World Library  
Flag as Inappropriate
Email this Article

Kolmogorov–Arnold–Moser theorem

Article Id: WHEBN0000016972
Reproduction Date:

Title: Kolmogorov–Arnold–Moser theorem  
Author: World Heritage Encyclopedia
Language: English
Subject: Theorems in dynamical systems, Vladimir Arnold, Dynamical system, Index of physics articles (K), Chaos theory
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Kolmogorov–Arnold–Moser theorem

The Kolmogorov–Arnold–Moser theorem (KAM theorem) is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics.

The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Kolmogorov in 1954. This was rigorously proved and extended by Vladimir Arnold (in 1963 for analytic Hamiltonian systems) and Jürgen Moser (in 1962 for smooth twist maps), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated, could not be applied directly as a whole to the motions of the solar system because of the presence of degeneracy in the unperturbed Kepler problem. However, it is useful in generating corrections of astronomical models, and to prove long-term stability and the avoidance of orbital resonance in solar system. Arnold used the methods of KAM to prove the stability of elliptical orbits in the planar three-body problem.

Statement

The KAM theorem is usually stated in terms of trajectories in phase space of an integrable Hamiltonian system. The motion of an integrable system is confined to a doughnut-shaped surface, an invariant torus. Different initial conditions of the integrable Hamiltonian system will trace different invariant tori in phase space. Plotting the coordinates of an integrable system would show that they are quasi-periodic.

The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and survive, while others are destroyed. The ones that survive are those that meet the non-resonance condition, i.e., they have “sufficiently irrational” frequencies. This implies that the motion continues to be quasiperiodic, with the independent periods changed (as a consequence of the non-degeneracy condition). The KAM theorem specifies quantitatively what level of perturbation can be applied for this to be true. An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasi-periodic motions, now known as KAM theory. Notably, it has been extended to non-Hamiltonian systems (starting with Moser), to non-perturbative situations (as in the work of Michael Herman) and to systems with fast and slow frequencies (as in the work of Mikhail B. Sevryuk).

The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom. As the number of dimensions of the system increases, the volume occupied by the tori decreases.

Those KAM tori that are not destroyed by perturbation become invariant Cantor sets, named Cantori by Ian C. Percival in 1979.

As the perturbation increases and the smooth curves disintegrate we move from KAM theory to Aubry-Mather theory which requires less stringent hypotheses and works with the Cantor-like sets.

See also

References

  • Arnold, Weinstein, Vogtmann. Mathematical Methods of Classical Mechanics, 2nd ed., Appendix 8: Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem. Springer 1997.
  • require('Module:No globals')

local function getCatForId( id )

   local title = mw.title.getCurrentTitle()
   local namespace = title.namespace
   if namespace == 0 then

return ''

   elseif namespace == 2 and not title.isSubpage then

return ''

   else

return ''

   end

end

local function viafLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'VIAF' )

end

local function kulturnavLink( id )

   return '.. id .. ' id' 

end

local function sikartLink( id )

   return '.. id .. '&lng=en ' .. id .. '' 

end

local function tlsLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'WIKI') end)

   return '.. id2 .. ' ' .. id .. '' 

end


local function ciniiLink( id )

   return '.. id .. '?l=en ' .. id .. '' 

end

local function bneLink( id )

   return '.. id .. ' ' .. id .. '' 

end


local function uscongressLink( id )

   return '.. id .. ' ' .. id .. '' 

end

local function narapersonLink( id )

   return '.. id .. ' ' .. id .. '' 

end

local function naraorganizationLink( id )

   return '.. id .. ' ' .. id .. '' 

end

local function botanistLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'PATH') end)

   return '.. id2 .. ' ' .. id .. '' 

end

local function mgpLink( id )

   -- TODO Implement some sanity checking regex
   return '.. id .. ' ' .. id .. '' 

end

local function rslLink( id )

