matlab-like notations  (MathWorks Inc)

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: The finite state machine for the inner loop of the Algorithm 1

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: The geometrical meaning of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\triangle _{i,j}^{o}$$\end{document} ▵ i , j o for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le i, j \le p -1$$\end{document} 0 ≤ i , j ≤ p - 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o \in \{T, L, D, R\}$$\end{document} o ∈ { T , L , D , R }

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: Left: Tiling for the square \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-\overline{r}_{3}, \overline{r}_{3}]^{2}$$\end{document} [ - r ¯ 3 , r ¯ 3 ] 2 where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{r}_{3} = \frac{5\sqrt{17}}{16} $$\end{document} r ¯ 3 = 5 17 16 . Right: Tiling for the square \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-3, 3]^{2}$$\end{document} [ - 3 , 3 ] 2

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: Left: An optimal configuration for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 3$$\end{document} n = 3 . Right: Triangles 7, 12 and 22 contain an optimal arrangement

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: Enclosures of the optimal arrangement for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 3$$\end{document} n = 3

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: Instances discarded in the first phase without processing

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: Statistics for the processed instances on phase 1

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: Statistics for the processed instances on phase 2

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques:

Journal: Journal of Global Optimization

Article Title: Rigorous packing of unit squares into a circle

doi: 10.1007/s10898-018-0711-5

Figure Lengend Snippet: Statistics for the processed instances on phase 3

Article Snippet: We employ the Matlab -like notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g := a:s:b$$\end{document} g : = a : s : b to denote the array with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k := \lfloor \frac{b - a}{s} \rfloor + 1$$\end{document} k : = ⌊ b - a s ⌋ + 1 elements where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{i} := a + i s$$\end{document} g i : = a + i s for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0,\ldots , k - 1$$\end{document} i = 0 , ... , k - 1 .

Techniques: