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Florida: You Can't Unintentionally Abandon A Nonconforming Use
The Town of Fort Myers Beach, Florida, barred the sale of alcohol on beaches in 1995. Turns out that a beachfront business was already (legally) selling alcohol on its property at the time of the ban. And we know what that usually means: a grandfathered nonconforming use.
Today's case from the Florida District Court of Appeal (Second District), Persaud Properties FL Investments, LLC v. Town of Fort Meyers Beach, No. 2D19-1282 (Dec. 11, 2020), is at the intersection of two of our favorite subjects, land use and inverse condemnation.
No one questioned whether Persaud possessed a nonconforming use. The issue was whether it had it abandoned it when it closed the establishment "to begin extensive renovations."
The Town was well aware of the renovations as various construction permits had to be issued and inspections had to occur; additionally, during the renovation period, multiple stop-work orders were issued by the Town. Upon completion of the renovations in October 2015, Persaud sought the necessary approval to reopen and begin selling alcohol on the premises, including in the EC zone. During the one-year period of closure and at the request of Persaud, the property's liquor license had been held in escrow or "inactive status" by the Department of Alcoholic Beverages and Tobacco, a fact which had been known to the Town and which was important to a Town planner for purposes of the abandonment issue.
Slip op. at 2-3.
No-go, said the Town, you abandoned.
Persaud sued for a taking, among other claims, but the trial court granted the Town summary judgment, concluding Persaud had abandoned the nonconforming use by non-use, and that it did not, therefore, possess property that was taken.
The court of appeal reversed. Reading the Town's land development code's definition of abandonment to mean non-use of the location for nine straight months, the court held that an owner must intentionally cease the use:
Strictly construing the Town's ordinance in favor of the property owner, as we must, abandonment of a nonconforming use requires more than the passage of nine months while the property was closed for renovations; it requires voluntary cessation of the nonconforming use with the intent that the cessation of such use be permanent. The trial court erred in failing to apply the common law and in determining, as a matter of law, that Persaud had abandoned the nonconforming use of its property where there was no evidence that Persaud intended to discontinue selling alcohol in the EC zone.
Persaud did not abandon or discontinue the nonconforming use of the
property during the one-year period of closure where renovations and construction were ongoing. Persaud is therefore entitled, under the applicable provisions of the Town's municipal code, to maintain the property's status as a grandfathered nonconforming use.
Slip op. at 9.
Persaud Properties FL Investments, LLC v. Town of Ft Myers Beach, No. 2D19-1282 (Fla. Dist. Ct. App. Dec. 1...
Posted on December 22, 2020 in ▪ Inverse condemnation, ▪ Land use law, ▪ Municipal & Local Govt law, ▪ Vested rights, ▪ Zoning & Planning | Permalink


Hjem til grden 'Hjem til Grden' Ls Esmaeils brev og se, hvem han vlger til nste uges storbonde.What's on TV? Find out quickly and easily using TV.co.uk one of the most liked TV listings sites in the UK..What's on TV? Find out quickly and easily using TV.co.uk one of the most liked TV listings sites in the UK.."Amy's Baking Company" is the sixteenth episode of the sixth season of Kitchen Nightmares, and the nd episode of the series. The episode first aired on and centered onon Ramsay attempting to help Amy and Samy Bouzaglo, owners of Amy's Baking Company in Scottsdale, Arizona..
"Amy's Baking Company" is the sixteenth episode of the sixth season of Kitchen Nightmares, and the nd episode of the series. The episode first aired on and centered onon Ramsay attempting to help Amy and Samy Bouzaglo, owners of Amy's Baking Company in Scottsdale, Arizona.. So i bought some KitKats which came from Japan and I decided to film myself eating the different tastes..


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\section*{Introduction}

   Natural phenomena, financial events and risks are usually modeled through random vectors or processes. In these fields, considering dependencies between random variables is necessary. In actuarial science, this issue has led to the development of multivariate risk theories. Indeed, an insurance company is generally exposed to several risks which cannot be assumed to be independent. It is therefore necessary to adopt a multivariate approach that takes into account both the marginal structures of risks and their dependence structure.\\

   Multivariate risk theory is based on dependence modeling and includes multivariate ruin probabilities and multivariate risk measures. In the univariate case, ruin probability has been widely studied since the beginning of the 20th century (Lundberg (1903) \cite{lundberg1903}, Cramer (1930) \cite{cramer1930l}). In multivariate contexts, several definitions are proposed for ruin probability. Hult and Lindskog (2006) \cite{Mbt} defined a multivariate ruin probability based on a notion of ruin sets. Cai and Li (2007) \cite{Cai} defined different finite-time ruin probabilities. Risk measure theory was also enriched by some multivariate risk measures definitions. Jouini et al. (2004) \cite{jouini2004} defined vector-valued coherent risk measures. Cousin and Di Bernardino (2013) \cite{Elena2013} introduced some multi-dimensional extensions of usual univariate risk measures such as the Value-at-Risk.
   These measure are multi-dimensional valued, and thus cannot take benefit from a full order. Dhaene et al.~(2012) introduced of real valued family for risk measures for mono-periodic multivariate processes (\cite{AR3}). In multi-periodic context, Cénac et al. (2012) \cite{AR2} defined new multivariate risk indicators as sum of expected local ruin amounts using penalty functions.\\

   Capital allocation may be seen as a direct application of multivariate risk theory. Allocating a capital in actuarial contexts means distributing a positive quantity representing an allocation capital in a simplex. For insurance industry the calculation of the regulatory economic capital, which is called the Solvency Capital Requirement, is well controlled and its methodology is almost imposed by the supervisory authorities of the sector. Nevertheless, the allocation of this capital may be considered as an internal exercise for each company, and constitutes a management choice whose success is a key factor for firm performance optimization. It can also be seen as an indicator of its good governance.\\

  Several allocation methods were proposed in the actuarial literature. Some are driven from ideas of risk allocation using Euler or Shapley principles (see as examples Tasche (2007) \cite{Euler2} for the Euler allocation principle and Denault(2001) \cite{Denault2001} for the Shapley allocation principle). Others are based on the minimization of some ruin probabilities or multivariate risk indicators.  In this context, Cénac et al. (2012) \cite{AR2} proposed a capital allocation by minimizing some multivariate risk indicators. They presented a numerical optimization algorithm to find the optimal allocation in some general cases. In \cite{AR1} properties and asymptotic behavior of the allocation are studied in some bivariate cases. The impact of dependence on capital allocation was studied for certain cases of Euler's method by Barg{\`e}s et al. (2009) \cite{MarceauTvar}  and Cossette et al. (2012) \cite{MarceauTvar2}.\\

  In this paper, we give a further study on the allocation method by minimizing some multivariate risk indicators. The idea of minimizing multivariate risk indicators seems from our point of view more suited for the ORSA\footnote{Own Risk and Solvency
    Assessment} approach of Solvency 2 European norms. In a recent paper \cite{P12015}, we show that this method satisfies a set of coherence axioms. In the present paper, we study the impact of marginal distributions and dependence structure on the optimal allocation. We generalize the results presented in bivariate cases by Cénac et al. (2014) \cite{AR1}. Moreover, we study the impact of the dependence structure on the allocation using some parametric copulas. Our main results are explicit formulas for various classes of multivariate processes, as well as the limit behavior of the allocation as the capital goes to infinity.\\

    The paper is organized as follows. In the first section, we recall the allocation method by minimizing multivariate risk indicators. The second section is a generalization in higher dimension of the results presented for the bivariate case in Cénac et al.(2014) \cite{AR1}. We present some explicit formulas obtained for some particular models, and we discuss the asymptotic behavior of the optimal allocation for these models. The impact of the dependence structure on the allocation is studied in Section 3 through the analysis of comonotonic cases and some examples of copulas (FGM, Marshall-Olkins...).

   \section{Optimal allocation}
   In this section, we recall one capital allocation principle that consists of minimizing some risk indicators. This method was introduced in Cénac et al. (2012) \cite{AR2}. It is based on the minimization of some multivariate risk indicators.\\

     In a multivariate risk framework, we consider an insurance group composed of $d$ branches or business lines. We denote by $u$ the initial capital of the group. Let $X^p$  be a vectorial risk process $X_p=(X^1_p,\ldots,X^d_p)$, where $X^k_p$ corresponds to the losses  of the $k^{th}$ business line during the $p^{th}$ period.  We denote by  $R^k_p$ the reserve of the $k^{th}$ line at time $p$, so: $R^k_p=u_k-\displaystyle\sum_{\ell=1}^{p}X^k_\ell$, where $u_k\in\mathbb{R}^+$ is the initial capital of the $k^{th}$ business line, then $u=u_1+\cdots+u_d$.\\

   Cénac et al. (2012) \cite{AR2} defined the two following multivariate risk indicators, given penalty functions $g_k$:
   \begin{itemize}
   \item The indicator $I$: \[ \mathit{I}(u_1,\ldots,u_d)=\sum_{k=1}^{d}{\mathbb{E}\left(\sum_{p=1}^{n}{g_k(R_p^k)1\!\!1_{\{R_p^k<0\}}1\!\!1_{\{\sum_{j=1}^{d}R_p^j>0\}}}\right)} \/,\]

   \item The indicator $J$:

    \[ \mathit{J}(u_1,\ldots,u_d)=\sum_{k=1}^{d}{\mathbb{E}\left(\sum_{p=1}^{n}{g_k(R_p^k)1\!\!1_{\{R_p^k<0\}}1\!\!1_{\{\sum_{j=1}^{d}R_p^j<0\}}}\right)} \/,\]
   \end{itemize}
   $g_k: \mathbb{R}^-\rightarrow \mathbb{R}^+$ are $C^1$, convex functions with $g_k(0)=0\/, ~g_k(x)\geq 0$ for $x < 0,~k=1,\ldots,d$. They represent the cost that each branch has to pay when it becomes insolvent while the group is solvent for the $I$ indicator, or while the group is also insolvent in the case of the $J$ indicator.\\
   These multivariate risk indicators are a generalization of the mono-periodic indicator family introduced in Dhaene et al.(2012)\cite{AR3}.\\

  The indicator $I$ represents the expected sum of penalty amounts of local ruins, knowing that the group remains solvent. In the case of the indicator $J$, the local ruin severities are taken into account only in the case of group insolvency.\\

  The idea of the optimal allocation is to allocate some capital $u$ by minimizing these indicators. This is finding an allocation vector $(u_1,\ldots,u_d)$ that minimizes the indicator such as $u=u_1+\cdots+u_d$, where $u$ is the initial capital that need to be shared among all branches.  In our paper \cite{P12015}, we explained why this capital allocation method can be considered as economically coherent.\\

   By using optimization stochastic algorithms, one can estimate the minimum of these risk indicators. Cénac et al. (2012) \cite{AR2} propose a Kiefer-Wolfowitz version of the mirror algorithm as a convergent algorithm under general assumptions to find optimal allocation minimizing the indictor $I$. This algorithm is effective to solve the optimal allocation problem, especially for a large number of business lines, and for allocation over several periods.\\

   Since the solvency capital requirement (SCR) is calculated for one year time horizon only, it seems more practical to focus on the case of mono-periodic allocation.\\
   The risk $X_k$ corresponds to the losses of the $k^{th}$ branch during one period. It is a positive random variable in our context. We denote by $u$ the initial capital of the firm, it represents the allocation capital and it can be, as an example, the SCR or another investment capital. The quantity $u_k$ represents the portion of capital allocated to the $k^{th}$ branch, then $\sum_{i=1}^{d}u_i=u$. The simplex $\mathcal{U}^d_u=\{v=(v_1, \ldots, v_d)\in[0,u]^d, \sum_{i=1}^{d}v_i=u\}$ is the set of all possible allocations. For all $i\in \{1,\ldots,d\}$ we denote $\alpha_i=\frac{u_i}{u}$, then, $\sum_{i=1}^{d}\alpha_i=1$. For $(u_1,\ldots,u_d)\in\mathcal{U}^d_u$, we define also the reserve of the $k^{th}$ business line at the end of the period as $R^k=u_k-X_k$.  The aggregate sum of risks is $ S=\sum_{i=1}^{d}{X_i} $, and let  $ S^{-i}=\sum_{j=1;j\neq i}^{d}{X_j} $ for all $i\in \{1,\ldots,d\}$. Finally, $F_Z$ is the distribution function of a random variable $Z$, $\bar{F}_Z$ is its survival function and $f_Z$ its density function.\\

   In order to ensure the existence and uniqueness of the indicators minimum in $\mathcal{U}^d_u$, we suppose that for at least one $k\in\{1,\ldots,d\}$, $g_k$ is strictly convex. We also assume that the joint density $f_{(X_k,S)}$ support contains $[0,u]^2$ (see \cite{AR2}). In this case, the indicators $I$ and $J$ are strictly convex and admit a unique minimum.\\