   -- TODO Implement some sanity checking regex
   return '.. id .. '&CON_LNG=ENG ' .. id .. ''

end

local function leonoreLink( id ) -- Identifiants allant de LH/1/1 à LH/2794/54 (légionnaires) -- Identifiants allant de C/0/1 à C/0/84 (84 légionnaires célèbres) -- Identifiants allant de 19800035/1/1 à 19800035/385/51670 (légionnaires décédés entre 1954 et 1977, et quelques dossiers de légionnaires décédés avant 1954)

   if not string.match( id, '^LH/%d%d?%d?%d?/%d%d?%d?$' ) and
      not string.match( id, '^C/0/%d%d?$' ) and
           not string.match( id, '^19800035/%d%d?%d?%d?/%d%d?%d?%d?%d?$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' 

end

local function sbnLink( id )

   if not string.match( id, '^IT\\ICCU\\%d%d%d%d%d%d%d%d%d%d$' ) and not string.match( id, '^IT\\ICCU\\%u%u[%d%u]%u\\%d%d%d%d%d%d$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'SBN' )

end

local function nkcLink( id ) return '.. id .. '&CON_LNG=ENG ' .. id .. '' end

local function nclLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. '&CON_LNG=ENG ' .. id .. '' 

end

local function ndlLink( id ) return '.. id .. ' ' .. id .. '' end

local function sudocLink( id )

   if not string.match( id, '^%d%d%d%d%d%d%d%d[%dxX]$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' 

end

local function hlsLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. '.php ' .. id .. ''

end

local function lirLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. '.450.0.html ' .. id .. ''

end

local function splitLccn( id )

   if id:match( '^%l%l?%l?%d%d%d%d%d%d%d%d%d?%d?$' ) then
       id = id:gsub( '^(%l+)(%d+)(%d%d%d%d%d%d)$', '%1/%2/%3' )
   end
   if id:match( '^%l%l?%l?/%d%d%d?%d?/%d+$' ) then
        return mw.text.split( id, '/' )
   end
   return false

end

local function append(str, c, length)

   while str:len() < length do
       str = c .. str
   end
   return str

end

local function lccnLink( id )

   local parts = splitLccn( id )
   if not parts then
       return false
   end
   local lccnType = parts[1] ~= 'sh' and 'names' or 'subjects'
   id = parts[1] .. parts[2] .. append( parts[3], '0', 6 )
   return '.. lccnType .. '/' .. id .. ' ' .. id .. '' .. getCatForId( 'LCCN' )

end

local function mbLink( id )

   -- TODO Implement some sanity checking regex
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'MusicBrainz' )

end

--Returns the ISNI check digit isni must be a string where the 15 first elements are digits local function getIsniCheckDigit( isni )

   local total = 0
   for i = 1, 15 do
       local digit = isni:byte( i ) - 48 --Get integer value
       total = (total + digit) * 2
   end
   local remainder = total % 11
   local result = (12 - remainder) % 11
   if result == 10 then
       return "X"
   end
   return tostring( result )

end

--Validate ISNI (and ORCID) and retuns it as a 16 characters string or returns false if it's invalid --See http://support.orcid.org/knowledgebase/articles/116780-structure-of-the-orcid-identifier local function validateIsni( id )

   id = id:gsub( '[ %-]', ):upper()
   if not id:match( '^%d%d%d%d%d%d%d%d%d%d%d%d%d%d%d[%dX]$' ) then
       return false
   end
   if getIsniCheckDigit( id ) ~= string.char( id:byte( 16 ) ) then
       return false
   end
   return id

end

local function isniLink( id )

   id = validateIsni( id )
   if not id then
       return false
   end
   return '.. id .. ' ' .. id:sub( 1, 4 ) .. ' ' .. id:sub( 5, 8 ) .. ' '  .. id:sub( 9, 12 ) .. ' '  .. id:sub( 13, 16 ) .. '' .. getCatForId( 'ISNI' )

end

local function orcidLink( id )

   id = validateIsni( id )
   if not id then
       return false
   end
   id = id:sub( 1, 4 ) .. '-' .. id:sub( 5, 8 ) .. '-'  .. id:sub( 9, 12 ) .. '-'  .. id:sub( 13, 16 )
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ORCID' )

end

local function gndLink( id )

   return '.. id .. ' ' .. id .. '' .. getCatForId( 'GND' )

end

local function selibrLink( id ) if not string.match( id, '^%d+$' ) then

       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'SELIBR' )

end

local function bnfLink( id )

