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%% Generated: Mon Mar 12 10:01:09 EST 2007
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\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-mi(""))}{
\[\]}
\end{maplelatex}}{}
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\begin{maplelatex}\begin{Maple Normal}{\textbf{MATH 392 -- Seminar in Computational Commutative Algebra}}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{Maximum Likelihood Estimation in a Linear Mixture Model}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{We wish to determine the values of \begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("and", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"))}{
\[_{}\mathop{\rm  }{\rm \: \wedge \:}\mathop{\rm  }_{}\mathop{\rm  }\mathop{\rm  }\]}
\end{maplelatex}that maximize}\end{Maple Normal}
\begin{Maple Normal}{the log-likelihood function }\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{   l = \begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-mi("log", italic = "false", mathvariant = "normal"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("A", italic = "true", mathvariant = "italic")), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("A", italic = "true", mathvariant = "italic")), subscriptshift = "0")), superscriptshift = "0", subscriptshift = "0", Typesetting:-msemantics = "[none,none]"), Typesetting:-msubsup(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("C", italic = "true", mathvariant = "italic")), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("C", italic = "true", mathvariant = "italic")), subscriptshift = "0")), superscriptshift = "0", subscriptshift = "0", Typesetting:-msemantics = "[none,none]"), Typesetting:-msubsup(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("G", italic = "true", mathvariant = "italic")), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("G", italic = "true", mathvariant = "italic")), subscriptshift = "0")), superscriptshift = "0", subscriptshift = "0", Typesetting:-msemantics = "[none,none]"), Typesetting:-msubsup(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("T", italic = "true", mathvariant = "italic")), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("T", italic = "true", mathvariant = "italic")), subscriptshift = "0")), superscriptshift = "0", subscriptshift = "0", Typesetting:-msemantics = "[none,none]")), mathvariant = "normal"))}{
\[\left(_{}^{_{}}_{}^{_{}}_{}^{_{}}_{}^{_{}}\right)\]}
\end{maplelatex}}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{where}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{\begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("A", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity")), subscriptshift = "0"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".15", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".27", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".25", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0")), mathvariant = "normal"))}{
\[_{\mathop{\rm  }}=\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot\left(\mathop{\rm  }-\mathop{\rm  }_{}\mathop{\rm  }-\mathop{\rm  }_{}\right)\]}
\end{maplelatex}}\end{Maple Normal}
\begin{Maple Normal}{\begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("C", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity")), subscriptshift = "0"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".33", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".24", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".25", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0")), mathvariant = "normal"))}{
\[_{\mathop{\rm  }}=\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot\left(\mathop{\rm  }-\mathop{\rm  }_{}\mathop{\rm  }-\mathop{\rm  }_{}\right)\]}
\end{maplelatex}}\end{Maple Normal}
\begin{Maple Normal}{\begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("G", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity")), subscriptshift = "0"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".36", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".23", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".25", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0")), mathvariant = "normal"), Typesetting:-mi(""), Typesetting:-mspace(height = "0.0ex", width = "0.0em", depth = "0.0ex", linebreak = "newline"), Typesetting:-msub(Typesetting:-mi("p", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("T", italic = "true", mathvariant = "italic")), subscriptshift = "0"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".16", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".26", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("+", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn(".25", mathvariant = "normal"), Typesetting:-mo("&sdot;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&uminus0;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "mediummathspace", rspace = "mediummathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0")), mathvariant = "normal"))}{