   Now we focus on the optimality condition for the indicators $I$ and $J$.\\
   For an initial capital $u$, and an optimal allocation minimizing the multivariate risk indicator $I$, we look for $u^*\in \mathbb{R}_+^d$ such that:
   \[\mathit{I}(u^*)=\underset{v_1+\cdots+v_d=u}{\inf}\mathit{I}(v), ~~v\in \mathbb{R}_+^d  \/.\]
    We assume that the functions $g_k$ are differentiable and such that for all $k\in\{1,\ldots,d\}$, the derivative $g^\prime_k(u_k-X_k)$ admits a moment of order one, and that the random vector $(X_k,S)$ has a joint density distribution denoted by $f_{(X_k,S)}$. Under these assumptions, the risk indicators $I$ and $J$ are differentiable, and in this case, we can calculate the following gradients:
         \begin{align*}
         (\nabla I(v))_i&=\sum_{k=1}^{d}\int_{v_k}^{+\infty}g_k(v_k-x)f_{X_k,S}(x,u)dx
         + \mathbb{E}[g^\prime_i(v_i-X_i)1\!\!1_{\{X_i>v_i\}}1\!\!1_{\{S\leq u\}}]\\
         \mbox{and,}~~ (\nabla J(v))_i&=\sum_{k=1}^{d}\int_{v_k}^{+\infty}g_k(v_k-x)f_{X_k,S}(x,u)dx
               + \mathbb{E}[g^\prime_i(v_i-X_i)1\!\!1_{\{X_i>v_i\}}1\!\!1_{\{S\geq u\}}]\/.
         \end{align*}
   Using the Lagrange multipliers method, we obtain an optimality condition verified by the unique solution to this optimization problem:
   \begin{equation}
   \mathbb{E}[g^\prime_i(u_i-X_i)1\!\!1_{\{X_i>u_i\}}1\!\!1_{\{S\leq u\}}]=\mathbb{E}[g^\prime_i(u_j-X_j)1\!\!1_{\{X_j>u_j\}}1\!\!1_{\{S\leq u\}}],~~\forall j\in\{1,\ldots,d\}^2
   \label{Optcg}\/.
   \end{equation}
   A natural choice for penalty functions is the ruin severity: $g_k(x)=|x|$. In that case, and if the joint density $f_{(X_k,S)}$ support contains $[0,u]^2$, for at least one $k\in\{1,\ldots,d\}$, our optimization problem  has a unique solution.\\

   In Maume-Deschamps et al. (2015) \cite{P12015}, we showed that in the case of penalty functions $g_k(x)=|x|~ \forall k\in \{1,\ldots,d\}$, and for continuous random vector $(X_1,\ldots,X_d)$ such that the joint density $f_{(X_k,S)}$ support contains $[0,u]^2$, for at least one $k\in\{1,\ldots,d\}$, the optimal allocation by minimization of the indicators $I$ and $J$ is a symmetric riskless full allocation. It satisfies also the properties of comonotonic additivity, positive homogeneity, translation invariance, monotonicity, and continuity.\\

  We may write the indicators as follows:
 \begin{align*}
 \mathit{I}(u_1,\ldots,u_d)&=\sum_{k=1}^{d}{\mathbb{E}\left({\arrowvert R^k \arrowvert1\!\!1_{\{R^k<0\}}1\!\!1_{\{\sum_{i=1}^{d}R^i\geq0\}}}\right)}\\
 &=\sum_{k=1}^{d}{\mathbb{E}\left({(X_k-u_k)1\!\!1_{\{X_k>u_k\}}1\!\!1_{\{\sum_{i=1}^{d}X_i\leq u\}}}\right)}=\sum_{k=1}^{d}{\mathbb{E}\left({(X_k-u_k)^+1\!\!1_{\{S\leq u\}}}\right)}\/,
 \end{align*}
 and,
 \begin{align*}
 \mathit{J}(u_1,\ldots,u_d)&=\sum_{k=1}^{d}{\mathbb{E}\left({\arrowvert R^k \arrowvert1\!\!1_{\{R^k<0\}}1\!\!1_{\{\sum_{i=1}^{d}R^i\leq0\}}}\right)}\\
 &=\sum_{k=1}^{d}{\mathbb{E}\left({(X_k-u_k)1\!\!1_{\{X_k>u_k\}}1\!\!1_{\{\sum_{i=1}^{d}X_i\geq u\}}}\right)}=\sum_{k=1}^{d}{\mathbb{E}\left({(X_k-u_k)^+1\!\!1_{\{S\geq u\}}}\right)}\/.
 \end{align*}
In the special case where $g_k(x)=|x|$, \ref{Optcg} becomes:
\begin{equation}\label{ZoneO}
\mathbb{P}\left( X_i>u_{i}, S\leq u\right) = \mathbb{P}\left( X_j>u_{j}, S\leq u\right) , \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2}\/.
\end{equation}
For the $J$ indicator, this condition can be written:
\begin{equation}\label{ZoneV}
\mathbb{P}\left( X_i>u_{i}, S\geq u\right) = \mathbb{P}\left( X_j>u_{j}, S\geq u\right) , \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2}\/.
\end{equation}
Some explicit and semi-explicit formulas for the optimal allocation can be obtained with this optimality condition. Our problem reduces to the study of this allocation depending on the nature of the distributions of the risk $X_k$ and on the form of dependence between them.
   \section{Some general results in the independence case}
   In this section we generalize the results presented in dimension 2 by Cénac et al. (2014) in the first section of their paper \cite{AR1} to higher dimension.\\
   The results presented here give explicit forms to the optimal allocation for some specific distributions. This could be used as a benchmark to test optimization algorithms convergence.\\

    We also get some asymptotic results, when the capital $u$ goes to infinity. We study both the exponential and the sub-exponential cases, and we determine the difference between their asymptotic behavior for exponential and Pareto distributions cases.\\

   We consider from now on that the penalty functions are identical and equal to the severity of local ruin $g_k(x)=g(x)=|x|, \forall k\in\{1,2,\ldots,d\}$. The optimality conditions for minimizing the multivariate risk indicators $I$ and $J$ are given respectively by Equations (\ref{ZoneO}) and (\ref{ZoneV}). In this section we focus on the independence case.\\

   Recall that $\alpha_i=\frac{u_i}{u}\in[0,1]$, so that when $u\rightarrow +\infty$, we may consider convergent subsequences in the proofs below. By abuse of notation, we consider $\underset{u\rightarrow+\infty}{\lim}\alpha_i$. In fact, we consider a convergent subsequence and get the existence of the limit by obtaining the uniqueness of the limit point.
   \subsection{Independent exponentials}~~\\
 ~~Assume $X_1,X_2,\ldots,X_d$ are independent exponential random variables with respective parameters $0<\beta_1<\beta_2<\cdots<\beta_d$. Remark that in the
 particular case where $\beta_1=\beta_2=\cdots=\beta_d$, the optimal allocation is $u_1=\cdots= u_d= u/d$.
 \begin{Proposition}[The optimal allocation for the indicator $I$]
 \label{prp21}
 The allocation minimizing the risk indicator $I$ is the unique solution in $\mathcal{U}^d_u$, of the following equations system:
  \begin{equation}
  h(\beta_i\alpha_i)-h(\beta_j\alpha_j)
   -\sum_{\ell=1}^{d}{A_\ell h(\beta_\ell)[h(\alpha_i\cdot(\beta_i-\beta_\ell) )-h(\alpha_j\cdot(\beta_j-\beta_\ell) )]}=0,\forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2}
   \label{EIZO}\/,
 \end{equation}
 where h is the function defined by $ h(x)= \exp(-u\cdot x)$, and $A_\ell$ denotes the constants \text{$ A_\ell=\displaystyle\prod_{j=1,j\neq \ell}^{d}{\dfrac{\beta_j}{\beta_j-\beta_\ell}}$}, for
  $\ell=1,\ldots,d$.
 \end{Proposition}
 \begin{proof}
 If $X_i\sim\mathcal{E}(\beta_i)$ are independent exponential random variables, then $S^{-i} =\displaystyle\sum_{j=1;j\neq i}^{d}{X_j}$ have a generalized Erlang distribution with parameters $(\beta_1,\beta_2,\ldots,\beta_{i-1},\beta_{i+1},\ldots,\beta_d)$, so we write:
 \begin{align*}
  \mathbb{P}\left( X_i>u_{i},\sum_{j=1}^{d}{X_j}\leq u\right) &= \mathbb{P}\left( X_i>u_{i}\right) -\mathbb{P}\left( X_i>u_{i},\sum_{j=1}^{d}{X_j}>u\right)\\
  & = \bar{F}_{X_i}(u_{i})-\bar{F}_{X_i}(u)-\int_{u_{i}}^{u}\bar{F}_{S^{-i}}(u-s)f_{X_i}(s) \mathrm{ds}\\
  &=h(\beta_i\alpha_i)-h(\beta_i)-\sum_{\ell=1}^{d}{A_\ell h(\beta_\ell)h(\alpha_i\cdot(\beta_i-\beta_\ell) )}+\sum_{\ell=1}^{d}{A_\ell h(\beta_i)} \\
  &=h(\beta_i\alpha_i)-\sum_{l=1}^{d}{A_\ell h(\beta_\ell)h(\alpha_i\cdot(\beta_i-\beta_\ell) )} \/,
  \end{align*}
  because, $\bar{F}_{X_i}(x)=e^{-\beta_ix}$,  $\bar{F}_{S^{-i}}(x)=\displaystyle\sum_{\ell=1,\ell\neq i}^{d}{(\displaystyle\prod_{j=1,j\neq \ell,j\neq i}^{d}{\dfrac{\beta_j}{\beta_j-\beta_\ell}})e^{-\beta_\ell x}}$ and $\displaystyle\sum_{l=1}^{d}{A_\ell}=1$.\\
  The survival function of the generalized Erlang distribution with parameters $(\beta_1,\beta_2,\ldots,\beta_d)$ is given by:
  \[ \bar{F}_{X}(x)=\sum_{\ell=1}^{d}{(\displaystyle\prod_{j=1,j\neq \ell}^{d}{\dfrac{\beta_j}{\beta_j-\beta_\ell}})e^{-\beta_\ell x}}=\sum_{\ell=1}^{d}{A_\ell e^{-\beta_\ell x}}\/. \]
The optimal allocation is the unique solution in $\mathcal{U}^d_u$, of the following equations system:
 \[
  \mathbb{P}\left( X_i>u_{i},\displaystyle\sum_{k=1}^{d}{X_k}\leq u\right) = \mathbb{P}\left( X_j>u_{j},\displaystyle\sum_{j=k}^{d}{X_k}\leq u\right) , \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2}
 \/,\]
which leads to (\ref{EIZO}).
 \end{proof}
 The resulting system is a system of nonlinear equations, which can be solved numerically.

 \begin{Proposition}[The asymptotic optimal allocation for the indicator $I$]
 \label{pro3.2} When the capital $u$ goes to infinity, the asymptotic optimal allocation satisfies:\[\underset{u\to\infty}\lim{\left(\frac{u_1}{u},\frac{u_2}{u},\ldots,\frac{u_d}{u}\right)}=
     \left(\dfrac{\dfrac{1}{\beta_i}}{\displaystyle\sum_{j=1}^{d}{\dfrac{1}{\beta_j}}}\right)_{i=1,2,\ldots,d}
     \/.\]
 \end{Proposition}
 \begin{proof}
 Equations system (\ref{EIZO}) is equivalent to:
 \begin{equation}
 \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2},~~h(\beta_i\alpha_i)[1-\displaystyle\sum_{\ell=1}^{d}{A_\ell h((1-\alpha_i)\beta_\ell)}] =
 h(\beta_j\alpha_j)[1-\displaystyle\sum_{\ell=1}^{d}{A_\ell h((1-\alpha_j)\beta_\ell)}]\/.
 \label{eq2}
 \end{equation}
 Firstly, remark that for all $i \in\{{1,2,\ldots,d}\}$, $\underset{u\to\infty}\lim{\sup\frac{u_i}{u}}<1 $ because if this result was not satisfied then there would exist $i\in\{1,\ldots,d\}$ such that  $\underset{u\to\infty}\lim{\frac{u_i}{u}}=\underset{u\to\infty}\lim{\alpha_i}=1 $, taking if necessary a convergent subsequence of $\alpha_i$.  For all $ j\neq i$,  $\underset{u\to\infty}\lim{\frac{u_j}{u}}=\underset{u\to\infty}\lim{\alpha_j}=0 $, and Equations system (\ref{ZoneO}) cannot be satisfied in this case.\\
 Equations system (\ref{eq2}) is equivalent to:
 \[\forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2},~~h(\beta_i\alpha_i-\beta_j\alpha_j)=\frac{1-\sum_{\ell=1}^{d}{A_\ell h((1-\alpha_j)\beta_\ell)}}{1-\sum_{\ell=1}^{d}{A_\ell h((1-\alpha_i)\beta_\ell)}}\/,
 \]
 the right side of the last equations system tends to 1 when $u$ tends to $\infty$, therefore, we deduce that
 $\underset{u\to\infty}\lim{h(\beta_i\alpha_i-\beta_j\alpha_j)}=1$ and consequently:
 \[ \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2},\underset{u\to\infty}\lim{\alpha_i}=\frac{\beta_j}{\beta_i}\underset{u\to\infty}\lim{\alpha_j}\/, \]
 then, for all $i \in\{{1,2,\ldots,d}\}$:
  \[ \underset{u\to\infty}\lim{\alpha_i}=\dfrac{\dfrac{1}{\beta_i}}{\sum_{j=1}^{d}{\dfrac{1}{\beta_j}}}\/. \]
 \end{proof}
 \begin{remark}
 Based on the above result, we can conclude that asymptotically:
  \begin{itemize}
   \item since $\beta_i<\beta_j$ we have $\alpha_i>\alpha_j$, this means that we allocate more capital to the most risky business line.
   \item $\alpha_i$ is a decreasing function of $\beta_i$. This observation is consistent with the previous conclusion.

