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\title{Local Sustainability Model based on Sustainable Harmonic Movement Analysis}
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             \author[1]{Mario Alberto Gonzales  Torres}

             \affil[1]{  San Marcos National University}

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\date{\small \em Received: 8 June 2021 Accepted: 30 June 2021 Published: 15 July 2021}

\maketitle


\begin{abstract}
        


The proposed Local Sustainability Model (LSM) is framed in the Sustainable Development Goals (SDGs) and focused on Sustainable Cities and Communities. The research is based on the initial praxis developed in the construction of an LSM (Gonzales, 2019) for the Artisanal Gold Mining Activities in southern Peru (AGMA), and is based on the application of the Sustainable Harmonic Movement (SHM) analysis and its integration to methods and techniques such as Canvas, Dolphy, Dynamic Programming, Statistical Validation to determine the sustainability indicators and achieve the measure of Sustainable Development. Finding the LSM implies taking advantage of the natural resource in an efficient and effective way, for which it should be considered to determine the scope limited by the systematized environment (environment), considering in its evaluation the indicators of Environmental, Economic, Social and Governance Sustainability.

\end{abstract}


\keywords{sustainability, local sustainability model, sustainable development, sustainability harmonic movement.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
n the vision of directing the development of man (UN, 1987) established the scope of Sustainable Development \hyperref[b0]{1} Local Sustainable Development is based on the concept of endogenous local development that grants a predominant role to companies, organizations, local alternative to economic development, its foundation being the sustainability that has as a premise ¨satisfying the needs of the present without compromising the capacity of future generations¨, guaranteeing the balance between economic growth, the environment and social well-being. Likewise, the Sustainable Development Goals (UN, 2015) have been established, which prioritize and lead to the consideration of new actions to confront poverty, protect the planet and improve the lives and prospects of people in the world. Currently as an alternative to develop economic activities oriented to the SDGs. institutions and civil society itself, identifying the economic dimension that allows local entrepreneurs to improve competitiveness, the social dimension and the institutional dimension and the basic values of the development process. (Stöhr, 1985). The vision of local economic development in addition to economic development in general, are added the fields of analysis of natural capital, social capital, importance of institutional strengthening, promotion of the participation of local communities, human capital, by improving access to education, nutrition and health, the role of women, the innovation needs of each territorial productive system in context, competitiveness, economic globalization and the assessment of environmental sustainability (ECLAC, 2011). 
\section[{II.}]{II.} 
\section[{Sustainability Model}]{Sustainability Model}\par
The Local Sustainability Model (LSM) is conceptualized within strategic planning (Ander, 1995) with the purpose of highlighting the importance of current policies and interaction of social actors. This is how the LSM is characterized by being dynamic, highlighting in the methodology the contribution of the opinion of experts, who have a scale of values of what is convenient and not convenient in the objective of finding Economic, Environmental, Social and Governance Sustainability \hyperref[b1]{2} The conceptualization of the LSM is established through the relationship of variables between the definition of Local Sustainable Development (LSD) and necessary for develop the model. The beginning of the evaluation of the problem is established with the baseline which includes the regulations and specifies the normative planning. The Sustainability Model is articulated at the strategic level with what it can be and at the operational level with the desire to make a flexible scheme. Thus, in a first stage, the selection of indicators is determined from a cause/ effect model or an analytical approach (Glave et al.,  1995) and then they are subjected to the logic in the construction of the model, evaluating the indicators. In a second stage, the indices for each proposed model are dimensioned and interact.  
\section[{Sustainability Indicators}]{Sustainability Indicators}\par
The Local Institutional Sustainability = S L I From the relationship established in (i) for Local Sustainability and the Sustainability of the proposed local activity, we have:[S i L ] =µ [S ij AX ]. [S i L ] (i) [S i L ] =µ´ [I ij AX ]. [S i L ] [S i L ] =µ´´ [I ij AX ] / ?I ij ?. [S i L ]\par
Being: ?Iij? the largest scalar value for Iij, with i = j and The Normalized Sustainability Matrix is constructed from the division of the largest scalar value for ?Iij? where Iij, with i = j, thus characterizing that the trace will take the Optimal value of 4 which represents the ideal Indicator Sustainability for LSM.\par
IV. 
\section[{Materials and Methods}]{Materials and Methods}\par