   --Add cb prefix if it has been removed
   if not string.match( id, '^cb.+$' ) then
       id = 'cb' .. id
   end
   return '.. id .. ' ' .. id .. ' .. id .. ' (data)' .. getCatForId( 'BNF' )

end

local function bpnLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'BPN' )

end

local function ridLink( id )

   return '.. id .. ' ' .. id .. '' .. getCatForId( 'RID' )

end

local function bibsysLink( id )

   return '.. id .. '&feltselect=bs.autid ' .. id .. '' .. getCatForId( 'BIBSYS' )

end

local function ulanLink( id )

   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ULAN' )

end

local function nlaLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'NLA' ) end

local function rkdartistsLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'RKDartists' ) end

local function getIdsFromWikidata( item, property )

   local ids = {}
   if not item.claims[property] then
       return ids
   end
   for _, statement in pairs( item.claims[property] ) do
         if statement.mainsnak.datavalue then
          table.insert( ids, statement.mainsnak.datavalue.value )
         end
   end
   return ids

end

local function matchesWikidataRequirements( item, reqs )

   for _, group in pairs( reqs ) do
       local property = 'p' .. group[1]
       local qid = group[2]
       if item.claims[property] ~= nil then
           for _, statement in pairs ( item.claims[property] ) do
                if statement.mainsnak.datavalue ~= nil then
                        if statement.mainsnak.datavalue.value['numeric-id'] == qid then
                        return true
                end
            end
           end
       end
   end
   return false

end

local function createRow( id, label, rawValue, link, withUid )

   if link then
       if withUid then
           return '* ' .. label .. ' ' .. link .. '\n'
       else
           return '* ' .. label .. ' ' .. link .. '\n'
       end
   else

return '* \n'

   end

end

--In this order: name of the parameter, label, propertyId in Wikidata, formatting function local conf = {

   { 'VIAF', 'VIAF', 214, viafLink },
   { 'LCCN', 'LCCN', 244, lccnLink },
   { 'ISNI', 'ISNI', 213, isniLink },
   { 'ORCID', 'ORCID', 496, orcidLink },
   { 'GND', 'GND', 227, gndLink },
   { 'SELIBR', 'SELIBR', 906, selibrLink },
   { 'SUDOC', 'SUDOC', 269, sudocLink },    
   { 'BNF', 'BNF', 268, bnfLink },
   { 'BPN', 'BPN', 651, bpnLink },
   { 'RID', 'ResearcherID', 1053, ridLink },
   { 'BIBSYS', 'BIBSYS', 1015, bibsysLink },
   { 'ULAN', 'ULAN', 245, ulanLink },
   { 'HDS', 'HDS', 902, hlsLink },
   { 'LIR', 'LIR', 886, lirLink },
   { 'MBA', 'MusicBrainz', 434, mbLink },
   { 'MGP', 'MGP', 549, mgpLink },    
   { 'NLA', 'NLA', 409, nlaLink },
   { 'NDL', 'NDL', 349, ndlLink },
   { 'NCL', 'NCL', 1048, nclLink },
   { 'NKC', 'NKC', 691, nkcLink },
   { 'Léonore', 'Léonore', 640, leonoreLink }, 
   { 'SBN', 'ICCU', 396, sbnLink },     
   { 'RLS', 'RLS', 947, rslLink },
   { 'Botanist', '[[Author ci 
  • require('Module:No globals')

local function getCatForId( id )

   local title = mw.title.getCurrentTitle()
   local namespace = title.namespace
   if namespace == 0 then

return ''

   elseif namespace == 2 and not title.isSubpage then

return ''

   else

return ''

   end

end

local function viafLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'VIAF' )

end

local function kulturnavLink( id )

   return '.. id .. ' id' 

end

local function sikartLink( id )

   return '.. id .. '&lng=en ' .. id .. '' 

end

local function tlsLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'WIKI') end)

   return '.. id2 .. ' ' .. id .. '' 

end


local function ciniiLink( id )

   return '.. id .. '?l=en ' .. id .. '' 

end

local function bneLink( id )

   return '.. id .. ' ' .. id .. '' 

end


local function uscongressLink( id )

   return '.. id .. ' ' .. id .. '' 

end

local function narapersonLink( id )

   return '.. id .. ' ' .. id .. '' 

end

local function naraorganizationLink( id )

   return '.. id .. ' ' .. id .. '' 

end

local function botanistLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'PATH') end)

   return '.. id2 .. ' ' .. id .. '' 

end

local function mgpLink( id )

