Please use this identifier to cite or link to this item: http://localhost/handle/Hannan/582999
 Title: Correction to A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities&x201D; Authors: Sangwoo Park;Erchin Serpedin;Khalid Qaraqe subject: Science & Technology Year: 2016 Publisher: IEEE Abstract: Several corrections are necessary in our paper [1]. We will next describe these corrections. On page (p.) 7134, in equation (21), a tilde is missing above $k$ in $k(x,y,f_{1})$. On p. 7135, in equations (22), (23) and two lines below equation (24), the minus sign − in front of constant lambda $should be changed into the plus sign +. Also, on p. 7135, the left-hand side (LHS) of equation (22) should be integrated with respect to$y. Thus, the correct form of (22) should be \begin{align*}&\hspace {-2pc} \int K'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2} ) + \lambda \tilde {L}'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2}) \\[2pt]&-\,\lambda (y) \tilde {k}'_{f^{*}_{1}}(x,y,f^{*}_{1}) dy =0 \qquad \qquad \qquad \text{(22)} \end{align*} Because of the correction in (22), several other equations in the paper have to be updated appropriately. Therefore, equation (67) will take the form: \begin{align*}&\hspace {-2.5pc}K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}}=\int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x})(- \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&\hspace {-1.5pc}+\, \log f_{\scriptscriptstyle X^{*}}(\mathbf {x})+\alpha _{0} +\boldsymbol {\zeta } \mathbf {x}^{\scriptscriptstyle T} + \mathbf {x}^{\scriptscriptstyle T}\boldsymbol {\Gamma } \mathbf {x}+1-\lambda (\mathbf {y})) d \mathbf {y} =0 \qquad \qquad \text{(67)}\end{align*} Similarly, equations (112) and (113) must be updated to: \begin{align*}&\hspace {-1pc}\int K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} d \mathbf {y}= \int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) [ -\mu \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&+\,(1-\alpha _{1})\log f_{\scriptscriptstyle X^{*}}(\mathbf {x}) +\mu \left ({\mu -1}\right ) \log f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) \\[2pt]&+\, \alpha _{0} + \sum _{i=1}^{n} \sum _{j=1}^{n} (\gamma _{ij} y_{i} y_{j} -\gamma _{ij} x_{i} x_{j} - \gamma _{ij} \left ({y-x}\right )_{i} \left ({y-x}\right )_{j} \\[2pt]&+\,\theta x_{i} x_{j} \xi _{i} \xi _{j} +\phi _{ij} y_{i} y_{j})- \lambda (\mathbf {y})+1-\alpha _{1}] d \mathbf {y}= 0.\qquad \qquad \text{(112)}\\[2pt]&\hspace {3pc}\int K'_{f_{\scriptscriptstyle Y}} d \mathbf {x}+ \tilde {K}'_{ f_{\scriptscriptstyle Y}} \Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} \\[2pt]&\hspace {4pc}= -\frac {\mu \int f_{\scriptscriptstyle X}(\mathbf {x})f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) d\mathbf {x}}{f_{\scriptscriptstyle Y}(\mathbf {y})} + \lambda (\mathbf {y}) \\[2pt]&\hspace {4pc} = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(113)}\end{align*} and the sentence following equation (113) should be read as: n‘The following functions satisfy the equalities in (112) and (113):’ URI: http://localhost/handle/Hannan/164994http://localhost/handle/Hannan/582999 ISSN: 0018-94481557-9654 volume: 62 issue: 7 Appears in Collections: 2016 Files in This Item: File Description SizeFormat 7469875.pdf106.57 kBAdobe PDF  Title: Correction to A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities&x201D; Authors: Sangwoo Park;Erchin Serpedin;Khalid Qaraqe subject: Science & Technology Year: 2016 Publisher: IEEE Abstract: Several corrections are necessary in our paper [1]. We will next describe these corrections. On page (p.) 7134, in equation (21), a tilde is missing abovek$in$k(x,y,f_{1})$. On p. 7135, in equations (22), (23) and two lines below equation (24), the minus sign − in front of constant lambda$ should be changed into the plus sign +. Also, on p. 7135, the left-hand side (LHS) of equation (22) should be integrated with respect to $y$. Thus, the correct form of (22) should be \begin{align*}&\hspace {-2pc} \int K'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2} ) + \lambda \tilde {L}'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2}) \\[2pt]&-\,\lambda (y) \tilde {k}'_{f^{*}_{1}}(x,y,f^{*}_{1}) dy =0 \qquad \qquad \qquad \text{(22)} \end{align*} Because of the correction in (22), several other equations in the paper have to be updated appropriately. Therefore, equation (67) will take the form: \begin{align*}&\hspace {-2.5pc}K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}}=\int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x})(- \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&\hspace {-1.5pc}+\, \log f_{\scriptscriptstyle X^{*}}(\mathbf {x})+\alpha _{0} +\boldsymbol {\zeta } \mathbf {x}^{\scriptscriptstyle T} + \mathbf {x}^{\scriptscriptstyle T}\boldsymbol {\Gamma } \mathbf {x}+1-\lambda (\mathbf {y})) d \mathbf {y} =0 \qquad \qquad \text{(67)}\end{align*} Similarly, equations (112) and (113) must be updated to: \begin{align*}&\hspace {-1pc}\int K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} d \mathbf {y}= \int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) [ -\mu \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&+\,(1-\alpha _{1})\log f_{\scriptscriptstyle X^{*}}(\mathbf {x}) +\mu \left ({\mu -1}\right ) \log f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) \\[2pt]&+\, \alpha _{0} + \sum _{i=1}^{n} \sum _{j=1}^{n} (\gamma _{ij} y_{i} y_{j} -\gamma _{ij} x_{i} x_{j} - \gamma _{ij} \left ({y-x}\right )_{i} \left ({y-x}\right )_{j} \\[2pt]&+\,\theta x_{i} x_{j} \xi _{i} \xi _{j} +\phi _{ij} y_{i} y_{j})- \lambda (\mathbf {y})+1-\alpha _{1}] d \mathbf {y}= 0.\qquad \qquad \text{(112)}\\[2pt]&\hspace {3pc}\int K'_{f_{\scriptscriptstyle Y}} d \mathbf {x}+ \tilde {K}'_{ f_{\scriptscriptstyle Y}} \Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} \\[2pt]&\hspace {4pc}= -\frac {\mu \int f_{\scriptscriptstyle X}(\mathbf {x})f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) d\mathbf {x}}{f_{\scriptscriptstyle Y}(\mathbf {y})} + \lambda (\mathbf {y}) \\[2pt]&\hspace {4pc} = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(113)}\end{align*} and the sentence following equation (113) should be read as: n‘The following functions satisfy the equalities in (112) and (113):’ URI: http://localhost/handle/Hannan/164994http://localhost/handle/Hannan/582999 ISSN: 0018-94481557-9654 volume: 62 issue: 7 Appears in Collections: 2016

Files in This Item:
File Description SizeFormat
 Title: Correction to A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities&x201D; Authors: Sangwoo Park;Erchin Serpedin;Khalid Qaraqe subject: Science & Technology Year: 2016 Publisher: IEEE Abstract: Several corrections are necessary in our paper [1]. We will next describe these corrections. On page (p.) 7134, in equation (21), a tilde is missing above $k$ in $k(x,y,f_{1})$. On p. 7135, in equations (22), (23) and two lines below equation (24), the minus sign − in front of constant lambda $should be changed into the plus sign +. Also, on p. 7135, the left-hand side (LHS) of equation (22) should be integrated with respect to$y\$. Thus, the correct form of (22) should be \begin{align*}&\hspace {-2pc} \int K'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2} ) + \lambda \tilde {L}'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2}) \\[2pt]&-\,\lambda (y) \tilde {k}'_{f^{*}_{1}}(x,y,f^{*}_{1}) dy =0 \qquad \qquad \qquad \text{(22)} \end{align*} Because of the correction in (22), several other equations in the paper have to be updated appropriately. Therefore, equation (67) will take the form: \begin{align*}&\hspace {-2.5pc}K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}}=\int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x})(- \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&\hspace {-1.5pc}+\, \log f_{\scriptscriptstyle X^{*}}(\mathbf {x})+\alpha _{0} +\boldsymbol {\zeta } \mathbf {x}^{\scriptscriptstyle T} + \mathbf {x}^{\scriptscriptstyle T}\boldsymbol {\Gamma } \mathbf {x}+1-\lambda (\mathbf {y})) d \mathbf {y} =0 \qquad \qquad \text{(67)}\end{align*} Similarly, equations (112) and (113) must be updated to: \begin{align*}&\hspace {-1pc}\int K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} d \mathbf {y}= \int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) [ -\mu \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&+\,(1-\alpha _{1})\log f_{\scriptscriptstyle X^{*}}(\mathbf {x}) +\mu \left ({\mu -1}\right ) \log f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) \\[2pt]&+\, \alpha _{0} + \sum _{i=1}^{n} \sum _{j=1}^{n} (\gamma _{ij} y_{i} y_{j} -\gamma _{ij} x_{i} x_{j} - \gamma _{ij} \left ({y-x}\right )_{i} \left ({y-x}\right )_{j} \\[2pt]&+\,\theta x_{i} x_{j} \xi _{i} \xi _{j} +\phi _{ij} y_{i} y_{j})- \lambda (\mathbf {y})+1-\alpha _{1}] d \mathbf {y}= 0.\qquad \qquad \text{(112)}\\[2pt]&\hspace {3pc}\int K'_{f_{\scriptscriptstyle Y}} d \mathbf {x}+ \tilde {K}'_{ f_{\scriptscriptstyle Y}} \Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} \\[2pt]&\hspace {4pc}= -\frac {\mu \int f_{\scriptscriptstyle X}(\mathbf {x})f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) d\mathbf {x}}{f_{\scriptscriptstyle Y}(\mathbf {y})} + \lambda (\mathbf {y}) \\[2pt]&\hspace {4pc} = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(113)}\end{align*} and the sentence following equation (113) should be read as: n‘The following functions satisfy the equalities in (112) and (113):’ URI: http://localhost/handle/Hannan/164994http://localhost/handle/Hannan/582999 ISSN: 0018-94481557-9654 volume: 62 issue: 7 Appears in Collections: 2016

Files in This Item:
File Description SizeFormat