text:
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Get to Know Springfield » Business Resources
Springfield, the 3rd largest City in Massachusetts and the 4th largest in all of New England, is a City of many attributes. It has been known as the "City of Firsts" – a moniker earned through a history of innovation, including America's first Armory and military arsenal and the first American made automobile. Springfield may be best known for two other innovations - the birthplace of basketball and of Theodor Geisel, better known as "Dr. Seuss".
The City is in the midst of a successful and long term effort to revitalize itself. The economy is growing, businesses are flourishing, and public spaces are being made more vibrant everyday. The ability for both residents and visitors to be able to walk or bike around the City with ease and comfort is an important factor in this resurgence and essential in promoting a healthy and safe community.
Business Resources / Development Services
DevelopSpringfield
DevelopSpringfield was formed in 2008 as a nonprofit, 501(c)(3) corporation to advance development and redevelopment projects, to stimulate and support economic growth, and to expedite the revitalization process within the City of Springfield. The organization completed the historic redevelopment of its headquarters at 83 Maple Street, and is directing several redevelopment projects throughout Springfield.
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Economic Development Council of Western MA
The EDC is a regional collaborative and non-governmental initiative that brings together the public and private leadership of the Pioneer Valley's three counties. Centralized coordination, effective collaboration, and comprehensive professional services are this organization's unifying foundation. Board members include mayors, major corporation CEOs and college and university leaders. Affiliates include six economic and business development organizations managed by over 50 economic development professionals.
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City of Springfield – Office of Planning & Economic Development
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small businesses throughout Massachusetts by providing high-quality, in-depth advising, training, and capital access.
www.msbdc.org/wmass
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www.massdevelopment.com
Massachusetts Office of Business Development
The Massachusetts Office of Business Development (MOBD) is the state's one-stop source for businesses seeking to relocate to Massachusetts and businesses wishing to expand their current operations here. They offer a range of expertise and services to help your business flourish in Massachusetts.
www.mass.gov/mobd
Common Capital
Common Capital is a community loan fund and non-profit organization that is committed to a thriving local economy in order to create positive social and community impacts. They accomplish this by providing financing and business assistance to small businesses and high-impact community projects.
www.common-capital.org
Massachusetts Latino Chamber of Commerce
The MLCC's mission is to develop, promote and protect Latino businesses and communities in Massachusetts. It supports the advancement of education and economic growth of Latino businesses & community and works closely with profit and non-profit businesses & organizations, local and state Chambers of Commerce, and the U.S. Hispanic Chamber of Commerce.
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MassMutual Financial Group
Named to FORTUNE ® Magazine's FORTUNE 500® in 2012 and recognized by FORTUNE ® Magazine on its annual "World's Most Admired Companies®" list. Founded in 1851 in Springfield and now celebrating its 160th anniversary, the Company, which is owned by the very policy holders it serves, was recently named a 2016 World's most ethical company. MassMutual is one of the top 100 companies on the FORTUNE 500® list – ranked 76 in 2016, its highest on record.
Baystate Health
With a workforce of over 12,000 employees, Baystate Health is the largest private employer in the region. They've been recognized by the Human Rights Campaign as a national leader in providing inclusive and equitable healthcare, and their Employee Resource Groups, which help unite diverse constituencies of our team members with the organization as a whole, have received national acclaim.
This 160+ year-old company continues to add new jobs and now boasts over 1,500 employees at its world headquarters in Springfield, adding $77 million in salaries as a community partner.
Big Y Foods, Inc.
Serving its community for 80 years and founded on the principle of delivering world class service and value by exceeding expectations. This family-owned supermarket chain employs over 10,000 people and is one of the largest independently owned supermarket chains in New England.
Peter Pan Bus Lines has been an iconic means of transportation for riders throughout the Northeast for more than 80 years. The family-owned company established in Springfield is one of the country's largest privately owned intercity bus companies. In 1933, the company began with four buses and today has a fleet of 300- plus environmentally friendly buses, and employs 1,000 people companywide.
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text:
Mark Lane, Carol Klein, Monty Don
BBC / career / garden design / Index / publishing / Mark Lane / Publications / News
See the article about Mark, his career and more in Kent's Index Magazine
A new article about Mark's career, garden design work, BBC work and personal life is revealed in Kent's range of Index magazines. 'Design for Life' looks at Mark's low points as well as his high points and charts Mark's change of career path into landscape and garden design. Several of Mark's designs are showcased in the magazines. The article also looks at Mark's passion for plants and how they helped him turn a car crash into a positive spin, launching a new way of life that feeds Mark's creativity.
"You never know what life has in store; events that can seemingly spell disaster can sometimes have a silver lining that isn't obvious at the time. Seventeen years ago Mark Lane was enjoying a successful career in publishing in London, until a car crash which led to him needing spinal surgery, which was unfortunately complicated by being born with spina bifida, resulted in him being unable to get around without the use of a wheelchair…"
The post See the article about Mark, his career and more in Kent's Index Magazine appeared first on Mark Lane Designs.

redpajama_set_name: RedPajamaCommonCrawl


text:
require 'green-button-data/enumerations/accumulation'
require 'green-button-data/enumerations/authorization_status'
require 'green-button-data/enumerations/commodity'
require 'green-button-data/enumerations/currency'
require 'green-button-data/enumerations/data_custodian_application_status'
require 'green-button-data/enumerations/data_qualifier'
require 'green-button-data/enumerations/espi_service_status'
require 'green-button-data/enumerations/flow_direction'
require 'green-button-data/enumerations/measurement'
require 'green-button-data/enumerations/phase_code'
require 'green-button-data/enumerations/quality_of_reading'
require 'green-button-data/enumerations/service'
require 'green-button-data/enumerations/third_party_application_status'
require 'green-button-data/enumerations/third_party_application_type'
require 'green-button-data/enumerations/third_party_application_use'
require 'green-button-data/enumerations/time_attribute'
require 'green-button-data/enumerations/time_period_of_interest'
require 'green-button-data/enumerations/unit_multiplier'
require 'green-button-data/enumerations/unit_symbol'

module GreenButtonData
  module Enumerations
  end
end

redpajama_set_name: RedPajamaGithub


text:
21 March 2014: Distinguished Yale professors visit Yale-NUS College
In March 2014, Yale-NUS College welcomed three distinguished Yale University professors, Professors Bryan Garsten, Deborah Davis and Amy Hungerford, who spoke at the College and interacted with Yale-NUS faculty and students during their visit.
On 11 March, J Y Pillay Visiting Professor Bryan Garsten captivated a crowded lecture hall with his lecture titled 'Rethinking Solitude'. At Yale-NUS, Professor Garsten was Chair of the Yale-NUS Curriculum Committee from 2012 to 2013, and is now the J Y Pillay Visiting Distinguished Professor.
Watch Professor Garsten's lecture here
On 12 March, Professor Deborah Davis, Tan Chin Tuan Visiting Professor in Chinese Studies, shared her expertise on urban studies, with an emphasis on urban China, in a lecture titled 'Re-Placing the Urban: Rural Binary'. Professor Davis has been involved in Yale-NUS' development since 2009, as part of the original committee to visit Singapore on the first exploratory trip from Yale, and subsequently in early curriculum development discussions.
Watch Professor Davis' lecture here
On 13 March, Professor Amy Hungerford spoke on the American Modernist poet, Wallace Stevens, as part of the President's Speakers Series. The English professor's talk was titled 'Wallace Stevens and the Art of the Empty Mind'. Professor Hungerford is known globally for her free, popular online course, 'The American Novel Since 1945', available on YouTube and Coursera. She is also the founder of Post•45, a professional association for scholars working in post-45 literary and cultural studies.
Watch Professor Hungerford's talk here


text:
\section{Introduction}

Let $G$ be a real, connected Lie group. A vector field $\XC$ on $G$ is called {\it linear} if its flow, which is denoted by $\varphi_t$, is a family of automorphisms of $G$. In this work, we assume that $G$ is a semisimple Lie group. Our wish is to study some aspects of stability of linear flow $\varphi_t$ and periodic orbits of linear and invariant flows.

Our first task is to study the stability in a fix point of linear flow $\varphi_t$. In natural way, we follow ideas presented in classical literature of dynamical systems on Euclidian space (see for instance \cite{colonius}, \cite{hirsch} and \cite{robinson}). In \cite{santana}, Da Silva, Santana and Stelmastchuk show that a necessary and sufficient condition to asymptotically and exponential stability of $\varphi_t$ at identity $e$ is that $\XC$ is hyperbolic. However, if a linear vector field $\XC$ is hyperbolic, then $G$ is a nilpotent Lie group. Because this obstruction, we choose restrict our study on compact, semisimple Lie groups.

Let $G$ be a real, connected, compact, semisimple Lie group. Consider a linear vector field $\XC$ on $G$ and its linear flow $\varphi_t$. We show that any fix point of linear flow $\varphi_t$ is stable (see Theorem \ref{stable}). Furthermore, we demonstrate that any periodic orbit of linear flow $\varphi_t$ is stable (see Theorem \ref{periodic}). Also, we proof that the derivation $\DC = -ad(\XC)$ associated to $\XC$ has only semisimple eigenvalues since the identity $e$ is stable. The last fact is the key to study periodic orbits of linear flow $\varphi_t$.

Our next purpose is to study periodic orbits of linear flows on compact, semisimple Lie groups. In semisimple Lie groups, for any linear vector field $\XC$ there is a right invariant vector field $X$ associated to it. Thus, our first step is to study the relation between periodic orbits of linear flow $\varphi_t$ and invariant flow $\exp(tX)$. In fact, we proof that they are equivalent. Done this, we show that an orbit of linear or invariant system is periodic if and only if the derivation $\DC$ of $\XC$ has as eigenvalues $0$ and $\mu = \pm \alpha i$ for an unique $\alpha \in \R$ (see Theorem \ref{teo1}). As a direct consequence, every orbit that is not a fix point of a linear flow $\varphi_t$ or invariant flow $\exp(tX)$ on a 3-dimensional, compact, semisimple Lie group $G$ is periodic. Other consequence is a version of Poincar\'e-Bendixon Theorem for linear or invariant flows. We remark that for a linear or invariant flow all orbits that is not a fix point of linear or invariant flows are periodic or not.

To end, we presents periodic orbits of linear and invariant flows on $SO(3)$ and $SU(2)$, and we classify periodic orbits on $SO(4)$ (see Theorem \ref{periodicSO(4)}).

This paper is organized as follows. Section 2 briefly reviews the notions of linear vector fields. Section 3 works with stability on compact, semisimple Lie groups. Section 4 develops results about periodic orbits. Finally, section 5 applies previous results on compact, semisimple Lie groups $SO(3)$, $SU(2)$ and $SO(4)$.

\section{Linear vector fields}

Let $G$ be a connected Lie group and let $\fg$ denote its Lie algebra. We call a vector field $\XC$ linear if its flow $(\varphi_t)_{t\in\R}$ are automorphisms of Lie group. It is known there is a derivation $\DC$ associated to $\XC$, which is given by
\[
  \DC(Y) = - [\XC,Y], \, Y \in \fg.
\]
In Euclidian case, it is true that $[Ax,b] = - Ab$. As a consequence, the derivation $\DC$ coincide with linear map induced by $A$; then, the dynamical system
\[
  \dot{g} = \XC(g), \ \ g \in G,
\]
is a generalization of dynamical system on $\R^{n}$ given by
\[
  \dot{x} = Ax,
\]
where $A \in \R^{n \times n}$ and $x \in \R^n$.