\[_{\mathop{\rm  }}=\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot\left(\mathop{\rm  }-\mathop{\rm  }_{}\mathop{\rm  }-\mathop{\rm  }_{}\right)_{}=\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot_{}\mathop{\rm  }+\mathop{\rm  }\cdot\left(\mathop{\rm  }-\mathop{\rm  }_{}\mathop{\rm  }-\mathop{\rm  }_{}\right)\]}
\end{maplelatex}}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{and from the data  \begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("A", italic = "true", mathvariant = "italic")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn("10", mathvariant = "normal"), Typesetting:-mo(",", mathvariant = "normal", fence = "false", separator = "true", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "verythickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("C", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity")), subscriptshift = "0"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mn("14", mathvariant = "normal"), Typesetting:-mo(",", mathvariant = "normal", fence = "false", separator = "true", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "verythickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("G", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity")), subscriptshift = "0"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mn("15", mathvariant = "normal"), Typesetting:-mo(",", mathvariant = "normal", fence = "false", separator = "true", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "verythickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("u", italic = "true", mathvariant = "italic"), Typesetting:-mrow(Typesetting:-mi("T", italic = "true", mathvariant = "italic")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn("10.", mathvariant = "normal"))}{
\[_{}\mathop{\rm  }=\mathop{\rm  },\mathop{\rm  }\mathop{\rm  }_{\mathop{\rm  }}=,\mathop{\rm  }\mathop{\rm  }_{\mathop{\rm  }}=,\mathop{\rm  }_{}\mathop{\rm  }=\mathop{\rm  }\]}
\end{maplelatex}}\end{Maple Normal}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%p[A] := `*`(15, 1/100)*theta[1]+27/100*theta[2]+`*`(25, 1/100)*(1-theta[1]-theta[2]); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%-1/10*theta[1]+1/50*theta[2]+1/4}{
\[-1/10\,\theta_{{1}}+{\frac {1}{50}}\,\theta_{{2}}+1/4\]}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%p[C] := 33/100*theta[1]+`*`(24, 1/100)*theta[2]+`*`(25, 1/100)*(1-theta[1]-theta[2]); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%2/25*theta[1]-1/100*theta[2]+1/4}{
\[{\frac {2}{25}}\,\theta_{{1}}-{\frac {1}{100}}\,\theta_{{2}}+1/4\]}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%p[G] := `*`(36, 1/100)*theta[1]+23/100*theta[2]+`*`(25, 1/100)*(1-theta[1]-theta[2]); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%11/100*theta[1]-1/50*theta[2]+1/4}{
\[{\frac {11}{100}}\,\theta_{{1}}-{\frac {1}{50}}\,\theta_{{2}}+1/4\]}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%p[T] := `*`(16, 1/100)*theta[1]+`*`(26, 1/100)*theta[2]+`*`(25, 1/100)*(1-theta[1]-theta[2]); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%-9/100*theta[1]+1/100*theta[2]+1/4}{
\[-{\frac {9}{100}}\,\theta_{{1}}+{\frac {1}{100}}\,\theta_{{2}}+1/4\]}
\end{maplelatex}\begin{maplelatex}\begin{Maple Normal}{The log-likelihood function is}\end{Maple Normal}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%l := ln(p[A]^10*p[C]^14*p[G]^15*p[T]^10); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%ln((-1/10*theta[1]+1/50*theta[2]+1/4)^10*(2/25*theta[1]-1/100*theta[2]+1/4)^14*(11/100*theta[1]-1/50*theta[2]+1/4)^15*(-9/100*theta[1]+1/100*theta[2]+1/4)^10)}{
$\ln  \left(  \left( -1/10\,\theta_{{1}}+{\frac {1}{50}}\,\theta_{{2}}+1/4 \right) ^{10} \left( {\frac {2}{25}}\,\theta_{{1}}-{\frac {1}{100}}\,\theta_{{2}}+1/4 \right) ^{14}\\
\mbox{} \left( {\frac {11}{100}}\,\theta_{{1}}-{\frac {1}{50}}\,\theta_{{2}}+1/4 \right) ^{15} \left( -{\frac {9}{100}}\,\theta_{{1}}+{\frac {1}{100}}\,\theta_{{2}}+1/4 \right) ^{10}\\
\mbox{} \right) $}
\end{maplelatex}\begin{maplelatex}\begin{Maple Normal}{To maximize, we apply the usual process from multivariable calculus:}\end{Maple Normal}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%eq1 := numer(simplify(diff(l, theta[1]))); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%388080*theta[1]^3-1338*theta[2]^3-179706*theta[1]^2*theta[2]-2825625*theta[1]+457500*theta[2]-151950*theta[1]^2+42525*theta[1]*theta[2]-3825*theta[2]^2+27156*theta[1]*theta[2]^2+1359375}{
$388080\,{\theta_{{1}}}^{3}-1338\,{\theta_{{2}}}^{3}-179706\,{\theta_{{1}}}^{2}\theta_{{2}}-2825625\,\theta_{{1}}\\
