text:
\section{Introduction}
When the linear approximation is zero, the homogeneous one can be used
for nonlinear systems analysis and control. Generalised definitions such as weighted homogeneity \cite{rosier1992homogeneous}, \cite{bacciotti2006liapunov} or in the bi-limit \cite{andrieu2008homogeneous} allow covering wider classes of nonlinear systems. The Lyapunov framework has produced significant results on stability \cite{zubov1964methods}, \cite{hermes1991homogeneous}, robustness \cite{rosier1992homogeneous} as well as observer and controller design \cite{hermes1991homogeneous}, to name a few. \\
To study homogeneous systems with delays, researchers have naturally resorted to the Lyapunov-Razumikhin framework \cite{efimov2011homogeneity}. Some general results on delay-independent and finite-time stability for the cases of positive and negative degree, respectively, as well as for stability of locally homogeneous systems are obtained in \cite{efimov2016weighted, efimov2016global}. For weighted homogeneous systems of positive degree, the delay-independent stability was established with the help of the Lyapunov function of the corresponding delay-free system \cite{aleksandrov2014weighted}. The approach has allowed to present the estimates for solutions \cite{aleksandrov2012asymptotic} and for the attraction region, as well as to analyze perturbed systems and complex systems describing the interaction between several subsystems \cite{aleksandrov2014delay}. Moreover, contrary to the often expressed view that the Razumikhin approach allows obtaining qualitative estimates only, it was shown recently \cite{portilla2020comparison} that the estimates of \cite{aleksandrov2012asymptotic} are close enough to the system response. A similar conclusion was made in \cite{efimov2020onestimation} for different kind of systems.\\
Recently, for the case of standard dilation and homogeneity degree strictly greater than one, a Lyapunov-Krasovskii functional was introduced in \cite{Voronezh}, \cite{alexandrova2019lyapunov}. It was inspired by the Lyapunov functional of complete type for delay linear systems \cite{kharitonov2003lyapunov}, \cite{kharitonov2013time}, and lead to stability and instability results \cite{zhabko2020automatica}, estimates of the region of attraction \cite{alexandrova2019lyapunov} and of the system response \cite{portilla2020comparison}, see also \cite{portilla2020thesis}.\\
In this contribution, we extend this functional to the case of weighted homogeneous time-delay systems of positive degree and use it to construct quantitative estimates of the region of attraction and of the system response.
Two approaches are developed. The first one is based on the classical ideas of the Lyapunov-Krasovskii method, whereas a combination of Lyapunov-Krasovskii and Razumikhin techniques is used in the second one. The idea is to construct a lower bound for the Lyapunov-Krasovskii functional which is only valid on a special set of functions inspired by the Razumikhin condition, see \cite{medvedeva2015synthesis}  and \cite{alexandrova2018junction} for the linear and nonlinear cases, respectively.
Exploiting the ideas in \cite{aleksandrov2012asymptotic}, we require the solutions of the comparison equation for the functional to satisfy the same condition, thus ensuring  that the final estimates hold for all solutions from a certain neighbourhood. This approach yields better estimates than the classical one.\\
Note that there exists a parallel work on the generalization of the functional of \cite{Voronezh} to the case of weighted dilation covering both asymptotic stability for the case of positive
degree and boundedness of solutions for that of negative degree \cite{efimov2020analysis}. The main difference between the functional we use and those of \cite{efimov2020analysis} is that to cover both cases the authors of \cite{efimov2020analysis} use a more complex construction with additional parameters, a more complicated bounding and non-standard norms, thus achieving a moderate computational performance. In contrast, we bound the functional and its derivative componentwise following naturally the componentwise definition of homogeneous functions, and use natural norm based on the homogeneous vector norm. Additionally, we present fully computed quantitative estimates of the response and of the attraction region, via this combined Lyapunov-Krasovskii-Razumikhin approach.
\\
The contribution is organised as follows. Previous results on homogeneous systems are reminded in section II. The Lyapunov-Krasovskii functional construction is presented in Section III.  The functional is applied to the estimation of the attraction region in Section~IV and of the homogeneous system solutions in Section~V. An illustrative example is given in Section~VI. \\
\textbf{Notation: } The space of $\mathbb{R}^n$ valued continuous functions on $[-h,0]$ endowed with the norm $ \|\varphi\|_h=\max_{\theta\in[-h,0]}\|\varphi(\theta)\|$ is denoted by $C([-h,0],\mathbb{R}^n)$. Here, $\|\cdot\|$ stands for the Euclidean norm. In computations it turns out to be more convenient to use the following homogeneous norm
$$ \|\varphi\|_\mathscr{H}=\max_{\theta\in[-h,0]}\|\varphi(\theta)\|_{r,p},$$
where $\|\cdot\|_{r,p}$ stands for the typical vector homogeneous norm defined below. The solution of a time delay system and the restriction of the solution to the segment $[t-h,t]$, corresponding to the initial function $\varphi\in C([-h,0],\mathbb{R}^n),$ are respectively denoted by $x(t)$ and $x_t$. If the initial condition is important, we write $x(t,\varphi)$ and $x_t(\varphi)$, respectively.

\section{Preliminaries}
We start with a brief reminder of the definitions related to the homogeneity concept \cite{bacciotti2006liapunov,zubov1962oxford}. Define the vector of weights \mbox{$r=(r_1,\ldots,r_n)^T,$} where $r_i>0,$ $i=\overline{1,n},$ and the dilation operator
\begin{equation*}
    \delta_\varepsilon^r(\textup{x})=(\varepsilon^{r_1}\textup{x}_1,\ldots,\varepsilon^{r_n}\textup{x}_n),\quad \varepsilon>0.
\end{equation*}
Here, $\textup{x}=(\textup{x}_1,\ldots,\textup{x}_n)^T.$
Then, function
\begin{equation*}
    \|\textup{x}\|_{r,p}=\left(\sum_{i=1}^n |\textup{x}_i|^{p/r_i}\right)^{1/p},
\end{equation*}
where $p\geq 1,$ is called \textit{the $\delta^r$-homogeneous norm}. Although it is not a norm in the usual sense, it has been shown to be equivalent to the Euclidean norm. A scalar function $V:\mathbb{R}^n\rightarrow \mathbb{R}$ is called \textit{$\delta^r$-homogeneous}, if there exists $\mu\in\mathbb{R}$ such that
\begin{equation*}
    V(\delta_\varepsilon^r(\textup{x}))=\varepsilon^\mu V(\textup{x})\quad\forall\,\varepsilon>0.
\end{equation*}
A vector function $f=f(\textup{x},\textup{y}):\mathbb{R}^{2n}\rightarrow\mathbb{R}^{n}$ is called \textit{$\delta^r$-homogeneous}, if there exists $\mu\in \mathbb{R}$ such that its component $f_i$ is a $\delta^r$-homogeneous function of degree $\mu+r_i,$ i.e.
\begin{equation*}
    f_i(\delta_\varepsilon^r(\textup{x}),\delta_\varepsilon^r(\textup{y}))=\varepsilon^{\mu+r_i}f_i(\textup{x},\textup{y})\quad\forall\,\varepsilon>0,\quad i=\overline{1,n},
\end{equation*}
where $\textup{x},\textup{y}\in\mathbb{R}^n.$
In both cases, the constant $\mu$ is called \textit{the degree of homogeneity}.
It is worth mentioning that the homogeneous norm is a $\delta^r$-homogeneous function of degree one: $\|\delta_\varepsilon^r(\textup{x})\|_{r,p}=\varepsilon\|\textup{x}\|_{r,p}.$
Assume that $\mu\geq -\min_{i=\overline{1,n}} r_i.$

\begin{lemma} \label{lemma:bound_f_i}
There exist $m_i>0$ such that the components of the $\delta^r$-homogeneous function $f(\textup{x},\textup{y})$ satisfy
\begin{equation*
    |f_i(\textup{x},\textup{y})|\leq m_i\left(\|\textup{x}\|_{r,p}^{\mu+r_i}+\|\textup{y}\|_{r,p}^{\mu+r_i}\right),\quad i=\overline{1,n}.
\end{equation*}
\end{lemma}
\begin{proof} If $\mu+r_i>0,$ then we take
$$
m_i=\max_{\|\textup{x}\|_{r,p}^{\mu+r_i}+\|\textup{y}\|_{r,p}^{\mu+r_i}=1}|f_i(\textup{x},\textup{y})|>0,$$
and $\varepsilon=(\|\textup{x}\|_{r,p}^{\mu+r_i}+\|\textup{y}\|_{r,p}^{\mu+r_i})^{-1/(\mu+r_i)}.$ It can be easily seen that
$$
\|\delta_\varepsilon^r(\textup{x})\|_{r,p}^{\mu+r_i} + \|\delta_\varepsilon^r(\textup{y})\|_{r,p}^{\mu+r_i}=1.
$$
This implies
$$
    |f_i(\delta_\varepsilon^r(\textup{x}),\delta_\varepsilon^r(\textup{y}))|\leq m_i.
$$
Using the definition of homogeneity, we arrive at the result. If \mbox{$\mu+r_i=0,$} then the same conclusion can be drawn with
$$
m_i=\max_{\|\textup{x}\|_{r,p}^{k}+\|\textup{y}\|_{r,p}^{k}=1}|f_i(\textup{x},\textup{y})|>0
$$
for any $k>0.$
\end{proof}
\begin{lemma} \label{lemma:bound_der_f_ij}
Assume that $f(\textup{x},\textup{y})$ is continuously differentiable with respect to $\textup{x}$ and $\delta^r$-homogeneous. Then, there exist \mbox{$\eta_{ij}>0$} such that
\begin{equation*
    \left|\frac{\partial f_i(\textup{x},\textup{y})}{\partial \textup{x}_j}\right|\leq \eta_{ij}\left(\|\textup{x}\|_{r,p}^{\mu+r_i-r_j}+\|\textup{y}\|_{r,p}^{\mu+r_i-r_j}\right),\ i,j=\overline{1,n},
\end{equation*}
if $\mu+r_i-r_j>0,$ and
\begin{equation*
    \left|\frac{\partial f_i(\textup{x},\textup{y})}{\partial \textup{x}_j}\right|\leq \frac{\eta_{ij}}{\left(\|\textup{x}\|_{r,p}^{-\mu-r_i+r_j}+\|\textup{y}\|_{r,p}^{-\mu-r_i+r_j}\right)},\ i,j=\overline{1,n},
\end{equation*}
if $\mu+r_i-r_j<0$ and
at least one of the vectors $\textup{x}$ and $\textup{y}$ is nonzero.
\end{lemma}
Now, consider a time delay system of the form
\begin{equation} \label{eq:time-delay_system}
\dot{x}(t)=f(x(t),x(t-h)),
\end{equation}
where $x(t)\in \mathbb{R}^n,$ $h>0$ is a constant delay. The following assumptions are made.
\begin{assumption}\label{as:hom_degree}
The vector function $f(\textup{x},\textup{y})$ is continuously differentiable with respect to $\textup{x}\in\mathbb{R}^n,$ locally Lipshitz with respect to $\textup{y}\in\mathbb{R}^n,$ and $\delta^r$-homogeneous of degree $\mu>0.$
\end{assumption}
\begin{assumption}\label{as:delay_free_asymp}
The delay free system
\begin{equation} \label{eq:delay-free_system}
    \dot{x}(t)=f(x(t),x(t))
\end{equation}
is asymptotically stable.
\end{assumption}
In \cite{rosier1992homogeneous}, \cite{zubov1962oxford} the existence of a homogeneous Lyapunov function for system \eqref{eq:delay-free_system} is established. More precisely, it is proven that for any $l\in\mathbb{N}$ and $\gamma\geq l\max_{i=\overline{1,n}}\{r_i\}$ there exists a positive definite $\delta^r$-homogeneous of degree $\gamma$ and of class $C^l$ Lyapunov function $V(\textup{x})$ such that its time derivative with respect to system \eqref{eq:delay-free_system} is a negative definite
$\delta^r$-homogeneous function of degree $\gamma+\mu,$ that is
\begin{equation}
\label{eq:bound_dot_V}
\left(\frac{\partial V(\textup{x})}{\partial \textup{x}}\right)^T f(\textup{x},\textup{x})\leq -\textup{w}\|\textup{x}\|_{r,p}^{\gamma+\mu},\quad \textup{w}>0.
\end{equation}
We set $l=2$ and use a Lyapunov function $V(\textup{x})$ of class $C^2$ and the homogeneity degree $\gamma\geq 2\max_{i=\overline{1,n}}\{r_i\}$ below. The following estimates hold \cite{bacciotti2006liapunov}, \cite{zubov1962oxford}:
\begin{gather}\label{eq:bound_V}
\alpha_0\|\textup{x}\|_{r,p}^\gamma\leq V(\textup{x})\leq \alpha_1\|\textup{x}\|_{r,p}^\gamma,\quad \alpha_0,\alpha_1>0,\\
\label{eq:bound_dV}
\left|\frac{\partial V(\textup{x})}{\partial \textup{x}_i}\right|\leq \beta_{i}\|\textup{x}\|_{r,p}^{\gamma-r_i},\quad \left|\frac{\partial^2 V(\textup{x})}{\partial \textup{x}_i \partial \textup{x}_j}\right|\leq \psi_{ij}\|\textup{x}\|_{r,p}^{\gamma-r_i-r_j},
\end{gather}
where $\beta_{i},\psi_{ij}>0,$ $i,j=\overline{1,n}.$