Da Silva, in \cite{dasilva}, write
\[
  \fg^+=\bigoplus_{\alpha; \mathrm{Re}(\alpha)>0}\fg_{\alpha}, \;\;\;\;\fg^0=\bigoplus_{\alpha; \mathrm{Re}(\alpha)=0}\fg_{\alpha}, \;\;\;\mbox{ and }\;\;\;\fg^-=\bigoplus_{\alpha; \mathrm{Re}(\alpha)<0}\fg_{\alpha}
\]
where $\alpha$ are eigenvalues of the derivation $\DC$ such that
\[
  \fg =\fg^+\oplus \fg^{0} \oplus \fg^{-} \ \ \mbox{and} \ \ [\fg_{\alpha},\fg_{\beta}] = \fg_{\alpha + \beta}
\
with $\alpha + \beta = 0$ if the sum is not an eigenvalue. Let us denote by $G^+$, $G^0$ and $G^-$  $\varphi_t$-invariant, connected Lie subgroups of Lie algebras $\fg^+$, $\fg^0$ and $\fg^-$, respectively. Lie subgroups $G^+$, $G^0$ and $G^-$ are called unstable, central and stable groups associated to $\varphi_t$, respectively.

For the convenience of the reader we resume some facts about a linear vector field $\XC$ and its flow $\varphi_t$. The proof of these facts can be found in \cite{cardetti}.
\begin{proposition}\label{linearproperties}
  Let $\XC$ be a linear vector field and let $\varphi_t$ denote its flow. The following assertions hold:
  \begin{description}
    \item{(i)} $\varphi_t$ is an automorphism of Lie groups for each $t$;
	  \item{(ii)} $\XC$ is linear iff $\XC(gh) = R_{h*}\XC(g) + L_{g*}\XC(h)$;
  	\item{(iii)} $(d\varphi_t)_e = e^{t\DC}$ for all $t \in \R$.
	\end{description}
\end{proposition}

\section{Stability of the linear flow}

Let $G$ be a semisimple Lie group and $\XC$ a linear vector field on $G$. In this section, our wish is to study stability of the linear flow $\varphi_t$ that is the solution of the differential equation on $G$ given by
\begin{equation}\label{odelinear}
  \dot{g} = \XC(g).
\end{equation}

Being $G$ semisimple, there is a right invariant vector field $X$ such that $\XC = X + I_*X$, where $I_{*}X$ is the left invariant vector field associated to $X$ and  $I_*$ is the differential of inverse map $\mathfrak{i}(g) = g^{-1}$ (more details is founded in \cite{sanmartin}). From this it follows that the linear flow can be written as
\[
  \varphi_t(g) = \exp(tX).g.\exp(t (I_*X)), \, \forall \, g \in G.
\]
According to above expression, we have that the identity $e$ is a fix point for the linear flow. However, it may to exist other fix point.

\begin{proposition}
  If $g$ is a point belongs to center of $G$, then $g$ is a fix point of the linear flow $\varphi_t$.
\end{proposition}
\begin{proof}
  Let $g$ be a point in the center of $G$. Then, for all $t \in \R$,
	\[
	  \varphi_t(g) = \exp(tX)\cdot g \cdot \exp(-tX) =  \exp(tX) \cdot \exp(-tX) \cdot g = g,
	\]
	which is the desired conclusion.
\end{proof}

Our next step is to presents the hyperbolic concept to linear vector fields. We remember that the stability in Euclidian space is obtained if a dynamical system is hyperbolic (see for instance \cite{robinson}). As one can see in \cite{santana}, it is also true if $\XC$ is hyperbolic on a Lie group $G$.

\begin{definition}
  Let $\mathcal{X}$ be a linear vector field on a Lie group $G$. We call $\XC$ {\bf hyperbolic} if its associated derivation $\DC$ is hyperbolic, that is, $\DC$ has no eigenvalues with zero real part.
\end{definition}

Let $\XC$ be a hyperbolic linear vector field on a semisimple Lie group $G$. Then $\DC$ has no eigenvalues with zero real part. Denoting by $\fg_{\alpha}$ the generalized eigenspace associated with an eigenvalue $\alpha$ of $\DC$ we get
\[
  [\fg_{\alpha}, \fg_{\beta}]\subset\fg_{\alpha+\beta},
\]
where $\alpha+\beta$ is an eigenvalue of $\DC$ and zero otherwise (see for instance Proposition 3.1 in \cite{sanmartin}). Since $\dim G<\infty$, it implies that the Lie algebra $\fg$ is nilpotent. In consequence, $G$ is nilpotent. We have thus proved

\begin{proposition}\label{noexistence}
  There no exists hyperbolic linear vector field on semisimple Lie groups.
\end{proposition}

We now begin the study of the stability of  linear flows on semisimple Lie groups. Firstly, we remember some concepts of stability.
\begin{definition}
  Let $g\in G$ be a fixed point of the linear vector field $\mathcal{X}$. We call $g$
  \begin{itemize}
    \item[1)]{\bf stable} if for all $g$-neighborhood $U$ there is a $g$-neighborhood $V$ such that $\varphi_t(V)\subset U$ for all $t\geq 0$;
    \item[2)] {\bf asymptotically stable} if it is stable and there exists a $g$-neighborhood $W$ such that $\lim_{t\rightarrow\infty}\varphi_t(x)=g$ whenever $x\in W$;
    \item[3)] {\bf exponentially stable} if there exist $c, \mu$ and a $g$-neighborhood $W$ such that for all $x\in W$ it holds that
    $$\varrho(\varphi_t(x), g)\leq c\rme^{-\mu t}\varrho(x, g),\;\;\;\;\mbox{ for all }\;\;t\geq 0;$$
    \item[4)] {\bf unstable} if it is not stable.
  \end{itemize}
\end{definition}

Since property 3) is local, it does not depend of the metric on $G$. Because of this, we will assume from now on that $\varrho$ is a left invariant Riemmanian metric.

In order to characterize the stability, let us work with the Lyapunov exponent. We follow \cite{santana} in assuming that the Lyapunov exponent can be written as
\[
  \lambda(e,v) = \limsup_{t \to \infty} \frac{1}{t} \rm log(\|e^{tD}(v)\|),
\]
where $v$ is $\fg$ and the norm $\| \cdot \|$ is given by the left invariant metric.

We will use $\lambda_1, \ldots, \lambda_k$ to denote $k$ distinct values of the real parts eigenvalues of the derivation $\DC$. Then, the Lie algebra $\fg$ can be written as
\[
  \fg=\bigoplus_{i=1}^k\fg_{\lambda_i} \;\;\mbox{ where }\;\;\fg_{\lambda_i}:=\bigoplus_{\alpha; \mathrm{Re}(\alpha)=\lambda_i}\fg_{\alpha}.
\]
Furthermore, from Theorem 4.2 in \cite{santana} we see that
\begin{equation}\label{lyapunovroots}
  \lambda(e, v)= \lambda \;\;\;\Leftrightarrow\;\;\;v\in\fg_{\lambda}:=\bigoplus_{\alpha; \mathrm{Re}(\alpha)=\lambda}\fg_{\alpha}.
\end{equation}

Using the Lyapunov exponent we make a first result about stability of the linear flow $\varphi_t$.

\begin{theorem}
  For any linear vector field $\XC$ on a semisimple Lie group $G$, any fix point is neither asymptotically nor exponentially stable to the linear flow $\varphi_t$.
\end{theorem}
\begin{proof}
	We first observe that Lyapunov exponents satisfy the following: $\lambda(g,v) = \lambda(e,v)$ for each $v \in \fg$. We need only consider the assertion at identity $e$. Suppose, contrary to our claim, that the identity $e$ is either asymptotically or exponentially stable. By Theorem 4.5 in \cite{santana}, it follows that all Lyapunov exponents of $\DC$ are negatives. From (\ref{lyapunovroots}) it follows that any eigenvalue of $\DC$ has the real part negative. It means that $\XC$ is hyperbolic, and this contradicts Proposition \ref{noexistence}.
\end{proof}

Despite any fix point is neither asymptotically nor exponentially stable, they are stable if $G$ is compact and semisimple as we will show. For this purpose, we begin by introducing an appropriate metric on $G$.

Let $G$ be a compact and semisimple. Then, we know that the Cartan-Killing form is negative defined. Thus, we adopt the metric $<,>$ given by negative of the Cartan-Killing form on $\fg$. Since $<,>$ satisfies
\[
  <\Ad(g)X,\Ad(g)Y> = <X,Y>, \,\, \forall \, g \in G \,\, \mbox{and}\,\, X,Y \in \fg,
\]
it follows that $<,>$ is an invariant Riemmanian metric on $G$ associated to Cartan-Killing form (see \cite{arvani} for more details). From now on we make the assumption: every compact, semisimple Lie group is equipped with the Riemannian metric given by Cartan-Killing form.

Adopting these invariant metric and using Lyapunov exponent leads us a algebraic characterization of linear vector fields on compact, semisimple Lie groups, namely, its eigenvalues have real part null.

\begin{proposition}\label{central}
  Let $\XC$ be a linear vector field on a compact, semisimple Lie group $G$. Then, $G$ is the central group of the linear flow $\varphi_t$.
\end{proposition}
\begin{proof}
  We begin by writing $\XC = X + I_*(X)$ with $X \in \fg$. It is clear that $\DC = \ad(X)$. Then, for any $v \in \fg$ we have
	\[
	  \|e^{t\DC}v\| = \|e^{t\, \ad(X)}v\| = \|\Ad (\exp(tX))v\|= \|v\|,
	\]
	where we used the $\Ad$-invariance of metric at last equality. Thus, Lyapunov exponents can be written as
	\[
	  \lambda(e,v) = \limsup_{t \to \infty} \frac{1}{t} \rm log(\|e^{t\DC}v\|) = \limsup_{t \to \infty} \frac{1}{t} \rm log(\|v\|) = 0.
	\]
	Therefore, $\lambda_1= \ldots= \lambda_k = 0$. Using the relation (\ref{lyapunovroots}) we conclude that $\fg =\fg_0$. Since $G$ is connected, $G= G_0$. It means that $G$ is the central group associated to linear flow $\varphi_t$.
\end{proof}

Despite Proposition above is presented in \cite{dasilva}, we want to prove this because our proof is done by dynamical concepts instead of algebraic concepts, as done there.

