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MathGroup Archive 2004

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Re: Estimating parameters p and q in y'' + p y' + q y = Tide(t)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46284] Re: Estimating parameters p and q in y'' + p y' + q y = Tide(t)
  • From: drbob at bigfoot.com (Bobby R. Treat)
  • Date: Fri, 13 Feb 2004 21:56:48 -0500 (EST)
  • References: <c0hhrh$lgd$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

There's no need to go so 'round about to get that into a notebook.
Just copy and paste it into Mathematica. The front end asks if you
want it to interpret the text; you say yes, and it's done.

Bobby

gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi) wrote in message news:<c0hhrh$lgd$1 at smc.vnet.net>...
> Dear Math User friends:
> I have two data sets; the first one corresponds to tide data, and the
> second one corresponds to
> water elevation data obtained from a groundwater monitoring well.  The
> tide affects the water level
> inside the well.  If we let the variable y(t) represent the height of
> the water column inside the pipe,
> and Tide(t) be a least square fit representation of our tide record,
> with t representing time,
> then we can form a Differential Equation:  y'' + p y' + q y = Tide(t),
> where Tide(t) acts as a forcing
> function.  Since I have a water elevation record; what I'm seeking is
> to find a way to estimate
> two values p and q, such that the solution y(t) to the above DE,
> becomes a model that fits my
> water elevation data; i.e. a model in the least square sense, showing
> a correlation (of say) 0.95,
> or above.  The following is an unevaluated Mathematica notebook to
> elaborate this question
> with the aid of a specific example. Please copy the following text and
> paste it into Wordpad, or
> Notepad and save it as DE.txt  Then change the name of this file to
> DE.nb, (ignore the "are you
> sure that you want to  change extention name" message) and open the
> new notebook using
> Mathematica (version 5.0, or version above 5.0) as usual.  Thank you
> for your help!
> 
> Start copying here:
> (************** Content-type: application/mathematica **************
>                      CreatedBy='Mathematica 5.0'
> 
>                     Mathematica-Compatible Notebook
> 
> This notebook can be used with any Mathematica-compatible
> application, such as Mathematica, MathReader or Publicon. The data
> for the notebook starts with the line containing stars above.
> 
> To get the notebook into a Mathematica-compatible application, do
> one of the following:
> 
> * Save the data starting with the line of stars above into a file
>   with a name ending in .nb, then open the file inside the
>   application;
> 
> * Copy the data starting with the line of stars above to the
>   clipboard, then use the Paste menu command inside the application.
> 
> Data for notebooks contains only printable 7-bit ASCII and can be
> sent directly in email or through ftp in text mode.  Newlines can be
> CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
> 
> NOTE: If you modify the data for this notebook not in a Mathematica-
> compatible application, you must delete the line below containing
> the word CacheID, otherwise Mathematica-compatible applications may
> try to use invalid cache data.
> 
> For more information on notebooks and Mathematica-compatible 
> applications, contact Wolfram Research:
>   web: http://www.wolfram.com
>   email: info at wolfram.com
>   phone: +1-217-398-0700 (U.S.)
> 
> Notebook reader applications are available free of charge from 
> Wolfram Research.