It is proved in \cite{aleksandrov2014weighted} that if Assumptions~\ref{as:hom_degree} and \ref{as:delay_free_asymp} hold, then the trivial solution of time delay system \eqref{eq:time-delay_system} is asymptotically stable for all $h>0.$ In the next section, we present the Lyapunov-Krasovskii functional validating this result, and further construct the estimates for the solutions of \eqref{eq:time-delay_system} and for the attraction region. An important step in the obtention of the estimates is the use of a lower bound for the functional on the special set
\begin{multline*}
S_\alpha = \Bigl\{\varphi\in C([-h,0],\mathbb{R}^n)\Bigl\arrowvert\\
\|\varphi(\theta)\|_{r,p}\leq \alpha\|\varphi(0)\|_{r,p},\;\theta\in[-h,0]\Bigr\},
\end{multline*}
where $\alpha>1.$ This set was introduced in the Lyapunov-Krasovskii analysis in \cite{medvedeva2015synthesis} and \cite{alexandrova2018junction} for linear and nonlinear time delay systems, respectively. In particular, it was shown that it is enough to construct the lower bound for the functional on the set $S_\alpha$ in order to prove asymptotic stability.
\section{The Functional Construction}
A natural generalization of the Lyapunov-Krasovskii functional introduced in \cite{Voronezh,alexandrova2019lyapunov} to the case of $\delta^r$-homogeneous time delay systems is
\begin{align}\label{eq:functional}
v(\varphi)&=V(\varphi(0))+\left.\left(\frac{\partial V(\textup{x})}{\partial \textup{x}}\right)^T \right|_{\textup{x}=\varphi(0)}\int_{-h}^{0}f(\varphi(0),\varphi(\theta))d\theta
\\
&+\int_{-h}^{0}(\mathrm{w_1}+(h+\theta)\mathrm{w_2})\|\varphi(\theta)\|_{r,p}^{\gamma+\mu}d\theta.\notag
\end{align}
Here, $\mathrm{w}_1, \mathrm{w}_2>0,$ and $\mathrm{w}_0=\mathrm{w}-\mathrm{w}_1-h\mathrm{w}_2>0$. In this section, we show that functional \eqref{eq:functional} satisfies the classical Lyapunov-Krasovskii theorem \cite{hale2013introduction}.
For the sake of brevity, the three summands of \eqref{eq:functional} are denoted by $I_1(\varphi)$, $I_2(\varphi)$ and $I_3(\varphi),$ respectively.

\begin{lemma}
\label{lemma:lower_bound}
There exist $\delta>0$ and $a_1,\,a_2>0$ such that
\begin{equation} \label{eq:lower_bound}
    v(\varphi)\geq a_1\|\varphi(0)\|_{r,p}^\gamma+a_2\int_{-h}^{0}\|\varphi(\theta)\|_{r,p}^{\gamma+\mu} d\theta
\end{equation}
in the neighbourhood $\|\varphi\|_\mathscr{H}\leq \delta.$
\end{lemma}
\begin{proof}
It is obvious that
\begin{equation*}
    I_1(\varphi)\geq \alpha_0\|\varphi(0)\|_{r,p}^\gamma,\;\, I_3(\varphi)\geq \mathrm{w}_1\int_{-h}^{0}\|\varphi(\theta)\|_{r,p}^{\gamma+\mu}d\theta.
\end{equation*}
Now, we use Lemma~\ref{lemma:bound_f_i} and the first of bounds \eqref{eq:bound_dV} for the second summand of the functional:
\begin{align*}
|I_2(\varphi)|
&\leq h\left(\sum_{i=1}^n \beta_i m_i\right)\|\varphi(0)\|_{r,p}^{\gamma+\mu} + \sum_{i=1}^n\beta_i m_i\chi^{\gamma-\mu-2r_i}\\
&\times\left(\frac{\|\varphi(0)\|_{r,p}}{\chi}\right)^{\gamma-r_i}
\int_{-h}^{0}(\chi\|\varphi(\theta)\|_{r,p})^{\mu+r_i}d\theta,
\end{align*}
where $\chi>0$ is a parameter. Using the standard inequality $A^{p_1} B^{p_2}\leq A^{p_1+p_2}+B^{p_1+p_2},$ where $p_1,p_2\geq0$ and $A,B\geq0$, we get
\begin{align*}
|I_2(\varphi)|&\leq h\left(\sum_{i=1}^n \beta_i m_i \left(1+\chi^{-2(\mu+r_i)}\right)\right)\|\varphi(0)\|_{r,p}^{\gamma+\mu}\\&+\left(\sum_{i=1}^n\beta_i m_i\chi^{2(\gamma-r_i)}\right)\int_{-h}^{0}\|\varphi(\theta)\|_{r,p}^{\gamma+\mu}d\theta.
\end{align*}
Combining all summands together and making use of $I_2(\varphi)\geq -|I_2(\varphi)|$ along with $\|\varphi\|_\mathscr{H}\leq \delta,$ we arrive at the lower bound \eqref{eq:lower_bound} with
\begin{align*}
    a_1&=\alpha_0-h\sum_{i=1}^n\beta_i m_i\left(1+\chi^{-2(\mu+r_i)}\right)\delta^{\mu},\\
    a_2&=\mathrm{w}_1-\sum_{i=1}^n\beta_i m_i\chi^{2(\gamma-r_i)}.
\end{align*}
Here, the parameter  $\chi>0$ is chosen such that $a_2>0,$ and
\begin{equation} \label{H1}
    \delta<H_1 =\left(\frac{\alpha_0}{h\sum_{i=1}^n\beta_i m_i\left(1+\chi^{-2(\mu+r_i)}\right)}\right)^{1/\mu}.
\end{equation}
\end{proof}

\begin{lemma}
\label{lemma:derivative}
There exist $\delta>0$ and $c_0,c_1,c_2>0$ such that the time derivative of functional~\eqref{eq:functional} along the solutions of system \eqref{eq:time-delay_system} satisfies
\begin{align} \label{eq:derivative}
\frac{\mathrm{d}v(x_t)}{\mathrm{d}t}&\leq -c_0\|x(t)\|_{r,p}^{\gamma+\mu}-c_1\|x(t-h)\|_{r,p}^{\gamma+\mu}\\&-c_2\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{\gamma+\mu}d\theta,\quad\text{if}\quad \|x_t\|_\mathscr{H}\leq \delta.\notag
\end{align}
\end{lemma}
\begin{proof}
Similarly to the case of standard dilation \cite{alexandrova2019lyapunov}, we differentiate the functional along the solutions of system \eqref{eq:time-delay_system}:
\begin{gather*}
\begin{split}
    \frac{\mathrm{d}v(x_t)}{\mathrm{d}t}
    &= -\mathrm{w}_0\|x(t)\|_{r,p}^{\gamma+\mu}-\mathrm{w}_1\|x(t-h)\|_{r,p}^{\gamma+\mu}\\
    &-\mathrm{w}_2\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{\gamma+\mu}d\theta + \sum_{j=1}^{2}\Lambda_j,\quad\text{where}
    \end{split}\\
    \begin{split}
    \Lambda_1&=\sum_{i,j=1}^n \left.\frac{\partial V(\textup{x})}{\partial \textup{x}_i} \right|_{\textup{x}=x(t)}\left.\int_{t-h}^{t}\frac{\partial f_i(\textup{x},x(s))}{\partial \textup{x}_j}\right|_{\textup{x}=x(t)} \mathrm{d}s\\
&\times f_j(x(t),x(t-h)),\quad
\Lambda_2=\sum_{i,j=1}^n f_i(x(t),x(t-h))\\&\times\left.\left(\frac{\partial^2 V(\textup{x})}{\partial \textup{x}_i\textup{x}_j}\right)\right|_{\textup{x}=x(t)}\int_{-h}^{0}f_j(x(t),x(t+\theta))\mathrm{d}\theta.
\end{split}
\end{gather*}
Next, we estimate $\Lambda_1$ and $\Lambda_2$ with the help of Lemmas~\ref{lemma:bound_f_i}, \ref{lemma:bound_der_f_ij} and inequalities \eqref{eq:bound_dV}. We introduce the sets of indices
\begin{align*}
    R_1&=\{(i,j):\;i,j=\overline{1,n},\;\mu+r_i-r_j\geq 0\},\\
    R_2&=\{(i,j):\;i,j=\overline{1,n},\;\mu+r_i-r_j< 0\}
\end{align*}
for the estimation of $\Lambda_1.$ Notice that Lemma~\ref{lemma:bound_der_f_ij} implies that
$$
\left|\frac{\partial f_i(\textup{x},\textup{y})}{\partial \textup{x}_j}\right|\leq \eta_{ij},\quad (i,j)\in R_2,
$$
hence,
\begin{gather*}
 \Lambda_1\leq \!\!\!\sum_{(i,j)\in R_1}\!\!\!\beta_i m_j \|x(t)\|_{r,p}^{\gamma-r_i}(\|x(t)\|_{r,p}^{\mu+r_j}+\|x(t-h)\|_{r,p}^{\mu+r_j})\\
 \times\int_{-h}^{0}\eta_{ij}(\|x(t)\|_{r,p}^{\mu+r_i-r_j}+\|x(t+\theta)\|_{r,p}^{\mu+r_i-r_j})d\theta\\
 +\!\!\!\!\!\sum_{(i,j)\in R_2}\!\!\!\!\! h\beta_i m_j \eta_{ij} \|x(t)\|_{r,p}^{\gamma-r_i}(\|x(t)\|_{r,p}^{\mu+r_j}+\|x(t-h)\|_{r,p}^{\mu+r_j}),\\
 \Lambda_2\leq \sum_{i=1}^n\sum_{j=1}^n m_i m_j \psi_{ij}(\|x(t)\|_{r,p}^{\mu+r_i}+\|x(t-h)\|_{r,p}^{\mu+r_i})
 \end{gather*}
 \begin{gather*}
 \times\|x(t)\|_{r,p}^{\gamma-r_i-r_j}\int_{-h}^{0}(\|x(t)\|_{r,p}^{\mu+r_j}+\|x(t+\theta)\|_{r,p}^{\mu+r_j})d\theta.
 \end{gather*}
Using the standard inequality $$A^{p_1} B^{p_2} C^{p_3}\leq A^{p_1+p_2+p_3}+B^{p_1+p_2+p_3}+C^{p_1+p_2+p_3},$$ where $p_1,p_2,p_3\geq0$ and $A,B,C\geq0$, and defining
\begin{align*}
s_{ij}&=\left\{
\begin{aligned}
&1,&\quad &(i,j)\in R_1,\\
&\delta^{r_j-r_i-\mu},&\quad &(i,j)\in R_2,
\end{aligned}\right.\\
L&=\sum_{i=1}^n\sum_{j=1}^n m_j\left(\beta_i\eta_{ij}s_{ij}+m_i\psi_{ij}\right),\\
g(x_t)&=4h\|x(t)\|_{r,p}^{\gamma+2\mu}+2h\|x(t-h)\|_{r,p}^{\gamma+2\mu}\\
&+2\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{\gamma+2\mu},
\end{align*}
we arrive at
$
\Lambda_1+\Lambda_2 \leq L g(x_t).
$
Since $\mu>0,$ we obtain bound \eqref{eq:derivative} with
\begin{align*}
    c_0=\mathrm{w}_0-4hL\delta^\mu,\;\, c_1=\mathrm{w}_1-2hL\delta^\mu,\;\,c_2=\mathrm{w}_2-2L\delta^\mu.
\end{align*}
It is enough to choose
\begin{equation} \label{H2}
\delta<H_2=\left(\min\left\{\frac{\mathrm{w}_0}{4hL},\frac{\mathrm{w}_1}{2hL},\frac{\mathrm{w}_2}{2L}\right\}\right)^{1/\mu}
\end{equation}
to end the proof.
\end{proof}