   -- TODO Implement some sanity checking regex
   return '.. id .. ' ' .. id .. '' 

end

local function rslLink( id )

   -- TODO Implement some sanity checking regex
   return '.. id .. '&CON_LNG=ENG ' .. id .. ''

end

local function leonoreLink( id ) -- Identifiants allant de LH/1/1 à LH/2794/54 (légionnaires) -- Identifiants allant de C/0/1 à C/0/84 (84 légionnaires célèbres) -- Identifiants allant de 19800035/1/1 à 19800035/385/51670 (légionnaires décédés entre 1954 et 1977, et quelques dossiers de légionnaires décédés avant 1954)

   if not string.match( id, '^LH/%d%d?%d?%d?/%d%d?%d?$' ) and
      not string.match( id, '^C/0/%d%d?$' ) and
           not string.match( id, '^19800035/%d%d?%d?%d?/%d%d?%d?%d?%d?$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' 

end

local function sbnLink( id )

   if not string.match( id, '^IT\\ICCU\\%d%d%d%d%d%d%d%d%d%d$' ) and not string.match( id, '^IT\\ICCU\\%u%u[%d%u]%u\\%d%d%d%d%d%d$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'SBN' )

end

local function nkcLink( id ) return '.. id .. '&CON_LNG=ENG ' .. id .. '' end

local function nclLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. '&CON_LNG=ENG ' .. id .. '' 

end

local function ndlLink( id ) return '.. id .. ' ' .. id .. '' end

local function sudocLink( id )

   if not string.match( id, '^%d%d%d%d%d%d%d%d[%dxX]$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' 

end

local function hlsLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. '.php ' .. id .. ''

end

local function lirLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. '.450.0.html ' .. id .. ''

end

local function splitLccn( id )

   if id:match( '^%l%l?%l?%d%d%d%d%d%d%d%d%d?%d?$' ) then
       id = id:gsub( '^(%l+)(%d+)(%d%d%d%d%d%d)$', '%1/%2/%3' )
   end
   if id:match( '^%l%l?%l?/%d%d%d?%d?/%d+$' ) then
        return mw.text.split( id, '/' )
   end
   return false

end

local function append(str, c, length)

   while str:len() < length do
       str = c .. str
   end
   return str

end

local function lccnLink( id )

   local parts = splitLccn( id )
   if not parts then
       return false
   end
   local lccnType = parts[1] ~= 'sh' and 'names' or 'subjects'
   id = parts[1] .. parts[2] .. append( parts[3], '0', 6 )
   return '.. lccnType .. '/' .. id .. ' ' .. id .. '' .. getCatForId( 'LCCN' )

end

local function mbLink( id )

   -- TODO Implement some sanity checking regex
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'MusicBrainz' )

end

--Returns the ISNI check digit isni must be a string where the 15 first elements are digits local function getIsniCheckDigit( isni )

   local total = 0
   for i = 1, 15 do
       local digit = isni:byte( i ) - 48 --Get integer value
       total = (total + digit) * 2
   end
   local remainder = total % 11
   local result = (12 - remainder) % 11
   if result == 10 then
       return "X"
   end
   return tostring( result )

end

--Validate ISNI (and ORCID) and retuns it as a 16 characters string or returns false if it's invalid --See http://support.orcid.org/knowledgebase/articles/116780-structure-of-the-orcid-identifier local function validateIsni( id )

   id = id:gsub( '[ %-]', ):upper()
   if not id:match( '^%d%d%d%d%d%d%d%d%d%d%d%d%d%d%d[%dX]$' ) then
       return false
   end
   if getIsniCheckDigit( id ) ~= string.char( id:byte( 16 ) ) then
       return false
   end
   return id

end

local function isniLink( id )

   id = validateIsni( id )
   if not id then
       return false
   end
   return '.. id .. ' ' .. id:sub( 1, 4 ) .. ' ' .. id:sub( 5, 8 ) .. ' '  .. id:sub( 9, 12 ) .. ' '  .. id:sub( 13, 16 ) .. '' .. getCatForId( 'ISNI' )

end

local function orcidLink( id )

   id = validateIsni( id )
   if not id then
       return false
   end
   id = id:sub( 1, 4 ) .. '-' .. id:sub( 5, 8 ) .. '-'  .. id:sub( 9, 12 ) .. '-'  .. id:sub( 13, 16 )
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ORCID' )

end

local function gndLink( id )

   return '.. id .. ' ' .. id .. '' .. getCatForId( 'GND' )

end

local function selibrLink( id ) if not string.match( id, '^%d+$' ) then

       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'SELIBR' )

end

local function bnfLink( id )