   \item $\alpha_j$ is an increasing function of $\beta_i$ for $j\neq i$.
  \end{itemize}
 \end{remark}
 \begin{Proposition}[The optimal allocation for the indicator $J$] The allocation minimizing the risk indicator $J$ is the unique solution in $\mathcal{U}^d_u$, of the following equations system:
 \begin{equation}
  \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2},~~~~
     \sum_{\ell=1}^{d}{A_\ell h(\beta_\ell)[h(\alpha_i\cdot(\beta_i-\beta_\ell) )-h(\alpha_j\cdot(\beta_j-\beta_\ell) )]}=0
     \/.\label{EIZV}
  \end{equation}
  \end{Proposition}
  \begin{proof}
  The proof is similar to that of Proposition \ref{prp21}.
  \end{proof}
  \begin{Proposition}[The asymptotic optimal allocation for the indicator $J$] When the capital $u$ goes to infinity, the optimal allocation minimizing the risk indicator $J$ is the following:
   \[\underset{u\to\infty}\lim{\frac{u_1}{u}}=1
         ~and~
        \underset{u\to\infty}\lim{\frac{u_{j}}{u}}=0
              ~\forall j \in \{2,3,\ldots,d\}\/.\]
    \end{Proposition}
   \begin{proof}
    Equations system (\ref{EIZV}) is equivalent to the following one:     \[ \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2},~~~~
         \sum_{\ell=1}^{d}{A_\ell\cdot h((1-\alpha_i)\cdot(\beta_\ell-\beta_i) )}=\sum_{\ell=1}^{d}{A_\ell\cdot h(\alpha_j\cdot(\beta_j-\beta_\ell)+\beta_\ell-\beta_i )}\/.
      \]
      If $\varlimsup\frac{u_1}{u}<1$, as $u$ goes to $+\infty$ in the equations of the previous system for $i=1$ we get, taking if necessary a convergent subsequence,
      \begin{equation}
      ~\forall j \in \{2,3,\ldots,d\}, ~~
      \underset{u\to\infty}\lim{\sum_{\ell=1}^{d}{A_\ell\cdot h(\alpha_j\cdot(\beta_j-\beta_\ell)+\beta_\ell-\beta_1 )}}=A_1\/.
      \label{veronique}
      \end{equation}
     The first terms of these equations can be decomposed into three parts as follows:
     \begin{align*}
     \underset{u\to\infty}\lim{\sum_{\ell=1}^{d}{A_\ell\cdot h(\alpha_j\cdot(\beta_j-\beta_\ell)+\beta_\ell-\beta_1 )}}&=A_1\cdot\underset{u\to\infty}\lim{ h(\alpha_j\cdot(\beta_j-\beta_1))}\\&+\sum_{\ell=2}^{j}{A_\ell}\cdot
             \underset{u\to\infty}\lim{ h(\alpha_j\cdot (\beta_j-\beta_\ell)+\beta_\ell-\beta_1 )}\\&+\sum_{\ell=j+1}^{d}{A_\ell}\cdot\underset{u\to\infty}\lim{ h(\alpha_j\cdot(\beta_j-\beta_\ell)+\beta_\ell-\beta_1 )}\/.
     \end{align*}
       For all,  $j>1$, $\displaystyle\sum_{\ell=2}^{j}{A_\ell}\cdot\underset{u\to\infty}\lim{ h(\alpha_j\cdot(\beta_j-\beta_\ell)+\beta_\ell-\beta_1 )}=0$, because $\beta_\ell-\beta_1>0$ and \text{$\beta_j-\beta_\ell\geq 0$} for $ \ell\in \{2,3,\ldots,j\}$. Moreover, $\displaystyle\sum_{\ell=j+1}^{d}{A_\ell}\cdot\underset{u\to\infty}\lim{ h(\alpha_j\cdot(\beta_j-\beta_\ell)+\beta_\ell-\beta_1 )}=0$, because for all \text{$\ell\in \{j+1,j+2,\ldots,d\},$} $\alpha_j\cdot(\beta_j-\beta_\ell)+\beta_\ell-\beta_1 = (\beta_\ell-\beta_j)(1-\alpha_j)+\beta_j-\beta_1>0$. So that \ref{veronique} leads to $\forall j\in\{1,\ldots,d\}$ \text{$\underset{u\to\infty}\lim{h(\alpha_j\cdot(\beta_j-\beta_1))}=1$}.
       We deduce that $~\forall j \in \{2,3,\ldots,d\}; ~~\alpha_j\overset{u\to\infty}{=}o(\frac{1}{u})$. This contradicts the necessary condition: $\underset{u\to\infty}{\lim}{\sum_{\ell=1}^{d}{\alpha_j}}=1$.
   \end{proof}