\begin{lemma}
\label{lemma:upper_bound}
There exist $b_1,b_2>0$ such that
\begin{equation} \label{eq:upper_bound}
    v(\varphi)\leq b_1\|\varphi(0)\|_{r,p}^{\gamma}+b_2\int_{-h}^{0}\|\varphi(\theta)\|_{r,p}^{\gamma}d\theta,
\end{equation}
if $\|\varphi\|_\mathscr{H}\leq \delta.$
\end{lemma}
\begin{proof}
The bound is obtained straightforwardly with
\begin{align*}
    b_1&=\alpha_1+2h\left(\sum_{i=1}^n \beta_i m_i\right)\delta^\mu,\\ b_2&=\left(\mathrm{w}_1+h\mathrm{w}_2+\sum_{i=1}^n\beta_i m_i\right)\delta^{\mu}.
\end{align*}
\end{proof}
\begin{corollary}
Functional \eqref{eq:functional} admits an upper bound
\begin{gather}\label{eq:sec_upper_bound_v}

    v(\varphi)\leq \alpha_1\|\varphi(0)\|^\gamma + b_3 \|\varphi\|_\mathscr{H}^{\gamma+\mu},
\end{gather}
where $$b_3=\left(\mathrm{w}_1+h\mathrm{w}_2+2h\sum_{i=1}^n \beta_im_i\right)h.$$
\end{corollary}

Now, we present a lower bound for the functional $v(\varphi)$ on the set $S_{\alpha}.$ This bound  will be useful for the construction the of the estimates in Sections~IV and V.
\begin{lemma}
\label{lemma:lower_bound_set_S}
There exist $\delta>0$ and $\tilde{a}_1=\tilde{a}_1(\alpha)>0$ such that
\begin{equation} \label{eq:lower_bound_set_S}
    v(\varphi)\geq \tilde{a}_1\|\varphi(0)\|_{r,p}^\gamma+\mathrm{w}_1\int_{-h}^{0}\|\varphi(\theta)\|_{r,p}^{\gamma+\mu} d\theta,
\end{equation}
if $\varphi\in S_\alpha$ and $\|\varphi\|_\mathscr{H}\leq \delta.$
\end{lemma}
\begin{proof}
Using $\|\varphi(\theta)\|_{r,p}\leq \alpha\|\varphi(0)\|_{r,p},$ \mbox{$\theta\in[-h,0],$} for the estimation of the second summand, we obtain that bound \eqref{eq:lower_bound_set_S} is satisfied with
\begin{gather}
\tilde{a}_1=\alpha_0-h\left(\sum_{i=1}^n (1+\alpha^{\mu+r_i})m_i\beta_i\right) \delta^\mu,\quad\text{where} \notag
\\
 \label{H3}
    \delta<H_3=\left(\frac{\alpha_0}{h\sum_{i=1}^n (1+\alpha^{\mu+r_i})m_i\beta_i}\right)^{1/\mu}.
\end{gather}
\end{proof}

\section{Estimates for the Attraction Region}
Lemmas~\ref{lemma:lower_bound}--\ref{lemma:lower_bound_set_S} allow us to present straightforwardly estimates of the domain of attraction of the trivial solution of system \eqref{eq:time-delay_system}. The proofs are very similar to that in \cite{alexandrova2019lyapunov} (see Corollary~10 and Remark~11). The estimates differ in the lower bound for the functional used: bound \eqref{eq:lower_bound} in Theorem~\ref{thm:attr_region_LK_w} and bound \eqref{eq:lower_bound_set_S} in Theorem~\ref{thm:attr_region_LK_set_S}.

\begin{theorem}\label{thm:attr_region_LK_w}
Let $\Delta$ be a positive root of equation
\begin{equation*}
\alpha_1\Delta^\gamma + b_3 \Delta^{\gamma+\mu} = a_1\delta^\gamma,
\end{equation*}
where $\delta<\min\{H_1,H_2\},$ and $H_1$ and $H_2$ are defined by \eqref{H1} and \eqref{H2}, respectively.
If system \eqref{eq:delay-free_system} is asymptotically stable, then the set of initial functions
\begin{equation*}
\Omega=\{\varphi\in C([-h,0],\mathbb{R}^n):\|\varphi\|_\mathscr{H}<\Delta\},  \end{equation*}
estimates the attraction region of the trivial solution of \eqref{eq:time-delay_system}.
\end{theorem}

\begin{theorem}\label{thm:attr_region_LK_set_S}
Let $\Delta_\alpha$ be a positive root of equation
\begin{equation*}
\alpha_1\Delta_\alpha^\gamma + b_3 \Delta_\alpha^{\gamma+\mu} = \tilde{a}_1\delta^\gamma,
\end{equation*}
where $\delta<\min\{H_2,H_3\},$ and $H_3$ is defined by \eqref{H3}.
If system \eqref{eq:delay-free_system} is asymptotically stable, then the set of initial functions
\begin{equation*}
\Omega_\alpha=\{\varphi\in C([-h,0],\mathbb{R}^n):\|\varphi\|_\mathscr{H}<\Delta_\alpha\},
\end{equation*}
estimates the attraction region of the trivial solution of \eqref{eq:time-delay_system}.
\end{theorem}

\begin{remark}
Proofs of Theorem~\ref{thm:attr_region_LK_w} and Theorem~\ref{thm:attr_region_LK_set_S} imply that $\|x(t,\varphi)\|_{r,p}<\delta$ $\,\forall\, t\geq 0,$ if $\|\varphi\|_\mathscr{H}<\Delta$ or $\|\varphi\|_\mathscr{H}<\Delta_\alpha.$
\end{remark}

\section{Estimates for the solutions}
For the standard dilation, the estimates for the solutions obtained with the help of the Lyapunov-Krasovskii functional \eqref{eq:functional} are presented in \cite{portilla2020comparison}. We extend straightforwardly this result to the case of weighted dilation in Section~\ref{subs_classical}. A novel approach which combines the use of functional \eqref{eq:functional} with ideas of the Razumikhin framework is presented in Section~\ref{subs_via_set_S}.

\subsection{Classical Approach}
\label{subs_classical}
Bounds \eqref{eq:derivative} and \eqref{eq:upper_bound} imply that if $\|x_t\|_\mathscr{H}\leq \delta,$ $t\geq0,$ then
\begin{align}
\label{eq:sec_bound_dot_v}
\frac{dv(x_t)}{dt}&\leq -c\left(\|x(t)\|_{r,p}^{\gamma+\mu}+\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{\gamma+\mu}d\theta\right),\\
\label{eq:first_upper_bound_v}
    v(x_t)&\leq b\left(\|x(t)\|_{r,p}^{\gamma}+\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{\gamma}d\theta\right),
\end{align}
where $c=\min\{c_0,c_2\},$ $b=\max\{b_1,b_2\},$ \mbox{$\delta<\min\{H_1,H_2\}.$} Define the values
$$
\rho_1=\bigl(2\max\{1,h\}\bigr)^{\frac{\mu}{\gamma}},\quad\rho_2=\frac{c}{\rho_1 b^{\frac{\gamma+\mu}{\gamma}}}.
$$
The following relation was established in \cite{portilla2020comparison} on the basis of Hölder's inequality:
\begin{multline} \label{inqx_w}
    \left(\|x(t)\|_{r,p}^{\gamma}+\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{\gamma}d\theta\right)^{\frac{\gamma+\mu}{\gamma}}\\ \leq  \rho_1\left(\|x(t)\|_{r,p}^{\gamma+\mu}+\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{\gamma+\mu}d\theta\right).
\end{multline}
Combining \eqref{eq:sec_bound_dot_v}, \eqref{eq:first_upper_bound_v} and \eqref{inqx_w} gives the following
connection between functional \eqref{eq:functional} and its derivative.
\begin{lemma} \label{lemma:connect_dv_v_w}
The following inequality is satisfied:
\begin{equation}\label{eq:connect_v_dot_v}
    \frac{dv(x_t)}{dt} \leq -\rho_2 v^{\frac{\gamma+\mu}{\gamma}}(x_t),\quad t\geq0,
\end{equation}
along the solutions of system \eqref{eq:time-delay_system} with $\|x_t\|_\mathscr{H}\leq \delta.$
\end{lemma}
Considering the comparison equation \cite{khalil1996nonlinear} of the form
\begin{equation}\label{eq:equ_comparison}
    \frac{du(t)}{dt} = -\rho_2 u^{\frac{\gamma+\mu}{\gamma}}
(t),
\end{equation}
with initial condition
$
   u(0)=u_0=(\alpha_1 + b_3 \Delta^{\mu})\|\varphi\|_\mathscr{H}^\gamma
$
and exploiting the classical ideas, we arrive at the following estimates for solutions in the homogeneous norm.
\begin{theorem}\label{th:estimation}
Let Assumptions~\ref{as:hom_degree} and \ref{as:delay_free_asymp} hold. Then, the solutions of system \eqref{eq:time-delay_system} with initial functions with $\|\varphi\|_\mathscr{H}<\Delta,$ where $\Delta$ is defined in Theorem~\ref{thm:attr_region_LK_w}, admit an estimate of the form
\begin{gather} \label{eq:estimate_final}
   \|x(t,\varphi)\|_{r,p} \leq \frac{\hat{c}_1 \|\varphi\|_\mathscr{H}}{\left(1+\hat{c}_2\|\varphi\|_\mathscr{H}^{\mu}t\right)^{1/\mu}},\quad\text{where}
\\
\begin{split}
\hat{c}_1&=\left(\frac{\alpha_1+b_3 \Delta^{\mu}}{a_1}\right)^\frac{1}{\gamma}=\frac{\delta}{\Delta},\notag
\\
\hat{c}_2&=\frac{c}{b}\left(\frac{\mu}{\gamma}\right)\left(\frac{\alpha_1+b_3 \Delta^{\mu}}{2b\max\{1,h\}}\right)^{\frac{\mu}{\gamma}}.\notag
\end{split}
\end{gather}
\end{theorem}
\begin{remark}
Note that $\|\varphi\|_\mathscr{H}<\Delta$ implies $\|x_t(\varphi)\|_\mathscr{H}<\delta$ for all $t\geq0$ according to Theorem~\ref{thm:attr_region_LK_w} thus making use of \eqref{eq:sec_bound_dot_v} and \eqref{eq:first_upper_bound_v} legal.
\end{remark}
\subsection{Novel Approach Using the Set $S_\alpha$}
\label{subs_via_set_S}
The lower bound (\ref{eq:lower_bound_set_S}) is expected to be less conservative than the original bound (\ref{eq:lower_bound}), since it should hold on the reduced set $S_{\alpha}$ instead of the set of all continuous functions.
Thus, a natural question appears: Can we replace the constant $a_1$  coming from the lower bound in \eqref{eq:estimate_final} with $\tilde{a}_1$? We give an affirmative answer to this question with some restrictions below.
Exploring the proof of Theorem~\ref{th:estimation}, one finds that the lower bound for the functional is used at the final step of the proof, namely, $$a_1\|x(t,\varphi)\|_{r,p}^\gamma\leq v(x_t)\leq u(t),$$
where $u(t)$ is the solution of \eqref{eq:equ_comparison}. It is crucial that the last formula is true for all solutions, not only those with each segment in $S_\alpha.$ A similar difficulty appears while constructing the estimates of solutions via the Razumikhin theorem \cite{aleksandrov2012asymptotic}. Here, we adapt the ideas of \cite{aleksandrov2012asymptotic} to reduce the conservatism of Theorem~\ref{th:estimation}.

