# link d_Qndfs 'economics, markets/Puetz - Universal Waves Series, model derivations.txt' # www.BillHowell.ca 20Apr2020 initial taken fromn 18Apr2020 'Puetz - Universal Waves Series, header.ndf' # view this file in a text editor, with [constant width font, tab = 3 spaces], no line-wrap #**************************** # List of derivations, generated with : # $ cat "$d_Qndfs""Puetz - Universal Waves Series, model derivations.txt" | grep "^#]" | sed 's/^#\]/ /' # #*********** #] References # the [formulae, data] come from : [1] Stephen Puetz, Glenn Borchardt 2011 "Universal Cycle Theory, Neomechanics of the hierarchically infinite universe" www.OutskirtsPress.com, ISBN 978-1-4327-8133-0 http://www.uct-news.com # see also : [2] Stephen Puetz 2009 "The Unified Cycle Theory, How cycles dominate the structure of the universe and influence life on earth" www.OutSkirtsPress.com, 489pp, ISBN 978-1-4327-1216-7 http://www.uct-news.com # the following table is copied-over from "$d_Qndfs""Puetz - Universal Waves Series.ods" # tables of cycle parameters # see p565-566 Tables C3.2.1-.6 for UWS constants # see p567 Tables C3.3.1-.3 for Double-UWS constants #************************* #] Derivation of t_lag_cycle(n) from θ(n) 20Apr2020 initial COMMENT : time lags are NOT the same as phase angles! blah, blah, blah # page 564h0.7 Equation A1.3.1, rename t_lag to t_lag_app (1) y(i) = A*sin((2*pi*(t(i) + t_scale + t_lag_app)/lambda(n)) + theta(n)) I will use [t_lag_cycle, t_lag_app] y(i) = A*sin((2*pi*(t(i) + t_scale + t_lag_cycle + t_lag_app)/lambda(n))) therefore 2*pi*(t(i) + t_scale + t_lag_app) /lambda(n) + theta(n) = 2*pi*(t(i) + t_scale + t_lag_cycle + t_lag_app)/lambda(n) ... theta(n) = 2*pi*(t(i) + t_scale + t_lag_cycle + t_lag_app)/lambda(n) - 2*pi*(t(i) + t_scale + t_lag_app)/lambda(n) ... theta(n)*lambda(n)/(2*pi) = (t(i) + t_scale + t_lag_cycle + t_lag_app) - (t(i) + t_scale + t_lag_app) = t_lag_cycle or ... t_lag_cycle = theta(n)*lambda(n)/(2*pi) #************************* #] Derivation of θn (initial attempts - not used) 18Apr2020 # page 564h0.7 Equation A1.3.1 (1) y(i) = A*sin((2*pi*(t(i) + t_scale + t_lag)/lambda(n)) + theta(n)) Example theta_1st_max(n), p465h0.8, use Table A1.4.2, p465h0.8 : y(i)/A = 1 t_lag = 0, is the lag time that depends on the process being investigate (here no time lag) t_scale = -2000, for BP/AP with year 0 Gregorian as basis t(i) = "present" = 2010 t_inc = 1/(3*365), time increment down column n = 3, for 28.7 day period lambda(n)= 0.0028970478728148 theta(n) = 3.391242799 Example : y(i) = A*sin((2*pi()*(t(i) + t_scale + t_lag)/lambda(n)) + theta(n)) y(i)/A = sin((2*pi()*(t(i) + t_scale + t_lag)/lambda(n)) + theta(n)) = sin( 2*pi()*(2010 - 2000 + 0 )/0.0028970478728148 + 3.391242799) OK - this worked just as in the text. I made many mistakes getting there! Derive formula for theta(n) : theta(n) = arcsin(y(i)/A) - { 2*pi*(t(i) + t_scale + t_lag) /lambda(n) } - not a function as it's many-valued = arcsin(y(i)/A) - 2*pi*(t(i) + t_scale + t_lag) /lambda(n) theta(n) : is an angle, so it can vary from +-infinity, but p565h0.6 Table C3.2.1 varies from +-2*pi, so it takes the first max? we typically want just the first max, for which arcsin(y(i)/A) = 1 : arcsin(y(i)/A) = arcsin(1) = pi/2 so take the modulus (remainder) after dividing by lambda(n) therefore : theta_1st_max(n) = modulus[pi()/2 - 2*pi()*(t(i) + t_scale + t_lag)/lambda(n), 2*pi()] Assumptions - follow the same parameters as for the example above, but use the updated table y(i)/A = 1 t_lag = 0, is the lag time that depends on the process being investigate (here no time lag) t_scale = -2000, for BP/AP with year 0 Gregorian as basis t(i) = "present" = 2010 n = formula will be use to calculate all n lambda(n)= as calculated from n=0 (multiples of 3) then : theta_1st_max(n) = modulus[pi()/2 - 2*pi()*(t(i) + t_scale + t_lag)/lambda(n), 2*pi()] = modulus[pi()/2 - 2*pi()*(2010 - 2000 + 0 )/lambda(n), 2*pi()] = modulus[pi()/2 - 2*pi()*10/lambda(n), 2*pi()] This does NOT work : theta_1st_max(n) : given = 1.910577326 result = 2.892167090 large n give constant theta_1st_max(n) = 1.570796327, which is wrong! but it makes sense given my assumptions! Try 2 Assumptions : y(i)/A = 1 t_lag = 0, is the lag time that depends on the process being investigate (here no time lag) t_scale = 0, for BP/AP with year 2000 as basis t(i) = "present" = 2010 n = formula will be use to calculate all n lambda(n)= as calculated from n=0 (multiples of 3) then : theta_1st_max(n) = modulus[pi()/2 - 2*pi()*(t(i) + t_scale + t_lag)/lambda(n), 2*pi()] = modulus[pi()/2 - 2*pi()*(2010 + 0 + 0 )/lambda(n), 2*pi()] = 3.272536818 vs 1.910577326 std NO - I must solve problem at higher n this seems to mean that the time basis has changed - look at abbreviated divide by the [day, yr, kyr, My, Gy, Ty] - yes, as per section titles OK - this works Now try to find t_lags to make it work find a t_lag for each section that works, these must align UWS to actual situation? Perhaps can't calculate? what is t(i) for each section? eg year 2000? NYET - A better approach would be to assume that he manually set the θn, so manually input all of them # enddoc