> *******************************************************************)
> 
> (*CacheID: 232*)
> 
> 
> (*NotebookFileLineBreakTest
> NotebookFileLineBreakTest*)
> (*NotebookOptionsPosition[     21225,        425]*)
> (*NotebookOutlinePosition[     21931,        449]*)
> (*  CellTagsIndexPosition[     21887,        445]*)
> (*WindowFrame->Normal*)
> 
> 
> 
> Notebook[{
> Cell[BoxData[
>     StyleBox[\( (*\(\(*\)\(\ \)\(I\)\(\ \)\(have\)\(\ \)\(two\)\(\
> \)\(data\)\
> \(\ \)\(sets\)\); \ the\ first\ one\ corresponds\ to\ tide\ data, \ 
>         and\ the\ second\ one\ corresponds\ to\n
>           water\ elevation\ data\ obtained\ from\ a\ groundwater\
> monitoring\ \
> well . \ \ The\ tide\ affects\ the\ water\ level\n
>           inside\ the\ well . \ \ If\ we\ let\ the\ variable\ y
> \((t)\)\ \
> represent\ the\ height\ of\ the\ water\ column\ inside\ the\ pipe, \n
>         and\ Tide \((t)\)\ be\ a\ least\ square\ fit\ representation\
> of\ our\
> \ tide\ record, \ with\ t\ reperesenting\ time, \n
>         then\ we\ can\ form\ a\ Differential\ \(Equation : \ \ y''\  +
> \
>                 p\ y'\  + \ q\ y\)\  = \ Tide \((t)\), \ 
>         where\ Tide \((t)\)\ acts\ as\ a\ forcing\n
>           function . \ \ Since\ I\ have\ a\ water\ elevation\ record;
> \
>         what\ I' m\ seeking\ is\ to\ find\ a\ way\ to\ estimate\n
>           two\ values\ p\ and\ q, \ 
>         such\ that\ the\ solution\ y \((t)\)\ to\ the\ above\ DE, \ 
>         becomes\ a\ model\ that\ fits\ my\nwater\ elevation\ data; \ 
>         i . e . \ a\ model\ in\ the\ least\ square\ sense, \ 
>         showing\ a\ correlation\ \((of\ say)\)\ 0.95, \n
>         or\ above . \ \ The\ following\ example\ is\ an\ attempt\ to\
> clarify\
> \ my\ \(question : \n\(\(Here\ is\ the\ tide\ record\ \((the\ values\
> are\ \
> measured\ in\ decimal\ meters)\)\)\(:\)\)\)\ **) \),
>       FormatType->StandardForm,
>       FontFamily->"Arial"]], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
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> \
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> {264, \(-0.845\)}};\)\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(L = Length[tide]\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(plt1 = 
>       ListPlot[tide, PlotJoined \[Rule] True, 
>         PlotStyle \[Rule] RGBColor[1, 0, 0]]\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(\(initparams = {{a\_0, \(-0.31278\)}, {a\_1, \(-0.21078\)},
> {a\_2, \
> \(-0.12503\)}, {a\_3, \(-0.03388\)}, {a\_4, \(-0.34959\)}, {b\_1, 
>             8.093077}, {b\_2, 10.51904}, {b\_3, 2.014087}, {b\_4, 
>             7.077294}, {c\_1, \(-0.05472\)}, {c\_2, \(-0.05678\)},
> {c\_3, \
> \(-0.1125\)}, {c\_4, \(-0.06463\)}, {d\_1, \(-5.01326\)}, {d\_2, \
> \(-6.27323\)}, {d\_3, \(-1.67511\)}, {d\_4, \(-10.6115\)}, {v\_1, 
>             0.126851}, {v\_2, 0.135713}, {v\_3, 0.089487}, {v\_4, 
>             0.130158}, {w\_1, 0.049697}, {w\_2, 
>             0.049183}, {w\_3, \(-0.00844\)}, {w\_4, 0.046353}};\)\)],
> "Input",\
> 
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(\(model\  = \ 
>         a\_0 + Sum[a\_i*Sin[v\_i*t - b\_i], {i, 1, 4}] + 
>           Sum[c\_i*Sin[w\_i*t - d\_i], {i, 1, 4}];\)\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(<< Statistics`NonlinearFit`\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     RowBox[{\(TIDE[t_]\), "=", 
>       RowBox[{"Chop", "[", 
