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An Introduction to Distance Geometry applied to Molecular GeometryFrom Continuous to Discrete

An Introduction to Distance Geometry applied to Molecular Geometry: From Continuous to Discrete [One approach that has been used to solve the DGP is to represent it as a continuous optimization problem [59]. To understand it, we consider a DGP with K = 2, V = {u, v, s}, E = {{ u, v}, {v, s}}, where the associated quadratic system is (xu1−xv1)2+(xu2−xv2)2=duv2(xv1−xs1)2+(xv2−xs2)2=dvs2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle\begin{array}{rcl} (x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2}& =& d_{ uv}^{2} {}\\ (x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2}& =& d_{ vs}^{2}, {}\\ \end{array}$$ \end{document} which can be rewritten as (xu1−xv1)2+(xu2−xv2)2−duv2=0(xv1−xs1)2+(xv2−xs2)2−dvs2=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle\begin{array}{rcl} (x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2} - d_{ uv}^{2}& =& 0 {}\\ (x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2} - d_{ vs}^{2}& =& 0. {}\\ \end{array}$$ \end{document} Consider the function f:ℝ6→ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f: \mathbb{R}^{6} \rightarrow \mathbb{R}$$ \end{document}, defined by f(xu1,xu2,xv1,xv2,xs1,xs2)=(xu1−xv1)2+(xu2−xv2)2−duv22+(xv1−xs1)2+(xv2−xs2)2−dvs22.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle\begin{array}{rcl} f(x_{u1},x_{u2},x_{v1},x_{v2},x_{s1},x_{s2})& =& \left ((x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2} - d_{ uv}^{2}\right )^{2} {}\\ & +& \left ((x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2} - d_{ vs}^{2}\right )^{2}. {}\\ \end{array}$$ \end{document} It is not hard to realize that the solution x∗∈ℝ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x^{{\ast}}\in \mathbb{R}^{6}$$ \end{document} of the associated DGP can be found by solving the following problem: 3.1minx∈ℝ6f(x).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{ \min _{x\in \mathbb{R}^{6}}f(x). }$$ \end{document} That is, we wish to find the point x∗∈ℝ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x^{{\ast}}\in \mathbb{R}^{6}$$ \end{document} which attains the smallest value of f.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

An Introduction to Distance Geometry applied to Molecular GeometryFrom Continuous to Discrete

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/lp/springer-journals/an-introduction-to-distance-geometry-applied-to-molecular-geometry-ioPIItffJS
Publisher
Springer International Publishing
Copyright
© The Author(s) 2017
ISBN
978-3-319-57182-9
Pages
13 –20
DOI
10.1007/978-3-319-57183-6_3
Publisher site
See Chapter on Publisher Site

Abstract

[One approach that has been used to solve the DGP is to represent it as a continuous optimization problem [59]. To understand it, we consider a DGP with K = 2, V = {u, v, s}, E = {{ u, v}, {v, s}}, where the associated quadratic system is (xu1−xv1)2+(xu2−xv2)2=duv2(xv1−xs1)2+(xv2−xs2)2=dvs2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle\begin{array}{rcl} (x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2}& =& d_{ uv}^{2} {}\\ (x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2}& =& d_{ vs}^{2}, {}\\ \end{array}$$ \end{document} which can be rewritten as (xu1−xv1)2+(xu2−xv2)2−duv2=0(xv1−xs1)2+(xv2−xs2)2−dvs2=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle\begin{array}{rcl} (x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2} - d_{ uv}^{2}& =& 0 {}\\ (x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2} - d_{ vs}^{2}& =& 0. {}\\ \end{array}$$ \end{document} Consider the function f:ℝ6→ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f: \mathbb{R}^{6} \rightarrow \mathbb{R}$$ \end{document}, defined by f(xu1,xu2,xv1,xv2,xs1,xs2)=(xu1−xv1)2+(xu2−xv2)2−duv22+(xv1−xs1)2+(xv2−xs2)2−dvs22.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle\begin{array}{rcl} f(x_{u1},x_{u2},x_{v1},x_{v2},x_{s1},x_{s2})& =& \left ((x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2} - d_{ uv}^{2}\right )^{2} {}\\ & +& \left ((x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2} - d_{ vs}^{2}\right )^{2}. {}\\ \end{array}$$ \end{document} It is not hard to realize that the solution x∗∈ℝ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x^{{\ast}}\in \mathbb{R}^{6}$$ \end{document} of the associated DGP can be found by solving the following problem: 3.1minx∈ℝ6f(x).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{ \min _{x\in \mathbb{R}^{6}}f(x). }$$ \end{document} That is, we wish to find the point x∗∈ℝ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x^{{\ast}}\in \mathbb{R}^{6}$$ \end{document} which attains the smallest value of f.]

Published: Jul 14, 2017

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