 \subsection{Some distributions of the sub-exponential family}~~\\
 In most cases of risk distributions, we cannot give explicit or semi-explicit optimal allocation formulas, the difficulty comes from the lack of a simple form of the risks sum $S$ and its joint distribution with each risk $X_i$. In this section, we present asymptotic results (as $u$ goes to infinity) for the optimal allocation in the case of some distributions of the sub-exponential family. In this way, we generalize results of \cite{AR1} to higher dimension.\\
 We recall the sub-exponential distributions family definition, consisting in distributions of positive support, with a distribution function that satisfies:
 \[ \frac{\overline{F^{*2}}(x)}{\overline{F}(x)}\overset{x\to+\infty}{\longrightarrow}2\/,\]
 where $\overline{F^{*2}}$ is the convolution of $\overline{F}$.\\
 In Asmussen (2000) \cite{RuinP}, it is proven that the sub-exponential distributions satisfy also the following relation, for all $d\in\mathbb{N}^*$:
 \[ \frac{\overline{F^{*d}}(x)}{\overline{F}(x)}\overset{x\to+\infty}{\longrightarrow}d\/,\]
 where $\overline{F^{*d}}$ is the $d^{th}$convolution of $\overline{F}$.\\
We shall use the following theorem proved in Cénac and al.
 \begin{theorem}[Sub-exponential distributions \cite{AR1}] \label{thZO}
 Let $X$ be a random variable with sub-exponential distribution $F_X$, $Y$ a random variable with support $\mathbb{R}^+$, independent of $X$, and $(u,v)\in (\mathbb{R}^+)^2$, such that:
 \begin{itemize}
 \item there exists $0<\kappa_1<\kappa_2<1$ such that for $u$ large enough, $\kappa_1\leq\frac{v}{u}\leq\kappa_2$,
 \item
 $ \frac{\bar{F}_X(y)}{\bar{F}_X(x)}\overset{x\to+\infty}{=}O(1)\/, \mbox{ if } y=\Theta(x)$\footnote{For $(x,y)\in\mathbb{R}^{2+}$, we shall denote $y=\Theta(x)$ if there exist $0<C_1\leq C_2<\infty$, such that for $x$ large enough, $C_1\leq\frac{y}{x}\leq C_2$ }.
 \end{itemize}
 Then, \[ \underset{u\to\infty}\lim{\frac{\mathbb{P}(X\geq v,~X+Y\geq u)}{\bar{F}_X(u)}}=1 \/.\]
 \end{theorem}
 \subsubsection{The asymptotic behavior}~~\\
 Here we examine the asymptotic behavior of the optimal allocation by minimizing the indicators $I$ and $J$ in the cases of some sub-exponential distributions.\\
  In what follows, $(u_1,\ldots,u_d)$ denotes the optimal allocation of $u$ associated to the risk indicator $I$.
  \begin{theorem}
   Let $(X_1,X_2,\ldots,X_d)$ be continuous positive and independent random variables.\\ Assume that $\forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2}$:
   \begin{enumerate}
   \item $\bar{F}_{X_i}(x)\stackrel{ x\rightarrow+\infty}{=}\Theta(\bar{F}_{X_j}(x))$,
   \item $\bar{F}_{X_i}(s)\stackrel{s\rightarrow +\infty}{=}o(\bar{F}_{X_i}(t))$, if $t=o(s)$,
   \end{enumerate}
   then, there exist $\kappa_1>0$ and $\kappa_2<1$ such that,
   \begin{align}
      \kappa_1\leq \frac{u_\ell}{u} \leq \kappa_2 ~~~ \forall \ell\in \{1,2,\ldots,d\}\/.
      \label{k1k22}
   \end{align}
   \label{th2.6}
  \end{theorem}
  Note that the first condition of Theorem \ref{th2.6} is not satisfied for exponential distributions. However, Pareto distributions satisfy the hypothesis of Theorem \ref{th2.6}.
  \begin{proof}
   Taking if necessary a convergent subsequence, we assume that $\exists i\in\{1,\ldots,d\}$ such that: $\frac{u_i}{u}\stackrel{u\rightarrow +\infty}{\longrightarrow}1$ or $\frac{u_i}{u}\stackrel{u\rightarrow +\infty}{\longrightarrow}0$, the first case implies that foll all $j\neq i$, $\frac{u_j}{u}\stackrel{u\rightarrow +\infty}{\longrightarrow}0$, then, it is sufficient to prove that the existence of an $i\in\{1,\ldots,d\}$ such that $\frac{u_i}{u}\stackrel{u\rightarrow +\infty}{\longrightarrow}0$ is impossible.\\Let us assume the existence of $i\in{\{1,\ldots,d\}}$, such that: $\frac{u_i}{u}\stackrel{u\rightarrow +\infty}{\longrightarrow}0$.\\
    Then, $\exists j\in{\{1,\ldots,d\}\backslash i}$ such that $\underset{u\rightarrow +\infty}{\lim}\frac{u_j}{u}\in]0,1]$, therefore, $u_j\stackrel{u\rightarrow +\infty}{\longrightarrow}+\infty$ and $u_i\stackrel{u\rightarrow +\infty}{=}o(u_j)$.\\
    Using Assumptions (1) and (2), we deduce that:
    \begin{align}
    \frac{\bar{F}_{X_j}(u_j)}{\bar{F}_{X_i}(u_i)}\stackrel{u\rightarrow +\infty}{\longrightarrow}0
    \label{eq21}\/.
    \end{align}
    The optimality condition \eqref{ZoneO} can also be written for all $j\neq i$ as follows:
    \begin{align}
    \bar{F}_{X_i}(u_{i})-\mathbb{P}\left( X_i>u_{i},S>u\right)=\bar{F}_{X_j}(u_{j})-\mathbb{P}\left( X_i>u_{j},S>u\right)
    \label{opti2}\/.
    \end{align}
    That presents a trivial contradiction if $u_i$ remains bounded.\\
   Now, assume that $u_i\rightarrow+\infty$. Recall that $S^{-i}=\displaystyle\sum_{k=1,k\neq i}^{d}{X_k}$, then: \[
   \mathbb{P}\left( X_i>u_{i},S>u\right)=\mathbb{P}\left( X_i>u_{i},S^{-i}>\sqrt{u},S>u\right)+\mathbb{P}\left( X_i>u_{i},S^{-i}<\sqrt{u},S>u\right)
    \/.\]
    We have:
   \[ \mathbb{P}\left( X_i>u_{i},S^{-i}>\sqrt{u},S>u\right)\leq \mathbb{P}\left( X_i>u_{i}\right)\mathbb{P}\left(S^{-i}>\sqrt{u}\right)\stackrel{u\rightarrow +\infty}{=}o(\bar{F}_{X_i}(u_{i}))\/. \]
   Using assumption (2) and since $u_i=o(u)$,
   \[ \mathbb{P}\left( X_i>u_{i},S^{-i}<\sqrt{u},S>u\right)\leq\bar{F}_{X_i}(u-\sqrt{u})\stackrel{u\rightarrow +\infty}{=}o(\bar{F}_{X_i}(u_{i})). \]
   We deduce that:\begin{align}
   \mathbb{P}\left( X_i>u_{i},S>u\right)  \stackrel{u\rightarrow +\infty}{=}o(\bar{F}_{X_i}(u_{i}))
   \label{eq22}\/.
   \end{align}
   We remark also that:
   \begin{align}
      \mathbb{P}\left( X_j>u_{j},S>u\right)  \stackrel{u\rightarrow +\infty}{=}O(\bar{F}_{X_j}(u_{j}))\stackrel{u\rightarrow +\infty}{=}o(\bar{F}_{X_i}(u_{i}))
      \label{eq23}\/.
      \end{align}
  Equation (\ref{opti2}) leads to:
  \[ 1-\underbrace{\frac{\mathbb{P}\left( X_i>u_{i},S>u\right)}{\bar{F}_{X_i}(u_{i})}}_{T_1}=\underbrace{\frac{\bar{F}_{X_j}(u_{j})}{\bar{F}_{X_i}(u_{i})}}_{T_2}-\underbrace{\frac{\mathbb{P}\left( X_j>u_{j},S>u\right)}{\bar{F}_{X_i}(u_{i})}}_{T_3} \/.\]
  Now, relations: (\ref{eq22}), (\ref{eq21}), and (\ref{eq23}), imply that $T_1$, $T_2$, and $T_3$, go to zero, and this is a contradiction.
  \end{proof}
  \begin{Proposition}
  \label{subth1}
Let $X_1\/,\ldots\/, X_d$ be continuous, positive and independent random variables such that the support of the density of $(X_i\/,S)$ is $({\mathbb R}^+)^2$. Let $(u_1\/,\ldots\/, u_d)$ be the optimal allocation of $u$ associated to the risk indicator $I$. We assume:
\begin{enumerate}
\item there exist $0<\kappa_1<\kappa_2<1$ such that for all $i=1\/,\ldots\/, d$ and for all $u\in{\mathbb R}^+$,
$$\kappa_1\leq \frac{u_i}u\leq \kappa_2\/,$$
\item for all $i = 1\/, \ldots\/, d$, if $y=y(x)$ is such that
$$0< \liminf_{x\rightarrow\infty}\frac{y}x\leq \limsup_{x\rightarrow\infty}\frac{y}x<1$$
then
$$\frac{\overline{F}_{X_i}(x)}{\overline{F}_{X_i}(y)} \stackrel{x\rightarrow\infty}{\longrightarrow} 0 \/.$$
\end{enumerate}
Then, for all $i\/,j=1\/,\ldots\/,d$,
$$\frac{\overline{F}_{X_i}(u_i)}{\overline{F}_{X_j}(u_j)}\stackrel{u\rightarrow \infty}{\longrightarrow} 1\/.$$
 \end{Proposition}
 Assumptions of Proposition \ref{subth1} are satisfied for distributions of exponential type (see remark~(\ref{rem3.9}) below). Its application gives another proof to Proposition \ref{pro3.2}. In contrast, Proposition \ref{subth1} cannot be used for Pareto distributions.
\begin{proof}
Following the lines of the proof of Theorem \ref{th2.6}, take $0<\gamma<1-\kappa_2$,
$$\mathbb{P}(X_i > u_i\/, \ S>u) = \mathbb{P}(X_i >u_i \/, \ S^{-i} > \gamma u \/, \ S>u) + \mathbb{P}(X_i >u_i \/, \ S^{-i} > \gamma u \/, \ S>u) \/.$$
As before,
$$\mathbb{P}(X_i >u_i \/, \ S^{-i} > \gamma u \/, \ S>u)\leq \mathbb{P}(X_i > u_i ) \mathbb{P}(S^{-i} > \gamma u) \stackrel{u\rightarrow +\infty}{=} o(\overline{F}_{X_i}(u_i)) \/.$$
On the other hand,
\begin{eqnarray*}
\mathbb{P}(X_i >u_i \/, \ S^{-i} > \gamma u \/, \ S>u) & \leq & \mathbb{P}(X_i> u-\gamma u)\\
	&=& \overline{F}_{X_i}((1-\gamma)u) \stackrel{u\rightarrow +\infty}{=} o(\overline{F}_{X_i}(u_i)) \\
	&& \mbox{because} \ 0<\frac{\kappa_1}{1-\gamma}\leq \frac{u_i}{(1-\gamma)u}\leq \frac{\kappa_2}{1-\gamma}<1 \/.\\
\end{eqnarray*}
So that, $\mathbb{P}(X_i > u_i\/, \ S>u) \stackrel{u\rightarrow +\infty}{=} o(\overline{F}_{X_i}(u_i))$ and the same computation gives $\mathbb{P}(X_j > u_j\/, \ S>u) \stackrel{u\rightarrow +\infty}{=} o(\overline{F}_{X_j}(u_j))$. Now, $u_i$ and $u_j$ satisfy Equation (1) and thus,
$$1 + o(1) \stackrel{u\rightarrow +\infty}{=} \frac{\overline{F}_{X_j}(u_j)}{\overline{F}_{X_i}(u_i)} + o(1) \frac{\overline{F}_{X_j}(u_j)}{\overline{F}_{X_i}(u_i)}\/.$$
This implies that $\frac{\overline{F}_{X_j}(u_j)}{\overline{F}_{X_i}(u_i)}$ is bounded from above and thus
$$ \frac{\overline{F}_{X_j}(u_j)}{\overline{F}_{X_i}(u_i)}  \stackrel{u\rightarrow +\infty}{=} 1 + o(1) \/.$$
\end{proof}
  \begin{remark}\label{rem3.9}
  We remark that the hypothesis of Proposition \ref{subth1} are satisfied for distribution of {\em exponential type}, that is distributions verifying:\[ \bar{F}_{X_i}(x)=\Theta(e^{-\mu_ix}) \/.\]
  Indeed, in this case, we have $0<\varliminf\frac{u_i}{u}\leq\varlimsup\frac{u_i}{u}<1,~~\forall i\in\{1,\ldots,d\}$. In fact, if $u_i\stackrel{u\rightarrow +\infty}{=}o(u)$, then, $\exists j\in\{1,\ldots,d\}\backslash\{i\}$ such that $\frac{u_j}{u}\rightarrow \kappa\in]0,1]$, so $u_i\stackrel{u\rightarrow +\infty}{=}o(u_j)$.\\
  Since $\mu_i,\mu_j>0$, $\mu_iu_i\stackrel{u\rightarrow +\infty}{=}o(\mu_ju_j)$.
  As in the proof of Theorem \ref{th2.6}, we get:
  \begin{align*}
     \mathbb{P}\left( X_i>u_{i},S>u\right)  \stackrel{u\rightarrow +\infty}{=}o(\bar{F}_{X_i}(u_{i}))
     \end{align*}
  and,
  \begin{align*}
       \mathbb{P}\left( X_j>u_{j},S>u\right)  \stackrel{u\rightarrow +\infty}{=}o(\bar{F}_{X_j}(u_{j}))\/.
       \end{align*}
  From the optimality condition, \[ \frac{\bar{F}_{X_i}(u_{i})}{e^{-\mu_iu_i}}-\underbrace{\frac{\mathbb{P}\left( X_i>u_{i},S>u\right)}{e^{-\mu_iu_i}}}_{T_1}=\underbrace{\frac{\bar{F}_{X_j}(u_{j})}{e^{-\mu_iu_i}}}_{T_2}-\underbrace{\frac{\mathbb{P}\left( X_j>u_{j},S>u\right)}{e^{-\mu_iu_i}}}_{T_3}\/, \]
  which is absurd because as $u\rightarrow+\infty$:\begin{itemize}
  \item $\bar{F}_{X_i}(u_{i})=\Theta(e^{-\mu_iu_i})$,
  \item $T_1=o(1)$,
  \item $T_2=\frac{\bar{F}_{X_j}(u_{j})}{e^{-\mu_ju_j}}e^{-\mu_ju_j+\mu_iu_i}\rightarrow 0$, since $\mu_iu_i=o(\mu_ju_j)$,
  \item and $T_3=o(1)e^{-\mu_ju_j+\mu_iu_i}\rightarrow 0$.
  \end{itemize}
  \end{remark}
 \begin{Proposition}[The asymptotic optimal allocation for the indicator $I$]
 \label{subth2}
  Under the same conditions of Theorem \ref{th2.6}, and if, for all $i\in\{1,\ldots,d\}$, $F_{X_i}$ is a sub-exponential distribution, that verifies:
  \[ \frac{\bar{F}_{X_i}(y)}{\bar{F}_{X_i}(x)}\stackrel{x\rightarrow+\infty}{=}O(1),~~\text{for}~~0<\kappa_1\leq\frac{y}{x}\leq\kappa_2<1 \/.\]
   Then, by minimizing the $I$ indicator, $u_i$ and $u_j$ satisfy:
  \begin{equation}
        \bar{F}_{X_i}(u_i)-\bar{F}_{X_i}(u)\stackrel{u\rightarrow +\infty}{=}\bar{F}_{X_j}(u_j)-\bar{F}_{X_j}(u)+o(\bar{F}_{X_i}(u))\/.
        \label{eqsub2}
        \end{equation}
   \end{Proposition}
   Proposition \ref{subth2} is applicable in the case of Pareto distributions. So, we will use it for determining the optimal asymptotic allocation, for independent risks of Pareto distributions in the next subsection.
   \begin{proof}
   The proof of this theorem is a direct application of Theorems \ref{thZO} and \ref{th2.6}.
    \end{proof}
Now, we focus on the asymptotic optimal allocation by minimizing the risk indicator $J$, and we study the case of sub-exponential distribution family.
\begin{Proposition}[The asymptotic optimal allocation for the indicator $J$] \label{prpZV}
Let $(X_1,X_2,\ldots,X_d)$ be continuous positive and independent random variables, such that there exists $i\in\{1,\ldots,d\}$ with a sub-exponential distribution, the optimal capital allocation by minimizing the $J$ indicator $(u_1,\ldots,u_d)$ verifies, for all $~j\neq i$:
\[ \underset{u\to\infty}\lim{\frac{\mathbb{P}(X_j\geq u_j,~S\geq u)}{\bar{F}_{X_i}(u)}}=1 \/.\]
\end{Proposition}
 \begin{proof}
 The solution to (\ref{ZoneV}) satisfies :
    \[
    \forall j\in\{{{1,2,\ldots,d}}\}, ~~~
    \frac{\mathbb{P}\left( X_i>u_{i},S\geq u\right)}{\mathbb{P}\left( X_i>u\right)} =
    \frac{\mathbb{P}\left( X_j>u_{j},S\geq u\right))}{\mathbb{P}\left( X_i>u\right)}
     \/.\]
       When $u$ goes to $+\infty$, and using Theorem \ref{thZO}, we obtain Proposition \ref{prpZV}.
 \end{proof}
 \subsubsection{Application to Pareto independent distributions}~~\\
 We consider $d$ independent random variables
  $(X_1,X_2,\ldots,X_d)$ of Pareto distributions, with parameters   $(a,b_i)_{\{i=1,2,\ldots,d\}}$ respectively, such that $b_1>b_2>\cdots>b_d>0$. Therefore, these distributions will be characterized by densities and survival functions of the following forms:
  \[ f_{X_i}(x)=\frac{a}{b_i}\left(1+\frac{x}{b_i}\right)^{-a-1} \/,\]
  and \[ \bar{F}_{X_i}(x)=\left(1+\frac{x}{b_i}\right)^{-a} \/.\]
   \begin{Proposition}[The asymptotic optimal allocation minimizing $I$]\label{prop3.12}Asymptotically, the unique solution to (\ref{ZoneO}) satisfies:

     \[ \forall (i,j)\in\{{{1,2,\ldots,d}}\}^2,~~~~\left(\frac{\underset{u\to\infty}\lim{\alpha_i}}{b_i}\right)^{-a}-\left(\frac{\underset{u\to\infty}\lim{\alpha_j}}{b_j}\right)^{-a}=\left(\frac{1}{b_i}\right)^{-a}-\left(\frac{1}{b_j}\right)^{-a} \/.\]
      \label{GPDO}
  \end{Proposition}
  \begin{proof}
  Follows from Proposition \ref{subth2}.
    \end{proof}
   \begin{Proposition}[The asymptotic optimal allocation minimizing $J$]\label{prop3.13}
      The unique solution to~(\ref{ZoneV}) satisfies:
                \[ \underset{u\to\infty}\lim{\alpha_1}=1
          ~~and~~ \underset{u\to\infty}\lim{\alpha_i}=0,\forall i\in\{{{2,3,\ldots,d}}\}
          \/. \]
    \end{Proposition}
    \begin{proof}
    We suppose that $\exists \in\{1,\ldots,d\}$ such that $0<\underset{u\to\infty}\lim{\alpha_j}<1$.\\
    From Theorem \ref{thZO} : \[ \underset{u\to\infty}\lim{\frac{\mathbb{P}(X_j\geq u_j,S\geq u)}{\bar{F}_{X_j}(u)}}=1 \/.\]
    On the other hand, and applying Proposition \ref{prpZV} in the Pareto distributions case, we get for $i\in\{1,\ldots,d\}\setminus\{j\}$:
     \[ \frac{\mathbb{P}(X_j\geq u_j,S\geq u)}{\bar{F}_{X_j}(u)}= \frac{\mathbb{P}(X_j\geq u_j,S\geq u)}{\bar{F}_{X_i}(u)}\cdot\frac{\bar{F}_{X_i}(u)}{\bar{F}_{X_j}(u)}\stackrel{ u\rightarrow+\infty}{\sim}\frac{\bar{F}_{X_i}(u)}{\bar{F}_{X_j}(u)}=\left(\frac{1+\frac{u}{b_i}}{1+\frac{u}{b_j}}\right)^{-a} \/,\]
  then, for $i\in\{1,\ldots,d\}\setminus\{j\}$: \[ \underset{u\to\infty}\lim{\frac{\mathbb{P}(X_j\geq u_j,S\geq u)}{\bar{F}_{X_j}(u)}}= \left(\frac{b_j}{b_i}\right)^{-a}\neq 1\/. \]
 That is absurd. We deduce that
  $\forall i\in\{{{1,2,\ldots,d}}\}$:\[\underset{u\to\infty}\lim{\alpha_i}\in\{0,1\}\/, \]
  and since $\sum_{i=1}^{d}{\alpha_i}=1$, then, there is a unique $i$ such that $~ \underset{u\to\infty}\lim{\alpha_i}=1$ and for all $~j\neq i$ $~ \underset{u\to\infty}\lim{\alpha_j}=0$.\\
  Let us recall the definition of the order stochastic dominance, as it is presented in Shaked and Shanthikumar (2007)\cite{OrdreSto}. For random variables $X$ and $Y$, $Y$ first-order stochastically dominates $X$ if and only if:
  \[\bar{F}_X(x)\leq\bar{F}_Y(x), ~~\forall x\in\mathbb{R}^+ \/,\]
  and in this case we denote: $X\preccurlyeq_{st}Y$.\\
  Now, clearly  $X_d\preccurlyeq_{st}\cdots\preccurlyeq_{st}X_2\preccurlyeq_{st}X_1$ because $b_1>b_2>\cdots>b_d>0$.\\
  In \cite{P12015} (Proposition 3.10) we have proved that the optimal capital allocation satisfies the monotonicity property. We deduce that $\alpha_1\geq\cdots\geq\alpha_d$, and thus $\underset{u\to\infty}\lim{\alpha_1}=1$.
    \end{proof}
    \subsection{Comparison between the asymptotic behaviors in exponential and sub-exponential cases}

