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| In [[mathematics]], '''time-scale calculus''' is a unification of the theory of [[difference equation]]s with that of [[differential equation]]s, unifying integral and differential [[calculus]] with the [[calculus of finite differences]], offering a formalism for studying hybrid discrete–continuous [[dynamical system]]s. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function which acts on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function acting on the integers then it is equivalent to the forward difference operator.
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| ==History==
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| Time-scale calculus was introduced in 1988 by the German mathematician [[Stefan Hilger]].<ref name=hilger>{{cite journal| last = Hilger | first = Stefan | authorlink = Stefan Hilger |title = Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten |publisher = Universität Würzburg | year = 1998}}</ref> However, similar ideas have been used before and go back at least to the introduction of the [[Riemann–Stieltjes integral]] which unifies sums and integrals.
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| ==Dynamic equations==
| | Everyone now a days desires to be ecologically friendly. Alongside the many advantages, this makes validating the statement a lot tough. The meaning of what environmentally aware actually entails varies and with this being said... which country is determined to be the most eco-friendly?<br><br>As the tops in working their portion to defend the earth, Costa Rica moves on to work this by catering pertaining to the group of eco-aware tourists.<br><br>The whole eco paradigm has given birth to a new age gen that are illuminated and tuned in to their footprint on this Globe.<br><br>A current study by the popular huge travel site, Tripadvisor, emphasized the developing trend among travelers who are engaged with their carbon bearing also of the bearing the different elements of their vacation packages might have. This does include paying surplus for a hotel who is aware of their bearing on the environment and who takes the essential steps to minimize any environmental effects.<br><br>In this report, Tripadvisor discovered that the #1 most environmentally aware tourism spot on the planet is...Costa Rica!<br><br>Being previously described [http://www.link286.com/site/vacationstocostarica.com/RK=0/RS=R403cXEmGijwMFB52Y03hpax9FE- VacationsToCostaRica] as most ecologically friendly country in the world has in a positive way impacted other sectors outside of the Costa Rica vacations sector.<br><br>An additional sector that reaped benefits from this new planetary eco-awareness is the real estate segment.<br><br>Before the global crisis struck the sector with the most hope was the Costa Rica real-estate sector -- MSNBC even referred to it as the hottest on the earth. Though the ride on top was interesting, eventually the world would reach Costa Rica.<br><br>And man did it hit like a tsunami.<br><br>That was then and this is now...<br><br>With the focus on getting "green" this day and age, informed investors are beginning to realize the possibilities the Costa Rica real estate segment holds.<br><br>A huge hook for many organizations is the ability to market their companies as "eco friendly" while getting based along the rainforest rimmed south Pacific shoreline of [http://CentralAmerica.net/ Central America].<br><br>The very best aspect for many is the fact that this is all viable in Costa Rica with it's unblemished rain forests, abounding fauna and continual initiatives at preserving our holistic habitats -- is it any surprise why Costa Rica is seen as the 1 most eco informed spot on the Globe? |
| Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their [[continuous function|continuous]] counterparts.<ref name=bp>{{cite book | author=Martin Bohner & Allan Peterson | title=Dynamic Equations on Time Scales | publisher=Birkhäuser | year=2001 | isbn=978-0-8176-4225-9 | url = http://www.springer.com/west/home/birkhauser?SGWID=4-40290-22-2117582-0 }}</ref> The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown [[function (mathematics)|function]] is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the [[Set (mathematics)|set]] of [[real number]]s or set of [[integer]]s but to more general time scales such as a [[Cantor set]].
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| The three most popular examples of [[calculus]] on time scales are [[differential calculus]], [[finite differences|difference calculus]], and [[quantum calculus]]. Dynamic equations on a time scale have a potential for applications, such as in [[population dynamics]]. For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non–overlapping population. | |
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| ==Formal definitions==
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| A '''time scale''' (or '''measure chain''') is a [[closed subset]] of the [[real line]] <math>\mathbb{R}</math>. The common notation for a general time scale is <math>\mathbb{T}</math>.