We start with Lemma~\ref{lemma:connect_dv_v_w}. To overcome the mentioned difficulty, we take $\rho<\rho_2,$ and consider the comparison initial value problem
\begin{gather}
    \frac{du(t)}{dt} = -\rho u^{\frac{\gamma+\mu}{\gamma}} (t),\\
    u(0)=\tilde{u}_0=(\alpha_1 + b_3 \Delta_\alpha^{\mu})\|\varphi\|_\mathscr{H}^\gamma,
\end{gather}
which admits the solution
\begin{equation*}
    u(t) = \tilde{u}_0\left[1+\rho\left(\frac{\mu}{\gamma}\right)\tilde{u}_0^{\frac{\mu}{\gamma}}t\right]^{-\frac{\gamma}{\mu}}.
\end{equation*}
Now, we present a set of auxiliary results to extend the approach of \cite{aleksandrov2012asymptotic} to the Lyapunov-Krasovskii framework. In Lemma~\ref{lemma:add_restric}, a choice of $\rho$ is made. Such choice is always possible due to $\alpha>1.$   Theorem~\ref{thm_set_S_prelim} allows us to switch from the bound on the set $S_\alpha$ to the bound which holds for all solutions in a certain neighbourhood of the trivial one. The proofs are omitted due to length limitations.
\begin{lemma}\label{lemma:strict_comparison}
If $\|\varphi\|_\mathscr{H}<\Delta_\alpha,$ then
\begin{equation*}
v(x_t)<u(t),\quad t\geq 0.
\end{equation*}
\end{lemma}
\begin{lemma} \label{lemma:add_restric}
If the condition
\begin{equation}
\label{add_restrict}
1+\rho h\left(\frac{\mu}{\gamma}\right)(\alpha_1 +b_3\Delta_\alpha^{\mu})^{\frac{\mu}{\gamma}}\Delta_\alpha^{\mu}\leq \alpha^{\mu}
\end{equation}
holds, then $u(t+\theta)<\alpha^\gamma u(t)$
for all $t\geq 0$ and $\theta\in[-h,0]$ such that $t+\theta\geq 0.$
\end{lemma}
\begin{theorem} \label{thm_set_S_prelim}
If $\|\varphi\|_\mathscr{H}<\Delta_\alpha$ and inequality \eqref{add_restrict} holds, then
$$
\tilde{a}_1\|x(t,\varphi)\|_{r,p}^\gamma< u(t),\quad t\geq 0.
$$
\end{theorem}
Based on Lemmas~\ref{lemma:strict_comparison}, \ref{lemma:add_restric} and Theorem~\ref{thm_set_S_prelim} we present the main result of the section.
\begin{theorem}\label{th:estimation_S}
Let Assumptions~\ref{as:hom_degree}, \ref{as:delay_free_asymp} and inequality \eqref{add_restrict} hold. Then, the solutions of system \eqref{eq:time-delay_system} corresponding to the initial functions with $\|\varphi\|_\mathscr{H}<\Delta_\alpha,$ where $\Delta_\alpha$ is defined in Theorem~\ref{thm:attr_region_LK_set_S}, admit an estimate of the form \eqref{eq:estimate_final}
with
\begin{align*}
\hat{c}_1&=\left(\frac{\alpha_1+b_3 \Delta_\alpha^{\mu}}{\tilde{a}_1}\right)^\frac{1}{\gamma}=\frac{\delta}{\Delta_\alpha},
\\
\hat{c}_2&=\frac{c}{b}\left(\frac{\mu}{\gamma}\right)\left(\frac{\alpha_1+b_3 \Delta_\alpha^{\mu}}{2b\max\{1,h\}}\right)^{\frac{\mu}{\gamma}}.
\end{align*}
\end{theorem}

\section{ILLUSTRATIVE EXAMPLE}
Consider the following system, which is
used to model complex interactions, either instantaneous or delayed, occurring amongst transcription
factors and target genes  \cite{efimov2016global}:
\begin{gather}
\begin{split} \label{example}
\dot{x}_1(t)&=-\kappa_1x_1^2(t)+\lambda_1x_2(t-h),
\\
\dot{x}_2(t)&=-\kappa_2x_2^{3/2}(t)+\lambda_2x_2(t)x_1(t-h).
\end{split}
\end{gather}
Here $x_1(t),x_2(t)\in\mathbb{R}^+$ represent interactions occurring in a genetic
network, $h>0$ is the transition delay in the network, and $\kappa_1,\kappa_2,\lambda_1,\lambda_2$ are positive parameters. System \eqref{example} is $\delta^r$-homogeneous of degree $\mu=1$ with $(r_1,r_2)=(1,2).$ Set $\gamma=4$ and consider the Lyapunov function
\begin{equation*
   V(x)=x_1^4+x_2^2,
\end{equation*}
which is positive definite. Its derivative along the trajectories of system \eqref{example} when $h=0$ is of the form
\begin{gather*}
    \dot{V}(x)=-4\kappa_1x_1^5+4\lambda_1x_1^3x_2-2\kappa_2x_2^{5/2}+2\lambda_2x_2^2x_1\\
    \leq -2\min\{2\kappa_1,\kappa_2\}(x_1^5+x_2^{5/2})+4\max\{2\lambda_1,\lambda_2\}\|x(t)\|_{r,p}^5.
\end{gather*}
Choosing $p=5$ for the homogeneous norm, we arrive at bound \eqref{eq:bound_dot_V} with
$\mathrm{w}=2\min\{2\kappa_1,\kappa_2\}-4\max\{2\lambda_1,\lambda_2\}.$
Compute the other constants:
$m_1=\max\{\kappa_1,\lambda_1\},$  $m_2=\kappa_2+\lambda_2,$  $\eta_{11}=2\kappa_1,$ $\eta_{12}=\eta_{21}=0,$  $\eta_{22}=\max\{1.5\kappa_2,\lambda_2\},$  $\beta_1=4,$
$\beta_2=2,$ $\psi_{11}=12,$ $\psi_{12}=\psi_{21}=0,$ $\psi_{22}=2,$ $\alpha_0=1$ and $\alpha_1=2^{1/5}$.
\begin{figure}[H]
     \centering
      \includegraphics[width=0.5\textwidth]{weighted_ex_1_2.eps}
      \caption{Estimation of the solution of system \eqref{example} for $p=5$}
      \label{figure_estimation_w_1_2}
  \end{figure}
For the parameters $(\kappa_1,\kappa_2,\lambda_1,\lambda_2)=(9,18,0.25,0.5)$ and the initial function $\varphi(\theta)=[5\cdot 10^{-11},5\cdot 10^{-11}]$, $\theta \in [-10,0],$ the system response (continuous line) and the estimates using Theorem~\ref{th:estimation} (dashed line) and  Theorem~\ref{th:estimation_S} (dashed-dot line) are depicted on Figure~\ref{figure_estimation_w_1_2}. We conclude that the use of the set $S_\alpha$ allows us to obtain a tighter estimate than those via the classical approach.

\section{Conclusion}
In this paper, we present a Lyapunov-Krasovskii functional for weighted homogeneous time delay systems of positive degree and show its potential as an analysis and design tool by computing the estimates of the domain of attraction and of the system solutions.
\begin{comment}
\appendices
\section{General Inequalities}
First, we remind some useful inequalities. Next, we prove Lemma 2 that allows connecting the bounds of the functional with those of its derivative.
\begin{lemma}
\textit{Let $a,b,\nu\in \mathbb{R}$, $a,b\geq 0$ and $\nu\geq 1$, then the following inequality is satisfied}
\begin{equation*}
    (a+b)^\nu\leq 2^{\nu-1}(a^\nu+b^\nu)
\end{equation*}
\begin{proof}
The mapping $f:x\rightarrow x^\nu$, for $x\geq 0$, is convex since $f''(x)\geq 0$, indeed
\begin{equation*}
f''(x)=\nu(\nu-1)x^{\nu-2}\geq 0.
\end{equation*}
Now, using the convexity property for $a,b>0$, we get
\begin{equation*}
f(\lambda a+(1-\lambda)b)\leq \lambda f(a)+(1-\lambda)f(b),\ \lambda\in [0,1].
\end{equation*}
For $\lambda=1/2$
\begin{equation*}
\left(\frac{a}{2}+\frac{b}{2}\right)^\nu\leq \frac{a^\nu}{2}+\frac{b^\nu}{2},
\end{equation*}
equivalently,
\begin{equation}\label{eq:le1}
(a+b)^\nu\leq 2^{\nu-1}(a^\nu+b^\nu).
\end{equation}
\end{proof}
\end{lemma}
\textbf{Proof of Lemma \ref{lemma:inequality}.}
Using \eqref{eq:le1} and taking $\nu=\frac{u}{q}$, $a=\|x(t)\|_{r,p}^q$, $b=\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^qd\theta
$, we get
\begin{equation} \label{inqx2}
\left(\|x(t)\|_{r,p}^{q}+\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{q}d\theta\right)^\frac{u}{q}
\end{equation}
\begin{equation*}
\leq
2^{\frac{u}{q}-1}\left(\|x(t)\|_{r,p}^{u}+\left(\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{q}d\theta\right)^{\frac{u}{q}}\right).
\end{equation*}
To conclude the proof, we use Holder's inequality,
\begin{equation}
\left|\int_{-h}^{0}f(x_t)g(x_t)d\theta\right|
\end{equation}
\begin{equation*}
\leq \left(\int_{-h}^{0}|f(x_t)|^rd\theta\right)^{\frac{1}{r}}\left(\int_{-h}^{0}|g(x_t)|^sd\theta\right)^{\frac{1}{s}},
\end{equation*}
for $r,s\geq 1$ such that $\frac{1}{r}+\frac{1}{s}=1$. For $s=\frac{u}{q}$, $r=\frac{u}{u-q}$, $f(x_t)=1$ and $g(x_t)=\|x(t+\theta)\|_{r,p}^q$, we have
\begin{equation*}
    \left(\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^q d\theta\right)^{\frac{u}{q}}\leq h^{\frac{u-q}{q}}\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^u d\theta.
\end{equation*}
Then, it follows from \eqref{inqx2} that
\begin{equation*}
\left(\|x(t)\|_{r,p}^{q}+\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{q}d\theta\right)^\frac{u}{q}\leq
\end{equation*}
\begin{equation*}
2^{\frac{u}{q}-1}\left(\|x(t)\|_{r,p}^{u}+h^{\frac{u-q}{q}}\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{u}d\theta\right)\leq
\end{equation*}
\begin{equation*}
2^{\frac{u}{q}-1}\max\left\{1,h^{\frac{u-q}{q}}\right\}\left(\|x(t)\|_{r,p}^{u}+\int_{-h}^{0}\|x(t+\theta)\|_{r,p}^{u}d\theta\right)
\end{equation*}
and the lemma is proved.
\end{comment}