 Let us consider the following risk indicator:
    \begin{align*}
     \mathit{I_{loc}}(u_1,\ldots,u_d)=\sum_{k=1}^{d}{\mathbb{E}\left({(X_k-u_k)1\!\!1_{\{X_k>u_k\}}}\right)}=\sum_{k=1}^{d}{\mathbb{E}\left({(X_k-u_k)^+}\right)}\/.
     \end{align*}
With this indicator, the impact of each branch on the group is not taken into account. This indicator thus takes only \textit{local} effects into account. If, for any $k=1\/,\ldots \/,d$, the random vector $(X_k\/,S)$ admits a density whose support contains $[0\/,u]^2$, then the optimality condition associated to $ \mathit{I_{loc}}$ gives:
\[ \mathbb{P}(X_i>u_i)=\mathbb{P}(X_j>u_j) \/.\]
We remark that this corresponds to the asymptotic result for indicator $I$ of Proposition \ref {subth1}, where $ \frac{\overline{F}_{X_j}(u_j)}{\overline{F}_{X_i}(u_i)}  \stackrel{u\rightarrow +\infty}{=} 1 + o(1) $. In other words, in the case of {\em exponential type} and independent risks, the group effect is asymptotically negligible. This behavior is also clear in Proposition \ref{pro3.2}, where we found the asymptotic capital allocation for $I$ for independent exponential distributions as:
$$u_i=\dfrac{\dfrac{1}{\beta_i}}{\sum_{j=1}^{d}{\dfrac{1}{\beta_j}}}u,~\text{for all}~i=1\/,\ldots\/,d\/,$$ as for $I_{loc}$.\\

Proposition \ref{subth2} shows that for independent sub-exponential distributions the asymptotic behavior is different. Indeed, in Equation (\ref{eqsub2}), the terms $\bar{F}_{X_i}(u)$ and $\bar{F}_{X_j}(u)$  lead to take into account the group effect. In the Pareto distributions case as example, recall that Proposition \ref{GPDO} gives that the asymptotic behavior of the optimal allocation for $I$ is described by :
$$\left(\frac{\underset{u\to\infty}\lim{\alpha_i}}{b_i}\right)^{-a}-\left(\frac{\underset{u\to\infty}\lim{\alpha_j}}{b_j}\right)^{-a}=\left(\frac{1}{b_i}\right)^{-a}-\left(\frac{1}{b_j}\right)^{-a}\/,$$
whereas the optimal allocation for $\mathit{I_{loc}}$ for independent Pareto distributions is given by $\alpha_i=\frac{b_i }{\sum_{\ell=1}^{d}b_\ell}$, for all $i\in\{1,\ldots,d\}$.\\

For optimal allocation by minimizing the $J$ indicator, the asymptotic behavior is identical for the two families of distributions, the riskiest branch is considered as first responsible of the overall ruin, and thus, the optimal solution is to allocate the entire capital $u$ to this business line.

\section{The impact of the dependence structure}
In this section, we focus on the impact of the dependence structure on the optimal allocation. We study at first the impact of mixture exponential-gamma to construct a correlated Pareto  distributions, compared to the independence case presented in the previous section. Then we analyze the optimal allocation in the case of comonotonic risks. The last sub-section is devoted to the study of the impact of the dependence nature on the optimal allocation, using some bivariate models with copulas.
\subsection{Correlated Pareto}~~\\
Let $(X_1,\ldots,X_d)$ be a mixture of exponential distributions such that for all $i\in \{1,2,\ldots,d\}$,
 $ X_i\sim\mathcal{E}(\beta_i\theta)$, with ($\beta_1<\beta_2\cdots<\beta_d$), and
$\theta\sim \Gamma(a,b)$. Therefore, $X_i$ have survival functions of the form:
\[ \bar{F}_{X_i}(x)=\int_{0}^{\infty}\bar{F}_{X_i|\Theta=\theta}f_\Theta(\theta)d\theta=\int_{0}^{\infty}e^{-\beta_i\theta x}f_\Theta(\theta)d\theta =\left(1+\frac{\beta_ix}{b}\right)^{-a}\/, \]
consequently, $X_i$ have Pareto distribution of parameters $\left(a,\frac{b}{\beta_i}\right)$. They are conditionally independents. So, the idea is conditioning on the random variable $\theta $ and then integrate the formulas found for the case of independent exponential distributions. This model has been studied in e.g.\cite{oakes1989bivariate, Yeh2007}. Recall $ A_\ell=\displaystyle\prod_{j=1,j\neq \ell}^{d}{\dfrac{\beta_j}{\beta_j-\beta_\ell}}$, $\ell=1,\ldots,d$.
  \begin{Proposition}[The optimal allocation for the indicator $I$]
  The optimal allocation minimizing the multivariate risk indicator $I$ is the unique solution in $\mathcal{U}^d_u$, of the following equations system:
   \[
      \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2}\/, \]
      \begin{equation}
      s(\beta_i\alpha_i)-s(\beta_j\alpha_j)
        -\sum_{\ell=1}^{d}{A_\ell [s(\alpha_i\beta_i+(1-\alpha_i)\beta_\ell) )-s(\alpha_j\beta_j+(1-\alpha_j)\beta_\ell) )]}=0
        \/,\label{eq10}
      \end{equation}
      where $s$ is the function defined by $s(x)=(1+x\frac{u}{b})^{-a}$ and $\alpha_i=\frac{u_i}{u}$ for all $i\in\{1,\ldots,d\}$.
     \end{Proposition}
   \begin{proof}
   It suffices to integrate Equations System (\ref{EIZO}), multiplied by the density function of $\theta$.
  \end{proof}
 \begin{Proposition}[The asymptotic optimal allocation for the indicator $I$]\label{prop4.2}  When the capital $u$ goes to infinity, the optimal allocation by minimization of the risk indicator $I$ is the unique solution in $\mathcal{U}^d_u$ of the following equations system:
 \[ \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2}\/, \]
 \begin{equation}
                                 (\beta_i\alpha_i)^{-a}-(\beta_j\alpha_j)^{-a}-\sum_{\ell=1}^{d}{A_\ell [(\alpha_i\beta_i+(1-\alpha_i)\beta_\ell)^{-a}- (\alpha_j\beta_j+(1-\alpha_j)\beta_\ell)^{-a} ]}=0
         \/.\label{eq11}
       \end{equation}
 \end{Proposition}
  \begin{proof}
  We divide Equations System (\ref{eq10}) by s(1), and let $u$ go to $+\infty$ to get Equations System~(\ref{eq11}).
  \end{proof}
  Proposition \ref{prop4.2} shows the impact of the dependence related to the mixture. Indeed, in the case of independent Pareto distributions, of parameters $\left(a,\frac{b}{\beta_i}\right)_{i=1,\ldots,d}$, the asymptotic optimal allocation for the indicator $I$ is given by Proposition \ref{prop3.12} as the solution of the equations system:\[ \forall (i,j)\in\{{{1,2,\ldots,d}}\}^2,~~~~\left(\beta_i\alpha_i\right)^{-a}-\left(\beta_j\alpha_j\right)^{-a}=\left(\beta_i\right)^{-a}-\left(\beta_j\right)^{-a} \/.\]
  Each equation in this system depends only on two risks, unlike the mixture case, where the equations of Equations System (\ref{eq11}), depend on all the risks.

  \begin{Proposition}[The optimal allocation for the indicator $J$]
  The optimal allocation minimizing the multivariate risk indicator $J$ is the unique solution in $\mathcal{U}^d_u$, of the following equations system:
  \begin{equation}
       \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2},
          \sum_{\ell=1}^{d}{A_\ell [s(\alpha_i\beta_i+(1-\alpha_i)\beta_\ell) )-s(\alpha_j\beta_j+(1-\alpha_j)\beta_\ell) )]}=0
            \/.\label{ZvDPGCorr}
          \end{equation}
   \end{Proposition}
   \begin{proof}
    It suffices to integrate Equations System (\ref{EIZV}), multiplied by the density function of $\theta$.
  \end{proof}
   \begin{Proposition}[The asymptotic optimal allocation for the indicator $J$] \label{prop4.4}When the capital $u$ goes to infinity, the optimal allocation by minimization of the risk indicator $J$, is the unique solution in $\mathcal{U}^d_u$ of the following equations system:
    \begin{equation}
          \forall (i,j)\in\{{{1,2,\ldots,d}}\}^{2},~~
                       \sum_{\ell=1}^{d}{A_\ell [(\alpha_i\beta_i+(1-\alpha_i)\beta_\ell)^{-a}- (\alpha_j\beta_j+(1-\alpha_j)\beta_\ell)^{-a} ]}=0
                       \label{asy10}\/.
             \end{equation}
   \end{Proposition}
   \begin{proof}
    We divide Equations system (\ref{ZvDPGCorr}) by s(1), and we let $u$ fo to $+\infty$ to get Equations system~(\ref{asy10}).
   \end{proof}
   Proposition \ref{prop4.4} shows that for the indicator $J$, the asymptotic behaviour of the optimal capital allocation takes into account the mixture effect. In fact, for independent Pareto distributions, we have proved in Proposition \ref{prop3.13}, that we allocate the entire capital $u$ to the riskiest branch $ X_1 $, while the asymptotic optimal allocation in the correlated Pareto distributions case is the solution of Equations system (\ref{asy10}).
\subsection{Comonotonic risks}~~\\
The concept of comonotonic random variables is related to the studies of Hoeffding (1940) \cite{Hoeff1940} and Fréchet (1951) \cite{Frechet1951}. Here we use the definition of comonotonic risks as it was first mentioned in the actuarial literature in Borch (1962) \cite{Borch1962}.\\
A vector of random variables $(X_1,X_2,\ldots,X_n)$ is comonotonic if and only if there exists a random variable $Y$ and non-decreasing functions $\varphi_1,\ldots,\varphi_n$ such that:
\[ (X_1,\ldots,X_n)\overset{d}{=}(\varphi_1(Y),\ldots,\varphi_n(Y)) \/.\]
In the case where the risks $X_1$,\ldots,$X_d$ are comonotonic, we may give explicit formulas for the optimal allocation minimizing the multivariate risk indicators $I$ and $J$, and for some risk models. For that, we use the existence of a uniform random variable $U$ such that $X_i=F^{-1}_{X_i}(U)$ for all $i\in\{1,\ldots,d\}$, and $S=\sum_{i=1}^{d}F^{-1}_{X_i}(U)=\varphi(U)$, where $\varphi(t) = \sum_{i=1}^{d}F^{-1}_{X_i}(t)$, $\varphi$ is a non-decreasing function.\\
The main result of this section is given below.
\begin{Proposition}\label{prop:comonotone}
Let $X_1$,\ldots,$X_d$ be comonotonic  risks, with increasing distribution functions and support containing $[0\/,u]$. The optimal allocations for indicators $I$, $J$ and $I_{loc}$ coincide, they are given by $(u_1\/,\ldots \/, u_d)\in\mathcal{U}_u^d$ and
$$F_{X_i}(u_i) = F_{X_j}(u_j) \ \forall \ i\/, j = 1\/, \ldots \/, d \/.$$
\end{Proposition}
\begin{proof}
Let us denote: $w_i=F_{X_i}(u_i)$, $v=\varphi^{-1}(u)$, $M_i=\max(w_i\/,v)$. The indicators $I$ and $J$ may be rewritten for $(u_1\/,\ldots \/, u_d)\in\mathcal{U}_u^d$:
\begin{eqnarray*}
I(u_1\/,\ldots\/, u_d) &=& \sum_{i=1}^d \mathbb{E}\left((F_{X_i}^{-1}(U) -u_i){\bf 1}_{\{U\geq w_i\/, \ U\leq v\}}\right)\\
J(u_1\/,\ldots\/,u_d) &=&\sum_{i=1}^d \mathbb{E}\left((F_{X_i}^{-1}(U) -u_i){\bf 1}_{\{U\geq M_i\}} \right)\/.
\end{eqnarray*}
We remark that since $(u_1\/,\ldots \/, u_d)\in\mathcal{U}_u^d$, and $F_{X_i}$ is strictly increasing for all $i\in\{1,\ldots,d\}$, we cannot have that $w_i<v$ for all $i$, so that, $I$ is not trivially equal to $0$. We use Lagrange multiplier to get that the minimum of $I$ and $J$ are reached in $\mathcal{U}_u^d$ respectively for:
\begin{itemize}
\item $\mathbb{P}(U\geq w_i\/, \ U\leq v) = \mathbb{P}(U\geq w_j\/, \ U\leq v)$, for $i\/,j =1\/,\ldots\/, d$,
\item $\mathbb{P}(U\geq M_i) = \mathbb{P}(U\geq M_j)$, for $i\/,j =1\/,\ldots\/, d$.
\end{itemize}
These equality are acheaved if and only if $w_i=w_j=v$ or in other words if $F_{X_i}(u_i)=F_{X_j}(u_j)$. We remark that the minimum of $I$ is then $0$.
\end{proof}
The following three corollaries are direct applications of Proposition \ref{prop:comonotone} to some particular cases.
\begin{Example}[Comonotonic exponential model]
For comonotonic risks of exponential distributions $X_i\sim \exp(\beta_i)$, the optimal allocation by minimization of the two risk indicators is: \[ \forall i\in \{1,\ldots,d\},~~u_i=\frac{1/\beta_i}{\sum_{j=1}^{d}1/\beta_j}u \/.\]
\end{Example}
It is noticeable that in this particular case, the optimal allocation for comonotonic risks coincide with the asymptotic allocation in the independence case (see Proposition \ref{pro3.2}).

 \begin{Example}[Comonotonic log-normal model]
For comonotonic risks of log-normal distributions $X_i\sim LN(\mu_i,\sigma)$, the optimal allocation by minimization of the two risk indicators is:
\[  \forall i\in \{1,\ldots,d\},~~u_i=\frac{\exp(\mu_i)}{\sum_{\ell=1}^{d}\exp(\mu_\ell)}u \/. \]
 \end{Example}

 \begin{Example}[Comonotonic Pareto model]
 For comonotonic risks of Pareto distributions of the same shape parameter $\alpha$: $X_i\sim Pa(\alpha, \lambda_i)$, the optimal allocation by minimization of the two risk indicators is: \[ \forall i\in \{1,\ldots,d\},~~u_i=\frac{\lambda_i}{\sum_{\ell=1}^{d}\lambda_\ell}u \/. \]
  \end{Example}
  Contrary to the exponential case, this result does not coincide with the asymptotic behavior obtained for  independent Pareto distributions (see Proposition \ref{prop3.12}).