>         RowBox[{"NonlinearFit", "[", 
>           RowBox[{"tide", ",", "model", ",", "t", ",", "initparams",
> ",",
>             FormBox[
>               FormBox[\(AccuracyGoal \[Rule] 2\),
>                 "TraditionalForm"],
>               "TraditionalForm"]}], "]"}], "]"}]}]], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(plt2 = 
>       Plot[TIDE[t], {t, 1, 264}, PlotStyle -> RGBColor[0, 1, 0]]\)],
> "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(Show[{plt1, plt2}, ImageSize\  \[Rule] \ 540]\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(Gravitational\)\(\ \)\(Constant\)\(\
> \)\(G\)\)\  = \
>         9.8\ m/sec\^2\ **) \)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \( (*\(*\)\(\ \)\(Tide\)\(\ \)\(acting\)\(\ \)\(as\)\(\ \)\(a\)\(\
> \
> \)\(Forcing\)\(\ \)\(Function\)\(\ \)\(term\)\(\ \)\(in\)\(\
> \)\(the\)\(\ \
> \)\(following\)\(\ \)\(Differential\)\(\ \)\(\(Equation\)\(:\)\)\ **)
> \)], \
> "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     RowBox[{"(*", 
>       RowBox[{\(\(\(*\)\(\ \)\(Using\)\(\ \)\("\<trial and
> error\>"\)\(\ \
> \)\(we\)\(\ \)\(are\)\(\ \)\(using\)\(\ \)\(values : \ p\)\)\  = \ 
>           9.8\ \((buoyancy\ factor)\)\), ",", " ", \(q = 1\), ",", 
>         " ", \(and\ two\ initial\ conditions; \ y[0] = 0\), ",", " ", 
>         RowBox[{
>           RowBox[{"and", " ", 
>             RowBox[{
>               SuperscriptBox["y", "\[Prime]",
>                 MultilineFunction->None], "[", "0", "]"}]}], "=",
> "1"}], ",",
>         " ", \(to\ set\ up\ our\ differential\
> \(\(equation\)\(:\)\)\)}],
>       "**)"}]], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     RowBox[{
>       RowBox[{"solution", "=", 
>         RowBox[{"NDSolve", "[", 
>           RowBox[{
>             RowBox[{"{", 
>               RowBox[{
>                 RowBox[{
>                   RowBox[{
>                     RowBox[{
>                       SuperscriptBox["y", "\[DoublePrime]",
>                         MultilineFunction->None], "[", "t", "]"}],
> "+",
>                     RowBox[{\((9.8)\), "*", 
>                       RowBox[{
>                         SuperscriptBox["y", "\[Prime]",
>                           MultilineFunction->None], "[", "t", "]"}]}],
>                     "+", \(y[t]\)}], "\[Equal]", \(TIDE[t]\)}], 
>                 ",", \(y[0] \[Equal] 0\), ",", 
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>                   RowBox[{
>                     SuperscriptBox["y", "\[Prime]",
>                       MultilineFunction->None], "[", "0", "]"}],
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>                   "1"}]}], "}"}], ",", "y", ",", \({t, 0, L}\)}],
> "]"}]}],
>       ";"}]], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(\(plt3 = 
>         Plot[y[t] /. solution, {t, 0, L}, 
>           PlotStyle \[Rule] RGBColor[1, 0, 1]];\)\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(Here\)\(\ \)\(is\)\(\ \)\(the\)\(\
> \)\(elevation\)\(\ \
> \)\(data\)\); \ \(\(i . e . \ 
>             the\ water\ elevation\ data\ obtained\ from\ a\
> groundwater\ \
> monitoring\ well\ \((the\ values\ are\ measured\ in\ decimal\ \
> meters)\)\)\(:\)\)\ **) \)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
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> \(-0.009\)}, {91, \(-0.005\)}, {92, \(-0.011\)}, {93, \(-0.007\)},
> {94, \