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
CA State Professor: Trump's Climate Views Disqualify Him From Using Twitter
"Such reforms would make it more difficult for networks to shove people like Cheney, Rove, and Fleischer down our throats."
A history professor at Cal State University, Sacramento, argued President-Elect Donald Trump should not be able to use Twitter because "people who dismiss science in one area shouldn't be able to benefit from science in others."
Newsbusters reports that Joseph A. Palermo (pictured above) believes Trump's views on "the science of climate change" disqualify him from using "the science of the Internet," and other technical achievements.
"I've always believed that people who dismiss science in one area shouldn't be able to benefit from science in others," he wrote for the Huffington Post on Dec. 27.
Accordingly, Palermo said that President Trump shouldn't be able to use "the science of global positioning" for drones or "the science of nuclear power" for weapons. Presumably he believes Trump also shouldn't be able to use a cell phone or electric toothbrush, either.
Newbusters also notes that in 2009, Palermo argued for "all sorts of reforms" by the federal government that he said would "open up the nation's political discourse." For him, opening up the nation's political discourse meant silencing former George W. Bush administration officials from participating:
Such reforms would make it more difficult for networks to shove people like [Dick] Cheney, [Karl] Rove, and [Ari] Fleischer down our throats because enhanced competition would mean that rivals might be broadcasting more attractive fare.
Another day, another totalitarian academic condemning dissent from leftist orthodoxy. Sorry Professor, but come January 20th there will be a new sheriff in town, and your anti-free speech days are numbered.
Yale Lecturer: 'Electoral College is Thwarting Our Ability to Battle Global Warming'

redpajama_set_name: RedPajamaCommonCrawl


During a recent google search, I found an interesting article that listed the top 50 questions asked to "Consultants in 2016". Literally 48/50 were related to some form of Marketing, Social Media or SEO. While those are very important reasons to look for an expert, I am concerned(and my business shows) that many businesses are missing out on other fundamental ways to help their business. Given National closure rates are so high for Small/Medium Businesses, do people think Marketing will save their business? What are the top reasons a business fails and how does Marketing help avoid that?
The number one concern a business owner should have is how to maximize their businesses ability to generate revenue.
Marketing is one of the 4 most important areas to develop and can not be overlooked. But not developing the other three can jeopardize the sustainability of a business. I have seen too many instances where an organization improved efforts to increase the customer base through Marketing, only to find the business couldn't capitalize on the new found popularity or volume. If your first reaction to this is "that's a good problem", you are wrong. It is a problem that may hurt your business more than it helps it. To increase revenue in a sustainable way, you need to consider building on all 4 key areas.
A classic example is the restaurant that creates a great coupon that is hard to say no to. People come in droves and crowd the restaurant. Tables are full, the Bar is crowded and there is a wait list as long as your arm.
Then things start to go wrong, the food comes out late, servers become overwhelmed, orders begin to get switched around and as a result the tables don't turn. Then the customers at tables start to become frustrated and before you know it, they have such a bad experience, they don't want to return. People waiting for a table end up waiting twice the time they were quoted.
Perhaps the house buys a drink or dessert to appease, this calms many people but that may not work on everyone. Customers take out frustration on your staff. Your staff takes out there frustration on each other. All in all, it turns out to be a tough night for that restaurant.
But the pain continues because customers who want to vent are just getting started.
Customers proceed to tell there friends what a bad time they had when they're at work or through social media and this can linger for some time. So while the coupon brought increased sales initially, the other areas of the business that weren't ready for this suffered. The servers don't make($) what they expected due to the restaurant failing, the kitchen crew is frustrated because they weren't set up for that kind of business, and the product quality suffered due to the volume. Marketing people call this a successful campaign and while sales were up temporarily, they then slumped even lower than before the coupon was sent out. Damage has been done.
We all know this story, and it can happen in all industries.
This problem was created by an over emphasis on Marketing and not insuring that other important components needed were addressed. Generally this is a type of ad hoc marketing, and don't confuse it with "strategy". A Business Strategist would not only look at ways to get more people through the door, but make sure the business was equipped to handled the increase. I have literally seen Groupon tactics destroy businesses, they will get people to your business, but it is unlikely you will make money off the new business or create repeat customers (which is what every owner that signs up for that service wants). I am all for Marketing. Marketing/Advertising is to increase demand or volume, just make sure your business can meet the demand you create.
So if Marketing won't save my business, what should I look at?
The SBA, SBDC, Dunn & Bradstreet, IRS & Census bureau will also show similar statistics.
Learn to evaluate your business. When you are in a position of growth, expansion or even for a limited promotion like the coupon in the case study above, make sure you look at the 4 key area's needed to generate revenue for your business. If any of those area's slip during a period of increased advertising, understand it will jeopardize the results you had planned. Maintaining product standards, Quality Assurance measures, and Brand management are the tip of the iceberg to insure daily execution. Daily execution is what your customer base will judge you by and ultimately determines everything from customer loyalty to profitability. Much more information on these 4 areas (Pillars)can be found on our website.
Don't confuse sales with growth. Sustainable businesses create strategies that take all 4 areas and develop each of them to meet desired goals. Top line sales don't always translate to growth especially if you can't sustain that level of revenue. Training everyone in the company to focus on all 4 areas can lead to rapid, organic growth for a company that is sustainable. In short, develop your people and they will develop the business. Give them systems and tools that help them achieve their job. When systems, tools, training, and goals are all focused on developing your products, then you have something to Market. Never Market under the "Strategy" these others will "fall in line". This is a common mistake and they seldom do.
Don't over rely on online development. Unless you are strictly E-Commerce. As mentioned above, 98% of the questions searched about consultants were for Marketing based consultants. Yet, Marketing usually doesn't make the top 5 reasons why businesses close. Marketing Consultants are important in today's business world and C.S.Simons Consulting works with several extremely valuable Marketing experts, but there is more to developing your business than a Marketing Expert can offer. They are part of the solution, not the whole solution.
Know how the pieces connect. The business "Pillars" I refer to have overlap and it can be confusing as to how they fit together to build a business. Further more, what is needed to support each pillar can change by industry, by business size and even business age. Knowing what is needed to get you to the next stage is the key to growth & sustainability. If you think all your organization needs to grow is better or different Marketing, I strongly recommend you look at the other pieces of the puzzle to.
All businesses need to Market. Many outsource this for a variety of reasons and many of them are good reasons. The point is to understand the impact of the marketing tactics to the other sides of your business before they are launched.
If you really want to increase your revenue or strengthen your business make sure you are defended against the top reasons business fail in the first place!
A couple of years ago while performing a "Market Study" for a well established Non Profit, I learned of yet another way to utilize the SMART Goal methodology. This Market Study Project had multiple steps that were designed equally for discovery and for future planning purposes. Discover who the current supporters were, identify opportunities to reach new audiences AND analyse the existing Communications Plan and aide in developing a new, more targeted approach to plan for messaging.
For the first phase, three surveys were designed and well received by the region which provided excellent information of current users, lapsed membership, and of non members. Multiple avenues were used to reach each base and we had nearly a 40% return rate (which is very good). Once compiled we had lots of information to use to determine Attitudes and Motivators.
Many reports and presentations later, our clients were still having a difficult time determining who to target and what to say. They had never looked at this from the scientific side and didn't know how to generate goals using this data. Long story short, I realized the group had a very difficult time creating goals because they had never used the "SMART Goal" technique.
Previous goals were to "increase Volunteers" or "Donations", but with no details on how to know when you achieved them, or even How you achieved them.
My Corporate Trainer background makes me assume everybody knows and uses the Smart Goal technique, but that simply is not so. Further more, the entire Board of Directors didn't use this either and as a result created some unattainable goals in past years. For those unfamiliar SMART is an acronym that stands for Specific, Measurable, Actionable, Relevant(realistic)& Time based. The methodology is that SMART is used as a project plan to create a goal. If you can't think of a way to answer one of these headers, then you risk setting a bad goal that is not achievable.
I offered to Facilitate a "Goal Setting" session with the Board to help create more concrete, achievable goals that the Marketing team could then build a campaign from. In one afternoon we were able to unite many different thoughts of what direction this NP should focus based on using the data provided by our firm AND by using the SMART methodology. As an exercise, I used some past goals and illustrated how if even one component is left out of SMART when forming a goal, that it is likely not achievable. Then I referred to the notes taken in our initial assessment and proved what they were presenting then as problems were the result of poorly written goals.