\subsection{The dependence impact with some copulas models}~~\\
~~
In this section, we study the impact of the dependence on the optimal capital allocation using some copulas (see Nelsen \cite{nelsen} for review on copulas). The idea is to find the optimal allocation as a function of the copula parameters in each case. We focus on the indicator $I$, the same kind of calculations can be done for the indicator $J$.
\subsubsection{FGM Bivariate Model}~~\\
Let $X_1$ and $X_2$ be two risks of marginal exponential distributions $X_i\sim \exp(\beta_i)$ and FGM bivariate dependence structure with $-1\leq\theta\leq 1$ as parameter (see Nelsen \cite{nelsen},Example 3.12., section 3.2.5). We assume that $\beta_1<\beta_2/2$.\\
In this case, the copula Pearson correlation coefficient is given by $\rho_P=\frac{\theta}{4}$, and the bivariate distribution function is:
\[ F_{X_1,X_2}(x_1,x_2)=(1-e^{-\beta_1x_1})(1-e^{-\beta_2x_2})+\theta(1-e^{-\beta_1x_1})(1-e^{-\beta_2x_2})e^{-\beta_1x_1}e^{-\beta_2x_2}\/. \]
\begin{Proposition}[The optimal capital allocation for the indicator $I$ in the FGM Model]
\label{efgm}
For the indicator $I$, the optimal allocation of a capital $u$ is given by $(\beta u,(1-\beta)u)$
such that $\beta=u_1/u$ is the unique solution in $[0,1]$ of the following equation:
\begin{align}\label{didier}
(1+2\theta)(h(\beta)-h(\alpha-\alpha\beta))&+2\theta(h(2\beta)-h(2\alpha-2\alpha\beta))\\\nonumber+(1+\theta)h(\alpha+\beta-\alpha\beta)+\theta h(2\alpha+2\beta-2\alpha\beta)-&\theta h(\alpha+2\beta-\alpha\beta)-\theta h(2\alpha+\beta-2\alpha\beta)\\\nonumber=\frac{1+\theta}{\alpha-1}(h(\alpha)+\alpha h(1))+\frac{\theta}{\alpha-1}(h(2\alpha)+\alpha h(2))-&\frac{\theta}{\alpha-2}(2h(\alpha)+\alpha h(2))-\frac{\theta}{2\alpha-1}(h(2\alpha)+2\alpha h(1))\/,
\end{align}
where, $h$ is the function $h(x)=\exp(-\beta_1ux)$,and  $\alpha=\beta_2/\beta_1$.
\end{Proposition}
\begin{proof}
The proof is postponed to Appendix \ref{FGM}.
\end{proof}
\begin{remark}
In the case of $\theta =0$, we find exactly Equation (\ref{EIZO}) given by Proposition \ref{prp21} for the independent exponential distributions model.
\end{remark}
 Equation (\ref{didier}) gives the behavior of the optimal allocation with respect to $\theta$. It may be solved numerically.\\
Figure \ref{EFGM-case1} presents an illustration of the optimal allocation variation with respect to the dependence parameter of the FGM copula.
 \begin{figure}[H]
                \center
                \includegraphics[width=14cm]{EFGM-Corr2.png}
              	\caption{$\beta$ as a function of $\theta$. Case : $\beta_1=0.05$, $\beta_2=0.25$ , and $u=50$}
               \label{EFGM-case1}
 \end{figure}
 For the illustration parameters, we remark that $\beta$ is an increasing function of $\theta$, this can be verified analytically using the implicit function theorem. It is important to remark that the optimal allocation is also a function of the capital $u$. The variations of $\beta$ with respect to the dependence parameter depends on $u$. Figure \ref{EFGM-case2} give what we obtain as result for an allocation with the same distributions parameters but with $u=100$ as allocation capital.
  \begin{figure}[H]
                 \center
                 \includegraphics[width=14cm]{corFGM100env.png}
               	\caption{$\beta$ as a function of $\theta$. Case : $\beta_1=0.05$, $\beta_2=0.25$ , and $u=100$}
                \label{EFGM-case2}
  \end{figure}
 The optimal allocation depends on the initial capital. As an example, with the same parameters, $\beta$ is a decreasing function of $\theta$ if $u<53$, and it is an increasing function of $\theta$ for $u\geq 53$.\\
The variation range size of $\beta$ is very small, and its size is function of the distributions parameters and the allocation capital. That is due to the dependence structure, the FGM copula present only a light dependence. For Clayton copula as example, which is stronger dependence structure, the variation range size of $\beta$ is more important.
 \subsubsection{Marshall-Olkin Model}~~\\
 Let $Y_i\sim \exp(\lambda_i)$, with $i=0,1,2$ be three independent random variables.\\
  We construct two random variables with common shock: $X_i=\min (Y_i,Y_0)$ for $i=1,2$. $X_i$'s have exponential marginal distributions of parameters $\beta_i=\lambda_i+\lambda_0$ (see e.g. Nelsen \cite{nelsen} section 3.1.1.).\\
  This dependence construction model has as Pearson correlation coefficient
   $\rho_P=\frac{\lambda_0}{\lambda_0+\lambda_1+\lambda_2}$.\\
  The joint distribution function is given by: \begin{align*}
  \bar{F}_{X_1,X2}(x_1,x_2)&=\mathbb{P}(X_1>x_1, X_2>x_2)= \mathbb{P}(Y_1>x_1, Y_2>x_2,Y_0>\max(x_1,x_2) )\\
  &=e^{-\lambda_1x_1}e^{-\lambda_2x_2}e^{-\lambda_0\max(x_1,x_2)}\\
  &=e^{-(\lambda_0+\lambda_1)x_1}e^{-(\lambda_0+\lambda_2)x_2}e^{\lambda_0\min(x_1,x_2)}\\
  &=\bar{F}_{X_1}(x_1)\bar{F}_{X_2}(x_2)e^{\lambda_0\min(x_1,x_2)}\/,
   \end{align*}
   and the joint density function is the following:
   \[  f_{X_1,X_2}(x_1,x_2)=\left\lbrace
   \begin{array}{lcl}
   f^1_{X_1,X_2}(x_1,x_2)=\beta_1e^{-\beta_1 x_1}(\beta_2-\lambda_0)e^{-(\beta_2-\lambda_0)x_2} & si & x_1>x_2\\
   f^2_{X_1,X_2}(x_1,x_2)=(\beta_1-\lambda_0)e^{-(\beta_1-\lambda_0)x_1}\beta_2e^{-\beta_2 x_2} & si & x_1<x_2\\
   f^0_{X_1,X_2}(x_1,x_2)=\lambda_0e^{-\beta_1x} e^{-\beta_2x}e^{\lambda_0x}& si & x_1=x_2=x
   \end{array}\right.\/.\]
   \begin{Proposition}[The optimal capital allocation for the indicator $I$ in the Marshall-Olkin Model]
   We suppose that $\lambda_1<\lambda_2$. The optimal allocation of a capital $u$ minimizing the indicator $I$ is given by $(\beta u,(1-\beta)u)$,
   such that $\beta=u_1/u$ is the unique solution in $[0,1]$ of the following equation:
   \label{MOeq}
   \begin{align*}
                g(\beta_2(1-\beta))-g(\beta_1\beta)+\frac{\beta_1}{\beta_1-\lambda_2}g((\beta_1-\lambda_2)\beta+\lambda_2)+\frac{\lambda_2}{\beta_1-\lambda_2}g((\lambda_2-\beta_1)(1-\beta)+\beta_1)\\-\frac{\lambda_1}{\lambda_1-\beta_2}g(\beta_2)
                   =\frac{\lambda_2}{\beta_1-\lambda_2}g(\beta_1)+g(\lambda_s/2)[\frac{\beta_1}{\beta_1-\lambda_2}-\frac{\lambda_1}{\lambda_1-\beta_2}]
                   \/, \end{align*}
       where, $\lambda_s=\lambda_0+\lambda_1+\lambda_2$, and $g$ is the function $g(x)= \exp(-ux)$.
  \end{Proposition}
  \begin{proof}
  The proof is postponed to Appendix \ref{MO}.
  \end{proof}
   \begin{remark}
   In the case $\lambda_0=0$ which is the independence case, we find again Equation~(\ref{EIZO}) given by Proposition \ref{prp21}.
   \end{remark}
 We can consider $\lambda_0$ as a dependence parameter in this model. Figure \ref{MO-Corr} presents an illustration of the variation of $\beta$ as a function of $\lambda_0$.
  \begin{figure}[H]
                       \center
                       \includegraphics[width=14cm]{MMO-Corr-en.png}
                     	\caption{$\beta$ as a function of $\lambda_0$ : $\beta_1=0.05$, $\beta_2=0.25$ and $u=50$. }
                      \label{MO-Corr}
        \end{figure}
  One can notice that $\beta$ is a decreasing function of $\lambda_0$. This is coherent with the increase of $\beta $ as a function of $\alpha =\frac{\beta_2}{\beta_1}=\frac{\lambda_2+\lambda_0}{\lambda_1+\lambda_0}$ demonstrated in \cite {AR1} in the independence case, since the two risks are independent, conditionally to  $Y_0 $.
 \section*{Conclusion}
  In this article, we have studied the allocation asymptotic behavior based on the level of the group capital. It has enabled us to build an idea of the capital level impact on the sensitivity of its allocation between branches. The comparison between the asymptotic optimal allocation, in the case of sub-exponential and exponential distributions, underscores the impact of the risks nature on the behavior of the allocation for a very large capital.\\

  Compared to the classical risk allocation methods, the allocation by minimizing multivariate risk indicator take into account the capital level. This seems to be a more acceptable behavior in a capital allocation operation.\\

  The risk aggregation transforms the global risk portfolio in a univariate risk. At the opposite, the allocation is based on a multivariate analysis. The goal of this paper is to highlight the importance of the dependence modeling in the success of an allocation capital operation. The capital allocation is also sensitive to the risks nature, we tried in this paper to make more evident the impact of distributions' nature on the allocation behavior. Since the capital allocation is an important financial decision, especially for groups, insurers must be very careful in their risk and dependence modeling choices to get an efficient allocation.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Fish Fry is a 1944 Andy Panda cartoon directed by James Culhane, and produced by Walter Lantz Productions. The plot centers around a street cats endless attempts to eat Andy's goldfish after ordering it from a pet shop.

Plot
Andy Panda is fascinated by a cute little goldfish in a pet shop window, buys it, and starts to take it home. However he's stalked by a mangy, hungry alley cat who tries to eat it. The big tomcat tries to get the baby fish by sneaking up and grabbing it, then by disguising himself as a thirst-crazed desert traveler dying for a drink of water, and finally, by crude by effective brute force. Andy's stuck in the middle of a guerrilla war between the ravenous cat and the goldfish. Guess who's more sadistic? In his haste, the cat loses the fish down the gutter, but retrieves it, only to lose it again. Andy catches the fish and is promptly chases back to the pet shop. The cat's ambush outside the shop is foiled by a big bulldog at Andy's side who disposes of the cat without lifting an eyebrow.

Notes
Production Number: D-6
The cartoon was nominated for the 1944 Academy Award for Best Animated Short Film, but lost to MGM's Mouse Trouble, Starring Tom and Jerry.
Andy Panda never speak in this cartoon, though Walter Tetley provides the line, "Poor little pussy cat" for the goldfish.

Home Media
Woody Woodpecker and His Friends: The Screwdriver (VHS; Goodtimes)
Woody Woodpecker and His Friends: Volume 2 (VHS and Laserdisc; Universal/MCA)
Woody Woodpecker and Friends: Volume 4 (DVD; Columbia House)
The Woody Woodpecker and Friends Classic Cartoon Collection (DVD; Universal/MCA)
Woody Woodpecker Favorites (DVD; Universal/MCA)

References

External links

1944 animated films
1944 films
1940s American animated films
Andy Panda films
Films directed by James Culhane

redpajama_set_name: RedPajamaWikipedia


text:
Die Borussenfront ist ein Zusammenschluss von Anhängern Borussia Dortmunds. Insbesondere ihre Nähe zur rechtsextremen Szene ließ sie in den frühen 80er Jahren zu einer der berüchtigtsten Gruppierungen von Fußballfans und Hooligans werden. Bundesweit bekannt wurde die Borussenfront im September 1983 durch einen Artikel im Stern, für den der investigative Journalist Gerhard Kromschröder sich unter die Extremisten gemischt hatte.

Ehemalige führende Mitglieder der Borussenfront wie Siegfried Borchardt wurden in der Vergangenheit immer wieder wegen Körperverletzung und Vandalismus zu Geld- und Haftstrafen verurteilt. Die Vereinsführung hatte sich früher mehrfach von der Front distanziert. Ihre Kleidung ist auch heute noch zu sehen, auf Runen und verbotene rechtsextreme Symbole wird dabei aber verzichtet. Die Stadionordnungen der BVB-Spielstätten Westfalenstadion und Stadion Rote Erde verbieten u. a. Kleidung mit Logos und Schriftzügen der Borussenfront in ihrem Geltungsbereich auch dann, wenn kein direkter strafrechtlicher Zusammenhang besteht.