   --Add cb prefix if it has been removed
   if not string.match( id, '^cb.+$' ) then
       id = 'cb' .. id
   end
   return '.. id .. ' ' .. id .. ' .. id .. ' (data)' .. getCatForId( 'BNF' )

end

local function bpnLink( id )

   if not string.match( id, '^%d+$' ) then
       return false
   end
   return '.. id .. ' ' .. id .. '' .. getCatForId( 'BPN' )

end

local function ridLink( id )

   return '.. id .. ' ' .. id .. '' .. getCatForId( 'RID' )

end

local function bibsysLink( id )

   return '.. id .. '&feltselect=bs.autid ' .. id .. '' .. getCatForId( 'BIBSYS' )

end

local function ulanLink( id )

   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ULAN' )

end

local function nlaLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'NLA' ) end

local function rkdartistsLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'RKDartists' ) end

local function getIdsFromWikidata( item, property )

   local ids = {}
   if not item.claims[property] then
       return ids
   end
   for _, statement in pairs( item.claims[property] ) do
         if statement.mainsnak.datavalue then
          table.insert( ids, statement.mainsnak.datavalue.value )
         end
   end
   return ids

end

local function matchesWikidataRequirements( item, reqs )

   for _, group in pairs( reqs ) do
       local property = 'p' .. group[1]
       local qid = group[2]
       if item.claims[property] ~= nil then
           for _, statement in pairs ( item.claims[property] ) do
                if statement.mainsnak.datavalue ~= nil then
                        if statement.mainsnak.datavalue.value['numeric-id'] == qid then
                        return true
                end
            end
           end
       end
   end
   return false

end

local function createRow( id, label, rawValue, link, withUid )

   if link then
       if withUid then
           return '* ' .. label .. ' ' .. link .. '\n'
       else
           return '* ' .. label .. ' ' .. link .. '\n'
       end
   else

return '* \n'

   end

end

--In this order: name of the parameter, label, propertyId in Wikidata, formatting function local conf = {

   { 'VIAF', 'VIAF', 214, viafLink },
   { 'LCCN', 'LCCN', 244, lccnLink },
   { 'ISNI', 'ISNI', 213, isniLink },
   { 'ORCID', 'ORCID', 496, orcidLink },
   { 'GND', 'GND', 227, gndLink },
   { 'SELIBR', 'SELIBR', 906, selibrLink },
   { 'SUDOC', 'SUDOC', 269, sudocLink },    
   { 'BNF', 'BNF', 268, bnfLink },
   { 'BPN', 'BPN', 651, bpnLink },
   { 'RID', 'ResearcherID', 1053, ridLink },
   { 'BIBSYS', 'BIBSYS', 1015, bibsysLink },
   { 'ULAN', 'ULAN', 245, ulanLink },
   { 'HDS', 'HDS', 902, hlsLink },
   { 'LIR', 'LIR', 886, lirLink },
   { 'MBA', 'MusicBrainz', 434, mbLink },
   { 'MGP', 'MGP', 549, mgpLink },    
   { 'NLA', 'NLA', 409, nlaLink },
   { 'NDL', 'NDL', 349, ndlLink },
   { 'NCL', 'NCL', 1048, nclLink },
   { 'NKC', 'NKC', 691, nkcLink },
   { 'Léonore', 'Léonore', 640, leonoreLink }, 
   { 'SBN', 'ICCU', 396, sbnLink },     
   { 'RLS', 'RLS', 947, rslLink },
   { 'Botanist', '[[Author ci 
  • Rafael de la Llave (2001) A tutorial on KAM theory.
  • KAM theory: the legacy of Kolmogorov’s 1954 paper
  • Kolmogorov-Arnold-Moser theory from Scholarpedia
  • H Scott Dumas. The KAM Story - A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory, 2014, World Scientific Publishing, ISBN 978-981-4556-58-3. Chapter 1: Introduction
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.