.45pm kick-off, and expects the R's, who are now under the guidance of Neil Warnock after Chris Ramsey's recent departure, to improve.
"It's another good test, especially because they were in the Premier League last season and I'm sure that they're better than how they're playing at the moment," Karanka said to mfc.co.uk.
We'll be holding a minute's silence ahead of Friday's game with @QPRFC. More here - https://t.co/omTu2Nfly2pic.twitter.com/oCcY1DeP1Y
— Middlesbrough FC (@Boro) November 18, 2015
"They have good players and right now they have an experienced manager. They will have had two weeks to work on the pitch and I'm sure it's going to be really difficult.
"But we have also had two weeks to work here with the players so I hope we have a good reaction after our defeat at Hull."
Leroy Fer determined to upset the odds
QPR midfielder Leroy Fer believes Middlesbrough are the favourites for Friday night's clash, but insists he and his team-mates are travelling north to claim a win.
"Middlesbrough have started the season well; very well," Fer told qpr.co.uk.
"They're higher in the league that we are at the moment and they play at home on Friday, so they'll be favourites.
"They're a very strong team, who reached the Play-Offs last season, but we are a strong team as well and we will be going there to try and win the game."
GALLERY: #QPR warmed-up for Friday's match against @Boro at Harlington earlier today: https://t.co/V6GOPBpj5jpic.twitter.com/E1sFtHx4Bz
— QPR FC (@QPRFC) November 18, 2015
Coverage of Boro versus QPR starts at 7pm, with kick-off at 7.45pm. Tune into Sky Sports 1 to watch the game, and follow @SkyBetChamp on Twitter for our match updates.


text:
<?php
/**
 * Created by PhpStorm.
 * User: gseidel
 * Date: 2020-06-08
 * Time: 14:49
 */

namespace Enhavo\Component\Type\Tests;

use Enhavo\Component\Type\Factory;
use Enhavo\Component\Type\Registry;
use Enhavo\Component\Type\TypeCompilerPass;
use PHPUnit\Framework\TestCase;
use Symfony\Component\DependencyInjection\ContainerBuilder;
use Symfony\Component\DependencyInjection\Definition;

class TypeCompilerPassTest extends TestCase
{
    public function testProcess()
    {
        $container = new ContainerBuilder();
        $container->addDefinitions([
            'typeOne' => $this->createTypeDefinition('TypeOne'),
            'typeTwo' => $this->createTypeDefinition('typeTwo')
        ]);

        $pass = new TypeCompilerPass('Test', 'test.tag', 'ConcreteTypeClass');

        $pass->process($container);

        $container->getDefinition('typeOne');
        $registryDefinition = $container->getDefinition('Enhavo\Component\Type\RegistryInterface[Test]');
        $factoryDefinition = $container->getDefinition('Enhavo\Component\Type\FactoryInterface[Test]');

        $this->assertNotNull($registryDefinition);
        $this->assertEquals(Registry::class, $registryDefinition->getClass());
        $this->assertTrue($container->getDefinition('typeOne')->isPublic());
        $this->assertTrue($container->getDefinition('typeTwo')->isPublic());

        $this->assertNotNull($factoryDefinition);
        $this->assertEquals(Factory::class, $factoryDefinition->getClass());
    }

    private function createTypeDefinition($class)
    {
        $definition = new Definition();
        $definition->setClass($class);
        $definition->setPublic(false);
        $definition->addTag('test.tag');
        return $definition;
    }
}