Our next step is to verify if the linear flow $\varphi_t$ satisfies some metric property. To do this, we recall what is a Riemmanian distance. Let $(M,g)$ be a Riemmanian manifold, a Riemmanian distance is $\rho$ associated to $g$ is defined by
\[
  \rho(x,y) = \displaystyle\inf_{\sigma}\{\int_0^1 g( \dot{\sigma}(s), \dot{\sigma}(s))^{1/2} ds\},
\]
where the infimun is taken over all smooth curves $\sigma$ such that $\sigma(0) = x$ and $\sigma(1) = y$.

\begin{proposition}\label{isometry}
  Let $\XC$ be a linear vector field on a compact, semisimple Lie group $G$. Then, $\varphi_t$ is an isometry for all $t$.
\end{proposition}
\begin{proof}
  For any $g, h \in G$ and any $t \in \R$ the Riemmanian distance between $\varphi_t(g)$ and $\varphi_t(h)$ is taken over all smooth curves $\sigma_t(s)$ such that
	\[
	  \sigma_t(0) = \varphi_t(g) \ \ \mbox{and}\ \ \sigma_t(1) = \varphi_t(h).
	\]
	It is clear that smooth curves $\phi_t^{-1}\sigma_t(s)$ satisfy
	\[
	  \varphi_t^{-1}\circ \sigma_t(0)=g\ \ \mbox{e}\ \ \varphi_t^{-1}\circ \sigma_t(1) = h.
	\]
	Using the invariance of metric $g$ we obtain
	\[
	  \int_0^1 g( \sigma'(s), \sigma'(s) )^{1/2} ds  = \int_0^1 g((\varphi_t^{-1}\circ \sigma)'(s), (\varphi_t^{-1}\circ \sigma)'(s) )^{1/2} ds.
	\]
  Taking the infimun over all smooth curves joining $\varphi_t(g)$ and $\varphi_t(h)$ yields
	\[
	  \rho(\varphi_t(g), \varphi_t(h)) =  \rho(g,h),
	\]
	which shows that $\varphi_t$ is an isometry on $G$ to the invariant metric given by Cartan-Killing form.
\end{proof}

Before our next result, we need to introduce some notations. For $r>0$ we will denote an sphere of radius $r$ with center $g$ by $\mathbb{S}_r(g) = \{ x \in G: \rho(x,g) =r\}$ and an open ball of radius $r$ with center $g$ by  $ B_r(g) =\{ x \in G; \rho(x,g) < r\}$.

\begin{proposition}\label{sphere}
  If $G$ is a compact,semisimple Lie group, then for each $g \in G$ the linear flow $\varphi_t(g)$ is in an sphere.
\end{proposition}
\begin{proof}
  We first choose an arbitrary point $g \in G$ and write $r=\rho(g,e)$. Then,
	\[
	  \rho(\varphi_t(g),e) = \rho(\varphi_t(g), \varphi_t(e)) = \rho(g,e)=r, \, \,  \forall\,\, t.
	\]
	It means that $\varphi_t(g) \in \mathbb{S}_r$ for all $t$, and the proof is complete.
\end{proof}

A first consequence of Propostion above is about $\omega$-limit and $\alpha$-limit sets.

\begin{corollary}
	If $G$ is a compact, semisimple Lie group, then $\omega$-limit and $\alpha$-limit sets of $g$ are in spheres.
\end{corollary}

We can now to prove our main result of our section.

\begin{theorem}\label{stable}
	Let $G$ be a compact, semisimple. Then, any fix point of linear flow $\varphi_t$ is an stable point.
\end{theorem}
\begin{proof}
	We begin by fixing an arbitrary fix point $g$ of $G$. We also remember that a Riemmanian distance induces the topology of Riemmanian manifold. So it is sufficient to consider as neighborhoods of $g$ open balls $B_r(g)$, where $r>0$ is arbitrary. Choose $r_0>0$ such that $r_0 \leq r$ and consider the ball $B_{r_0}(g)$. Taking any $y \in B_{r_0}(g)$ we see that
	\[
	  \rho(\varphi_t(y),g) = \rho(y,g) < r_0 \leq r,
	\]
	where we used Proposition (\ref{isometry}) at first equality. It shows that $\varphi_t(B_{r_0}(g)) \subset B_{r}(g)$. Consequently, by definition, $g$ is a stable point to the linear flow $\varphi_t$.
\end{proof}

Hereafter, we give a characterization of derivations on compact, semisimple Lie groups. Before we need to introduce some concepts. Following \cite{colonius}, if for an eigenvalue $\mu$ all complex Jordan blocks are one-dimensional, i.e., a complete set of eigenvectors exists, it is called semisimple. Equivalently, the corresponding real Jordan blocks are one-dimensional if $\mu$ is real, and two-dimensional if $\mu$, $\mu$ and $\bar{\mu} \in \mathbb{R}$.

\begin{theorem}\label{derivation}
  On a compact, semisimple Lie group $G$, every derivation has only semisimple eigenvalues.
\end{theorem}
\begin{proof}
  Let $\DC$ be a derivation on $G$. From Theorem \ref{stable} we see that $e$ is a stable point of the linear flow $\varphi_t$ associated to $\DC$. Since $(d\varphi_t)_e = e^tD$, it follows that the linearization of $\dot{g} = \XC(g)$ is $X = \DC X$. Being $exp$ local diffeomorphism and $e$ stable, it follows that $0$ is stable. From Proposition \ref{central} we know that eigenvalues of $\DC$ has real part null. Then, Theorem 1.4.10 in \cite{colonius} assures that every eigenvalue of $\DC$ is semisimple, which gives the proof.
\end{proof}

Theorem above is fundamental to study periodic orbits of linear flows.

To end this section, we study the stability of periodic orbits to linear flow $\varphi_t$. We begin by recalling what means a stable periodic orbit. A periodic orbit $\Gamma$ of the linear flow $\varphi_t$ is stable if for each open set $V$ that contains $\Gamma$, there is an open set $W \subset V$ such that every solution, starting at a point in $W$ at $t = 0$, stays in $V$ for all $t \geq 0$.

Before to presents our next result, we need to introduce the following notation. Take $g \in G$ and  consider the orbit $\varphi_t(g)$.Write $Tube_r (\varphi_t(g)) = \{ h \in G: \rho(h,\varphi_t(g))<r$ for some $t \}$  for any $r>0$,.

\begin{proposition}
	Let $\XC$ be a linear vector field on a compact, semisimple Lie group $G$. If $h \in Tube_r(\varphi_t(g))$, then $\varphi_s(h) \in Tube_r(\varphi_t(g))$ for any $s \in \R$.
\end{proposition}
\begin{proof}
	Let $h \in Tube_r(\varphi_t(g))$. Then, for some $t$ we have $\rho( h, \varphi_t(g))< r$. From Proposition \ref{isometry} it follows that
	\[
		\rho(\varphi_s(h), \varphi_{t+s}(g)) = \rho(h, \varphi_{t}(g)) <r,
	\]
	which implies that $\varphi_s(h) \in Tube_r(\varphi_t(g))$.
\end{proof}

\begin{theorem}\label{periodic}
	Let $\XC$ be a linear vector field on a compact, semisimple Lie group $G$. Then every periodic orbit is stable.
\end{theorem}
\begin{proof}
	Let $g \in G$ such that $\varphi_t(g)$ is a periodic orbit of linear flow $\varphi_t$. We consider a open set $V$ such that $\varphi_t(g) \subset V$. Take $r_0 = \inf \{ r : B_r(\varphi_t(g)) \subset U, \ \forall\,\, t\geq 0 \}$. Thus it is sufficient to take $U =Tube_{r_0}(\varphi_t(g))$ and to apply Proposition above.
\end{proof}

\section{Periodic orbits}

In this section, we study periodic orbits of a linear flow in a compact, semisimple Lie group $G$. The key of our study is Theorem \ref{derivation} because it describes all eigenvalues of any derivation on $G$.

We begin by recalling that a linear vector field $\XC$ can be written as $\XC = X + I_*X$, where $X$ is a right invariant vector field, $I_{*}X$ is the left invariant vector field associated to $X$, and $I_*$ is the differential of inverse map $\mathfrak{i}(g) = g^{-1}$. In this way, we can rewrite the differential equation (\ref{odelinear}) as
\[
  \dot{g} = X(g) + (I_*X)(g).
\]
It implies that there exists a relation between flows of linear dynamical system $\dot{g} = \XC(g)$ and of invariant one $\dot{g} = X(g)$. In fact, a direct accounts shows that, for all $g \in G$, $\varphi_t(g)$ is solution of (\ref{odelinear}) if, and only if, $\varphi_t(g)\cdot \exp(tX)$ is solution of $\dot{g} = X(g)$. It suggests us that there exists a relation between periodic orbits of invariant and linear system. Therefore, our next step is to investigate this fact.

\begin{proposition}\label{equivalentperiodic}
	Let $\XC$ be a linear vector field on a compact, semisimple Lie group $G$. The following sentences are equivalent:
	\begin{description}
		\item[i)] for every $g$, the invariant flow $\exp(tX)g$ is periodic;
		\item[ii)]$e$ is a periodic point of invariant flow $\exp(tX)$;
		\item[iii)] any $g \in G$ is a periodic point of linear flow $\varphi_t$;
		\item[iv)] for every $g$, $Ad(g)$ is a periodic point of the flow $e^{tD}$.
	\end{description}
\end{proposition}
\begin{proof}
  i) $\Leftrightarrow$ ii)
	If $\exp(tX)g$ is periodic, then $e$ is a periodic point of the curve $\exp(tX)$. On contrary, suppose that $e$ is a periodic point of the flow $\exp(tX)$, that is, there is a $s>0$ such that $\exp((t+s) X) = \exp (tX)$. Then, for any $g \in G$
	\[
	  \exp((t+s)X)g = (\exp((t+s)X)\cdot e) \cdot g = \exp(tX)g.
	\]
  ii) $\Leftrightarrow$ iii)
  If $e$ is a periodic point of the flow $\exp(tX)$, then there is a $s>0$ such that $\exp(tX) = \exp((t+s)X)$. Thus
	\[
	  \varphi_{t+s}(g) = \exp((t+s)X)\cdot g \cdot \exp(-(t+s)X) = \exp(tX)\cdot g \cdot \exp(-tX) = \varphi_t(g).
	\]
	On contrary, if every $g$ is a periodic point of $\varphi_t$, then it is clear that $\exp(tX)$ is periodic.\\
	i) $\Leftrightarrow$ iv)
	Since $G$ is a semisimple Lie group, it follows
	\[
	  \Ad(\exp(tX)\cdot g) = e^{t\ad(X)} \Ad(g) = e^{tD}\Ad(g).
	\]
	We thus get the equivalence.
\end{proof}

The interest of the Proposition above is that periodic orbits of linear or invariant flows are equivalents on compact, semisimple Lie groups.

\begin{theorem}\label{teo1}
	Let $G$ be a compact, semisimple Lie group. It is equivalent
	\begin{description}
		\item[i)] if there exists a periodic orbit for the linear flow $\varphi_t$; or
		\item[ii)] if there exists a periodic orbit for the right invariant flow $\exp(tX)$;
		\item[iii)] the derivation $\DC$ of $\XC$ has only eigenvalues of the form $0$ or $\mu = \pm \alpha i$ with unique $\alpha \in \mathbb{R}^*$.
	\end{description}
	Furthermore, if there exists a periodic orbit, then its period is $T= 2\pi/\alpha$.
\end{theorem}
\begin{proof}
	We first observe that i) is equivalent to  ii) by Proposition \ref{equivalentperiodic}. We are going to show that ii) is equivalent to iii). For this, it is sufficient to consider $e$ as a periodic point to the flow $\exp(tX)$ with period $T>0$. Then, for all $t \in \mathbb{R}$,
	\[
		\exp((t+T)X) = \exp(tX) \Leftrightarrow \exp(TX) = e \Leftrightarrow e^{T\DC} = Id.
	\]
	Take the Jordan form $J$ of $\DC$. A simple account shows that $e^{TJ} =Id$.  Since any eigenvalues of $\DC$ is semisimple, its real Jordan Block has dimension 1 or 2 if it is real or complex, respectively. If $0$ is eigenvalue of $\DC$, then its real Jordan block is written as $J_0 =[0]$. Thus $e^{tJ_0}$ is constant. It implies that in direction of $0$ the $e^tJ$ is constant. Consequently, solutions associated to $0$ are trivially periodic. Suppose that there are eigenvalues different of $0$. From Proposition \ref{central} they are of the form $\pm\alpha i$ and $\pm\beta i$. By Theorem \ref{derivation}, its real Jordan blocks are, respectively,
	\[
		\left(
	  \begin{array}{cc}
			\cos(t\alpha) & -\sin(t\alpha) \\
			\sin(t\alpha) & \cos(t\alpha)
		\end{array}
		\right)
		 \ \ \mbox{and} \ \
		\left(
	  \begin{array}{cc}
			\cos(t\beta) & -\sin(t\beta) \\
			\sin(t\beta) & \cos(t\beta)
		\end{array}
		\right).
	\]
	As $e^{TJ} =Id$ we have $T = \frac{2\pi}{\alpha}$ and $T = \frac{2\pi}{\beta}$. This clearly forces $\alpha = \beta$. Consequently, eigenvalues different of $0$ are only $\alpha i$ and $-\alpha i$.

