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realizations that naturally occurring phenomena \ (river flow, atmospheric patterns, telecommunications, financial markets, \ etc) exhibit correlations that do not decay at a sufficiently fast rate. Significant correlations in observations separated by great periods of time. Long memory time series require different approaches to modeling than the \ short memory time series. \ \>", "Text", CellChangeTimes->{{3.475525979705353*^9, 3.475525999449602*^9}, { 3.475526413767523*^9, 3.475526417765077*^9}, {3.475526448467842*^9, 3.4755264892100573`*^9}, 3.477594433379613*^9, 3.477600668528789*^9, { 3.477629186757373*^9, 3.477629187438449*^9}, {3.477629361925931*^9, 3.477629403943083*^9}, {3.4776294754717493`*^9, 3.477629498016314*^9}, 3.477629595732176*^9, {3.477641995956359*^9, 3.477641997104498*^9}, 3.539285847946004*^9}], Cell[CellGroupData[{ Cell["Definitions of long memory", "Subsection", CellChangeTimes->{{3.467980611731907*^9, 3.467980635184805*^9}, { 3.475526060849709*^9, 3.475526070807515*^9}, {3.477340673934833*^9, 3.477340687566299*^9}}], Cell[TextData[{ "There are various definitions of long memory!\n\nWe focus on a covariance \ structure of the process:\n \n", StyleBox["Definition:", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Let ", Cell[BoxData[ FormBox[ SubscriptBox["X", "t"], TraditionalForm]]], " be a stationary process for which the following holds. 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", StyleBox["\n\nDefinition", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[":", FontWeight->"Bold"], " Let ", Cell[BoxData[ FormBox[ SubscriptBox["Y", "t"], TraditionalForm]]], " be a stochastic process with continuous time parameter ", Cell[BoxData[ FormBox["t", TraditionalForm]]], ". ", Cell[BoxData[ FormBox[ SubscriptBox["Y", "t"], TraditionalForm]]], " is called self-similar with self-similarity parameter ", Cell[BoxData[ FormBox["H", TraditionalForm]]], ", if for any positive stretching factor ", Cell[BoxData[ FormBox["c", TraditionalForm]]], ", the rescaled process with time scale ", Cell[BoxData[ FormBox["ct", TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["c", RowBox[{"-", "H"}]], SubscriptBox["Y", "ct"]}], TraditionalForm]]], ", is equal in distribution to the original process ", Cell[BoxData[ FormBox[ SubscriptBox["Y", "t"], TraditionalForm]]], ". (Beran 1994)\n\n\t", StyleBox["In other words: for any positive constant ", FontSize->12], StyleBox["c", FontSize->12, FontSlant->"Italic"], StyleBox[" and sequence of time points ", FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "1"], ",", "...", ",", SubscriptBox["t", "k"]}], TraditionalForm]], FontSize->12], StyleBox[", ", FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["c", RowBox[{"-", "H"}]], "(", RowBox[{ SubscriptBox["Y", SubscriptBox["ct", "1"]], ",", SubscriptBox["Y", SubscriptBox["ct", "2"]], ",", "...", ",", SubscriptBox["Y", SubscriptBox["ct", "k"]]}], ")"}], TraditionalForm]], FontSize->12], " ", StyleBox["has the same distribution as ", FontSize->12], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["Y", SubscriptBox["t", "1"]], ",", SubscriptBox["Y", SubscriptBox["t", "2"]], ",", "...", ",", SubscriptBox["Y", SubscriptBox["t", "k"]]}], ")"}], TraditionalForm]], FontSize->12], StyleBox[".", FontSize->12] }], "Text", CellChangeTimes->{{3.467984368044409*^9, 3.467984394655982*^9}, 3.4679844753430557`*^9, 3.467984952365877*^9, {3.4679850008022757`*^9, 