redpajama_set_name: RedPajamaGithub


text:
HomeBlogPay Per ClickAdWords Trademark Policy: Unraveling the Complexities
AdWords Trademark Policy: Unraveling the Complexities
Chris Bross
It seems like just about everything is trademarked these days, doesn't it? From the obvious big brands such as 'Sony,' to seemingly random common words such as 'face' (thanks, Facebook), and 'Chicago' (belonging to Apple), you could start wondering if we have any words left that are without a legal owner. If you've ever had your AdWords ads disapproved for Trademark (TM) Policy, you may have found yourself sifting through a lot of Help Center articles and Support emails, trying to figure out how to navigate what one would think to be a simple policy.
I have had some experience with this policy, I'd like to do my best to provide a quick and dirty overview, hopefully simplifying things a bit for those who feel lost. Please note, however, that Google's TM Policy is fairly complex and differs by region, so I'll be providing links to the official policy pages throughout. This is not meant to be a comprehensive guide, and you should always refer back to Google's official pages or reach out to their Support Team with questions.
In the United States (meaning ads targeted to the US), the first thing to note about TM policy is that it only governs ad text, and does not affect keywords. As they mention frequently, Google wishes to allow advertisers as much freedom as possible in the selection of their keywords, and as such in this country they do not monitor trademarked terms in keywords. This means that if you are a trademark owner and are upset that a competitor's ad is showing up when your trademarked term is entered into the Google search bar—sorry, you're out of luck. That competitor is allowed to use your term as their keyword. Whether you agree with this or not, keep in mind that were this policy different in this regard, all advertisers would have a lot of limitations when it comes to their keyword lists, and as a result the user may suffer due to lack of choice and/or variety in sponsored results.
So, the next question to answer is that of how Google monitors trademarks in ad text. Before we jump into this, there is one very important point that should be noted:
If you have not registered your trademarked term with Google and requested that they monitor it, your trademark will not be monitored. Period. Due to what I assume is a limitation of resources, Google does not proactively search out trademarked terms to keep on file for monitoring, and therefore you must submit a TM complaint in order to begin the monitoring process.
Once Google is aware of your trademark and is actively monitoring it, you should know that there will be people who are allowed to use it in their ads. Whether you're a trademark owner or an advertiser who is using a trademarked term in ad text, you should familiarize yourself with the rules regarding who is allowed to use a trademark in ads:
Ads using the term in a generic way, not referencing the TM owner or goods associated with the TM.
Ads leading to a landing page that allows and clearly demonstrates the ability to purchase the goods/services corresponding to a TM, or that facilitates their sale.
Ads leading to a landing page that clearly demonstrates the ability to purchase components, replacement parts, or compatible products corresponding to the TM, or that facilitates their sale.
Ads leading to a landing page providing informative details about the goods/services corresponding to a TM, and that does not allow or facilitate their sale.
Basically, if a site is legitimately selling or facilitating the sale of the goods associated with the trademarked term (i.e. a Ford dealership's website, and the trademarked term 'Ford'), parts or replacements for said goods, or products closely related to these goods, it's OK to use that term in ads. Here's the caveat: the landing page must make it very clear that the above criteria are met. Using the same Ford example, if the dealership has a landing page that is nothing but a promotional video, and there is little to no mention of the word 'Ford' on that page (since all of that content is within the video itself), that page would most likely not qualify. Therefore ads using the term 'Ford' that lead to that landing page would be correctly disapproved for the Trademark Policy.
So, as an advertiser what are the key takeaways here?
Essentially, if you're using a trademarked term in your ads you simply need to ensure that your landing page makes it as clear as possible that it is either allowing the user to purchase the goods or services associated with a trademarked term, or that it is providing information about said goods or services. If you feel as though you should be authorized to use a trademarked term but don't meet the criteria, you have the option of contacting the TM owner and requesting that they submit an authorization on your behalf. Google will be able to provide you with the contact information for that person.
If you're a TM owner, what should you walk away understanding?
That you need to submit a TM complaint in order to ensure monitoring of your TM by Google
Once you've done this your TM will only be monitored in ad text and not keywords
There will still be folks who are allowed to use your TM. You should also probably familiarize yourself with your options in terms of filing complaints, since whether you submit a general or specific complaint will matter. But that's another story for another day…
Once again, this information is in reference only to Google's TM Policy in the U.S. There is a lot more information available here, here, and more information about other regions here.
AdWords PolicyPay Per Click Account ManagementPPC Account Management
Journey to PPC Conversion Optimization
PPC Disaster Preparedness
Information Harvesting Policy Change: Think You're Immune? Think Again
Tips For Boosting Your Quality Scores
Quality Scores – What It Is, How It Works & Why It's Important
Google AdWords Policy: Online Pharmacy
July 1 Huge Date in EverEffect History
Best Client Ever – TulsaChain.com, LLC
You Need a YouTube Strategy. Here's Why.
Best Client Ever – Classic Auto Insurance Agency, Inc.
Agency-Client Relationship
Creative Team Management
EverEffect News
Landing Page Testing
Local SEO Optimization
Online Marketing ROI


text:
Aile Est BAT2, Geneva Airport
New wing in the east
Genève-Cointrin Airport is increasingly becoming relevant as the number of destinations and passengers continues to increase. This is due to the popular short-haul flights of low-cost providers, on the one hand, and the growth in international organisations in Geneva, such as the WHO, on the other. Since 2011, the airport has been preparing its infrastructure for further growth in various projects. In 2014 the Federal Office for the Environment, Transport, Energy and Communications gave the go-ahead for the construction of a new east wing.
Made of steel and glass
The Aile Est is the contemporary terminal for long-haul flights at Geneva Airport. The open, light-flooded new building replaces the previous temporary building erected in 1975. The new terminal will eventually have direct parking spaces for six aircraft - previously this was only possible for half the aircraft, and bus gates were built for the other three. The glass and steel construction, which is inclined by 26 degrees to the asphalt, is 520 metres long and 20 metres wide.
Aéroport International, Geneva
General contractor with full cost, quality and deadline guarantee
Rogers Stirk Harbour + Partners, Châtelaine
Jacques Bugna SA, Geneva
194,000 sqm
Overall cost
CHF 321 million
Duration of construction
Tamina thermal spa, Bad Ragaz
SwissTech Convention Center, Lausanne - Ecublens