\mbox{}+457500\,\theta_{{2}}-151950\,{\theta_{{1}}}^{2}+42525\,\theta_{{1}}\theta_{{2}}-3825\,{\theta_{{2}}}^{2}+27156\,\theta_{{1}}{\theta_{{2}}}^{2}\\
\mbox{}+1359375$}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%eq2 := numer(simplify(diff(l, theta[2]))); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%-60100*theta[1]^3+196*theta[2]^3+27332*theta[1]^2*theta[2]+457500*theta[1]-77500*theta[2]+21150*theta[1]^2-6550*theta[1]*theta[2]+650*theta[2]^2-4052*theta[1]*theta[2]^2-218750}{
$-60100\,{\theta_{{1}}}^{3}+196\,{\theta_{{2}}}^{3}+27332\,{\theta_{{1}}}^{2}\theta_{{2}}+457500\,\theta_{{1}}\\
\mbox{}-77500\,\theta_{{2}}+21150\,{\theta_{{1}}}^{2}-6550\,\theta_{{1}}\theta_{{2}}+650\,{\theta_{{2}}}^{2}-4052\,\theta_{{1}}{\theta_{{2}}}^{2}\\
\mbox{}-218750$}
\end{maplelatex}\begin{maplelatex}\begin{Maple Normal}{To solve the equations\begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mo("&PartialD;", mathvariant = "normal", fence = "unset", separator = "unset", stretchy = "unset", symmetric = "unset", largeop = "unset", movablelimits = "unset", accent = "unset", form = "", lspace = "", rspace = "", minsize = "", maxsize = ""), Typesetting:-mrow(Typesetting:-mi(""), Typesetting:-mo("&PartialD;", mathvariant = "normal", fence = "unset", separator = "unset", stretchy = "unset", symmetric = "unset", largeop = "unset", movablelimits = "unset", accent = "unset", form = "", lspace = "", rspace = "", minsize = "", maxsize = ""), Typesetting:-mspace(height = "0.0ex", width = "0.2em", depth = "0.0ex", linebreak = "auto"), Typesetting:-mi(""), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mi("")), linethickness = "1", denomalign = "center", numalign = "center", bevelled = "false"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mi("l", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mfrac(Typesetting:-mo("&PartialD;", mathvariant = "normal", fence = "unset", separator = "unset", stretchy = "unset", symmetric = "unset", largeop = "unset", movablelimits = "unset", accent = "unset", form = "", lspace = "", rspace = "", minsize = "", maxsize = ""), Typesetting:-mrow(Typesetting:-mi(""), Typesetting:-mo("&PartialD;", mathvariant = "normal", fence = "unset", separator = "unset", stretchy = "unset", symmetric = "unset", largeop = "unset", movablelimits = "unset", accent = "unset", form = "", lspace = "", rspace = "", minsize = "", maxsize = ""), Typesetting:-mspace(height = "0.0ex", width = "0.2em", depth = "0.0ex", linebreak = "auto"), Typesetting:-mi(""), Typesetting:-msub(Typesetting:-mi("&theta;", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mi("")), linethickness = "1", denomalign = "center", numalign = "center", bevelled = "false"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mi("l", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mn("0", mathvariant = "normal"), Typesetting:-mo(",", mathvariant = "normal", fence = "false", separator = "true", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "verythickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"))}{
\[\frac{\partial }{\partial _{}}\mathop{\rm  }\mathop{\rm  }=\mathop{\rm  }\frac{\partial }{\partial _{}}\mathop{\rm  }\mathop{\rm  }=\mathop{\rm  },\mathop{\rm  }\]}
\end{maplelatex}we will use our Grobner basis tools:}\end{Maple Normal}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%with(Groebner); -1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%B := Basis([eq1, eq2], plex(theta[1], theta[2])); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%[15059072*theta[2]^9-204388812773437500*theta[2]^4-302623360342382812500*theta[2]^3-3313234506368408203125*theta[2]^2+14248392817565917968750*theta[2]-2936000335693359375000+67847907642187500*theta[2]^5+202379922115625*theta[2]^6-2101960071750*theta[2]^7-3429945120*theta[2]^8, 779561911612527954533492789860607679554680597045898437500000*theta[1]+1143860978095040918551172086560432234583517824*theta[2]^8+5339977152447099985278623003643801908484503306662265625*theta[2]^4+40162789792827709797250228787649725530614274117458984375*theta[2]^3-23423362144514565485131811977539152418946067767083544921875*theta[2]^2-620940871147968061708026086268620360893238024177277832031250*theta[2]-268685817354617105102776149703221477505851547181396484375000+13616713092144043710912659018096645167388689847096875*theta[2]^5-162695540456494558305637643153494478554261764306750*theta[2]^6-247728437227096996819990448303546357011869879840*theta[2]^7]}{
$[15059072\,{\theta_{{2}}}^{9}-204388812773437500\,{\theta_{{2}}}^{4}-302623360342382812500\,{\theta_{{2}}}^{3}\\
\mbox{}-3313234506368408203125\,{\theta_{{2}}}^{2}+14248392817565917968750\,\theta_{{2}}\\
\mbox{}-2936000335693359375000+67847907642187500\,{\theta_{{2}}}^{5}+202379922115625\,{\theta_{{2}}}^{6}\\