	Reciprocally, suppose that $\DC$ has only eigenvalues $0$  or $\pm\alpha i$. In the case of $0$ as discussed above the solution is constant. In the case of an eigenvalue $\pm \alpha i$ with $\alpha \neq 0$. Being $\pm \alpha i$ semisimple, every real Jordan block associated to it has the dimension two and the solution applied at this block gives the following matrix
	\[
		\left(
	  \begin{array}{cc}
			\cos(t\alpha) & -\sin(t\alpha) \\
			\sin(t\alpha) & \cos(t\alpha)
		\end{array}
		\right).
	\]
	It is clear that matrix above is periodic with period $T= \frac{2 \pi}{\alpha}$. It implies that $e$ is periodic point of $e^{TJ}$ with period $T= \frac{2\pi}{\alpha}$, which is equivalent $Ad(e) = e$ to be periodic point of $e^{TD}$. Consequently, by Proposition \ref{equivalentperiodic}, the right invariant flow $\exp(tX)$ is periodic with period of $T= \frac{2 \pi}{\alpha}$.
\end{proof}

\begin{remark}
	The theorem above shows that all orbit that is not a fix point of a linear or invariant flows are periodic or not.
\end{remark}

\begin{corollary}\label{periodorbits3}
	Let $G$ be a compact, semisimple Lie group with dimension 3. Then
	\begin{description}
		\item [i)] every orbit of some invariant flow $\exp(tX)$ is periodic;
		\item [ii)] every orbit of some linear flow $\varphi_t$ is periodic.
	\end{description}
\end{corollary}
\begin{proof}
	It is sufficient to observe that the derivation $\DC=\operatorname{ad}(X)$ has only eigenvalues $0$, $\alpha i$, and $ -\alpha i$ with $\alpha \in \R^*$.
\end{proof}

An immediate consequence of Theorem  and is version of Poincar\'e-Bendixson's Theorem for compact, semisimple Lie groups.

\begin{theorem}[Poincar\'e - Bendixson]
	Let $G$ be a compact, semisimple Lie group. If a derivation $\DC$ has only eigenvalues $0$ or $\pm \alpha i$, if $\Omega$ is a nonempty compact $\omega$-limit set for the linear flow $\varphi_t$, and if $Ω$ does not contain a fix point point of $\varphi_t$, then $\Omega$ is a periodic orbit.
\end{theorem}

\section{Applications}

In this section, our wish is to study the periodic orbits on compact, semisimple Lie groups of lower dimension. In fact, we are interested to describe linear flows, that are periodic orbits, on $SO(3)$ and $SU(2)$, and we are interested to classify the periodic orbits on $SO(4)$.

\subsection{Linear flows on $SO(3)$ and $SU(2)$}

Our first case is to study  the linear flow on stability on the orthogonal group
\[
  SO(3) = \{g \in \R^{3\times 3}: gg^{T} = 1, \, \det g = 1 \}.
\]
It is well known that its Lie algebra is
\begin{eqnarray*}
  \mathfrak{so}(3) & = &
	\left\{ \left[
	\begin{array}{ccc}
		0  & -z & y \\
		z  & 0  & -x \\
		-y & x  & 0
	\end{array}
	\right]:
	x,\, y, \, z \in \R
	\right\}.
\end{eqnarray*}

Let $\XC$ be a linear vector field on $SO(3)$. Then, there exists a right invariant vector field $X$ such that $\XC = X + I_*X$. A direct calculus shows that eigenvalues of $\DC = \ad(X)$ are
\[
  \left\{0,-\sqrt{-x^2-y^2-z^2}, \sqrt{-x^2-y^2-z^2}\right\}.
\]

Write $\lambda_1 = -\sqrt{-x^2-y^2-z^2}$ and $\lambda_2 = \sqrt{-x^2-y^2-z^2}$. Then, using functional calculus we obtain
\[
  \exp(tX) = \frac{\cosh(t \lambda_1)-1}{ \lambda_1^2}X^2+ \frac{\sinh(t\lambda_1)}{\lambda_1}X + Id.
\]
From this, it is possible to give the solution of the linear flow $\varphi_t$ on $SO(3)$.

\begin{proposition}
  Let $\XC$ be a linear vector field on $SO(3)$. The solution of linear flow $\varphi_t(g)$ associated to $\XC$ is
	\[
    \left(\frac{\cosh(t \lambda_1)-1}{ \lambda_1^2}X^2+ \frac{\sinh(t\lambda_1)}{\lambda_1}X + Id\right) \cdot g \cdot \left(\frac{\cosh(t \lambda_2)-1}{ \lambda_2^2}X^2+ \frac{\sinh(t\lambda_2)}{\lambda_2}X + Id\right),
	\]
	where $X$ is the right invariant vector field associated to $\XC$ and $\lambda_1 = -\sqrt{-x^2-y^2-z^2}$ and $\lambda_2 = \sqrt{-x^2-y^2-z^2}$.
\end{proposition}

Corollary \ref{periodorbits3} now assures the characterization of periodic orbits of the linear flow $\varphi_t$.

\begin{proposition}
	On $SO(3)$,
\begin{description}
		\item [i)] every orbit of some invariant flow $\exp(tX)$ is periodic;
		\item [ii)] every orbit of some linear flow $\varphi_t$ is periodic.
	\end{description}
	Furthermore, the period is $T = 2\pi /\sqrt{x^2+y^2+z^2}$.
\end{proposition}

Our other case is the unitary group $SU(2)$, which is a matrix group given by
\[
  \mathrm{SU}(2) = \{ g \in \C^{2\times 2} : gg^T = 1, \,\mathrm{det} g = 1\}.
\]
The Lie algebra associated to $SU(2)$ is described as
\[
  \mathfrak{su}(2) =
	  \left\{ \left[
	  \begin{array}{cc}
		  \frac{i}{2}x   & \frac{1}{2}(iz+y) \\
		  \frac{1}{2}(iz-y) & -\frac{1}{2}x \\
		\end{array}
		\right]:
		x,\, y, \, z \in \R
		\right\}.
\]

Let $\XC$ be a linear vector field on $SU(2)$ and $X$ the right invariant vector field associated to it. In analogous way to case $SO(3)$, it is easily to see that eigenvalues of a derivation $\DC = \ad(X)$ are
\[
  \left\{0,-\sqrt{-x^2-y^2-z^2}, \sqrt{-x^2-y^2-z^2}\right\}.
\]
In consequence,

\begin{proposition}
	On $SU(2)$,
	\begin{description}
		\item [i)] every orbit of some invariant flow $\exp(tX)$ is periodic;
		\item [ii)] every orbit of some linear flow $\varphi_t$ is periodic.
	\end{description}
	Furthermore, the period is $T = 2\pi /\sqrt{x^2+y^2+z^2}$.
\end{proposition}

\subsection{periodic orbits on $SO(4)$}

In this subsection, our wish is to give a condition for orbits of invariant or linear flow be periodic or not on $SO(4)$. Let $\mathfrak{so}(4)$ be the Lie algebra of $SO(4)$ given by
\[
	\left\{ \left[
		\begin{array}{cccc}
		0 & -x & -y & -z\\
		x & 0  & -u & -v \\
		y & u  &  0 & -w \\
		z & v  &  w &  0
	\end{array}
	\right]:
	x,\, y,\, z, \, u, \, v, \, w \in \R
	\right\}.
\]
Consider the basis $\beta$ for $\mathfrak{so}(4)$ that consists of $4 \times 4$ matrices $e_{12},e_{13},e_{14},e_{23},e_{24},e_{34}$ that have $1$ in the $(i,j)$ entry, $-1$ in the $(j,i)$ entry, and $0$ elsewhere ($1\leq i < j \leq 4$). A computation of Lie brackets gives
\[
	[e_{12},e_{13}] = e_{23},\, [e_{12},e_{14}] = e_{24},\, [e_{12},e_{23}] = -e_{13},\, [e_{12},e_{24}] = -e_{14},\, [e_{12},e_{34}] = 0,
\]
\[
	[e_{13},e_{14}] = e_{34},\,[e_{13},e_{23}] = e_{12},\, [e_{13},e_{24}] = 0,\, [e_{13},e_{34}] = -e_{14},\, [e_{14},e_{23}] = 0,
\]
\[
	[e_{14},e_{24}] = e_{12},\, [e_{14},e_{34}] = e_{13},\, [e_{23},e_{24}] = e_{34}\, [e_{23},e_{34}] = - e_{24},\, [e_{24},e_{34}] = e_{23}.
\]
Let $\XC$ be a linear vector field on $SO(4)$. Let us denote by $\DC = \ad(X)$ the associated derivation to $\XC$, where $X$ is an right invariant vector field on $SO(4)$. Our next step is to describe the derivation $\DC$. To do this, write
\[
	X = a e_{12} +b e_{13} + c e_{14} + d e_{23} + e e_{24} + f e_{34}, \ \ a,b,c,d,e,f \in \R.
\]
By Lie brackets above, we compute
\[
	\DC = \ad(X) =
	\left(
	\begin{array}{cccccc}
		0 & -d & -e & b & c & 0 \\
		d & 0 & -f & -a & 0 & c \\
		e & f & 0 & 0 & -a & -b \\
		-b & a & 0 & 0 & -f & e \\
		-c & 0 & a & f & 0 & -d \\
		0 & -c & b & -e & d & 0
	\end{array}
	\right).
\]
Some calculus show that eigenvalues of $\DC = \ad(X)$ are
\[
	 \{ 0 ,\pm \sqrt{-(a+f)^2-(b-e)^2-(c+d)^2}, \pm \sqrt{-(a+f)^2-(b+e)^2-(c-d)^2}\}.
\]
We observe that eigenvalues are according to Theorem \ref{derivation}. We now are in position to give a condition that characterize periodic orbits of an invariant or linear flow.

\begin{theorem}\label{periodicSO(4)}
	Let $\XC$ be a linear vector field on $SO(4)$. Consider the derivation $\DC = \ad(X)$ of $\XC$, where $X$ is a right invariant vector field such that
	\[
		X = a e_{12} +b e_{13} + c e_{14} + d e_{23} + e e_{24} + f e_{34}, \ \ a,b,c,d,e,f \in \R.
	\]
	A necessary and sufficient condition to every orbit that is not a fix point of linear flow $\varphi_t$ or invariant flow $\exp(tX)$ be periodic is that $bc=ed$. Furthermore, if there are periodic orbits, then their periods are $T = 2\pi /\sqrt{(a+f)^2+(b+e)^2+(c-d)^2}$.
\end{theorem}
\begin{proof}.
	It is a simple application of Theorem \ref{teo1}. In fact, orbits of linear flow $\varphi_t$ or invariant flow $\exp(tX)$ are periodic if, and only if, eigenvalues of $\DC=\ad(X)$ are $0$ or $\pm \alpha i$ for a unique $\alpha \in \R$. Last condition is equivalent to
	\[
		\sqrt{-(a+f)^2-(b-e)^2-(c+d)^2} = \sqrt{-(a+f)^2-(b+e)^2-(c-d)^2},
	\]
	which is equivalent to $bc = ed$.
\end{proof}

As direct application of Theorem above, any right invariant vector field $e_{12}$, $e_{13}$, $e_{14}$, $e_{23}$, $e_{24}$ or $e_{34}$ yields periodic orbits for linear or invariant flows.