Take advantage of one of the best golf clubs in Oxfordshire with The Springs Golf Club Corporate Golf Packages.
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Enjoy a working day like no other with a round (or two!) of golf at our well renowned course whilst taking in the stunning scenery which surrounds you. Then feel free to visit our club house for a beverage and to continue conversations after a day on the course or dine at our popular on-site restaurant for an unforgettable working day.
Take advantage of all of the benefits that come with the Platinum Package as well as added extras when opting for the plus package.
Have a question about our new corporate membership packages? Get in touch by calling 07976 223 963 or email golf@darwinbusiness.co.uk and a member of the team will be in touch as soon as possible.


There are a few different ways to design a notebook so that it can also be used as a tablet. You can offer a detachable keyboard, a hinge that lets you twist the screen and push it down over the keyboard, or a design that lets the screen rotate within its frame.
Asus opted for a different option with its new TAICHI series ultrabooks. There are two displays: one on each side of the laptop lid.
Close the laptop and you're looking at a touchscreen tablet. Open the lid and you're looking at a normal laptop-style computer.
Interestingly, you can display separate content on each screen — which means that you can also share the computer with a friend sitting across the tablet from you while one of you plays a game and the other surfs a web. I imagine this would also be a killer setup for some Battleship-style video game.
While you'd think a computer with two touchscreen LCD displays would be a bit on the thick side, the Asus TAICHI doesn't look any thicker than any of the company's Zenbook ultrabooks.
The TAICHI will come with an 11.6 inch or 13.3 inch screen. Both models are available with high resolution 1920 x 1080 pixel Super IPS+ displays.
The notebooks feature Intel Ivy Bridge processors, 4GB of RAM, solid state disks, and front-and rear facing cameras, as well as USB 3.0, micro VGA and mini DisplayPort. The computer also features a gyroscopic sensor, NFC chip, and an e-compass.
There's no word on the price or release date, but since the TAICHI notebooks appear to be running Windows 8, it's likely that they won't be available until this fall at the earliest.
Can it be undocket ?
No, seriously, what's the point in this innovation?
Typical ingeneer product. No market for it.
And I dont wanna think about the price nor the life expentency of the screen on the lid.
And the battery if the 2 screens are used at the same time (wich is kinda the purpose of 2 screens instead of one we can rotate, I guess).
what's the point in this innovation?
Obviously you are in laggard adopter territory as I see this is a great way for those that own both an Ultrabook and a Tablet. This highly increases portability for those users (whom are the majority of the actual tech users). Please excuse yourself if you do not relate….
This is a great idea – I wondered when it was coming. Apple has been using the backlight to illuminate their logo for years – why not do something more interesting with the lid = back of the lcd panel?
No need to run both displays at same time – the idea is very quick and easy conversion from tablet to laptop with full keyboard. Swivel hinges are cumbersome and add weight and thickness. So are docking hinges, requiring alignment and leaving a wobble.
As screens get cheaper and cheaper, dual is the way to go.
Maybe you're right. But I really dont see the point.
A hybrid, why not… But this is just for showing off and geeks imo.
Oubviously, I'm not the target. But I don't see the real added value in it beside the wow factor (which is not bad in itself for some people I guess).
Still, one thing we can praise Asus for it's its innovation(s) mentality.
But if you do good market research, you don't need to throw everything and anything you can produce out there. Just stick with what you know will work.
Otherwise that's a lot of ressources wasted for experiments at this scale.
Ressources they could allocate on lowering price on other products for example.
Still, it's not my money and, in terms of image and brand equity/awareness, it seems to be a success.
They are on their way to become the 'new Sony'.
But Asus is not bad at all on the premium pricing either.
The difference is that Samsung create technology and Asus concepts.
So, we could indeed say that Samsung is closer to what Sony once was but the product mentality and the desire to innovate is definetely on Asus side. At least, we can say it's more an 'in your face' (plays on the wow factor, the eyes (there are good looking products), and the geek side that is in all men -don't know about women on this one-) approach.
It does tend to confuse or raise some questions as far as I'm concerned but all in all that's a good approach and a risky one at that (in termes of ressources and dilution of their image). But it will pay off.
I personally love this idea, I'm holding off buying the Transformer Infinity and waiting on this one. Sick and tired and carrying 2 portable electronics on the go (MBA w/Windows 7 & Transformer Prime tablet). It's a hassle to integrate datas on both machine as I use both depending on where I am. All in one ultra-tablet is a clear winner to me along with many other tech users in similar situation….If you own both an ultrabook & a tablet, you will love the TaiChi….great for business/work portability. Just hurry up and release it already so I can get rid of this Macbook air & Prime hassle….
Should I replace my mid 2010 MacBook pro? Been thinking about just getting a surface or transformer 800. But the 800 runs on intel atom. Not a big fan of windows 7. Maybe the transformer book would be better? I can see where the combination of touch and mouse will be usefull. Too bad only one is touchscreen. This will probably be to expensive. Looking for around $800. I wonder if you can hook up a mouse and keyboard and use the other side as a laptop too. That would be awesome.