3.467985054986305*^9}, {3.46798510923538*^9, 3.467985130112021*^9}, { 3.4679853686276503`*^9, 3.467985406751441*^9}, 3.4680019024060793`*^9, { 3.4755261270140457`*^9, 3.4755261273018007`*^9}, 3.475527608647297*^9, { 3.47552764443042*^9, 3.475527651179214*^9}, {3.477551297686142*^9, 3.4775514212940817`*^9}, {3.477551502734304*^9, 3.4775515588256483`*^9}, { 3.477551597831357*^9, 3.477551626240625*^9}, {3.477551666048917*^9, 3.477551825259779*^9}, {3.539203297875683*^9, 3.539203297875791*^9}, { 3.539203351319611*^9, 3.539203351319708*^9}, {3.539242993035795*^9, 3.539242993654249*^9}, {3.539243027076823*^9, 3.5392430432113237`*^9}, { 3.539244509153946*^9, 3.539244641782249*^9}, {3.539244678537426*^9, 3.5392447777939*^9}, 3.539244900656887*^9, {3.539244987483267*^9, 3.5392450193414717`*^9}, {3.539245102286283*^9, 3.53924512691965*^9}, { 3.539246339787119*^9, 3.53924634217638*^9}, {3.5392465264174128`*^9, 3.5392465459079447`*^9}, {3.539281607667377*^9, 3.539281608135263*^9}, { 3.5392821715626087`*^9, 3.539282172661004*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Stationary increments of self-similar process", "Subsection", CellChangeTimes->{{3.475527653518879*^9, 3.475527670073838*^9}, 3.4775943869232063`*^9, 3.5392449006571417`*^9}], Cell[TextData[{ StyleBox["Definition", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[":", FontWeight->"Bold"], " If for any k \[GreaterEqual] 1 and any time points ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "1"], ",", "...", ",", SubscriptBox["t", "k"]}], TraditionalForm]]], ", the distribution of ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["Y", RowBox[{ SubscriptBox["t", "1"], "+", "c"}]], "-", SubscriptBox["Y", RowBox[{ SubscriptBox["t", "1"], "+", "c", "-", "1"}]]}], ",", "...", ",", RowBox[{ SubscriptBox["Y", RowBox[{ SubscriptBox["t", "k"], "+", "c"}]], "-", SubscriptBox["Y", RowBox[{ SubscriptBox["t", "k"], "+", "c", "-", "1"}]]}]}], ")"}], TraditionalForm]]], " does not depend on ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"c", "\[Element]", "R"}], ","}], TraditionalForm]]], " then we say that ", Cell[BoxData[ FormBox[ SubscriptBox["Y", "t"], TraditionalForm]]], " has stationary increments. (Beran 1994)\n\nAll self-similar processes with \ stationary increments and ", Cell[BoxData[ FormBox[ RowBox[{"H", ">", "0"}], TraditionalForm]]], " can be obtained by partial sums\n\n", Cell[BoxData[ FormBox[ RowBox[{"\t", RowBox[{ SubscriptBox["S", "n"], "=", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["X", "1"], "+"}], "..."}], "+", SubscriptBox["X", "n"]}]}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"n", "=", "1"}], ",", "2", ",", "..."}], TraditionalForm]], FormatType->"TraditionalForm"], ".\n\n", Cell[BoxData[ FormBox[ RowBox[{"\t", RowBox[{ SubscriptBox["S", "n"], "=", RowBox[{ RowBox[{ SubscriptBox["Y", "n"], "-", SubscriptBox["Y", "0"]}], OverscriptBox["=", "d"], RowBox[{ RowBox[{ SuperscriptBox["n", "H"], "(", RowBox[{ SubscriptBox["Y", "1"], "-", SubscriptBox["Y", "0"]}], ")"}], "=", RowBox[{ SuperscriptBox["n", "H"], SubscriptBox["S", "1"]}]}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ 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3.477555358383642*^9}, 3.539244900659878*^9}], Cell[TextData[{ "We assume ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"E", "(", SubscriptBox["Y", "t"], ")"}], "=", "0"}], TraditionalForm]]], ". Let ", Cell[BoxData[ FormBox[ RowBox[{"s", "<", "t"}], TraditionalForm]]], " and denote the variance of the increment process ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["X", "t"], "=", RowBox[{ SubscriptBox["Y", "t"], "-", SubscriptBox["Y", RowBox[{"t", "-", "1"}]]}]}], TraditionalForm]]], " as \n\t\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Sigma]", "2"], "=", RowBox[{ RowBox[{"E", "[", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["Y", "t"], "-", SubscriptBox["Y", RowBox[{"t", "-", "1"}]]}], ")"}], "2"], "]"}], " ", "=", RowBox[{"E", "[", SuperscriptBox[ SubscriptBox["Y", "1"], "2"], "]"}]}]}], TraditionalForm]]], ". \n\nThen \n\t", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"E", "[", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["Y", "t"], "-", SubscriptBox["Y", "s"]}], ")"}], "2"], "]"}], " ", "=", RowBox[{ RowBox[{"E", "[", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["Y", RowBox[{"t", "-", "s"}]], "-", SubscriptBox["Y", "0"]}], ")"}], "2"], "]"}], "=", SuperscriptBox[ RowBox[{ SuperscriptBox["\[Sigma]", "2"], "(", RowBox[{"t", "-", "s"}], ")"}], RowBox[{"2", "H"}]]}]}]}], TraditionalForm]]], ". (*)\n\nWe can also write\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"E", "[", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["Y", "t"], "-", SubscriptBox["Y", "s"]}], ")"}], "2"], "]"}], " ", "=", RowBox[{ RowBox[{ RowBox[{"E", "[", SuperscriptBox[ SubscriptBox["Y", "t"], "2"], "]"}], "+", RowBox[{"E", "[", SuperscriptBox[ SubscriptBox["Y", "s"], "2"], "]"}], "-", RowBox[{"2", RowBox[{"E", "[", RowBox[{ SubscriptBox["Y", "t"], SubscriptBox["Y", "s"]}], "]"}]}]}], "=", RowBox[{ RowBox[{ SuperscriptBox["\[Sigma]", "2"], SuperscriptBox["t", RowBox[{"2", "H"}]]}], "+", RowBox[{ SuperscriptBox["\[Sigma]", "2"], SuperscriptBox["s", RowBox[{"2", "H"}]]}], "-", RowBox[{"2", RowBox[{ SubscriptBox["\[Gamma]", "y"], "(", RowBox[{"t", ",", "s"}], ")"}]}]}]}]}], TraditionalForm]]], ". (**)\n\nFrom (*) and (**) we get the covariance of ", Cell[BoxData[ FormBox[ SubscriptBox["Y", "t"], TraditionalForm]]], " as\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Gamma]", "y"], "(", RowBox[{"t", ",", "s"}], ")"}], "=", RowBox[{ FractionBox[ SuperscriptBox["\[Sigma]", "2"], "2"], "[", RowBox[{ SuperscriptBox["t", RowBox[{"2", "H"}]], "+", SuperscriptBox["s", RowBox[{"2", "H"}]], "-", SuperscriptBox[ RowBox[{"(", RowBox[{"t", "-", "s"}], ")"}], RowBox[{"2", "H"}]]}], "]"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4755277915811043`*^9, 3.4755278154132338`*^9}, { 3.477555382458005*^9, 3.477555717234432*^9}, {3.477555784082715*^9, 3.477555923654855*^9}, {3.477563707036107*^9, 3.477564270790596*^9}, 3.539244900662805*^9, {3.53928325397108*^9, 3.539283261454831*^9}, { 3.539283370207716*^9, 3.539283371292252*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The covariance and correlation of the increment sequence ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["X", "i"], "=", RowBox[{ SubscriptBox["Y", "i"], "-", SubscriptBox["Y", RowBox[{"i", "-", "1"}]]}]}], TraditionalForm]], "None"] }], "Subsection", CellChangeTimes->{{3.475527839821374*^9, 3.475527848196353*^9}, { 3.477564445058385*^9, 3.4775645348757277`*^9}, {3.477566159618863*^9, 3.47756616358638*^9}, 3.539244900663636*^9}], Cell[TextData[{ "The covariances of the increment sequence ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["X", "i"], "=", RowBox[{ SubscriptBox["Y", "i"], "-", SubscriptBox["Y", RowBox[{"i", "-", "1"}]]}]}], TraditionalForm]], "None"], " are