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| The two most commonly encountered examples of time scales are the real numbers <math>\mathbb{R}</math> and the [[Discrete time|discrete]] time scale <math>h\mathbb{Z}</math>.
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| A single point in a time scale is defined as:
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| :<math>t:t\in\mathbb{T}</math>
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| === Operations on time scales===
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| [[File:Timescales jump operators.png|thumb|upright=2.0|The forward jump, backward jump, and graininess operators on a discrete time scale]]
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| The ''forward jump'' and ''backward jump'' operators represent the closest point in the time scale on the right and left of a given point <math>t</math>, respectively. Formally:
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| :<math>\sigma(t) = \inf\{s \in \mathbb{T} : s>t\}</math> (forward shift operator / forward jump operator)
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| :<math>\rho(t) = \sup\{s \in \mathbb{T} : s<t\}</math> (backward shift operator / backward jump operator)
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| The ''graininess'' <math>\mu</math> is the distance from a point to the closest point on the right and is given by: | |
| :<math>\mu(t) = \sigma(t) -t.</math>
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| For a right-dense <math>t</math>, <math>\sigma(t)=t</math> and <math>\mu(t)=0</math>.<br />
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| For a left-dense <math>t</math>, <math>\rho(t)=t.</math>
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| ===Classification of points===
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| [[File:Timescales point classifications.png|thumb|upright=2.0|Several points on a time scale with different classifications]]
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| For any <math>t\in\mathbb{T}</math>, <math>t</math> is:
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| * ''left dense'' if <math>\rho(t) =t</math>
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| * ''right dense'' if <math>\sigma(t) =t</math>
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| * ''left scattered'' if <math>\rho(t)< t</math>
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| * ''right scattered'' if <math>\sigma(t) > t</math>
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| * ''dense'' if both left dense and right dense
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| * ''isolated'' if both left scattered and right scattered
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| <br />
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| As illustrated by the figure at right:
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| * Point <math>t_1</math> is ''dense''
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| * Point <math>t_2</math> is ''left dense'' and ''right scattered''
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| * Point <math>t_3</math> is ''isolated''
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| * Point <math>t_4</math> is ''left scattered'' and ''right dense''
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| ===Continuity===
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| [[Continuous function|Continuity]] on a time scale is redefined as equivalent to density. A time scale is said to be ''right-continuous at point <math>t</math>'' if it is right dense at point <math>t</math>. Similarly, a time scale is said to be ''left-continuous at point <math>t</math>'' if it is left dense at point <math>t</math>.
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| ==Derivative==
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| Take a function:
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| :<math>f: \mathbb{T} \rightarrow \mathbb{R}</math>,
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| (where R could be any normed [[Banach space]], but set it to be the real line for simplicity).
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| Definition: The ''delta derivative'' (also Hilger derivative) <math>f^{\Delta}(t)</math> exists if and only if:
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| For every <math>\epsilon > 0</math> there exists a neighborhood <math>U</math> of <math>t</math> such that:
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| :<math>|f(\sigma(t))-f(s)- f^{\Delta}(t)(\sigma(t)-s)|\le \varepsilon|\sigma(t)-s|</math>
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| for all <math>s</math> in <math>U</math>.
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| Take <math>\mathbb{T} =\mathbb{R}.</math> Then <math>\sigma(t) = t</math>, <math>\mu(t) = 0</math>, <math>f^{\Delta} = f'</math>; is the derivative used in standard [[calculus]]. If <math>\mathbb{T} = \mathbb{Z}</math> (the [[integer]]s), <math>\sigma(t) = t + 1</math>, <math>\mu(t)=1</math>, <math>f^{\Delta} = \Delta f</math> is the [[forward difference operator]] used in difference equations.