> \(-0.007\)}, {95, \(-0.006\)}, {96, \(-0.003\)}, \n\ \ \ \ \ \ \ \
> {97, \
> \(-0.002\)}, {98, 0. }, {99, \(-0.001\)}, {100, 0.001}, {101, 
>               0.002}, {102, \(-0.004\)}, {103, 0.003}, {104, 0.003},
> {105,
>               0.004}, \n\ \ \ \ \ \ \ \ {106, 0.004}, {107, 0.004},
> {108,
>               0.004}, {109, 0.005}, {110, 0.005}, {111, 0.004}, {112, 
>               0.003}, {113, 0.002}, {114, 
>               0.001}, \n\ \ \ \ \ \ \ \ {115, \(-0.001\)}, {116,
> \(-0.002\)}, \
> {117, \(-0.003\)}, {118, \(-0.004\)}, {119, \(-0.004\)}, {120,
> \(-0.004\)}, \
> {121, \(-0.005\)}, {122, \(-0.007\)}, \n\ \ \ \ \ \ \ \ {123,
> \(-0.008\)}, \
> {124, \(-0.01\)}, {125, \(-0.011\)}, {126, \(-0.011\)}, {127,
> \(-0.012\)}, \
> {128, \(-0.013\)}, {129, \(-0.014\)}, {130, \(-0.014\)}, \n\ \ \ \ \ \
> \ \ \
> {131, \(-0.013\)}, {132, \(-0.014\)}, {133, \(-0.014\)}, {134,
> \(-0.014\)}, \
> {135, \(-0.015\)}, {136, \(-0.014\)}, {137, \(-0.014\)}, {138,
> \(-0.015\)}, \n\
> \ \ \ \ \ \ \ \ {139, \(-0.014\)}, {140, \(-0.014\)}, {141,
> \(-0.012\)}, \
> {142, \(-0.011\)}, {143, \(-0.011\)}, {144, \(-0.009\)}, {145,
> \(-0.009\)}, \
> {146, \(-0.007\)}, \n\ \ \ \ \ \ \ \ {147, \(-0.007\)}, {148,
> \(-0.006\)}, \
> {149, \(-0.004\)}, {150, \(-0.004\)}, {151, \(-0.004\)}, {152,
> \(-0.004\)}, \
> {153, \(-0.003\)}, {154, \(-0.003\)}, \n\ \ \ \ \ \ \ \ {155,
> \(-0.003\)}, \
> {156, \(-0.004\)}, {157, \(-0.004\)}, {158, \(-0.004\)}, {159,
> \(-0.004\)}, \
> {160, \(-0.004\)}, {161, \(-0.004\)}, {162, \(-0.005\)}, \n\ \ \ \ \ \
> \ \ \
> {163, \(-0.006\)}, {164, \(-0.007\)}, {165, \(-0.007\)}, {166,
> \(-0.008\)}, \
> {167, \(-0.01\)}, {168, \(-0.011\)}, {169, \(-0.011\)}, {170,
> \(-0.013\)}, \n\
> \ \ \ \ \ \ \ \ {171, \(-0.015\)}, {172, \(-0.017\)}, {173,
> \(-0.017\)}, \
> {174, \(-0.018\)}, {175, \(-0.018\)}, {176, \(-0.018\)}, {177,
> \(-0.018\)}, \
> {178, \(-0.019\)}, \n\ \ \ \ \ \ \ \ {179, \(-0.02\)}, {180,
> \(-0.021\)}, \
> {181, \(-0.021\)}, {182, \(-0.021\)}, {183, \(-0.021\)}, {184,
> \(-0.022\)}, \
> {185, \(-0.019\)}, {186, \(-0.021\)}, \n\ \ \ \ \ \ \ \ {187,
> \(-0.02\)}, \
> {188, \(-0.018\)}, {189, \(-0.017\)}, {190, \(-0.017\)}, {191,
> \(-0.011\)}, \
> {192, \(-0.014\)}, {193, \(-0.011\)}, {194, \(-0.011\)}, \n\ \ \ \ \ \
> \ \ \
> {195, \(-0.01\)}, {196, \(-0.01\)}, {197, \(-0.005\)}, {198,
> \(-0.008\)}, \
> {199, \(-0.008\)}, {200, \(-0.006\)}, {201, \(-0.004\)}, {202,
> \(-0.004\)}, \n\
> \ \ \ \ \ \ \ \ {203, \(-0.006\)}, {204, \(-0.005\)}, {205,
> \(-0.003\)}, \
> {206, \(-0.004\)}, {207, \(-0.003\)}, {208, \(-0.003\)}, {209,
> \(-0.003\)}, \
> {210, \(-0.004\)}, \n\ \ \ \ \ \ \ \ {211, \(-0.004\)}, {212,
> \(-0.004\)}, \
> {213, \(-0.005\)}, {214, \(-0.005\)}, {215, \(-0.006\)}, {216,
> \(-0.008\)}, \
> {217, \(-0.008\)}, {218, \(-0.01\)}, \n\ \ \ \ \ \ \ \ {219,
> \(-0.011\)}, \
> {220, \(-0.013\)}, {221, \(-0.013\)}, {222, \(-0.015\)}, {223,
> \(-0.017\)}, \
> {224, \(-0.018\)}, {225, \(-0.018\)}, {226, \(-0.018\)}, \n\ \ \ \ \ \
> \ \ \
> {227, \(-0.018\)}, {228, \(-0.018\)}, {229, \(-0.02\)}, {230,
> \(-0.02\)}, \
> {231, \(-0.019\)}, {232, \(-0.023\)}, {233, \(-0.02\)}, {234,
> \(-0.021\)}, \n\
> \ \ \ \ \ \ \ \ {235, \(-0.02\)}, {236, \(-0.019\)}, {237,
> \(-0.018\)}, {238, \
> \(-0.016\)}, {239, \(-0.016\)}, {240, \(-0.014\)}, {241, \(-0.013\)},
> {242, \
> \(-0.013\)}, \n\ \ \ \ \ \ \ \ {243, \(-0.012\)}, {244, \(-0.011\)},
> {245, \