obtained in a similar way\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Gamma]", "(", "k", ")"}], "=", RowBox[{ RowBox[{"cov", "(", RowBox[{ SubscriptBox["X", "i"], ",", SubscriptBox["X", RowBox[{"i", "+", "k"}]]}], ")"}], "=", RowBox[{"cov", "(", RowBox[{ SubscriptBox["X", "1"], ",", SubscriptBox["X", RowBox[{"1", "+", "k"}]]}], ")"}]}]}], TraditionalForm]]], "\n\t\n\t ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Gamma]", "(", "k", ")"}], "=", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[Sigma]", "2"], "2"], "[", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}], RowBox[{"2", "H"}]], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"k", "-", "1"}], ")"}], RowBox[{"2", "H"}]], "-", RowBox[{"2", SuperscriptBox["k", RowBox[{"2", "H"}]]}]}], "]"}], "."}]}], TraditionalForm]]], " \n\t \n The correlations are given by\n\n", Cell[BoxData[ FormBox[ RowBox[{"\t", RowBox[{ RowBox[{"\[Rho]", "(", "k", ")"}], "=", RowBox[{ FractionBox["1", "2"], "[", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}], RowBox[{"2", "H"}]], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"k", "-", "1"}], ")"}], RowBox[{"2", "H"}]], "-", RowBox[{"2", SuperscriptBox["k", RowBox[{"2", "H"}]]}]}], "]"}]}]}], TraditionalForm]]], ".\n\nAs ", Cell[BoxData[ FormBox[ RowBox[{"k", "\[LongRightArrow]", "\[Infinity]", " "}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Rho]", "(", "k", ")"}], "\[Tilde]", RowBox[{ RowBox[{"H", "(", RowBox[{ RowBox[{"2", "H"}], "-", "1"}], ")"}], SuperscriptBox["k", RowBox[{ RowBox[{"2", "H"}], "-", "2"}]]}]}], TraditionalForm]]], ", i.e.,\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"\[Rho]", "(", "k", ")"}], RowBox[{ RowBox[{"H", "(", RowBox[{ RowBox[{"2", "H"}], "-", "1"}], ")"}], SuperscriptBox["k", RowBox[{ RowBox[{"2", "H"}], "-", "2"}]]}]], "\[LongRightArrow]", "1"}], TraditionalForm]]], " as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"k", "\[LongRightArrow]", "\[Infinity]"}], "."}], TraditionalForm]]], "\n\nFor ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", "H", "<", RowBox[{"1", "/", "2"}]}], TraditionalForm]], FontWeight->"Bold"], " the correlations are summable. The process has short-range dependence. \n\n\ \t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", RowBox[{"-", "\[Infinity]"}]}], "\[Infinity]"], RowBox[{"\[Rho]", "(", "k", ")"}]}], "=", "0"}], TraditionalForm]]], ".\n\t\t\nFor ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"H", "=", RowBox[{"1", "/", "2"}]}], FontWeight->"Bold"], TraditionalForm]]], " correlations at all non-zero lags are zero i.e., the observation of the \ process ", Cell[BoxData[ FormBox[ SubscriptBox["X", "i"], TraditionalForm]]], " are uncorrelated.\n\nFor ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"1", "/", "2"}], ">", "H", ">", "1"}], FontWeight->"Bold"], TraditionalForm]]], " the process ", Cell[BoxData[ FormBox[ SubscriptBox["X", "i"], TraditionalForm]]], " has long memory. Correlations decay to zero very slowly and are not \ summable:\n\t\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", RowBox[{"-", "\[Infinity]"}]}], "\[Infinity]"], RowBox[{"\[Rho]", "(", "k", ")"}]}], "=", "\[Infinity]"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.477564537150467*^9, 3.477564576772419*^9}, { 3.4775646080465813`*^9, 3.4775647284357557`*^9}, {3.477564787277244*^9, 3.4775648664940443`*^9}, {3.47756491385106*^9, 3.477565017024911*^9}, { 3.4775650504985437`*^9, 