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| ==Integration==
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| The ''delta integral'' is defined as the antiderivative with respect to the delta derivative. If <math>F(t)</math> has a continuous derivative <math>f(t)=F^\Delta(t)</math> one sets
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| :<math>\int_r^s f(t) \Delta(t) = F(s) - F(r).</math>
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| ==Laplace transform and z-transform==
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| A [[Laplace transform]] can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal<ref name=bp/> to a modified [[Z-transform]]:
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| <math>\mathcal{Z}'\{x[z]\}=\frac{\mathcal{Z}\{x[z+1]\}}{z+1}</math>
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| ==Partial differentiation==
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| [[Partial differential equation]]s and [[partial difference equation]]s are unified as partial dynamic equations on time scales.<ref>[http://dx.doi.org/10.1016/S0377-0427(01)00434-4 Partial differential equations on time scales], Calvin D. Ahlbrandt, Christina Morian</ref><ref>[http://marksmannet.com/TimeScales/Papers/partial.pdf Partial dynamic equations on time scales], B Jackson – Journal of Computational and Applied Mathematics, 2006</ref><ref>[http://web.mst.edu/~bohner/papers/pdots.pdf Partial differentiation on time scales], M Bohner, GS Guseinov, Dynamic Systems and Applications 13 (2004) 351–379</ref>
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| ==Multiple integration==
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| [[Multiple integration]] on time scales is treated in Bohner (2005).<ref>{{cite journal | id = {{citeseerx|10.1.1.79.8824}} | title = Multiple integration on time scales | first = M | last1 = Bohner | first2 = GS | last2 = Guseino | journal = Dynamic Systems and Applications | year = 2005 }}</ref>
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| ==Stochastic dynamic equations on time scales==
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| [[Stochastic differential equation]]s and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.<ref>[http://scholarsmine.mst.edu/thesis/pdf/Sanyal_09007dcc80519030.pdf STOCHASTIC DYNAMIC EQUATIONS], SUMAN SANYAL, 2008</ref>
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| ==Measure theory on time scales==
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| Associated with every time scale is a natural [[Measure (mathematics)|measure]]<ref>{{cite journal | doi = 10.1016/S0022-247X(03)00361-5 | title = Integration on time scales | first = GS | last = Guseinov | journal = J. Math. Anal. Appl. | volume = 285 | year = 2003 | pages = 107–127 }}</ref><ref>{{cite web | url = http://library.iyte.edu.tr/tezler/master/matematik/T000568.pdf | title = Measure theory on time scales | first = A | last = Deniz | year = 2007 }}</ref> defined via
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| :<math>\mu^\Delta(A) = \lambda(\rho^{-1}(A)),</math>
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| where <math>\lambda</math> denotes [[Lebesgue measure]] and <math>\rho</math> is the backward shift operator defined on <math>\mathbb{R}</math>. The delta integral
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| turns out to be the usual [[Lebesgue–Stieltjes integral]] with respect to this measure
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| :<math>\int_r^s f(t) \Delta t = \int_{[r,s)} f(t) d\mu^\Delta(t)</math>
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| and the delta derivative turns out to be the [[Radon–Nikodym derivative]] with respect to this measure<ref>{{cite journal | arxiv = 1102.2511 | title = On the connection between the Hilger and Radon–Nikodym derivatives | first1 = J | last1 =Eckhardt | authorlink2 = Gerald Teschl | first2 = G | last2 = Teschl | journal = J. Math. Anal. Appl. | volume = 385 | year = 2012 | pages = 1184–1189 }}</ref>
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| :<math>f^\Delta(t) = \frac{df}{d\mu^\Delta}(t).</math>
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| ==Distributions on time scales==
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| The [[Dirac delta]] and [[Kronecker delta]] are unified on time scales as the ''Hilger delta'':<ref>[http://www.marksmannet.com/RobertMarks/REPRINTS/2007_TheLaplaceTransformOnTimeScales.pdf The Laplace transform on time scales revisited], John M. Davis, Ian A. Gravagne , Billy J. Jackson ,
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| Robert J. Marks II , Alice A. Ramos, J. Math. Anal. Appl. 332 (2007) 1291–1307</ref><ref>[http://marksmannet.com/RobertMarks/REPRINTS/short/BLaplaceOct2009.pdf Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series], John M. Davis, Ian A. Gravagne and Robert J. Marks II</ref>