Yes, it is very important when first using this model to write down all thoughts and to then present it to the group for discussion. SMART is an exercise that helps embed this process. After time, it will become an integrated behavior or way of thinking, but at first give it it's due diligence and write them down.
Closer investigation showed that certain goals had been too "lofty" and not Realistic. The end result sounded nice, but how to get there hadn't really been planned out. As a group we broke these down to smaller goals that could be accomplished, once that happened it was easier to attach Measurements to them. Specifying action steps to help achieve each goal were discussed and they realized that "SMART" is also a management tool to determine what progress they were making.
Our interviews showed that both the ED and the Board had shown some frustration to each other because the previous goal hadn't been accomplished.
By following the SMART technique; Management has direction from the Board and the Board has a way to gauge progress. There had also been a confusion when setting goals for the organization on the relationship between time and resources. Most Executive Directors I have met take on far more than they should. If the Organization needs it, well, they will find a way to get it done. This creates a problem over time and literally empowers a BOD to take the Time factor out of goal setting. If the Board doesn't know how to assess the ED's ability to manage time vs resources, then how can they apply it to goal setting?
This may have been the biggest takeaway of our meeting; the need to insure a goal is achievable with in a certain time frame. Many do not put timelines with goals for a variety of reasons, and I think I have heard them all. Understanding that time lines must be part of the foundation of a goal is what makes it a goal. Otherwise it is simply a dream. Time lines and accountability make it real and coincidentally, they also make them happen.
Over the next month, utilizing the SMART goal method this group was able to develop 3 distinct goals to then ask the Executive Director and the Marketing team to complete. Each goal gave Specific details of what would be accomplished, it defined how it would be Measurable with both metrics and timelines, it outlined Actionable steps similar to a project plan, the entire team felt these were Realistic steps given the mission of the organization and the new data they had from our Market Study and lastly , the Marketing team felt they could accomplish this goal given the Time they were given and resources that were made available for it.
This Board session also highlighted the need for the organization to budget money on an annual basis to support the ongoing needs of Marketing. Previously no money was budgeted(common), but the ED spent it when needed(also common). How is this good planning? Budgeting and goal setting should be done at the same time. Money is a key resource when considering Marketing tactics. General guideline is 5-10% of annual Revenue.
Almost 3 years later that Non Profit has a dedicated Marketing Manager with an annual budget. Other non profits in the area are looking to them for advice on Marketing and Membership has grown. They have an annual process of Goal setting before the Budget is determined for the following year(yeah!). This has also opened communications between the ED and the BOD. Our facilitation that afternoon featured the SMART Goal methodology which certainly had an impact on the way this group does business.
C.S.Simons Consulting specializes in Management & Business Strategy, offering 25 years of successes for you to benefit from. Offering a Shared Risk Platform – Results Guaranteed or you won't be billed!
Content Management, Frequency, and Target Audience are but a few of the components in a Communications Plan. Non Profit(s) and Small Businesses need to be very strategic on what messaging they need and which tactics to use when creating a Marketing Campaign due to the amount of time needed and financial investment it takes for these to work.
Knowing the attitudes and motivators of your buyers/supporters/followers has a direct impact on measuring a plans effectiveness. The following are excerpts from a white paper published by C.S.Simons Consulting in 2012 designed as a FREE resource for those businesses who recognize the need for better effectiveness in their communications but have limited funds to hire a professional Marketing firm. There is also 2 worksheets available to expedite your Communication Planning, feel free to contact me if you would like a copies.
Why develop a strategy around a Communications Plan?
Planning contributes greatly to success.
Strategic communication is planned and accomplishes a specific outcome. It is essentially a project plan for everything you may need to "Market" a product, service or business effort. Designing one will ask the right questions of you as to how you want it presented(see below). It is designed ahead of time so your company controls the exposure and narrative that puts your product/service/effort in the best light to the people or businesses you want to see it. Otherwise you risk the reputation and message being delivered. Below is a checklist to help you organize a Communication Plan of your own.
Strategic communication is targeted to a particular audience or audiences utilizing known demographics about your customer.
Strategic communication aims for results with the best possible use of time and resources.
Strategic communications should be tracked, with measurable performance.
What will you need to do to get there?
What role can communication, education and training play to achieve your goals?
How will you learn from your experiences?
Communication is part of the entire Policy and Strategy implementation process. To implement a Communication Strategy there are many projects to be prepared and undertaken. There must be an understanding of what your customer wants from you, who your customer is and where to find them. Demographics, Attitudes and Motivators is how it is referred to with in the Marketing world. These are addressed first prior to forming a Communications Strategy around how to effectively have that group pay attention to you. Most business people or Executive Directors would agree with this yet this stage often receives little attention until much later on. Often communication is considered only after plans, polices or projects are prepared which strongly reduces the potential for successful project implementation.
Ad hoc communication is not effective. There is an enormous difference between communication strategically planned and that without strategic planning. It will miss the target audience or deliver the wrong message. You may have very little time to get the attention of your audience, so every little effort needs to be effective.
Message: This will comprise a combination, of WHO you are trying to speak to, and WHAT you want them to remember or HOW you want them to act.
Timeframes & Frequency: You have to clarify if you are designing a communication strategy with long term goals, a communication plan with short term goals or a communication plan for a specific project. A Strategic Communication Plan will likely utilize all three types of communication plans and is a comprehensive approach for long term messaging.
Target Audience: To create a master strategy, match the goals of your strategic plan to groups identified in the recent membership survey. The membership survey results provide detailed information on demographics such as age, proximity to lake, length of membership, and communication preferences. Use this information to choose which medium to communicate to each demographic. To broadcast a particular message, many organizations choose multiple mediums and alter their message slightly to improve connection with a particular demographic. Much of a communication program success relies upon the content connecting with its target audience.
Content: The body of the communication and the vehicle for reaction. Utilize demographic information to "speak in a language" that the reader will understand. Incorporate grabbing headlines, pictures or graphs whenever possible as this is a proven to increase attention and action.
Tools & Format; what method or medium would be best suited to deliver message and achieve results. Consider the target audience, and how to best reach them based on prior success. If trying to reach a new target audience, which design works best for the market? Is capital expense needed for a first impression? Is graphic designing needed for a mailing or email blast? Print ready advertisement design? Pictures needed for social media or press release?
Instrumental style communication – You need to be conscious of whether you are dealing with a communication campaign that is organized to raise the interest of the public, politicians and other special groups for a particular issue, or to generate support for policies or plans.
Maintenance and Accountability; to assist with daily management of any communication campaign, the creation of tools to help monitor message, frequency, placements and responsibility should be utilized. Examples of such tools are provided in the tools section of this report.
When implementing, regularly check how feasible the plan is, and what disasters may occur.
What will affect the success of the message? Which stakeholder is it designed for?
Be flexible in adapting the plan in case of shortages in money or time.
What will people's reaction be – What do they want in the communication?
If the communication is announcing an event or action item, how much time is given for members to react?
A campaign can be designed for virtually any application, after all, more is often better when done right. The primary goal behind a multi-tactic campaign is to leverage each individual tactic/method to achieve greater impact with the desired message.
Single source messaging which is a one time event through standard media such as direct mail, email, website or online posting and is designed as a "one and done". It doesn't have a direct connection to either the message that LSPA has agreed to or a direct connection to the mission of the organization. While it is important and informational, it may not lead to a call for action like those in a campaign will from its recipients.
What is the current Knowledge, Attitude and Practices (behaviors) of the target groups and stakeholders involved?
What reactions do you want the target groups and stakeholders have?
In these cases the different states of knowledge, attitudes and practices need different communication solutions, and communication may be used as a sole instrument.
e.g. is knowledge needed? new skills and practices?
e.g. do we need an attitude change from them?
Indicators are not defined for the communication targets/objectives, making evaluation of the outcome difficult.
For more on this subject or to discuss additional ways to help your organizations Ability to Generate Revenue, contact C.S.Simons Consulting.