3.477565234357901*^9}, {3.477565794587377*^9, 3.477566049441276*^9}, 3.4775661269272203`*^9, {3.477566174621188*^9, 3.4775663609094067`*^9}, {3.477566392045712*^9, 3.4775666149617023`*^9}, { 3.477566645131666*^9, 3.477566666185505*^9}, {3.477567836635085*^9, 3.47756795746659*^9}, {3.47756799920411*^9, 3.477568049820005*^9}, { 3.4775696043264713`*^9, 3.477569618249526*^9}, {3.477594366797723*^9, 3.4775943676348133`*^9}, 3.5392449006678057`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Remark", "Subsection", CellChangeTimes->{ 3.477588730269085*^9, {3.4775887704121*^9, 3.477588783410355*^9}, 3.539244900668001*^9}], Cell[TextData[{ "\nThe sample mean of stationary increments of self-similar process:\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["X", "_"], " ", "=", RowBox[{ RowBox[{ SuperscriptBox["n", RowBox[{"-", "1"}]], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "n"], SubscriptBox["X", "i"]}]}], "=", RowBox[{ RowBox[{ SuperscriptBox["n", RowBox[{"-", "1"}]], "(", RowBox[{ SubscriptBox["Y", "n"], "-", SubscriptBox["Y", "0"]}], ")"}], OverscriptBox["=", "d"], RowBox[{ SuperscriptBox["n", RowBox[{"-", "1"}]], RowBox[{ SuperscriptBox["n", "H"], "(", RowBox[{ SubscriptBox["Y", "1"], "-", SubscriptBox["Y", "0"]}], ")"}]}]}]}]}], TraditionalForm]]], "\n\nand the variance\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"var", "(", OverscriptBox["X", "_"], ")"}], "=", RowBox[{ SuperscriptBox["\[Sigma]", "2"], 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"q"]}]}], ")"}], SubscriptBox["\[Epsilon]", "t"]}]}]}], TraditionalForm]]], "\n\n", StyleBox["ARIMA (p, d, q)", FontWeight->"Bold"], ":\nSeries are said to be ARIMA(\[ScriptP],1,\[ScriptQ]), when the process ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["y", "t"], "-", SubscriptBox["y", RowBox[{"t", "-", "1"}]]}], "=", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "L"}], ")"}], SubscriptBox["y", "t"]}]}], TraditionalForm]]], " follows a stationary and invertible ARMA(\[ScriptP],\[ScriptQ]). Parameter \ \[ScriptD] is equal to 1, and means, that ARMA(\[ScriptP],\[ScriptQ]) series \ are differenced once (parameter ", Cell[BoxData[ FormBox["d", TraditionalForm]]], " is an integer, ", Cell[BoxData[ FormBox[ RowBox[{"d", "\[GreaterEqual]", "0"}], TraditionalForm]]], ").\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"\[Phi]", "(", "L", ")"}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "\[Minus]", "L"}], ")"}], "d"], SubscriptBox["X", "t"]}], "=", RowBox[{ RowBox[{"\[Theta]", "(", "L", ")"}], SubscriptBox["\[CurlyEpsilon]", "t"]}]}], TraditionalForm]]], "\n\n", StyleBox["ARFIMA", FontWeight->"Bold"], " generalize ARIMA by allowing ", Cell[BoxData[ FormBox["d", TraditionalForm]]], " to assume any real value.\n\n", StyleBox["Definition", FontWeight->"Bold"], ": Let ", Cell[BoxData[ FormBox[ SubscriptBox["X", "t"], TraditionalForm]]], " be a stationary process such that\n\n\t ", Cell[BoxData[ FormBox[ TagBox[ RowBox[{ RowBox[{ RowBox[{"\[Phi]", "(", "L", ")"}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "\[Minus]", "L"}], ")"}], "d"], SubscriptBox["X", "t"]}], "=", RowBox[{ RowBox[{"\[Theta]", "(", "L", ")"}], SubscriptBox["\[CurlyEpsilon]", "t"]}]}], "MathMLPresentationTag", AutoDelete->True], TraditionalForm]]], "\n\t \nfor some ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "<", "d", "<", FractionBox["1", "2"]}], TraditionalForm]]], ". 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