{\small

redpajama_set_name: RedPajamaArXiv


Members of the Capterra team at one of their gatherings that encourage innovation.
Innovation should be important to all organizations. And it is. It's more a matter of execution for many organizations, according to CB Insights State of Innovation report (registration required). According to the report, 84.9 percent of corporate strategy leaders said innovation is very important, but 78 percent can only focus on incremental changes. Further, about 60 percent of companies said it takes a year or longer to produce new products.
So, where is your organization along the innovation spectrum and how can your company encourage innovation in the workplace? We talked with some experts who offered the following advice.
Iteration: Incremental change to an existing product or process.
Evolution: Significant change to an existing product or process.
Revolution: Something new or an historic change to a product or process.
Disruption: Something new that creates an entirely different market.
Creativity and innovation are incredibly important to business growth, but in practice can often feel like distractions to getting "real work" done, according to Claire Alexander, general manager for Capterra. This, she said, leads to the creation of separate departments with "innovative" talent. "Money is spent and the impact on the business is frequently … debatable," Alexander added.
Capterra, instead of siloing off innovation, holds a 24-hour hackathon (they call it a "Capathon") to remind everyone they are "creative beings with agency to imagine new ways of approaching work." For some, Alexander said, this takes the form of exercising a skill that has nothing obvious to do with work: an account manager taking headshots, engineers and marketers discovering they all play instruments and holding an impromptu jam session. For others, it involves applying the skills they already have to solve pain points in their day-to-day jobs.


text:
Il porto di Reggio Calabria (classe II) è uno dei maggiori porti della Calabria, sede della direzione marittima della regione. Secondo l'aggiornamento ISTAT del 2006, con oltre 10 milioni di passeggeri trasportati all'anno è il secondo porto italiano nel settore dopo quello di Messina.

Il porto di Reggio, che si apre sulla sponda orientale dello Stretto, è costituito da un bacino artificiale protetto dalla lunga Banchina di Ponente. La città di Reggio si affaccia sul porto con il quartiere Santa Caterina.

Sul lato interno si trova la banchina di levante con il mercato ittico, la banchina Margottini e, più a sud, il molo di sottoflutto destinato alle imbarcazioni da diporto. Una banchina è invece destinata alla pesca.

La bocca di entrata è larga 110 m, mentre le banchine misurano circa 2,5 km. I fondali sono profondi mediamente 7,50 m, mentre la superficie complessiva del bacino occupa  e la differenza di livello-marea è di 60 cm.

Presso il porto si trovano gli uffici della sanità marittima e aerea, e l'ufficio veterinario. Dal molo di ponente si può osservare un ampio e suggestivo panorama della città di Reggio.

Collegamenti

Il porto di Reggio Calabria è collegato con Messina, con le isole Eolie e con Malta:
 i collegamenti veloci con Messina sono effettuati con linee su monocarena Blu Jet;
 il collegamento veloce con le isole Eolie è effettuato attraverso gli aliscafi di Liberty Lines nel solo periodo estivo;
 il trasporto di mezzi gommati per Messina e Tremestieri viene effettuato da Meridiano Lines;
 il collegamento con Malta effettuato con navi di Ma.Re.Si. e della compagnia maltese SeaMalta.
 è inoltre attivo, da giugno 2007, un collegamento veloce anche con Taormina.

Servizi portuali
Dal 1929, la darsena del porto di Reggio è gestita dalla Compagnia portuale "Tommaso Gulli" che si occupa del traffico merci e del sollevamento portuale ed extraportuale. La compagnia portuale è stata istituita, sotto forma di ente con il diretto controllo del Ministero della Navigazione, fra i lavoratori dei porti di Reggio Calabria e Villa San Giovanni, nel 1993 si trasformò in una s.r.l.

Fari del porto

Caratteristiche dei fari

Mercato ittico
Costruito nel 1958 e ristrutturato nel 1977, il mercato ittico, orientato al consumo, è gestito dal comune di Reggio. I suoi acquirenti sono essenzialmente dettaglianti, grossisti, ambulanti, ristoratori e pescherie. Il prezzo del prodotto è fissato con trattativa diretta.

Il pescato del mercato ittico di Reggio Calabria proviene dallo stesso porto di Reggio e da quelli di Mazara del Vallo, Bari, Ancona e Trieste, mentre i prodotti d'importazione provengono da Norvegia, Spagna, Portogallo, Marocco e Senegal.

I prodotti commercializzati sono il pesce azzurro, il pesce spada, le orate, le spigole, le triglie; quelli d'importazione sono salmoni, pesci spada e merluzzi.

Porticciolo turistico
Sul molo settentrionale sorge il porticciolo turistico che tra l'altro ospita anche mezzi militari della capitaneria di porto, della guardia di finanza e della polizia.

Con una capacità di circa 100 posti barca, gestito da anni dalla Compagnia Portuale "Tommaso Gulli", offre servizi a terra: telefono, rifornimento, acqua ed energia, docce, autobus per la città.

Venti
 La zona del porto è interessata dai venti che appartengono al I e II quadrante, da nord nord-ovest e ovest nord-ovest sono di traversia.

Note

Voci correlate
Porti d'Italia per flusso di passeggeri
Aeroporto di Reggio Calabria
Porto di Gioia Tauro
Porto di Messina
Meridiano Lines

Altri progetti

Collegamenti esterni

Porto
Reggio Calabria

redpajama_set_name: RedPajamaWikipedia


text:
A library that uses the Java OpenGL port LWJGL to render UI elements and other 2D sprites much like Java Swing. It features customizable elements housed in a singular sprite-sheet for maximum performance. It also included a variable multi-threaded game loop that can be altered to run differently on the fly. The goal is to create a versatile engine that will allow quick creation of complex yet efficient UI's and 2D games.

#### Current UI Elements
  - Window Frames
  - Window Panels
  - Buttons
  - Check Boxes

#### Backend Features
  - Variable Multi-Threaded Game Loop
  - Textures
  - TrueTypeFonts

#### Planned UI Elements
  - Radio Buttons
  - Combo Boxes
  - Dialogues
  - Text Fields
  - Text Areas
  - Scroll Bars
  - Scrollable Lists
  - Password Fields
  - Progress Bars
  - Sliders
  - Spinners
  - Tool Tips
  - Tool Bars
  - Color Chooser ?

#### Planned Backend Features
  - Dubugging/Console Interface
  - Rendering Batches
  - Entities
  - Sprites
  - Sprite-Sheets
  - OpenAL
  - OpenCL ?
  - File IO
  - Physics ?
  - AI (A*) ?
  - Networking

### TODO
  - GLFrame needs events rather than actually calling the OpenGL functions from its methods. Ex. frame.setBackground should be able to be called anywhere, not just in the render or initialize method where there is OpenGL context. It should trigger an event and the next time render runs it should set the background.
  - Figure out why the TrueTypeFont class does not generate accurate widths and heights
  - SpriteSheetUtil that will allow multiple spritesheet objects of the same size to be loaded. Such that only one group of texture coordinates must be supplied to GLButton. The method will calculate the others that are directly to the right of it.
  - Create global variables for each GLObject for the ActionEvents that are fired when things happen.


The Bronco's long gone. Land Rovers are expensive. Chrysler's got no money. Seems like a perfect time for an automaker to take a run at the rock-crawler segment. Hyundai US CEO John Krafcik told us they may do just that.
We asked Hyundai US CEO John Krafcik over dinner at Slow's BBQ (go eat there, like, now) the other night during the Detroit Auto Show what new segments the Korean automaker is looking at for future expansion. More on some of that shortly, but what blew our mind was the answer he gave when pressed about other types of vehicles — say, 4WD SUVs.
Krafcik responded by telling us that "every time our designers get together and start looking at concepts and the future, the first thing that comes out of those meetings — what everyone gets excited about — is the prospect of a Bronco-like, highly-capable off-roader." As a follow-up on that answer, I asked if he meant a Wrangler-fighter. He answered only by smiling and repeating himself — "highly-capable."
Given Chrysler's sorry fiscal state, don't expect the R&D monies for a new Jeep Wrangler to materialize anytime soon — which means the Wrangler's at its weakest cycle point in its product lifetime. That makes right now the perfect time for an automaker to come out with something — anything — to compete with the old man of 4WD vehicles.
We'd love to see it. Our bet is this means we'll be seeing a concept in the near-term future — maybe Chicago or New York — to gauge interest. So, let us know below if it's something you'd be really interested in seeing out of Hyundai.
FYI to FoMoCo forum-hounds: The image up top? It's not real. It's a photoshop of the ol' Bronco concept. I want to stop reading in the forum posts that you think it's an actual image of an actual Hyundai concept, OK?


text:
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NewLeaf Home Medical offers Bariatric Bathroom & Safety products to specifically meet the needs of larger patients. Bariatric Commodes and Toilet Frames, as well as raised toilet seats are durable and constructed to accommodate people of all sizes. Features vary by item and include buckets, arm rests, adjustable height and the ability to fold for convenient storage and travel.
As with any product NewLeaf Home Medical offers, we have a team of devoted and patient advisors available to help you by providing answers and guidance based on your specific needs. Please do not hesitate to call us at 888.739.4330 for help.


After purchasing the rights to the film, Netflix has revealed the new trailer for Mowgli by Andy Serkis.
Warner Bros was planning to release Mowgli in October this year, but the company eventually sold the rights to the film to Netflix in July. The streaming giant estimated that the movie would be released in 2019. But everything changed this week, when Netflix published an unseen trailer, announcing the release of the film on December 7 this year.
The feature film, which mixes live action and motion capture, is directed by Andy Serkis, better known as the actor who played Gollum in The Lord of the Rings. Serkis is not only working behind the camera, and will also make an appearance as Baloo, the bear. The five-star cast features the voices of Benedict Cumberbatch, Christian Bale, Cate Blanchett, and Naomie Harris.
"I wanted to make an emotional version of this story, which has already been told many times", Andy Serkis explained in Singapore, as Variety reports. This adaptation of Rudyard Kipling's Jungle Book focuses on the difficulties Mowgli faces in living between two worlds: the animal world where he was brought up and the human community which takes him in.
Mowgli will be released on Netflix on December 7 and screened at several cinemas in Los Angeles, New York, San Francisco and London from November 29.