Das Fanprojekt Dortmund versuchte damals, den Mitgliedern der Gruppe neue oder andere Perspektiven zu vermitteln, insbesondere durch Arbeitsprojekte im Vereinsumfeld. Von der Dortmunder Sozialverwaltung wird dieser Ansatz sehr gelobt. Das Fanprojekt wird staatlich finanziert und lebt noch heute vom Ruf, die "Borussenfront beseitigt" zu haben. Mindestens ebenso wichtig war die aktive Dortmunder Fanszene, die sich über Jahre hinweg massiv gegen Rassismus im Stadion gewehrt hat.

Seit einigen Jahren ist die Gruppe allerdings wieder deutlich aktiver und feiert sich selbst als wieder auferstandene "Legende". Aufgrund der Stadionverbote einiger Mitglieder tritt die Borussenfront bei Spielen in der Bundesliga nicht offen auf, sondern weicht auf unterklassige Spiele in Ober- und Verbandsliga aus. In der Saison 2005/06 war sie vor allem bei Spielen des DSC Wanne-Eickel anzutreffen, ebenso war sie im Juni 2006 beim Spiel der zweiten Dortmunder Mannschaft bei Westfalia Herne für massive Krawalle verantwortlich. Im Jahr 2012 feierte die Front ihr 30-jähriges Bestehen und ist seitdem wieder aktiver. Sie versucht, vor allem unter jungen Fans mit Stadionverbot Nachwuchs zu rekrutieren. Ihre Mitglieder skandierten beim Freundschaftsspiel von Borussia Dortmund in Marl "Hoch die nationale Solidarität". Vor dem Champions-League-Finale 2013 in London wurden 50 randalierende Dortmund-Fans festgenommen, von denen einige zuvor ein Transparent mit der Aufschrift 30 Jahre Borussenfront Dortmund entrollt hätten.

Weblinks
 Dokumentation über die Borussenfront aus den 1980er Jahren (PDF; 1,35 MB)

Literatur und Videos

 Landeszentrale für politische Bildung (Hrsg.): "Die sind eben so". 1983. VHS-Videofilm, 14 Minuten. (Inhaltsangabe)

Einzelnachweise

Fußball (Nordrhein-Westfalen)
Borussia Dortmund
Organisation (Rechtsextremismus)
Hooliganismus


text:
##Binary Tree Paths

	Given a binary tree, return all root-to-leaf paths.

	Have you met this question in a real interview? Yes
	Example
	Given the following binary tree:

	   1
	 /   \
	2     3
	 \
	  5
	All root-to-leaf paths are:

	[
	  "1->2->5",
	  "1->3"
	]

####Tags Expand
Binary Tree Binary Tree Traversal Facebook Google

####思路
- 唯一需要注意的是,每进入新一层,就要重新生命string,不然有可能会改变string值,导致错误

```java
/**
 * Definition of TreeNode:
 * public class TreeNode {
 *     public int val;
 *     public TreeNode left, right;
 *     public TreeNode(int val) {
 *         this.val = val;
 *         this.left = this.right = null;
 *     }
 * }
 */

public class Solution {
    /**
     * @param root: the root of the binary tree
     * @return: all root-to-leaf paths
     */
    public List<String> binaryTreePaths(TreeNode root) {
        // write your code here
        List<String> results = new ArrayList<String>(0);
        if (root == null) {
            return results;
        }
        StringBuilder sb = new StringBuilder();
        divide(root, sb, results);

        return results;

    }

    public void divide(TreeNode root, StringBuilder sb, List<String> results) {

        if (root.left == null && root.right == null) {
            sb.append(root.val);
            String result = new String(sb.toString());
            results.add(result);
            return;
        }

        sb.append(root.val);
        sb.append("->");
        if (root.left != null) {
            StringBuilder sbLeft = new StringBuilder(sb);
             divide(root.left, sbLeft, results);
        }

        if (root.right != null) {
            StringBuilder sbRight = new StringBuilder(sb);
            divide(root.right, sbRight, results);
        }
    }
}
```

redpajama_set_name: RedPajamaGithub


Description: All In One Cleaning is a business categorized under building component cleaning service, which is part of the larger category building maintenance services. All In One Cleaning is located at the address 1321b East 39th Street North in Sioux Falls, South Dakota 57104-0833. The Owner is Teresa Matthis who can be contacted at (605)357-8490.


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If you are looking for making DIY bath bombs or a pair of baby gear guide then surely you will get some options in here, on our pages. To know all about baby gears stay with iteenworld.com and for any other information, query, complaint get in touch via our contact us page.
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Explore the remains of a past era on a tour of Chichen Itza, a Mayan site full of ruins and history. Then, dive into Ik Kil, a sacred cenote, to cool off in its crystal-clear waters before visiting the colonial town of Valladolid.
Visit Chichen Itza and Ek Balam to discover the beauty of the Mayan world. Learn about this lost civilization's astronomical, architectural, and mathematical knowledge as you tour these ancient ruins.
Explore Chichen Itza and Cabo, two icons of the Mayan world, on a tour of ancient Mayan ruins. Marvel at this culture's astronomical, architectural, and mathematical knowledge.


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Biografia
E' figlio di Franky Vanhooren, anche lui pattinatore di short track di caratura internazionale: rappresentò il  ai Giochi olimpici invernali di .

Intrattiene una relazione sentimentale con la pattinatrice di short track Alexandra Danneel, sua compagna in nazionale.

Ha studiato communicazione e marketing all'Hogeschool VIVES di Kortrijk.

Carriera
Ha provato per la prima volta a pattinare sul ghiaccio all'età di due anni. Ha iniziato a praticare lo short track nel 2003 a Bruges.

La sua squadrea di club è il Boudewijn Schaats Club Brugge.

Ha partecipato a diverse edizioni dei campionati mondiali ed europei, senza mai riuscire a vincere medaglie.

Vanta il 5º posto nella staffetta 5000 m ai mondiali di Montréal 2022, con i connazionali Stijn Desmet, Ward Pétré, Warre Van Damme ed Enzo Proost.

I suoi migliori piazzamenti continentali sono stati il 5º posto nella staffetta 5000 m, a Danzica 2021 e Danzica 2023 e il 6º posto a Dordrecht 2019.

Note

Collegamenti esterni

Pattinatori figli d'arte


Introduction: secuential measurement of intra-abdominal pressure are of paramount importance for an early detection and appropriate therapeutic management of intra-abdominal hypertension and abdominal compartment syndrome. Objective: to validate a device and technique developed to measure intra-abdominal pressure (an innovation of the Foley urinary catheter named intra-abdominal pressure catheter).
Methods: three different sets of measurements where done to test the intra-abdominal pressure catheter device I. 50 measurements where done with the intra-abdominal pressure catheter device and compared against those measurements done using a manometer graded in cm H2O. II. Direct measurement of intra-abdominal pressure in five patients during elective laparoscopy vs the intraabdominal pressure catheter device and III. Measurement of intra-abdominal pressure by the Kron method (Gold standard) vs intra-abdominal pressure catheter device in three patents with intra-abdominal hypertension/abdominal compartment syndrome. Measurements where compared with Pearson correlation test and Bland Altman statistics.
Results: I. Intra-abdominal pressure catheter vs graded manometer: r = 0.99, with a mean pressure difference of 0.27 ± 0.23 mmHg, CI (0.039 to 0.092 mmHg). II. Direct measurement of intra-abdominal pressure during laparoscopy vs intra-abdominal pressure catheter device: r = 0.93, with a mean pressure difference of 0.18 ± 0.84 mmHg, CI (-1.46 to 1.83 mmHg) and III. Measurement of intra-abdominal pressure by the Kron Method vs intra-abdominal pressure catheter device: r = 0.81, with a mean pressure difference of -0.41 ± 0.87, CI (-2.12 mmHg to 1.30 mmHg).
Conclusions: the intra-abdominal pressure catheter device is a safe and reliable instrument to measure intra-abdominal pressure.
Key words: Intra-abdominal pressure, intra-abdominal hypertension, abdominal compartment syndrome, intra-abdominal pressure catheter.
De Waele J, Desender L, De Laet I, Ceelen W, Pattyn P, Hoste E. Abdominal decompression for abdominal compartment syndrome in critically ill patients: a retrospective study. Acta Clin Belg 2010;65(6):399-403.
Malbrain MLNG, De laet IE, De Waele JJ. IAH/ACS: The Rationale for Surveillance. World J Surg 2009;33(6):1110-1115.
Sugrue M, Buhkari Y. Intra-Abdominal Pressure and Abdominal Compartment Syndrome in Acute General Surgery. World J Surg 2009;33(6):1123-1127.
Verzilli D, Constantin JM, Sebbane M, Chanques G, Jung B, Perrigault PF, et al. Positive end-expiratory pressure affects the value of intra-abdominal pressure in acute lung injury/acute respiratory distress syndrome patients: a pilot study. Crit Care 2010;14:R137.
Raeburn CD, Moore EE, Biffl WL, Johnson JL, Meldrum DR, Offner PJ, et al. The abdominal compartment syndrome is a morbid complication of postinjury damage control surgery. Am J Surg 2001;182(6):542-546.
Chen H, Li F, Sun JB, Jia JG. Abdominal compartment syndrome in patients with severe acute pancreatitis in early stage. World J Gastroenterol 2008;14(22):3541-3548.
Daugherty EL, Hongyan L, Taichman D, Hansen-Flaschen J, Fuchs BD. Abdominal Compartment Syndrome Is Common in Medical Intensive Care Unit Patients Receiving Large-Volume Resuscitation. J Intensive Care Med 2007;22(5):294-299.
Regueira T, Bruhn A, Hasbun P, Aguirre M, Romero C, Llanos O, et al. Intra-abdominal hypertension: Incidence and association with organ dysfunction during early septic shock. J Crit Care 2008;23(4):461-467.
Malbrain MLNG, Cheatham ML, Kirkpatrick A, Sugrue M, Parr M, De Waele J, et al. Results from the International Conference of Experts on Intra-abdominal Hypertension and Abdominal Compartment Syndrome. I. Definitions. Intensive Care Med 2006;32(11):1722-1732.
Cheatham ML, Malbrain MLNG, Kirkpatrick A, Sugrue M, Parr M, De Waele J, et al. Results from the International Conference of Experts on Intra-abdominal Hypertension and Abdominal Compartment Syndrome. II. Recommendations. Intensive Care Med 2007;33(6):951-962.
Mailbrain MLNG. Different techniques to measure intra-abdominal pressure (IAP): time for a critical re-appraisal. Intensive Care Med 2004;30(3):357-371.
Kron IL, Harpman PK, Nolan SP. The measurement of intraabdominal pressure as a criterion of abdominal reexploration. Ann Surg 1984;199(1):28-30.
Foley FE. A hemostatic bag catheter: one piece latex rubber structure for control of bleeding and constant drainage following prostatic resection. J Urol 1937;38:134-139.
Collee GG, Lomax DM, Ferguson C, Hanson GC. Bedside measurement of intra-abdominal pressure (IAP) via an indwelling naso-gastric tube: Clinical validation of the technique. Intensive Care Med 1993;19(8):478-480.
Kaplan LJ, Shaw AD. Standards for Education and Credentialing in Critical Care Medicine. JAMA 2011;305(3):296-297.


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Martin Holman
Published by Lund Humphries, March 2011
Includes 80 b&w illustrations
This is the first illustrated overview of the forty-year career of British sculptor Richard Rome (b.1943), who has made an important contribution to the development of modernist metal sculpture since the 1960s.
A leading artist whose first one-person show took place at London's Serpentine Gallery in 1975, Rome is most commonly associated with bold and open abstract shapes, often in public spaces and often in steel. His contemporaries are sculptors Phillip King, Tim Scott, Katherine Gili and Justin Knowles, and Rome himself cites as fundamental to his outlook the example of Anthony Caro, a generation older and a common factor in the careers of all these sculptors. Rome's work is also visibly influenced by post-war modernist American art, in particular the sculpture of David Smith.
For more information and to purchase: http://www.lundhumphries.com/.
Fine Art Metal Casting, An Illustrated Guide To Mould Making and Lost Wax
Richard Rome and Hamish Young
Published by Robert Hale Ltd, 2003
Includes 1061 colour illustrations
Fine Art Metal Casting is the first truly comprehensive and up-to-date guide to mould making and fine art lost wax casting. Mould making and the lost wax casting of metals for fine art are long-established techniques, which involve both artistic and material science considerations. Techniques are currently taught by the process of practical demonstration, which is both time and facility consuming. Furthermore, since an understanding of elements of the casting process requires an understanding of the whole process, teaching is often laboured. Whilst there is no substitute for practical experience, access to this demonstrative source will substantially increase understanding of the processes and inevitably free up valuable teaching time.
For more information and to purchase: http://www.halebooks.com/.
RICHARD ROME AT CANARY WHARF
Curated by Ann Elliott for Canary Wharf Group
Illustrated Essay and List of Works
"[The] collection of cast, forged and welded bronze and steel sculptures by Richard Rome forms the largest exhibition of his work to be seen in London since the 1970s. A number of monumental sculptures are displayed in external spaces at Canary Wharf - in Cabot Square and Westferry Circus - with numerous pieces in the Lobby of One Canada Square, which together cover nearly forty years of the artist's dedication to making sculpture."
view pdf of essay

redpajama_set_name: RedPajamaCommonCrawl


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La isla Pioneer (en ruso: стров Пионе́р, Ostrow Pioner) es una de las islas del archipiélago de Severnaya Zemlya, localizada en el Ártico, en aguas del mar de Kara.