The research was based on the praxis developed of the construction of an LSM for artisanal gold mining activities in southern Peru (Gonzales, 2019), and is based on the use of the Canvas and Dolphi Methods, for the determination of objectives and goals, expert opinion and Dynamic Programming techniques for logical sequence and statistical analysis withthe Friedman statistician with p value that allows to identify the sustainability indicators and indices and their dimension to the activity and availability of the resource to later build the Description Sheets of Sustainability Indicators associated with their dimensionality and then submit them to the analysis of the Sustainable Harmonic Movement (SHM) in order to evaluate the optimal MSL of Indicators.  
\section[{Table Nº1: Models and Index}]{Table Nº1: Models and Index} 
\section[{Analysis and Discussion of Results}]{Analysis and Discussion of Results}\par
Based on the results of the AGMA research supported by the operational dimensionality of the activity of 340 participants, the SHM analysis was carried out based on the AHM \hyperref[b2]{3}   \hyperref[tab_1]{Nº1 and Nº2}.\par
Using the Adapted Holzer Method (AHM) \hyperref[b3]{4} the natural frequencies of sustainability were determined under the condition initial unit S value, and iterated through the following relationship:     From the graphs, the natural frequencies of sustainability are determined and are presented in Table \hyperref[tab_2]{Nº3}. Likewise, the tabulation of Sustainability Frequencies complies with the Variation and Trend Tests of Natural Sustainability Frequencies. See Table \hyperref[tab_3]{Nº4} and Figure  {\ref Nº5}.   \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{N°1}\includegraphics[]{image-6.png}
\caption{\label{fig_4}Figure N°1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{N°2}\includegraphics[]{image-7.png}
\caption{\label{fig_5}Figure N°2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{N°3}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Figure N°3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{N°4}\includegraphics[]{image-9.png}
\caption{\label{fig_7}Figure N°4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{N°5}\includegraphics[]{image-10.png}
\caption{\label{fig_8}Figure N°5 :B}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.746969696969697\textwidth}P{0.020606060606060607\textwidth}P{0.020606060606060607\textwidth}P{0.050227272727272725\textwidth}P{0.0038636363636363634\textwidth}P{0.007727272727272727\textwidth}}
\multicolumn{6}{l}{LSD = SLD A1 + SLD A2 + SLD A3 + ??+SLD An}\\
\tabcellsep \tabcellsep \tabcellsep \multicolumn{3}{l}{LSD ? SLD AX}\\
\tabcellsep \tabcellsep \tabcellsep \multicolumn{3}{l}{SL= B(S AX )}\\
\multicolumn{6}{l}{Involving the advance of other economic}\\
\multicolumn{6}{l}{activities in the local geographical area, with which in}\\
\multicolumn{6}{l}{this context Local Sustainability is measured in function}\\
\multicolumn{6}{l}{to Activity Sustainnability by the following recursive}\\
equation:\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Where:\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\multicolumn{3}{l}{B: Between 0 and 1}\tabcellsep \tabcellsep \\
\multicolumn{6}{l}{B = 1 greater predominance of activity X}\\
Having,\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\multicolumn{4}{l}{S L = f (S L E , S L S , S L A, S L I )}\tabcellsep \\
S AX = f (S AX\tabcellsep E , S AX\tabcellsep S , S AX\tabcellsep A, S AX\tabcellsep \multicolumn{2}{l}{I )}\\
\multicolumn{6}{l}{In unambiguous correspondence there is:}\\
S AX = f (S AX\tabcellsep E , S AX\tabcellsep S , S AX\tabcellsep \multicolumn{2}{l}{A , S AX}\tabcellsep I )\\
\multicolumn{6}{l}{The following relationship is established in a matrix:}\\
\multicolumn{3}{l}{[S i L ] =µ [S ij AX ]. [S i L ]}\tabcellsep \tabcellsep \tabcellsep (i)\\
\multicolumn{3}{l}{For i=j i=1,2,3,4}\tabcellsep \tabcellsep \\
Where:\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\multicolumn{6}{l}{[S ij AX ]: Sustainability Matrix}\\
III.\tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{Nº} \par 
\begin{longtable}{P{0.85\textwidth}}
V.\\
90\\
Volume XXI Issue IV Version I\\
)\\
(\end{longtable} \par
 
\caption{\label{tab_1}Table Nº 2}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{Nº3} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_2}Table Nº3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{Nº4} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_3}Table Nº4 :}\end{figure}
 			\footnote{The definition of Sustainable Development is established according to the Rio Declaration 1992.} 			\footnote{According to the Earth Summit, Johannesburg 2002 the 4th component of Sustainability was established as Governance. In this scope, its correspondence with Institutional Sustainability is considered.} 			\footnote{Year 2021 B © 2021 Global Journals Local Sustainability Model based on Sustainable Harmonic Movement Analysis} 			\footnote{The Sustainable Harmonic Movement (MAS) is defined in reference to a physical system of masses subjected to the Oscillatory Harmonic Movement for the sustainability components.\hyperref[b3]{4} The Adapted Holzer Method (MHA) is based on the adaptation of the solution of the mass motion equation by the Holzer Method submitting the 4 Sustainability components.} 			\footnote{© 2021 Global Journals} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Economic Commission for Latin America and the Caribbean ()]{b3}\label{b3} 	 		\textit{Economic Commission for Latin America and the Caribbean},  		2011.  	 	 (Social Innovation and local development) 
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\end{document}