text:
Stefan Mittnik is a German economist, currently holds the Chair of Financial Econometrics at the  Ludwig Maximilian University of Munich. He is a fellow of the Center for Financial Studies and known for his work on financial market and financial risk modeling as well as macroeconometrics. He is also a co-founder of the German-British robo-advisor Scalable Capital.

Biography
Stefan Mittnik received a degree in business and engineering in 1981 from the Technical University Berlin in Germany. He continued his studies in the UK, earning an MA in development economics at the University of Sussex, and the U.S., earning his Ph.D. in economics and applied mathematics from Washington University in St. Louis in 1987.

Research
Mittnik's main research contributions have been in econometrics, time series analysis, finance, and risk management. Influenced by Benoit Mandelbrot, who was the first to criticize financial economists for relying on the normal distribution and ignoring fat tails in asset returns, he has developed methods for more realistic financial risk modeling, portfolio optimization and option pricing.

References

External links
 Website at LMU
 IDEAS RePEc page
 Deutsche Digitale Bibliothek (DDB)

German economists
Washington University in St. Louis alumni
Academic staff of the University of Kiel
1954 births
Living people

redpajama_set_name: RedPajamaWikipedia


text:
Elementary: 4.5 The Games Afoot Review
Last week, we saw Sherlock and Joan solve a case while Joan was being investigated by the police because of a certain Detective Cortes' personal grudge.
This week's episode is tied a little more into the seasonal arc. And by that I mean ALFREDO IS BACK!
And Sherlock seems to be skipping meetings, or at least avoiding him ever since last season's finale. Ostensibly, he feels guilty because of Alfredo's prior kidnapping. He claims it's not guilt, but a need for a fresh start after his relapse.
But while all this not-guilt is marinating, Sherlock and Joan get to investigate the murder of an archaeologist, killed by a spade to the chest. His ex-girlfriend's spade, to be exact.
The archaeologist, Eddie Ross, had been searching for something in landfills and former landfills, going so far as to impersonate city employees. He was looking for games, for one in particular called "Nottingham Knights". In the Elementary universe, it's the worst game ever, buried in an undisclosed landfill and probably worth quite a bit of money.
It's clearly aping from the tale of the ill-fated E.T. Atari game, thrown into a landfill after being notoriously awful.
Ross wasn't the only one looking for it. Every once in a while, rumors of the game resurface, until he thought he found it, chronicled his exploits, and abruptly stopped. Perhaps the killer found the games, took them, and killed him.
Sherlock can't help but note the similarities between digging for nostalgia in video games and Alfredo's restoring of classic cars, a way to regain a former age. So he visits Alfredo to explain himself, that his scarcity hasn't been about guilt but about shame. Going to his old meetings, seeing all the familiar faces and locations serves not as a strengthening agent for Sherlock's sobriety but as a reminder of his relapse. That extends to Alfredo: in the context of Narcotics Anonymous meetings, Sherlock's former sponsor is no longer his friend but a reminder of the fact that he fell off the wagon.
He needs a new sponsor, but also wants to keep Alfredo's friendship. The problem there, however, is a lack of commonality. Alfredo and Sherlock hardly share anything—sure, Sherlock can pick damn near any lock and solve any crime, and considering Alfredo is a former criminal, they have that bond… but not much else. Their connection was largely about being sober, which presents them with the conundrum of how to continue.
Not a conundrum, though, is how to connect the dots between Eddie Ross and other members of a gaming forum: "Everyone" is always there to help Sherlock. Provided, of course, that he plays a video game to completion. And we get to see Sherlock and Joan attempt to play games, which is hilarious. They're better at pretending to play games than they are pretending to drink coffee, that's for sure; Sherlock's frustrated expression definitely mirrors my own when I'm losing. Joan's expression at winning the game herself, though, is fantastic and made the episode worth it all on its own.
Ross wasn't killed by someone wanting the games, though—those had been found years prior. Instead, he found proof of a chemical company illegally dumping toxic waste into landfills. They weren't at fault, though—instead, it was the owner of the land in which Ross was digging. Toxic waste is a property value killer, you see.
So the property owner hired an arsonist to kill Eddie Ross.
And with the mystery out of the way, Sherlock visits Alfredo, having realized that Alfredo had been attending many more meetings than usual. It wasn't because of the kidnapping, but a hard time nonetheless, and the lack of communication made Sherlock worry. He offers to help Alfredo, who refused to come clean about his problems in an attempt to save Sherlock's renewed efforts toward staying sober. But therein lies the solution to their friendship: give and take.
Sure, it still relies on sobriety.
But it's friendship.
Image courtesy of CBS
In this article:cbs,Elementary,Joan Watson,jonny lee miller,lucy liu,sherlock holmes
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I can speak French, which is really no big deal since a lot of people speak French. But I learned this beautiful language from 2nd graders at a private school in Portland, OR and I'm guessing not too many people can say that. I am fluent in French kids games and 2nd grader jokes.
During my homeschool high school years I wanted to learn French. No one in my family spoke the language. Since this was in a day and age when we did not have computers in our homes and the internet was not yet invented, I was limited to borrowing cassette tape language courses from the local library and reading a Berlitz book. Basically I was learning French on my own and had no idea whether I was getting it right or not. Some people told me I'd never been able to master the language unless I took a French class. It was the only way. The only problem was that, since I was homeschooled, I was considered truant from school and, hence, unable to participate in any classes at the local schools.
When I was 16 years old, I saw an article in the Oregonian about the French-American International School (Ecole Francaise). My mother suggested that I write a letter to the directrice of the school, asking if there were any volunteer positions to be had. If I could be in the French environment on a daily basis it would help with my language learning endeavors. So I wrote the letter, sealed it in an envelope, and mailed it off with the hope that my offer to volunteer my time would be accepted.
A few weeks later, I received a reply offering me a job! The school needed a teacher's assistant and wondered if I would be available to work Monday through Friday during normal school hours. Being homeschooled, this was an option. I could work during the day and do my studies at night. I could hardly believe my good fortune!
During the next school year I worked each week assisting Martine, the second grade teacher. All of the teachers and students were French – like actually from France. (So much better than being in a class of Americans trying to learn French from their American teacher!) The children's parents had brought their families to the Portland area for mostly job-related reasons and wanted their kids to continue with their French schooling while in America. This was the student body of Ecole Francaise. So Monday through Friday, between he hours of 7:30-3:30, I packed up a lunch and headed off to work in Paris in Portland. It was wonderful. The kids were fantastic teachers of French. The other teachers were very patient with me as I learned the correct way to roll the R and cut off the N. Once I received the nicest compliment: a parent of one of the students told me I spoke English very well! Ha! I even fooled a native French speaker!
After that school year ended, I worked as a nanny for two precious little French girls living in Portland for the summer while their mother visited family and did some professional photography work. The following fall, I left for Europe with my friend Wayne, spending quite a bit of time in France as well as the French-speaking part of Switzerland. By that time, I was fairly fluent and could have conversations as well as accurately navigate Wayne and I around Europe with a bit of Franglais thrown in when visiting other Latin-based language parts of the world.
This past week my husband took me to a preview of "Eddie the Eagle," a movie set to open at the end of February. The movie depicts the true story of Eddie Edwards, an Olympic ski jumper from Great Britain. When Eddie was a kid he dreamed of being an Olympian. Only Eddie was not athletic and Eddie had problems with his legs that caused him to have extended stays in the hospital and wear leg braces for much of his childhood. Still, though, despite all the impossibilities, Eddie was determined to be an Olympian. The movie shows Eddie's amazing journey to the 1988 Olympics in Calgary. It is a truly remarkable story. Totally unbelievable really. Insane almost. Eddie was more determined that anyone else I've ever known. His story is so incredibly inspiring. I recommend that everyone go see "Eddie the Eagle" when it opens. Watch it twice and then #flylikeEddie.
This brings me around to #6 on my list of 10 things you probably didn't know about me: I love to prove people wrong. It's an amazing thing to do that which other people tell you can't be done. It's even better when the outcome helps people all over the world. Go see the movie….. you'll see what I mean.
When I was 18 years old I found out I could run faster than the police. At least in Italy. It's a good thing, too, because I've heard it's nearly impossible to get out of the Italian prison system once you're in it.
In 1987 I spent three months backpacking around Europe with a good friend of mine. Wayne, my traveling companion for those months, was more like a brother to me while we were growing up. It only made sense when I decided to take a few months between high school graduation and wedding day to galavant around Europe that Wayne would make a great choice for some added braun to keep me safe. We were both engaged at the time, only not to each other.
With passports, Eurail passes, and youth hostel memberships in hand, we slung bags over our youthful shoulders and headed off to see the world. We had many wonderful adventures along the way, stopping wherever and whenever our hearts desired. I remember one occasion stopping off in a small town in Holland (now the Netherlands) simply because it smelled like chocolate when the train rolled into the station. As it turned out, there was a chocolate factory nearby, and we were able to get a cheap room for the night, the delicious scent of chocolate wafting in through the open window.
I still have the pair of purple Converse hightops I wore most days during our 3-month excursion. I put many kilometers on those treads! Maybe that's why I've kept the shoes after all these years…..the memories in the soles.
I distinctly remember we were in Huemoz, Switzerland near Villars on my 19th birthday. I remember that because I got lost in the Alps that day. The kind of lost that leaves one completely alone in the dark on the side of a mountain, wondering if they will make it through the night or ever see their loved ones again. Just as the search party was gathering to head off to my last known whereabouts, I limped into town, down the one road that ran through the quaint village of Huemoz, and into the chalet where we were staying. It turns out I have survival skills. At least when it's not terribly cold out and freezing to death is an unlikely possibility.
I also distinctly remember the day Wayne and I were in Brindisi, Italy and decided to hop a boat to Greece. A misunderstanding that, to this day, I still don't understand. left us running for our lives when an angry Italian man called the police on us. I swear we did nothing wrong. Fortunately for us, Wayne and I were fast and strong and extremely motivated to NOT be arrested. Friends had jokingly (or maybe not so jokingly) told us to be sure not to get caught up with the PoPo in Italy because once you are locked up in an Italian jail you might never get out. I have no idea why these warnings were given to us. But, I can tell you, the unsolicited advise provided that extra bit of motivation needed to get our legs pumping, our hearts racing, and our minds determined to outrun the police that day. Just as we jumped onto a bus pulling away from the curb, our assailant grabbed Wayne by the backpack and nearly succeeded in stopping our getaway. Wayne prevailed, the bus pulled away in the nick of time, and we spent the next several hours awaiting the ship departure from port looking over our shoulders and fearing for our lives. When the ship finally pulled away from the land of all things Italian, we breathed out sighs of relief, letting the wind whip them away into the night out over the Adriatic Sea. We never looked back. Instead we went on to Greece and had a marvelous time taking in the sights, sampling the foods, and visiting an island with no cars on it.
So #5 on my list of 10 things you probably didn't know about me: I can outrun the police if need be. And I don't like Italy. It's amazing how a bad experience can leave an unsavory taste in your mouth decades later. I know we saw many lovely things during our time in Venice, Florence, and southward toward Brindisi, but for me most of it is overshadowed by the angry Italian dude screaming at us and then running for our lives, certain we were going to end up like Jason Bourne (whom we hadn't yet heard of…but you get the idea).


Amishi's beautiful crystal drop earrings are encrusted with light-catching cubic zirconia in a silver setting and finished with a black onyx crystal. This pair has a light feel that won't weigh you down. Wear it with evening looks, keeping them in focus with pulled-back hair.


text:
Synergus rubricornis é uma espécie de insetos himenópteros, mais especificamente de vespas pertencente à família Cynipidae.

A autoridade científica da espécie é Tavares, tendo sido descrita no ano de 1920.

Trata-se de uma espécie presente no território português.

Referências
 Synergus rubricornis - de Jong, Y.S.D.M. (ed.) (2013) Fauna Europaea version 2.6. Web Service available online at http://www.faunaeur.org (consultado em 12 de janeiro de 2014).

Ligações externas

 Synergus rubricornis - Biodiversity Heritage Library - Bibliografia
 Synergus rubricornis - NCBI Taxonomy Database
 Synergus rubricornis - Global Biodiversity Information Facility
 Synergus rubricornis - Encyclopedia of Life

Himenópteros de Portugal
rubricornis
Animais descritos em 1920


text:
Q: Replacing "/" in URL querystring to avoid 404 in htaccess I have a url which has two parameters and I have rewritten them in the htaccess file with this

RewriteRule ^subscriber/([^/.]+)/([^/.]+)/?$ subscriber/index.php?id=$1&name=$2 [NC,L]

The name in some cases contains a "/" which the browser is treating like another parameter and is giving me a 404 page. Is there a way to replace "/" to something else using the htaccess file?

A: Try putting this in the htaccess file in your document root:
RewriteEngine On

RewriteCond %{QUERY_STRING} ^(.*)name=([^&]*)/(.*)$
RewriteRule ^ %{REQUEST_URI}?%1name=%2-%3 [L]

# Change your existing rule a bit to account for the slash (plus a possible trailing slash):
RewriteRule ^subscriber/([^/.]+)/(.+?)/?$ /subscriber/index.php?id=$1&name=$2 [NC,L]

This will make the first "folder" in the URI after subscriber/, the id param, and everything after that except for a trailing slash be the name param. The first set of rules then repeatedly clean out the name param, replacing / with -.

A: You can use urlencode() function when setting the name variable
$name = 'Michael/';
$name = urlencode($name);
// will return you Michael%08 or something like this

Accordingly with this question the solution for .htaccess might be:
RewriteEngine On
RewriteCond %{REQUEST_URI} (.*)\\(.*)
RewriteRule .* %1/%2 [R=301]

redpajama_set_name: RedPajamaStackExchange


The Gold Nugget Triathlon would not be possible without our volunteers! Volunteers make a difference for each and every racer and are part of what make the GNT so special. Over 70% of the volunteers are racers. Family members and friends make great volunteers too.
Did you also know that you could win a guaranteed entry to the 2020 GNT just by volunteering this year? 10 lucky winners will be chosen from all volunteers who complete their shifts. Can you imagine not having to stress about registration? It's a Win-Win!
* Not all volunteer jobs are appropriate or safe for those younger than 14.


Sophie Vlaming for F Magazine July 2016.
Sophie Vlaming for F Magazine April 2016.
Sophie Vlaming for Grazia Italy October 2015.
Sophie Vlaming for Elle Germany Novembre 2014.
Sophie Vlaming for Grazia Germany September 2014.
Sophie Vlaming for L'Officiel Paris August 2014.
Sophie Vlaming for Marie Claire Netherlands May 2014.
Sophie Vlaming feautures in Marie Clare Netherlands.
Sophie Vlaming features in Marie Claire Netherlands, May 2013.
Sophie Vlaming for Blumarine gloves campaign, 2013.
Sophie Vlaming for Italian Marie Claire, December issue.