Kate Connelly, a new mother and a private investigator in training, gets a mysterious case from a friend. This only being her 4th case and she is still learning with the help of an official P.I./ ex-cop.
As she looks for a killer she has to watch her step or she will be the next in a growing pile of bodies.
This series by author Diana Orgain is the first cozy mysteries I have ever read. I did not even know cozy mysterious were exactly what they were called but yet cozy was what I immediately felt with them. This is a book you just want to curl up with some tea or cocoa and read until there is no tomorrow. The characters well defined. I never get confused at who is who, like other mysteries I have read and I am always cheering for the good guys.
There is not much wrong with the series or this book. One thing I was not happy about was that our apprentice in investigating reacted as if this were her first case and not 4th. There were things I swear she had done in previous books but she seemed to have forgotten.
I love the series. I hope it continues and I have found other mysteries written by Mrs. Orgain that I hope to get too soon.


text:
IRIS TORINO
07-ALTRO PRODOTTO SCIENTIFICO
07P-Working Paper
Alternative Food Networks in Piedmont: determinants of on-farm and off-farm direct sales by farmers
Direct sales are a widespread and important typology of the so-called Alternative Food Networks. The direct links between producers and consumers can take two basic forms: consumers going to buy agricultural products at the farm (on-farm sales), and farmers selling their products in urban areas. These practices are an alternative to traditional organizations of the agro-food chains, that typically involve several operators between producers and consumers. It is therefore important to analyse the reasons pushing farmers to adopt this new organization of their marketing chain. This research aims at analysing the territorial distribution of direct links between urban consumers and farmers in Piedmont (Italy), and to assess the main determinants of their choice. Firstly, the territorial distribution of direct sales practices (on-farm or elsewhere) is analysed. This is made possible by the access to micro-data from the 2010 Agricultural Census for Piedmont, a region whose agriculture is characterized by a strong emphasis on quality products. The farms that chose direct sales, both on- and off-farm, are mostly concentrated in specific clusters, such as the hilly wine-growing areas of Langhe and Monferrato, the hilly belt surrounding Torino, and some low Alpine valleys. Secondly, we analyse the determinants of the choice to sell directly to consumers, separately for on-farm and off-farm sales, with probit models. Explanatory variables comprise the structural characteristics of the farms (farm size, type of farming, etc.), the personal characteristics of the operators and of the farm households, and the proximity to urban and commercial areas. Operators' and farm characteristics are found to affect the choice of selling directly, but rather weakly. The most important factors affecting these choices are farm location and, for on-farm direct sales, the complementarity with agro-tourism and recreational activities.
Titolo: Alternative Food Networks in Piedmont: determinants of on-farm and off-farm direct sales by farmers
Autori Riconosciuti:
CORSI, Alessandro
NOVELLI, Silvia
PETTENATI, Giacomo
Autori: Corsi A.; Novelli S.; Pettenati G.
Abstract: Direct sales are a widespread and important typology of the so-called Alternative Food Networks. The direct links between producers and consumers can take two basic forms: consumers going to buy agricultural products at the farm (on-farm sales), and farmers selling their products in urban areas. These practices are an alternative to traditional organizations of the agro-food chains, that typically involve several operators between producers and consumers. It is therefore important to analyse the reasons pushing farmers to adopt this new organization of their marketing chain. This research aims at analysing the territorial distribution of direct links between urban consumers and farmers in Piedmont (Italy), and to assess the main determinants of their choice. Firstly, the territorial distribution of direct sales practices (on-farm or elsewhere) is analysed. This is made possible by the access to micro-data from the 2010 Agricultural Census for Piedmont, a region whose agriculture is characterized by a strong emphasis on quality products. The farms that chose direct sales, both on- and off-farm, are mostly concentrated in specific clusters, such as the hilly wine-growing areas of Langhe and Monferrato, the hilly belt surrounding Torino, and some low Alpine valleys. Secondly, we analyse the determinants of the choice to sell directly to consumers, separately for on-farm and off-farm sales, with probit models. Explanatory variables comprise the structural characteristics of the farms (farm size, type of farming, etc.), the personal characteristics of the operators and of the farm households, and the proximity to urban and commercial areas. Operators' and farm characteristics are found to affect the choice of selling directly, but rather weakly. The most important factors affecting these choices are farm location and, for on-farm direct sales, the complementarity with agro-tourism and recreational activities.
Collana: Dipartimento di Economia e Statistica "Cognetti de Martiis" Working Paper Series
Volume: 39/14
URL: http://www.est.unito.it/unitoWAR/ShowBinary/FSRepo/D031/Allegati/WP2014Dip/WP_39_2014.pdf
Appare nelle tipologie: 07P-Working Paper
Corsi_Novelli_Pettenati_AFN_WP_Econ_2014.pdf MATERIALE NON BIBLIOGRAFICO Accesso aperto Open Access Visualizza/Apri
http://hdl.handle.net/2318/1507243


The history of high-end sound began with the HD 414. The world's first open headphones revolutionised the headphones market in 1968. For the first time, the ears could hear in a way that they are accustomed to: open and free in space. The sound can travel to the ear drum much more naturally and clearly than in closed models. With its new principle, the HD 414 triggered an absolute boom in headphones sales. More than 100,000 sets of headphones had already been sold by the end of 1969. Even today, the HD 414, which has long since gained cult status, is still the bestselling headphones in the world, with sales of over 10 million units to date. The open principle has also become firmly established in the development of high-end headphones.
A further milestone followed when Sennheiser engineers for the first time combined the principle of open headphones with that of electrostatic transducer technology. Sennheiser's launch of the unipolar 2000 model in 1977 caused a sensation in the audio world. Its principle was once again an innovation in audio technology. In the electrostatic transducer, a high voltage is applied to two mesh-like electrodes. As in a loudspeaker, this voltage causes a coated foil diaphragm to vibrate in time with the audio signal. This results in an extremely fine resolution and low total harmonic distortion. "Its frequency curve has a relatively deep curvature from 2 to 4 kHz. This is one of the reasons why these headphones generate an extremely spacious, detailed, transparent, airy and nevertheless powerful sound image in unprecedented quality," was how one audio magazine 'Hifi-Stereofonie', sang the praises of the new headphones.
From a veritable flood of new headphones being introduced, one in particular stood out like a beacon in 1991: the electrostatic Orpheus HE 90/HEV 90. With the boldly ambitious mission to build the best headphones in the world, Sennheiser developers once again set to work – and once again exceeded all expectations of the audio sector with these electrostatic headphones and their impressive tube pre-amplifier. Two platinum-coated diaphragms vibrate between gold-coated glass electrodes. A 500 volt tube amplifier provides enough power to allow the Orpheus to fully unleash its sound. Due to the extremely complex technology involved, the Orpheus was produced in a limited edition of just 300 units. These legendary headphones still remain the ultimate benchmark for audiophile listening and for all high-end models that audio specialist Sennheiser continues to develop.
The HD 700 features a perfect combination of outstanding acoustic properties and sophisticated product design. Its styling and material selection have been thought through down to the finest detail in order to fully exploit the sound potential of the acoustic unit. The ear cups are designed in such a way that the sound waves are directed to the ears at a slight angle. This results in an impressively natural listening experience. The ear cups themselves have a completely open design. This not only ensures a highly transparent sound but also clearly displays the "heart" of these exciting headphones: the 40 mm Duofol transducer. Its powerful neodymium magnet systems guarantee detailed, lifelike audio reproduction from 10 to 42,000 Hz. To ensure that the acoustic properties of the headphones are not impaired by any partial vibrations, the transducer is mounted in a high-precision gauze made of stainless steel, as is the case in the HD 800 reference headphones. A new patent-pending feature is the special shape of the gauze, which continues the curved lines of the diaphragm.
True classics in the Sennheiser high-end world are the HD 600 and HD 650. Both models are open, dynamic, hi-fi stereo headphones with highly optimised transducer systems. Extremely light aluminium drive coils ensure excellent transient response, while neodymium ferrous magnet systems guarantee maximum efficiency. The HD 600 generates an exceptionally lifelike, spatial and accurate sound image with a frequency response of 12 to 39,000 Hz. With its deeper bass and pleasantly natural trebles, the HD 650 emphasis is on even greater expressiveness and emotion than the HD 600. The frequency response is 10 to 39,500 Hz.
With its new RS 220 wireless headphones, Sennheiser is opening up a new class of wireless listening. The digital wireless headphones guarantee a high-end sound image while offering total freedom of movement. The RS 220 transmits the audio signals without compression via a 2.4 GHz connection. To prevent interference during transmission, the transmitter uses what is known as frequency spreading. For a perfect sound experience, the RS 220 can be connected via various inputs: an analogue, a coaxial digital and an optical digital input. Discerning users can therefore decide themselves on the type and quality of the audio signal being fed in.
The IE 800 headphones from Sennheiser are the ones with the most innovations per square millimetre. The sound of the IE 800 can easily compete with that of the Sennheiser high-end portfolio. These dynamic in-ear phones offer fascinatingly brilliant trebles, precise bass response and a detailed, lifelike sound image with a frequency response of 5 to 46,500 Hz. The centrepiece is the specially developed Extra Wide Band (XWB) driver. With a diameter of just 7 mm, it is the smallest wide-band sound transducer currently available in dynamic headphones.
This year Sennheiser is launching its first amplifier for dynamic headphones, thus making the Sennheiser sound experience perfect. The outstanding features of the HDVD 800 include its balanced sound image, maximum precision and impressive spatiality. The digital HDVD 800 has a fully symmetrical layout and sampling frequencies of up to 24bit/192kHz to ensure an unequalled listening experience. It also harmonises perfectly with Sennheiser's HD 800, HD 700, HD 650 and HD 600 high-end headphones. A glass panel embedded in the aluminium housing of the amplifier gives a clear view of the top-quality interior of the HDVD 800. Selected components and sophisticated circuitry promise the ultimate in listening pleasure. The headphone amplifier is a further milestone in the history of development of premium audio products at the renowned audio specialist Sennheiser.


text:
import PropTypes from 'prop-types';
import React from 'react';

import languageShape from './shapes/languageShape';

class Filter extends React.PureComponent {
    static propTypes = {
        controlKey: PropTypes.string.isRequired,
        disabled: PropTypes.bool.isRequired,
        filterPlaceholder: PropTypes.string.isRequired,
        filterValue: PropTypes.string.isRequired,
        id: PropTypes.string.isRequired,
        lang: languageShape.isRequired,
        onFilterChange: PropTypes.func.isRequired,
    };

    render() {
        const {
            controlKey,
            disabled,
            filterPlaceholder,
            filterValue,
            id,
            lang,
            onFilterChange,
        } = this.props;

        return (
            <div className="rdl-filter-container">
                <label className="rdl-control-label rdl-sr-only" htmlFor={`${id}-filter-${controlKey}`}>
                    {lang[`${controlKey}FilterHeader`]}
                </label>
                <input
                    className="rdl-filter"
                    data-control-key={controlKey}
                    disabled={disabled}
                    id={`${id}-filter-${controlKey}`}
                    placeholder={filterPlaceholder}
                    type="text"
                    value={filterValue}
                    onChange={onFilterChange}
                />
            </div>
        );
    }
}

export default Filter;


text:
// LambLexer
// by James 2017/04/06

#include "util.h"

/*
 * ELFHash, used for access varible in Symbol Table faster
 */
unsigned int ELFHash(std::string str) {
    unsigned int hash = 0;
    unsigned int x = 0;
    unsigned int i = 0;
    unsigned int len = str.length();

    for (i = 0; i < len; i++) {
        hash = (hash << 4) + (str[i]);
        if ((x = hash & 0xF0000000) != 0) {
            hash ^= (x >> 24);
        }
        hash &= ~x;
    }
    return hash;
}

/*
 * Help menu
 */
void help_menu() {
    std::cout << "This compiler has two modes: test mode and normal mode\n"
                 "In test mode, it just uses the test data inside\n"
                 "In normal mode, you can compile file outside\n"
                 "You can modify the macro in config.h to change the mode\n\n"
                 "Usage:\n"
                 "\t./compiler [filename]\n"
                 "\t./compiler -v see the version of this compiler\n"
                 "\t./compiler -h see the help menu of this compiler\n"
              << std::endl;
}

/*
 * Show version
 */
void show_version() {
    std::cout << "LambCompiler, Version: " << version << std::endl << std::endl;
    ;
}


Synchrotron Radiation And Structural Proteomics Pechkova Eugenia Riekel Christian is good choice for you that looking for nice reading experience. We hope you glad to visit our website. Please read our description and our privacy and policy page.
Finally I get this ebook, thanks for all these Synchrotron Radiation And Structural Proteomics Pechkova Eugenia Riekel Christian can get now!