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| : <math>\delta_{a}^{\mathbb{H}}(t) = \begin{cases} \frac{1}{\mu(a)}, & t = a \\ 0, & t \neq a \end{cases}</math>
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| ==Integral equations on time scales==
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| [[Integral equation]]s and [[summation equation]]s are unified as integral equations on time scales.<ref>[http://web.maths.unsw.edu.au/~cct/tis-tomasia-IJDE-rev.pdf Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains], Tomasia Kulik and Christopher C. Tisdell, 2007</ref>
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| ==Fractional calculus on time scales==
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| [[Fractional calculus]] on time scales is treated in Bastos, Mozyrska, and Torres.<ref>{{cite paper | arxiv = 1012.1555 | title = Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform | first1 = Nuno R. O. | last1 = Bastos | first2 = Dorota | last2 = Mozyrska | first3 = Delfim F. M. | last3 = Torres }}</ref>
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| ==See also==
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| *[[Analysis on fractals]] for dynamic equations on a [[Cantor set]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *[http://web.mst.edu/~bohner/papers/deotsas.pdf Dynamic equations on time scales: a survey], Ravi Agarwal, Martin Bohner, Donal O’Regan, Allan Peterson, Journal of Computational and Applied Mathematics 141 (2002) 1–26
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| ==Further reading==
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| * [http://www.timescales.org The Baylor University Time Scales Group]
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| * [http://web.mst.edu/~bohner/tisc.html Dynamic Equations on Time Scales] Special issue of ''Journal of Computational and Applied Mathematics'' (2002)
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| * [http://www.hindawi.com/journals/ade/volume-2006/si.1.html Dynamic Equations And Applications] Special Issue of ''Advances in Difference Equations'' (2006)
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| * [http://www.e-ndst.kiev.ua/v9n1.htm Dynamic Equations on Time Scales: Qualitative Analysis and Applications] Special issue of ''Nonlinear Dynamics And Systems Theory'' (2009)
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| {{DEFAULTSORT:Time Scale Calculus}}
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| [[Category:Dynamical systems]]
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| [[Category:Calculus]]
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| [[Category:Recurrence relations]]
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Everyone now a days desires to be ecologically friendly. Alongside the many advantages, this makes validating the statement a lot tough. The meaning of what environmentally aware actually entails varies and with this being said... which country is determined to be the most eco-friendly?
As the tops in working their portion to defend the earth, Costa Rica moves on to work this by catering pertaining to the group of eco-aware tourists.
The whole eco paradigm has given birth to a new age gen that are illuminated and tuned in to their footprint on this Globe.
A current study by the popular huge travel site, Tripadvisor, emphasized the developing trend among travelers who are engaged with their carbon bearing also of the bearing the different elements of their vacation packages might have. This does include paying surplus for a hotel who is aware of their bearing on the environment and who takes the essential steps to minimize any environmental effects.
In this report, Tripadvisor discovered that the #1 most environmentally aware tourism spot on the planet is...Costa Rica!
Being previously described VacationsToCostaRica as most ecologically friendly country in the world has in a positive way impacted other sectors outside of the Costa Rica vacations sector.
An additional sector that reaped benefits from this new planetary eco-awareness is the real estate segment.
Before the global crisis struck the sector with the most hope was the Costa Rica real-estate sector -- MSNBC even referred to it as the hottest on the earth. Though the ride on top was interesting, eventually the world would reach Costa Rica.
And man did it hit like a tsunami.
That was then and this is now...
With the focus on getting "green" this day and age, informed investors are beginning to realize the possibilities the Costa Rica real estate segment holds.
A huge hook for many organizations is the ability to market their companies as "eco friendly" while getting based along the rainforest rimmed south Pacific shoreline of Central America.
The very best aspect for many is the fact that this is all viable in Costa Rica with it's unblemished rain forests, abounding fauna and continual initiatives at preserving our holistic habitats -- is it any surprise why Costa Rica is seen as the 1 most eco informed spot on the Globe?