Admit it. When you see someone with a beautiful smile, you notice it.
Our smiles positively impact our social, business, and romantic situations.
When you smile, you look good. When others see you smile, they smile too.
Some studies have even borne out that smiling can be good for your health! Smiling has been associated with reducing stress hormone levels and increasing levels of mood enhancing hormones. It has also been connected to lowering blood pressure.
It goes without saying that if you're not happy with the way your smile looks, you won't smile as much. If you don't smile, you miss out on some of those health benefits.
The beauty of your smile depends on a lot more than just how your teeth look. Your gums — whether they rest too high or too low on your teeth — can have a big impact on your smile and how you feel about it.
Most people think of improving the look of their teeth when they think of what they can do to make their smile better. But did you know that even problems with the gumline can be fixed?
At Natural Smiles in Louisville KY, we can fix those problems with gum contouring.
Let's face it you could have straight, white teeth, but if your gumline is uneven, you might not be happy with your smile. Ideally, gums should appear smooth and even and act as a frame for your teeth. It won't matter how good your teeth look if the frame around them is off.
Gums that cover too much of your teeth can make your teeth look too small. Gums that cover too much of your teeth could just be a hereditary issue or a result of a health problem.
Gums that are too high can make your teeth look long. This condition is also called gum recession. It can be a sign of gum disease.
Does your smile show more gum than teeth when you smile? If so, you have what we call a gummy smile. A gummy smile happens when your lip sits just above the top of the teeth. It gives you a disproportionate gum-to-tooth ratio.
You don't have to live with a smile you don't like! We can contour your gums and give you a smile you can be proud of. You'll be amazed at the change.
If you have a gummy smile, gum reshaping, also called gum contouring, treatment can help.
When you come in to see us, Dr. Gill will take a look at your teeth and gums and go over with you what she recommends as the right treatment.
Using a special dental laser, Dr. Gill can painlessly remove excess gum tissue. It's a painless procedure because we use a diode laser to sculpt your gums.
To start the process, Dr. Gill will numb your mouth with a local anesthetic. She'll use the laser to remove excess gum tissue and create a more uniform gumline.
The laser means less bleeding and a reduced chance for infection. The whole procedure can be done in just one visit to our office.
The discomfort after a gum reshaping procedure should be minimal. Your gums may be tender for about a week, but you shouldn't have any problems speaking, eating, or drinking. Ibuprofen should be all you need to manage the discomfort.
The opposite of a gummy smile, of course, is the toothy smile. If needed, Dr. Gill can graft gum tissue to your existing gums, covering more of the teeth so your smile looks great.
There may be some minor swelling and tenderness after this procedure as well. It's normal and will go away in a few days. If needed, you can apply an ice bag to the surgical area for quick relief.
For quicker healing after a gum reshaping procedure, it's best to eat softer foods. Don't smoke, and don't brush your teeth near the location of the surgery for at least one day.
Call us today at 502-785-8695 for an appointment or use our online form. Let us show you what we can do for your smile!


Morocco (المغرب in Arabic) is a country in northern Africa on the Mediterranean coast. The capital is Rabat and the currency is the Moroccan dirham. The current king of Morocco is Mohammed VI of Morocco.
Morocco shares borders with Spain along the coast with Spain's exclaves: Ceuta, Melilla, and Peñón de Vélez de la Gomera, Algeria to the east and Western Sahara to the south. Western Sahara is claimed by Morocco as a part of the country so technically Morocco also shares a border with Mauritania in the south.
The highest mountain in Morocco is Toubkal at 4,167 m.
Arabic and Berber are the country's official languages. Islam is the main religion in Morocco and is practised by almost all of the population.
This page was last edited on 12 March 2017, at 23:23.