text:
Category / 2022 Awards
2022 Awards, Coming of Age, Documentaries, Interviews, LIVE Q&A
CUSP: Q&A with directors Parker Hill + Isabel Bethencourt and more!
October 25, 2021 by Rebecca Martin 1 comment
Filmmakers Parker Hill and Isabel Bethencourt join their subjects (Autumn and Brittney), to talk about their coming of age documentary CUSP. Cinema Femme Managing Editor Rebecca Martin moderates the Q&A. We discuss several important topics including teenage life, rape culture, how it's "ok to not […]
2022 Awards, Documentaries, Interviews
Mariem Pérez Riera on her inspiring documentary about the first Latina EGOT winner, "Rita Moreno: Just a Girl Who Decided to Go for It"
June 22, 2021 by Dawn Borchardt 1 comment
"Rita Moreno: Just a Girl Who Decided to Go for It" is the latest documentary from filmmaker Mariem Pérez Riera ("Lovesickness", "Croatto", "la huella de un emigrante"). The film uses clips from Rita Moreno's acting career, combined with intimate interviews and stop-motion animation to tell […]
2022 Awards, Film Festivals, Interviews, LGBTQ+, Tribeca
Lauren Hadaway, master of cinematic sound, on her directorial debut "The Novice" starring Isabelle Fuhrman
"The Novice" is the feature directorial debut from accomplished sound artist Lauren Hadaway. The film follows college student Alex (Isabelle Fuhrman), who finds herself obsessing over whatever she puts her mind toward. Her new focus becomes being a collegiate rower and wanting to move up […]
2022 Awards, Indie Films, Interviews
"Holler" director Nicole Riegel and star Jessica Barden talk about the beauty and the grit of Middle America
June 11, 2021 by Rebecca Martin 2 comments
I was fortunate to speak with "Holler" director Nicole Riegel and star Jessica Barden ("The End of the F***ing World", "Pink Skies Ahead") about their film that comes to select theaters and On Demand today. I talked to them separately, so the format of the […]
2022 Awards, Film Festivals, indigenous filmmaker, Indigenous in Film, Interviews, producer, Sexual Assault Awareness, Tribeca
Co-producers Mollye Asher ("Nomadland") and Kimberly Parker on their Tribeca thriller "Catch the Fair One"
"Catch the Fair One" premieres at Tribeca on June 13th, and Cinema Femme was fortunate to speak with co-producers Kimberly Parker ("The Last Black Man in San Francisco") and Academy Award-Winner Mollye Asher ("Nomadland") about their film. "Catch the Fair One" is about former pro […]
Marilyn Agrelo on Her Must-See Documentary "Street Gang: How We Got to Sesame Street"
May 4, 2021 by Matt Fagerholm 1 comment
Sixteen years after her debut feature, 2005's crowd-pleasing documentary, "Mad Hot Ballroom," showed us how the art of dance transformed the lives of elementary schoolers throughout New York City, filmmaker Marilyn Agrelo has returned with the definitive cinematic portrait of the most beloved children's television […]
2022 Awards, April Showcase, Film Festivals, Now Playing
"Shiva Baby" Q&A with director Emma Seligman
April 12, 2021 by Cinema Femme 0 comments
Last week we had an exclusive screening of "Shiva Baby" followed by a 30-minute Q&A with director Emma Seligman. The Q&A will be moderated by Cinema Femme contributor Dawn Borchardt, podcast host of Faux Reel. Emma Seligman is a Canadian filmmaker based in New York. […]
2022 Awards, Awards, Interviews, Oscars
Writer/director Elvira Lind on Her Oscar-Nominated Short Film, "The Letter Room"
April 6, 2021 by Matt Fagerholm 0 comments
One of the joys of covering the movie beat is seeing a great artist whose work you believe in receive the recognition they deserve. When the 2021 Oscar nominations were announced on March 15th, I jumped out of my chair and cheered when Elvira Lind's […]


Acadia Pharmaceutical has struck a collaboration deal with Tokyo-based firm Meiji Seika Kaisha to develop treatments for schizophrenia and separate neurological disorders for the Japanese market and other parts of Asia, according to a press release. San Diego-based Acadia (NASDAQ:ACAD), which has discovered a product candidate related to the deal, says it could receive up to $25 million in upfront and milestone payments from Meiji Seika Kaisha as well as royalties on potential sales. See Luke's San Diego Biotech Survival Index, which shows that Acadia finished 2008 with $60.1 million in the bank.


text:
Q: WPF TextAlignment doesn't work inside a TextBlock in a Button I am trying to create a button in WPF similar as the one in Amazon's checkout cart as shown below:

My attempted button is generated dynamically, but it's equivalent to the following XAML
<Button Name="UpDownBtn">
    <Grid >
      <Grid.ColumnDefinitions>
        <ColumnDefinition Width="3*"/>
        <ColumnDefinition Width="*"/>
      </Grid.ColumnDefinitions>
      <Grid.RowDefinitions>
        <RowDefinition Height="*" />
      </Grid.RowDefinitions>

      <Viewbox Stretch="Uniform" Grid.Row="0" Grid.Column="0">
        <TextBlock x:Name="numCopiesTxt" Padding="3" FontSize="20">123</TextBlock>
      </Viewbox>
      <Viewbox Grid.Row="0" Grid.Column="1">
        <TextBlock TextAlignment="Right" FontFamily="Marlett" FontSize="20" Text="v"/>
      </Viewbox>
  </Grid>
</Button>

where v in Marlett Font represents the upAndDown arrow.
However, the TextAlignment="Right" doesn't seem to make it right (no pun intended) whether Viewbox is used or not. Why is it this? Shown below
With Viewbox

without Viewbox

(Updated: PLEASE IGNORE Question #2. This was because the height of the Grid was set to Auto, not *. The XAML above does not cause this problem.)
*2. When a Button (with Viewbox) is added to a Grid, the Button is so big that only part of the content is shown inside its designated grid cell; the rest stops at the grid line. Note that the Viewbox does make the TextBlocks fit to the grid cells inside the Button, but the Button doesn't fit to its designated grid cell. This only happens with the use of Viewbox.*
thanks!

redpajama_set_name: RedPajamaStackExchange


text:
Q: A question about eigenvalues of $5\times5$ matrix $A\in M_5({R})$
and $A^2-4A-I=0$ If $a_1,a_2,a_3,a_4,a_5$ are eigenvalues of $A$ what is value of $$\left(a_1-\dfrac{1}{a_1}\right)+\left(a_2-\dfrac{1}{a_2}\right)+\cdots+\left(a_5-\dfrac{1}{a_5}\right)$$
I tried to work with $A^2-4A-I=0$ but get nothing because $A_{5\times5}$ characteristic polynomial is something like $A^5+k_1A^4+...$
Any hint will be appreciated .

A: If $\lambda$ is an eigenvalue of $A$ then:
$$\lambda^2-4\lambda-1=0\to \lambda=2\pm\sqrt{5}\to \frac{1}{\lambda}=-2\pm\sqrt{5}$$
which means that
$$\lambda-\frac{1}{\lambda}=2\pm\sqrt{5}-(-2\pm\sqrt{5})=4$$
so,
$$\left(a_1-\dfrac{1}{a_1}\right)+\left(a_2-\dfrac{1}{a_2}\right)+\cdots+\left(a_5-\dfrac{1}{a_5}\right)=4+4+4+4+4=5\cdot4 =20$$

A: EDITED:
If $a_i$ are the eigenvalues of $A$, $a_i - 1/a_i$ are the eigenvalues of $A - A^{-1}$.  Since $A^2 - 4 A - I = 0$, $A^{-1} = A^{-1}(A^2 - 4 A) = A - 4 I$
and $A - A^{-1} = 4 I$.
Thus the result is $\text{trace}(4I) = 20$.


St. Mary's Medical Center will present a Masala Bhangra® Workout Master Class on Friday, Feb. 25 from 7 to 9 p.m. at Veterans Memorial Field House in Huntington. The class will be taught by Sarina Jain, creator and founder of Masala Bhangra Workout. She has been featured on Fit TV's "All-Star Workouts", the Today Show and CBS Early Show.
The goal of the Masala Bhangra® Workout is to offer a fun way for participants to become more physically active and stay fit, while introducing them to the music and dance of the Indian culture. The workout introduces high-energy Bhangra and Bollywood dance movements in an easy-to-follow fitness format.
Doors will open at 5:30 p.m. on Feb. 25. The event will include free health screenings provided by St. Mary's Medical Center, complimentary massages, guided yoga stretching, prize drawings, complimentary water, samples of Indian cuisine and Henna painting.
Tickets are $25 in advance ($30 at the door) and may be purchased at the Veterans Memorial Field House box office from 10 a.m.-Noon and 1-4 p.m. Monday through Friday. To charge by phone, call (304) 528-5183, ext. 300.
The following groups will receive a 20-percent discount on tickets: Students under the age of 18, college students (must present ID) and seniors over the age of 60.
For more information, call Toni Karle at (304) 208-4525, e-mail tksaints@zoominternet.net or visit www.masaladance.com <http://www.masaladance.com> .
Russell Middle School, Russell, Ky.


text:
Q: Dependent or Independent clause? Is "The way people write has changed" an independent clause or dependent clause? In any case please explain in detail. What is the subject, verb and object if there is or if there isn't. Thanks in advance.

A: First of all I would like to say that clause is a verb along with its subject and their modifiers. That is, any clause has only one subject and only one associated with this subject verb (you should understand that a verb can be compound, that is, it can be composed from some verbs by a formula:

auxillary verb(s) + main verb

but still there is only one subject and only one associated with it compound verb in any clause).
Now let's take a look at your example:

The way people write has changed.

You should understand that this sentence is slightly shortened. The "full version" of this sentence would be:

The way that people write has changed.

So, I will talk about this second version, because it is more understandable and everything that I will tell about it is absolutely true for your shortened version too. It is neither independent nor dependent clause. This fragment cannot be called clause just because it has two pairs of subjects and their corresponding verbs, these pairs are: {the way - subject, has changed - compound verb} and {people - subject, write - verb}.
It is rather an example of a complex sentence.
Complex sentence consists from one or more independent clauses and zero or more dependent clauses. In your example the complex sentence has one independent clause ("The way has changed") and one dependent clause ("people write"). To be more precise, this dependent clause is called relative clause, and there is a rule that describes why the word "that", which connects the dependent clause to the independent clause, can be omitted from this relative clause, but I won't write it here since it isn't directly connected to the question.

A: This clause is independent.
"The way people write" is the subject.
"has changed" is the verb.
To highlight the subject and verb, we can reduce this sentence to "way" and "changed":
The way changed.

Now this sentence does sound weird. Normally, there would be something after it in normal usage. Here's some examples that expand on the sentence and make it sound better:
The way people write has changed since the 1900s.
The way people write has changed after the invention of the computer.
The way people write has changed in recent times.

A: "Way" is the subject.  "(That) people write" is a dependent clause.  "Has changed" is the verb.  There is no object.