The form "30th UofSC High School Math Contest. January 30th, 2016" is no longer accepting responses.
Please, communicate with mcontest@math.sc.edu if you need to perform any changes to your previous application.


text:
Trapcode Mir Compilation
Uploaded on Saturday 27 July 2013
IainGoodyear
Shortly after producing Left of the Border Films Collaboration One I realised that, as an experimental film maker, motion graphics could be a powerful and creative way of producing the visuals for my films. I had produced Collaboration One using After Effects and during this process became aware of companies that specialized in producing motion graphics generating plug ins. After a little research I discovered Trapcode Mir. What this plug-in is capable of was to me very mesmerizing. The colours, textures and shapes it was possible to create and manipulate with this plug-in captivated me. What was so exciting to me was that here was motion graphics production suite that seemed geared completely toward experimental visual production. So I dived in and began to explore the possibilities. From this experimentation came an idea about moving forward. This seemed to be a concept that was particularly relevant to where I currently seemed to be as an artist and experimental film maker. I was literally moving forward into a new stage of experimental production that allowed me to reference themes I have previously explored and also, further develop them in a new visual format. Trapcode Mir Compilation is a series of sketches that reference themes I explored in Tunnel 8. I think the main difference between these two very visually different films is that Trapcode Mir Compilation has a far less oppressive, more hopeful feel to it. This film also continued my passion for collaboration. Trapcode Mir Compilation features a soundtrack produced by Christopher Gladwin. Chris is undoubtedly a truly exceptional sound designer and musician, producing music and experimental sound works under a range of guises, most notably as one half of the influential and truly excellent Team Doyobi and also as The Wyrding Module. You can find out more about Christopher Gladwin and Team Doyobi at http://www.psybertron.co.uk/.
#abstract
#compilation
#experimental_short
#graphics
#abstract_video
#christopher_gladwin
#experimental_animation
#mir
#iain
#goodyear
#trapcode


Bizarre Foods America – Portland!
I've always been a fan of Andrew Zimmern, he has a supreme likability, even when he's digging into something we'd rather not eat.
He came through Portland last summer to shoot an episode for his series Bizarre Foods America, and it recently aired on the Travel Channel.
It was cool to see him visit some of the best (and gluten-free) restaurants in the city and share in his experiences. Some of the spots included Salt & Straw, Olympic Provisions, Lincoln, Laurelhurst Market and Le Pigeon.
You can find a link to the Portland episode here, and below is a video countdown of his Portland Top Five.
This entry was posted in Cooking, Culture, restaurants, Reviews, Travel, Uncategorized, Video and tagged Andrew Zimmern, Bizarre Foods America, Bizarre Foods Portland, Portland Andrew Zimmern, Travel Channel, Travel Channel Portland on April 27, 2013 by glutenfreepdx.


text:
Van Buren teachers removed in cheating probe sue district
The teachers deny the allegations of cheating and say they want to return to their jobs.
Van Buren teachers removed in cheating probe sue district The teachers deny the allegations of cheating and say they want to return to their jobs. Check out this story on Freep.com: https://on.freep.com/1TsHkEn
Lori Higgins, Detroit Free Press Published 7:43 p.m. ET May 20, 2016 | Updated 7:20 a.m. ET May 21, 2016
Classroom chars with desk at dusk(Photo: mygueart, Getty Images/iStockphoto)
Four Van Buren Public Schools teachers are suing the district, saying they were falsely accused of cheating on a state exam and their rights were violated when they were removed from the classroom and placed on paid administrative leave.
The four teachers — Pamela Bradley, Michelle Komaromi, Brent Held and Rebecca Tennis — filed the federal lawsuit in U.S. District Court in Detroit today. They say the district has yet to officially charge them with any wrongdoing.
Named in the lawsuit: the school district, former Van Buren Superintendent Michael Van Tassel and Shonta Langford-Green, who up until late April was the district's interim superintendent. Also named are six of the seven members of the district's Board of Education: Brent Mikulski, Martha Toth, Kathy Kovach, Kevin English, Alison Bennett and Kelly Owen. Board member Sherry Frazier was not named.
Van Buren superintendent is out amid test-results probe
The suit alleges the defendants violated the teachers' freedom of speech, freedom of expression, freedom of association, freedom to petition and due process rights.
The suit stems from an investigation into alleged testing improprieties at Savage Road Elementary School that has had the district embroiled in controversy for months and led to the exit earlier this year of Van Tassel.
State: Van Buren teachers fed kids answers on M-STEP
District officials were not available for comment late Friday.
The plaintiffs in the suit were among five teachers at the school who were put on administrative leave in January after the district said an investigation found they inappropriately coached students during the M-STEP exam, in some cases alerting students if they answered a problem incorrectly and in other cases helping them come to the right answer.
While many students statewide struggled on the new exam, the Savage Road students posted what state officials said were abnormally high scores. They were sixth in the state on the math portion of the exam. And students at the school scored higher than a number of schools that educate only gifted and talented students.
Savage has a gifted program, but district officials have said that program accounts for 40% of its population.
Van Tassel alerted the state to what he thought were suspicious results last fall. The Michigan Department of Education told Van Tassel to conduct an investigation.
The teachers deny they cheated, and are asking to return to their jobs, have any disciplinary records removed from their personnel files, and be granted compensatory and punitive damages.
Deborah Gordon, the Bloomfield Hills attorney representing the teachers, said they've been out of the classroom for five months and have yet to be told what rule they broke.
"They've never been given anything in writing saying why they're out. They have asked on several occasions what their status is and when they can return. They get no response. They have been left with no choice. They can continue to sit home at the taxpayer expense or they can try to get back into the classroom."
Gordon said she believes district officials have backed themselves into "a very strange corner."
"How do you keep people out who have not been disciplined or done anything wrong?"
The lawsuit said Van Tassel and all other defendants "intentionally misled the public," stating publicly that defendants were engaged in an ongoing investigation into what was being referred to by January 2016 as the cheating scandal.
The lawsuit says the district will only allow the teachers to return to their jobs if they sign a document in which they would agreed to no further litigation against the district. The four rejected the offer. But the fifth teacher, who wasn't named in the suit, signed an agreement and returned to work.
Contact Lori Higgins: 313-222-6651, lhiggins@freepress.com or @LoriAHiggins
Read or Share this story: https://on.freep.com/1TsHkEn


text:
In Hope Stories
Just 1% of the national research spend has been allocated to this devastating disease
James Crossley
Life was turned upside down in August 2000 when James, aged nine, was diagnosed with a brain tumour and underwent two huge operations. After the last operation he was left with weakness down his right side, severe speech problems, as well as educational and visual difficulties. Today, James' story is one of hope as he overcomes his disabilities and looks to a more independent future.
Looking back, it was how James' behaviour changed that we know now were the first indications of his disease. Over a few months he became very lethargic and was just not himself. After his birthday, I sat James down to write thank you letters and was shocked to find his writing looking like a spider had crawled across the page.
After a visiting specialist teacher observed that James was no longer able to hold a pen properly, I spoke to a GP regarding his changes. A blood test was carried out and came back indicating that James' iron levels were slightly low.
We then saw a paediatrician who didn't seem unduly concerned about James until we mentioned a little tick which had developed in his hand. It was only when a piece of paper was laid on James' hands and also on his brother's hands that the paediatrician could see a distinct difference, but he was still reluctant about sending James for a scan. Even then it seemed that no one believed me that something was wrong.
We persisted however and the subsequent scan revealed a very rare brain tumour and a cyst the size of a mandarin. Despite all our concerns, we were totally unprepared for such devastating news.
James went through two huge operations – the first to remove the cyst and alleviate the pressure which had built up in his brain, the second to remove the tumour and almost the whole of his left temporal lobe. At that time his particular brain tumour couldn't be treated with chemo or radiotherapy. The treatments left James with considerable weakness down his right side and severe speech problems, as well as visual and learning difficulties.
Time stood still for months on end, but through sheer determination James got back on a skateboard within six months and his bike after two years. He rarely complained and was always cheerful. He was spurred on over the years by such a wonderful, dedicated team of specialists in all areas, especially in physio and speech therapy.
Through sheer determination, James defied all the 'managed' expectations and achieved seven GCSEs, employing mind-mapping techniques which helped him to learn in a visual way. James went on to achieve an AS level at Grade A, a BTEC National Diploma with three distinctions and then a Foundation Degree in Art and Design.
James has an acquired brain injury which is a hidden disability and no one can see that he has lost a huge part of his brain. Due to his visual impairment, he is unable to drive and has to rely on others to get about, and still needs support in many other areas. The job market for someone like James is practically non-existent and the search for suitable employment is quite soul-destroying. James has applied for numerous jobs, has had one interview and only two replies to all the applications so far. He is desperate to find a job and "wants to do something with his life".
It has been an incredible journey. We are so proud of James and what he has achieved through his remarkable strength of mind. He loves challenges and is now focusing on para cycling!
His life has enriched us and has shown us how much good has come out of this "evil tumour". We live in hope and live life to the full, savouring every moment that we have as a family. But it has also left us with anger and frustrations at the "not knowing": the chronic lack of investment in brain tumour research, the lack of knowledge about how and why brain tumours are formed, and the fact that there is still no known cure. Life still goes on hold at every scan.
Despite all the ongoing challenges, James is a cheerful young man with a strong determination to succeed beyond his limitations. He knows that he is a very lucky and is thankful every day for the life he has.
Our hearts go out to those families who are going through worse than us or have lost their fight against such a dreadful disease. There is a desperate need to raise money to fund research which will combat this "beast".
Amanda Crossley


The Food as Medicine series was created as a way to educate the community on how food affects your body and how making healthy choices can lead to a healthier life. Presented by professional nutrition experts Vibeke Weiland & Randi Miller, this lecture will cover how adding cruciferous vegetables like broccoli, cauliflower, and cabbage to your family's diet can help prevent cancer, detoxify harmful chemicals from your body, and positively change the way your genes affect your health.
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This entry was posted on January 18, 2019 by Hailey Phelan.


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Lloyd on the hunt for points at Oulton Park
Latest News posted 17th May 2017
MG Racing RCIB Insurance's Daniel Lloyd heads to Cheshire on the points trail as the Dunlop MSA British Touring Car Championship arrives at Oulton Park this weekend.
Lloyd has an excellent track record on the 2.23-mile Island layout. The MG Racing RCIB Insurance racer sealed a first BTCC front row start here last season in the Eurotech Racing Honda Civic Type R – missing out on pole by just 0.019 seconds.
"I'm looking forward to bouncing back from a tough weekend at Thruxton," said Lloyd. "Oulton Park is one of my favourite circuits. It's one of the closest tracks to home and we always have fantastic local support there from fans and sponsors – many of whom are coming along at the weekend."
Lloyd has endured a tough start to his 2017 BTCC campaign that has seen the Yorkshireman secure just two points finishes from nine races, with the Triple-Eight-run squad striving to unlock the MG6.
"It's no secret that we've been struggling this season, but with the change over to the new suspension kit this year combined with the new Dunlop tyre and limited pre-season testing it's been a really steep learning curve for us all," explained Lloyd.
"Hopefully the things that we picked up from our recent test at Croft will be more relevant here and we'll be able to stay out of trouble and make some progress.
"The MG Racing RCIB Insurance crew have done a brilliant job to get the car ready for this weekend and have been working non-stop to get us back on the front foot;" the Yorkshireman added.
"The guys have been busy on the rig since Thruxton, working on the dampers and we've made some pretty big changes to the MG6 for Oulton, so hopefully that work will start to pay off in results this weekend."


Each year there is an analysis of how far ISO is spreading and other factors they deem important. I believe you should take a read through the latest survey to get an idea how far and wide ISO 9K series has spread.
The ISO "Magical Demystifying Tour of ISO 9000 and ISO 14000" is on the web and gives a somewhat simplistic overview of the standard. It's mostly a promo, but does contain some pertinent information.
But watch out what you get from the ISO folks. Well meaning, there is a slight delay from time to time. For example, as of April 2001 their document meant to help companies decide which standard to apply (Selection and Use) is still posted, though it is obsolete (disk file name is 9000-selusee.pdf).
As with any other Miracle Cure, ISO registration is not the answer to all. If a business is being run poorly, ISO 9001 registration is not likely to significantly improve things. Companies have personalities which are typically set by upper management. They rarely 'Find Religion', so to speak, because of an ISO implementation process. So - don't expect miracles.
The ISO organization publishes a yearly assessment of the ISO 9001 distribution.


NYT Book Review's "By the Book" Interview with Ian Rankin!
Ian Rankin's "Rebus" series is one of the best known contemporary mystery series from the U.K. He has received the Crime Writers Association's Diamond Dagger for lifetime achievement in the genre. He has also been nominated a number of times for the Edgar award and won it in 2004. Click on our link here to read the New York Time Book Review's "By the Book" interview with this wonderful Scottish author!
Click here to go to By The Book's interview with Ian Rankin!


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