\mbox{}-2101960071750\,{\theta_{{2}}}^{7}-3429945120\,{\theta_{{2}}}^{8},779561911612527954533492789860607679554680597045898437500000\\
\mbox{}\,\theta_{{1}}+1143860978095040918551172086560432234583517824\,{\theta_{{2\\
\mbox{}}}}^{8}+5339977152447099985278623003643801908484503306662265625\\
\mbox{}\,{\theta_{{2}}}^{4}+40162789792827709797250228787649725530614274117458984375\\
\mbox{}\,{\theta_{{2}}}^{3}\\
\mbox{}-23423362144514565485131811977539152418946067767083544921875\\
\mbox{}\,{\theta_{{2}}}^{2}-620940871147968061708026086268620360893238024177277832031250\\
\mbox{}\,\theta_{{2}}-268685817354617105102776149703221477505851547181396484375000\\
\mbox{}+13616713092144043710912659018096645167388689847096875\\
\mbox{}\,{\theta_{{2}}}^{5}-162695540456494558305637643153494478554261764306750\\
\mbox{}\,{\theta_{{2}}}^{6}-247728437227096996819990448303546357011869879840\,{\theta_{{2\\
\mbox{}}}}^{7}]$}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%nops(B); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%2}{
\[2\]}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%factor(B[1]); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%(8*theta[2]-25)*(7*theta[2]-500)*(theta[2]+15)*(2*theta[2]+525)*(theta[2]+75)*(theta[2]-425)*(134456*theta[2]^3-10852275*theta[2]^2-4304728125*theta[2]+935718750)}{
$ \left( 8\,\theta_{{2}}-25 \right)  \left( 7\,\theta_{{2}}-500 \right)  \left( \theta_{{2}}+15 \right) \\
\mbox{} \left( 2\,\theta_{{2}}+525 \right)  \left( \theta_{{2}}+75 \right)  \left( \theta_{{2}}-425 \right)  \left( 134456\,{\theta_{{2}}}^{3}-10852275\,{\theta_{{2}}}^{2}-4304728125\,\theta_{{2}}+935718750\\
\mbox{} \right) $}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%rts := [fsolve(B[1], theta[2], complex)]; 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%[-262.5000000, -143.2006009, -75.00000000, -15.00000000, .2172513326, 3.125000000, 71.42857143, 223.6958131, 425.]}{
$[- 262.5000000,- 143.2006009\\
\mbox{},- 75.0\\
\mbox{},- 15.0, 0.2172513326\\
\mbox{}, 3.125000000, 71.42857143\\
\mbox{}, 223.6958131, 425.0\\
\mbox{}]$}
\end{maplelatex}\begin{maplelatex}\begin{Maple Normal}{Of these, note that only one is in the range  \begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-mn("0", mathvariant = "normal"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&le;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mi("r", italic = "true", mathvariant = "italic"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("&le;", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mn("1.", mathvariant = "normal"))}{
\[\mathop{\rm  }\le \mathop{\rm  }\mathop{\rm  }\le \]}
\end{maplelatex}}\end{Maple Normal}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%fsolve(subs(theta[2] = rts[5], B[2]), theta[1]); 1}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%.5191263946}{
\[ 0.5191263946\]}
\end{maplelatex}\begin{maplelatex}\begin{Maple Normal}{We have exactly one root in the "probability simplex"}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{\begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(",", mathvariant = "normal", fence = "false", separator = "true", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "verythickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("2", mathvariant = "normal")), subscriptshift = "0")), mathvariant = "normal"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mo("=", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "thickmathspace", rspace = "thickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn("0.5191263946", mathvariant = "normal"), Typesetting:-mo(",", mathvariant = "normal", fence = "false", separator = "true", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "infix", lspace = "0em", rspace = "verythickmathspace", minsize = "1", maxsize = "infinity"), Typesetting:-mn("0.2172513326", mathvariant = "normal")), mathvariant = "normal"))}{
\[\left(_{},\mathop{\rm  }_{}\right)\mathop{\rm  }=\mathop{\rm  }\left(,\right)\]}
\end{maplelatex}}\end{Maple Normal}
\begin{Maple Normal}{}\end{Maple Normal}
\begin{Maple Normal}{Since \begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi("θ", italic = "false", mathvariant = "normal"), Typesetting:-mrow(Typesetting:-mn("1", mathvariant = "normal")), subscriptshift = "0"), Typesetting:-mo(" ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric = "false", largeop = "false", movablelimits = "false", accent = "false", form = "", lspace = "0em", rspace = "0em", minsize = "1", maxsize = "infinity"))}{
\[_{}\mathop{\rm  }\]}
\end{maplelatex}is relatively large here, the data would indicate}\end{Maple Normal}
\begin{Maple Normal}{that this region is a CG-rich area.  }\end{Maple Normal}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{maplelatex}\mapleinline{inert}{2d}{%Typesetting:-mrow(Typesetting:-mi(""))}{
\[\]}
\end{maplelatex}}{}
\end{mapleinput}
\end{document}