Quality, new homes in Molalla. Hard to find single level with a 3 car garage. Open floor plan with vaulted ceilings, plant shelves, cozy fireplace in family room and plenty of light. Large kitchen with laminate flooring, granite or quartz counters, ss appliances, island and a pantry. Front yard is landscaped with sprinklers. Fully fenced with a gate. Very energy efficient and solar ready. Still time to choose colors and make this home your own!
Showing by appt, text or call agent (Jacob - 971-533-4304). 2 hour notice. RMLS Lockbox to the right of front door near water faucet.No sign in yard.


text:
Vodafone pledges 24mn GB of data to the National Databank to connect 200,000 people across the UK
12 January, 2022 at 11:07 AM
Posted by: Anasia D'mello
Vodafone has joined Good Things Foundation's National Databank to provide free data for the charity's network of community groups across the UK and the people they support. It has pledged 24 million gigabytes of data, enough to provide connectivity to 200,000 people for six months.
Vodafone has worked with Good Things Foundation on digital exclusion projects since 2020 and this initiative is part of its commitment to connect one million people living in digital poverty by the end of 2022.
Good Things Foundation's National Databank gives community groups across the UK access to free data either through SIM cards or vouchers that they can share with those who are digitally excluded and need support. Following a successful three-month pilot that saw over 400 data vouchers distributed to community partners, the programme is now live in 34 centres across the UK with plans to extend it to hundreds more during 2022. Community groups who are already a part of the Good Things Foundation's network can apply for the connectivity later this month using the Online Centres Network section of the charity's website.
The pandemic has highlighted the scale of digital exclusion, with the greatest impact felt by the most vulnerable in society. 1.5m households in the UK, or approximately 6% of the population, live without internet access. A recent report from Vodafone highlighted how the digital divide most impacts those who are already at a disadvantage, either financially or socially.
Ahmed Essam, Vodafone's UK CEO, comments, "We've put tackling digital exclusion at the heart of our business with our pledge to connect one million people by the end of 2022. Joining Good Things Foundation's National Databank is an important milestone in achieving this target and another great way to provide connectivity to grassroots organisations who can have immediate impact in the communities in which they work.
"Together, we can tackle digital exclusion."
Helen Milner OBE, group chief executive, Good Things Foundation, says, "We're delighted to welcome Vodafone to the National Databank. The pandemic has exposed and exacerbated the digital divide, highlighting a huge pressing social issue and leaving people with multiple social challenges left behind. It is not OK to leave millions of people locked out of the digital world.
"Vodafone has done some fantastic work over the last year to help these digitally excluded people get online; and with this generous pledge we can now reach many more organisations and the communities they support.
"We're excited to see the National Databank grow in scale in 2022, increasing from 34 participating centres to hundreds, possibly thousands, more.
"Our aim is to drive collaboration between public, private and community sector organisations and deliver sustainable solutions that will ultimately end data poverty in the UK by 2024. This is a real possibility. Together, we can all come together to break down the barriers causing data poverty in the UK once and for all."
Vodafone also announced that its charities.connected initiative has received applications for more than 112,000 SIM cards from 1,180 organisations, once again highlighting the extent of digital exclusion across the UK.
Charities.connected launched in August and gives every registered charity the opportunity to apply for free connectivity to improve its digital capability or help the individuals and families it supports to get online. Organisations who have applied to date support a huge range of social causes, from domestic abuse and homelessness to healthcare and learning disabilities. Charities can continue to apply for the free connectivity.
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category: Connectivity, Digital transformation, Network optimisation, News
Tags: Ahmed Essam, charities.connected, Good Things Foundation, Helen Milner OBE, National Databank, Vodafone


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#American
Frankie Valli Biography, Age, Height, Weight, Family, Wiki & More
Updated On : May 4, 2017
3 May,1934 (Thursday)
Newark, Essex, New Jersey
Frankie Valli Photos
Frankie Valli Popularity on Social Media
Frankie Valli was born on 03-05-1934 in Newark, Essex in the state of New Jersey, United States. He is an American Singer, Musician & Television Actor.
Francesco Stephen Castelluccio, Frankie Valley
Musician Television Actor
Frankie Valli Wiki Link
Frankie Valli Complete Bio & Career
Frankie Valli is a lot more popular for his most compelling sense of being himself and also is a very spiritual person. He is always said to be a seeker and searcher of veracity. According to his loved one's his life is basically devoted to finding the unknown and mysteries of life or it can be said as monumental. Frankie's birth name is Francesco Stephen Castelluccio but later he got popular among his fans with name Frankie Valli. His basic genres were Rock and pop, and so he gave some really good hits in his time. He started his career in 1953 and still giving his best shots. Frankie is not only a singer but also a musician and an artist. He has always been a vocalist in instruments. As he was a member of The four seasons, his most popular hits include Sherry in 1962, big girls don't cry in 1962, walk like a man in 1963, the rag doll in 1964 and most of all December 1963 (oh what a night) in 1975. His another creation "Can't Take My Eyes Off You" which also reached number second hit in 1967.
In 1966 Valli recorded a solo song became U.K's surprise hit and also become part of North Soul and that made a place of number 11 on the British Pop chart in the year of 1970. Later in 1999, he became the original member of season's Rock and Roll Hall of Fame in 1990 and also in Vocal group hall of Fame (1999).
Life's Important Dates Of Frankie Valli
Frankie Valli Age, Birthday Facts and Birthday Countdown
87 years, 8months, 16 days old age Frankie Valli will turn 88 on 03 May, 2022. Only 3 months, 13 days, 19 hours,54 minutes has left for his next birthday.
Frankie Valli Born On Thursday
Frankie Valli has celebrated the total number of 88 birthdays till date. See the analysis by days count and bar graph.
Salman Khan Biography, Age, Height, Weight, Girlfriends, Family, Facts, Wiki & More


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Q: Creating a View that is a UNION of a few tables A client of mine would like to create a view that is a UNION of a few tables.
From the client:

These tables are populated by streaming data sources and are pretty
  sizable already. Because of how the set operations get applied, any
  queries of this view are causing performance issues, since any
  filters/predicates get applied after the UNIONs. I know that you
  can't materialize a view with UNION operations, so I was wondering
  if Snowflake recommends any other solution short of building a
  separate table which unions the constituent tables.

A: I created a view as follows and pruning worked as expected, prior to the UNION.
I wonder if their problem is due to UNION (has the performance penalty of de-duping) versus UNION ALL
create or replace view test_unions
(
  mycol1,
  mycol2,
  mycol3
) as (
  (select mycol1,mycol2,mycol3 from tableA)
  union all (select  mycol1,mycol2,mycol3 from tableB)
  union all (select  mycol1,mycol2,mycol3 from tableC)
)
;


text:
Sea level changes
Future sea-level commitment
For each degree of warming above pre-industrial, sea- level will rise by about 2.3m within a period of 2000 years (Levermann et al. (2013).
For projections of the 21st century we propose a linear response function R(t) = Γ⋅tα with positive α for Antarctic ice discharge from surface warming & negative α for other contributions (Winkelmann & Levermann, 2012).
More on Sea Level Page.
A. Levermann, R. Winkelmann, S. Nowicki, J. Fastook, K. Frieler, R. Greve, H.H. Hellmer, M.A. Martin, M. Meinshausen, M. Mengel, A.J. Payne, D. Pollard, T. Sato, R. Timmermann, W.L. Wang, R.A. Bindschadler
Projecting Antarctic ice discharge using response functions from SeaRISE ice-sheet models
Earth System Dynamics, 5 (2014), 271-293, DOI:10.5194/esd-5-271-2014.
entry - entry
Data download: All data (8Mb) - Ascii time series (80Kb) - Readme
M. Mengel & A. Levermann
Ice plug prevents irreversible discharge from East Antarctica
Nature Climate Change 4 (2014), doi 10.1038/nclimate2226.
» Highlighted in Nature 509 (2014), 136, doi:10.1038/509136c.
B. Marzeion & A. Levermann
Loss of Cultural World Heritage and currently inhabited places to sea-level rise
Environmental Research Letters 9 (2014), 034001. doi:10.1088/1748-9326/9/3/034001.
» Supplementary information
J.A. Church, P.U. Clark, A. Cazenave, J.M. Gregory, S. Jevrejeva, A. Levermann, M.A. Merrifield, G.A. Milne, R.S. Nerem, P.D. Nunn, A.J. Payne, W.T. Pfeffer, D. Stammer, A.S. Unnikrishnan
Letter to the Editor: Sea-Level Rise by 2100
Science 342 (2013), 1445, doi: 10.1126/science.342.6165.1445-a.
A. Levermann, P.U. Clark, B. Marzeion, G. Milne, D. Pollard, V. Radic, A. Robinson
The multi-millennial sea-level commitment of global warming
Proceedings of the National Academy of Sciences (2013), published in early edition online.
» Commentary by B. Strauss in PNAS
A. Levermann, R. Winkelmann, S. Nowicki, J. Fastook, K. Frieler, R. Greve, H.H. Hellmer, M.A. Martin, M. Mengel, A.J. Payne, D. Pollard, T. Sato, R. Timmermann, W.L. Wang, R.A. Bindschadler
Earth System Dynamics Discussions 4 (2013), 1117-1168, doi:10.5194/esdd-4-1117-2013.
R. Winkelmann, A. Levermann, K. Frieler, M.A. Martin
Increased future ice discharge from Antarctica owing to higher snowfall
Nature 492 (2012), 239-242.
S. Nowicki, R.A. Bindschadler, A. Abe-Ouchi, A. Aschwanden, E. Bueler, H. Choi, J. Fastook, G. Granzow, R. Greve, G. Gutowski, U. Herzfeld, C. Jackson, J. Johnson,C. Khroulev, E. Larour, A. Levermann, W.H. Lipscomb, M.A. Martin, M. Morlighem, B. Parizek, D. Pollard, S.F. Price, D. Ren, E. Rignot, F. Saito, T. Sato, H. Seddik, H. Seroussi, K. Takahashi, R. Walker, W.L. Wang
Insights into spatial sensitivities of ice mass response to environmental change from the SeaRISE ice sheet modeling project II: Greenland
Journal of Geophysical Research: Earth Surface 118 (2013), 1025-1044, doi:10.1002/jgrf.20076.
S. Nowicki, R.A. Bindschadler, A. Abe-Ouchi, A. Aschwanden, E. Bueler, H. Choi, J. Fastook, G. Granzow, R. Greve, G. Gutowski, U. Herzfeld, C. Jackson, J. Johnson, A. Levermann, W.H. Lipscomb, M.A. Martin, M. Morlighem, B. Parizek, D. Pollard, S.F. Price, D. Ren, E. Rignot, F. Saito, T. Sato, H. Seddik, H. Seroussi, K. Takahashi, R. Walker, W.L. Wang
Insights into spatial sensitivities of ice mass response to environmental change from the SeaRISE ice sheet modeling project I: Antarctica
R.A. Bindschadler, S. Nowicki, A. Abe-Ouchi, A. Aschwanden, E. Bueler, H. Choi, J. Fastook, G. Granzow, R. Greve, G. Gutowski, C. Jackson, J. Johnson, A. Levermann, W.H. Lipscomb, M.A. Martin, M. Morlighem, B. Parizek, D. Pollard, S.F. Price, D. Ren, F. Saito, H. Seroussi, K. Takahashi, R. Walker, W.L. Wang, U. Herzfeld, H. Seddik, T. Sato
Ice-sheet model sensitivities to environmental forcing and their use in projecting future sea-level (The SeaRISE Project)
Journal of Glaciology 59 (2013), 195-224. DOI:10.3189/2013JoG12J125
R. Winkelmann and A. Levermann
Linear response functions to project contributions to future sea level
Climate Dynamics 38 (2012), DOI: 10.1007/s00382-012-1471-4, online first.
Uncertainty in future solid ice discharge from Antarctica
The Cryosphere Discussions 6 (2012), 673-714; DOI 10.5194/tcd-6-673-2012.
Projections of Antarctica's sea-level contribution (Levermann et al. (2014).
Ice plug in East Antarctica (Mengel & Levermann (2014).
Sea-level contribution from (a) ocean warming, (b) mountain glaciers, (c) Greenland, (d) Antarctica and (e) the total sea-level commitment after 2000 years (Levermann et al. (2013).


text:
Alfredo da Graça Couto (São Luiz, Maranhão,  – Rio de Janeiro, ) foi um médico brasileiro.

Foi eleito membro da Academia Nacional de Medicina em 1898, ocupando a Cadeira 52, que tem Paulo de Figueiredo Parreiras Horta como patrono.

Membros da Academia Nacional de Medicina
Naturais de São Luís (Maranhão)