Administrativamente, la isla depende del Krai de Krasnoyarsk.

Geografía

Con una superficie estimada de 1.527 km², por tamaño, es la cuarta isla del archipiélago. Está situada en la parte central del grupo, en la parte oeste, a tan solo 4,5 km de isla Komsomolets y a 9 km de la isla Revolución de Octubre. La isla alberga el glaciar homónima, al que da nombre, Pioneer.

Al suroeste de la isla se encuentra la isla Krupskoy, una isla relativamente grande, de más de 20 km de longitud y unos 11 km de ancho. El estrecho que la separa de Pioneer tiene solo unos 500 m de ancho.

Historia
La isla fue descubierta en 1913 por Boris Wilkizki y llevó el nombre de 'Svyataya Tatiana hasta 1926. Ha habido una petición para cambiar el nombre de esta isla por el Svyataya Tatiana (Santa Tatiana).

Esta isla no debe confundirse con la isla Pioneer en Canadá (coordenadas: 76°57'0" N, 96°49'60" O). Asimismo, no tiene nada que ver con la isla de Pioneer que aparece en la comedia de televisión Tom Goes to the Mayor''.

Véase también
 Severnaya Zemlya

Referencias

Enlaces externos

 Datos y fotos de satélite de las islas del grupo de las Severnaya Zemlya, disponibles en:: http://www.oceandots.com/arctic/russia./severnaya-zemlya.htm.

Islas del archipiélago Tierra del Norte


Whether it's siding damage from a recent storm, an unexpected home intrusion, or a randomly leaking shower, your home improvement needs don't always work around your schedule. After all, no one plans to need emergency roofing or a sturdier entry door—sometimes things just happen! And even if you have been preparing for a CT home remodel, it can be difficult to cover all the costs all at once.
At U.S. Window & Siding, we understand. We realize that home remodeling is just one of many things you have to balance, and we offer the flexible financing you need to make any project affordable. We offer a range of financing opportunities from local financiers, as well as a simple application process. Whether you'd like a quick replacement tub or a set of custom-built windows, our financing specialists will work with you to design a plan you can feel comfortable with.
We don't just offer financing so you'll choose us as your Connecticut remodeling company; we offer financing so you can afford to have the highest-quality products in your home, ones that won't need repair and replacement every single year. So let us help you invest in your home—call U.S. Window & Siding today to find a financing plan that's right for you!


text:
Q: Mac app In-App Purchase: restore purchases after each update? I have a Mac app that offers In-App Purchases. After releasing an update through the Mac App Store, a number of customers contacted me complaining that their purchased feature was not available anymore.
In all cases I would have the user restore their purchases by hand via a menu that I added.
However, this leaves me wondering if this happened for all users. Shouldn't the In-App-Purchases be available in the receipt as soon as an app was updated from the App Store?

redpajama_set_name: RedPajamaStackExchange


text:
Co sa stalo nové je 9. řadové album skupiny Čechomor. Album je na rozdíl od alba Proměny mnohem tvrdší, elektrická kytara a bicí dostaly mnohem více prostoru. Na albu je celkem 13 písní, které jsou nové a na žádném předchozím albu se neobjevily a jedna starší v novém podání. Na CD opět jako host vystoupila Lenka Dusilová a zazpívala si ve třech písních. Jako další host na albu vystoupil irský zpěvák Iarla O´Lionaird a japonský bubeník a hráč na flétnu šakuhači Joji Hirota. Album bylo v médiích podpořeno skladbou Zločin, která je zpívána na půl česky a na půl irskou gaelštinou. Album bylo pokřtěno na koncertu v Praze v červnu 2005. Koncertu se mimo jiné zúčastnil i producent Proměn Jaz Coleman. K albu se vázalo i turné Krušovice Tour 2005. Co sa stalo nové bylo nominováno na Anděla za nejlepší zvukovou nahrávku roku.

Skupina
 František Černý – kytara a zpěv
 Karel Holas – pětistrunné housle a zpěv
 Radek Pobořil – akordeon a trubka
 Michal Pavlík – violoncello a české dudy
 Roman Lomtadze – bicí, perkuze
 Lenka Dusilová – zpěv, vystupovala jako host
 Iarla O´Lionaird - zpěv, vystupoval jako host
 Joji Hirota - bicí, flétna šakuhači, vystupoval jako host

Seznam stop
 Hej, bystrá voda
 Z večera je jasno
 Neber sobě
 Zločin
 Co sa stalo nové
 Až půjdou cikáni
 Katuša
 Hezký Janek
 Na Vsetíně páni
 Ej, dolu Váhom
 Do kostela zvonili
 Malovaný kostel
 Teče voděnka
 Za gorum za vodum

Alba Čechomoru
Alba z roku 2005
Alba Columbia Records


text:
Q: Secondary Use case scenario I know how to write primary use case scenarios for documenting the use case diagrams. But i couldn't find a way to include secondary scenarios. Is there a format, or way to include it? I googled everywhere but there's no mention about secondary use case scenarios. any help would be appreciated. I just want a format or example.
Thank you.


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Christopher G. Pickering (9 November 1842 – 22 December 1920) was a British businessman and philanthropist. He made his fortune as a merchant and ship owner, particularly in the Kingston upon Hull fish trade. In 1914 he founded a park, almshouses, church and children's home in west Hull. The park and almshouses still bear his name.

Biography
Christopher Pickering was born in 1842, the son of a tailor. In 1861 his occupation was that of a fish curer in Kingston upon Hull, and subsequently he became a fish merchant, and by 1881 a ship owner. He controlled together with a Mr. Haldane Pickering & Haldane's Steam Trawling Co. and the fish and ice merchants Pickering, Haldane & Co.

He married Rachael Blakestone, lived in 114 Coltman Street, Hull, from 1874 to 1889, and moved to Hornsea in 1889 when he purchased The Hall.

He founded six almshouses in Hornsea in 1908, and in 1914 he built almshouses, a church, park and children's home in the west of Kingston upon Hull.

He was presented with the Freedom of the City of Hull in 1920.

He died in December 1920 aged 78.

Legacy
Both the Hull Almshouses (Christopher Pickering Lodge is now a grade II listed building) and the park (Pickering Park, Kingston upon Hull) he built still bear his name, as does a local council ward (Pickering Ward). The restoration of his Coltman Street home was featured in the third season of the BBC television series Restoration Home.

References

1842 births
1920 deaths
Businesspeople from Kingston upon Hull


text:
VARIETY: TrustNordisk Unveils Trailer for Danish Drama 'Unruly' (EXCLUSIVE)
By Elsa Keslassy
TrustNordisk has unveiled the trailer for "Unruly," the critically acclaimed sophomore outing of Danish director Malou Reymann that played at Toronto and Zurich film festivals.
The company has also sold the film to several territories, including Former Yugo (Cinemania Group) and Benelux (September Film). The heart-wrenching drama is set in the 1930s and tells the story of Maren, a rebellious teen who is sent away to an institution, the real-life Sprogø Women's Home. While there, she is being trained to behave in a more disciplined way but her refusal to comply with the rules leads to terrible consequences.
Penned by Reymann and Sara Isabella Jønsson, the film is inspired by events that unfolded at a women's institution in the 1930s. Raymann told Variety that "Denmark was actually the first European country to make eugenic-based sterilization laws." The filmmaker added that when "Nazi Germany made their laws they looked to the Danish laws."
"Unruly" stars Emilie Kroyer Koppel, Jessica Dinnage, Lene Maria Christensen, Anders Heinrichsen and Danica Curcic. It was produced by Matilda Appelin at Nordisk Film A/S.
Please read the original article and watch the trailer here.


Fantastic stay in Marrakech with great hosts. Very good breakfast Great location with a lovely Riad Highly recommended!
We had some very nice days at this beautiful Riad. The staff is very kind and warmly welcomed us. We directly felt at home. They also gave us some very useful tips to get around in Marrakech. You get a great value for the price. The breakfast is very good and you can also order some food there which is also delicious. Overall a great experience! We would love to stay there again.
Our stay at Hna Ben Saleh couldn't have been better. The rooms are very beautiful! Aymen and the other stuff are super nice and helpful. The breakfast is also very good, we would definitely stay there again! ++It was the most clean place we stayed in Morocco (we stayed also at 5 other hotels).
Great ready. Really kind staff and a very good service. We total enjoyed the time there. You can ask for everything and they try to help you. The roof terrace is also very nice.
Die Mitarbeiter und die Sauberkeit war einbahnfrei.Den Weg nach Hause muss mann sich halt gut merken wie überall in Marrakesh ,sonst verläuft man sich.Weiter zu Empfehlen!!!!
Obwohl das Hostel mitten in der lauten Medina ist, ist es absolut ruhig, hat schönen kleinen Empfangs-Aufenthaltsraum, hat tolle Terasse auf zwei Ebenen, je nach Wetterlage kann man da in der Sonne oder Schatten frühstücken. Es gibt superfreundliche Mitarbeiter, die einem jeden Wunsch erfüllen. Goldbarren würde ich aber nicht auf dem kleinen nicht richtig absperrbaren Zimmer lassen, sondern an der Rezeption zur Verwahrung geben.
Great Experience! Cosy Hostel, very friendly and helpful Staff and nice Breakfast. The Hostel is centrally located but has a totally calm and relaxed atmosphere. Thank you for the nice stay, Aymen and Hamid!


At work on a new album after her sophomore effort Fearless sold more than 3 million copies last year, the 19-year-old country-pop wunderkind talks to ELLE about what happens post–high school graduation.
ELLE: You started writing music when you were 12, and your first record came out when you were 16. How long were you actually in high school?
Taylor Swift: I was in high school freshman and sophomore year, and then I went on the road. I homeschooled the rest of the way. My brother's always calling me the dropout of the family.
ELLE: Do you even have a driver's license?
TS: I did get my driver's license in tenth grade, but I failed my permit test so many times that I ended up just bursting out crying the last time I went to the DMV. They gave [me a permit] because they felt bad, and I only missed it by one question.
ELLE: Do you ever feel like you missed out on typical teenage experiences?
TS: I actually did get high school experiences. I had a double life. During the day I walked around, talked to people, went to class, studied for tests, and had crushes on boys, and then after school I would go downtown to Music Row in Nashville and I would write songs about those experiences.
ELLE: About that: Do you ever think boys might be scared to date you because you'll call them out in a song?
TS: Well, I figure that if he's heading into the relationship knowing that he's going to do something bad for me to write a song about, then that's probably not someone I want to be interested in anyway.
ELLE: Do you have any boy drama to write about on your next album?
TS: There are definitely breakup songs on this record, but not too many. I like to balance out the amount of happy songs, breakup songs, sentimental songs, I-miss-you songs, angry songs. I don't want to try and harp on the same emotion too much because I feel like if you make the "angry" album, that's going to lose people.
ELLE: Who would you love to work with?
TS: John Mayer. He Twittered the other day that he wrote a song and he wants to make it a duet with me. I freaked out when I heard, because I've been such a big fan of John for such a long time. I'm really excited about just the idea that he would even mention me in his Twitter!
In Appreciation of the Layered, Mad Genius of Taylor Swift's "Blank Space"
Can You Decipher Taylor Swift's Teeny Tiny Romper?
Kate Middleton's Hat? Or Another Taylor Swift Needlepoint?


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Volvo to reveal its first V8-powered wagon at January's Detroit auto show
By Autoweek
Volvo will reveal its first V8-powered wagon, likely to be called V90, in concept form at January's Detroit auto show. Given Volvo's recent record of putting concepts into production—the new XC90 off-roader was thinly disguised as the ACC concept, for instance—the go-ahead for the V90 is likely. An early 2005 on-sale date is penciled into Volvo's product plans.
Jaguar's 4.0-liter V8 engine is the likely powerplant of choice. Volvo engineers evaluated the 4.6-liter Lincoln Town Car engine and other Ford V8s, but all were ruled out as insufficiently refined and too bulky to fit under the V90's hood.
Engineers have solved a major hurdle in shoehorning the Volvo-tweaked Jag V8 across the engine bay for Volvo's transverse/front-drive installation, opening the door for V8-engined XC90s and S80s in the future.
Electric Vehicle Sales in the United States
All the Costs Involved in Owning an Electric Car


DISTRIBUTION Scattered records in southern England (Devon, Wiltshire, Dorset, Sussex, Surrey, Kent, Essex, Oxfordshire, Suffolk). The vast majority of records are coastal, though it is recorded well inland at sites in Wiltshire and Oxfordshire.
HABITAT Preferences unclear. Known sites include a grassy flood embankment adjacent to coastal levels and a shell beach with sparse vegetation.
ECOLOGY Larvae aphidophagous and abroad have been found on the thistle Cirsium arvense feeding on aphids. Adults recorded from May to August.
STATUS Infrequent and declining. Only three known post 1960 sites: Blackgrounds Marsh, Essex (1983) and two sites in North Kent. Habitat loss probably places this species in a vulnerable position. Status revised from RDB3 (Shirt 1987).
THREAT Main ones likely to be management of flood embankments, including raising height with fresh soil/sediment; coastal development and recreational pressure on beaches and sand or shingle areas nearby.
MANAGEMENT Maintain a full succession of vegetation types on coastal sites, paying special attention to the continuity of dry sparsely vegetated ground.