> \(-0.01\)}, {246, \(-0.009\)}, {247, \(-0.009\)}, {248, \(-0.007\)},
> {249, \
> \(-0.005\)}, {250, \(-0.004\)}, \n\ \ \ \ \ \ \ \ {251, \(-0.004\)},
> {252, \
> \(-0.004\)}, {253, \(-0.003\)}, {254, \(-0.004\)}, {255, \(-0.005\)},
> {256, \
> \(-0.004\)}, {257, \(-0.005\)}, {258, \(-0.006\)}, \n\ \ \ \ \ \ \ \
> {259, \
> \(-0.007\)}, {260, \(-0.007\)}, {261, \(-0.008\)}, {262, \(-0.009\)},
> {263, \
> \(-0.01\)}, {264, \(-0.011\)}, {265, \(-0.011\)}, {266, \(-0.012\)},
> \n\ \ \ \
> \ \ \ \ \ {267, \(-0.013\)}, {268, \(-0.017\)}, {269, \(-0.017\)}};\),
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>       FontSize->12]], "Input",
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> Cell[BoxData[
>     \(Length[Elev]\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(To\)\(\ \)\(compare\)\(\ \)\(our\)\(\ \
> \)\(solution\)\(\ \)\(model\)\(\ \)\(to\)\(\ \)\(the\)\(\
> \)\(elevation\)\(\ \
> \)\(record\)\); \ 
>       it\ is\ necessary\ to\ magnify\ the\ elevation\ record\ by\ a\
> factor\ \
> of\ 50. \ \ There\ is\ also\ a\ time\ lag\ between\ the\ solution\
> model\ and\
> \ the\ elevation\ record\ of\ 6\ time\ \(\(units\)\(:\)\)\ **) \)],
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> 
> Cell[BoxData[
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>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \( (*\(*\)\(\ \)\(The\)\(\ \)\(abbreviation\)\(\
> \)\("\<rsElev\>"\)\(\ \)\
> \(stands\)\(\ \)\(for\)\(\ \)\("\<re-scaled Elevation\>"\)\ **) \)],
> "Input"],
> 
> Cell[BoxData[
>     \(\(rsElev = 
>         Table[{i - Lag, M*\(Elev[\([i]\)]\)[\([2]\)]}, {i, 1, 
>             L}];\)\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \(plt4 = 
>       ListPlot[rsElev, PlotJoined \[Rule] True, 
>         PlotStyle \[Rule] RGBColor[0, 0, 1]]\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(Here\)\(\ \)\(we\)\(\ \)\(compare\)\(\
> \)\(our\)\(\ \
> \)\(solution\)\(\ \)\(model\)\(\ \)\(with\)\(\ \)\(our\)\(\
> \)\(well\)\(\ \
> \)\(elevation\)\(\ \)\(data . \ \ "\<Not close, and no cigar\>"\)\); \
>       because\ we\ are\ attempting\ two\ arbitrary\ values\ for\ p\
> and\ \
> \(\(q\)\(:\)\)\ \ **) \)], "Input"],
> 
> Cell[BoxData[
>     \(Show[{plt3, plt4}]\)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     \( (*\(\(*\)\(\ \)\(Remark : \ \ One\ can\ see\ a\ downward\
> trend\ in\ \
> the\ elevation\ record . \ \ I\ don' 
>             t\ know\ what\ physical\ factor\ "\<out there\>"\ causes\
> this\ \
> trend . \ \ If\ we\ could\ identify\ it\)\), \ 
>       and\ incorporate\ it\ into\ the\ Differential\ Equation; 
>       perhaps\ our\ "\<Elevation DE Model\>"\ could\ be\ more\
> accurate\ in\ \
> accounting\ for\ "\<the behavior\>"\ of\ the\ elevation\
> \(\(record\)\(.\)\)\ \
> **) \)], "Input",
>   FontFamily->"Arial"],
> 
> Cell[BoxData[
>     StyleBox[\(\( (*\(\(*\)\(\ \)\(Again\)\); \ 
>         what\ I' m\ looking\ for\ is\ to\ find\ a\ way\ to\ estimate\n
>           the\ values\ p\ and\ q, \ 
>         such\ that\ the\ solution\ y \((t)\)\ to\ the\ above\ DE, \
> \(becomes\
> \ a\ model\ that\ fits\ my\)\n\(water\ elevation\ data; \ 
>           i . e . \ a\ model\ in\ the\ least\ square\ sense\), \ 
>         showing\ a\ correlation\ \((of\ say)\)\ 0.95, \nor\ above, \ 
>         without\ having\ to\ engage\ into\ trial\ and\
> \(\(error\)\(.\)\)\ \ \
> **) \)\(\ \)\),
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