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  • Author or Editor: Robert Connelly x
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Summary  

Dedication to Kroly Bezdek

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A bicycle ( n , k )-gon is an equilateral n -gon whose k -diagonals are equal. S. Tabach-nikov proved that a regular n -gon is first-order flexible as a bicycle ( n , k )-gon if and only if there is an integer 2 ≦ rn -2 such that tan (π/ n ) tan ( kr π/ n ) = tan ( k π/ n ) tan ( r π/ n ). In the present paper, we solve this trigonometric diophantine equation. In particular, we describe the family of first order flexible regular bicycle polygons.

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Abstract  

Let 0 < c < s be fixed real numbers such that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${c \mathord{\left/ {\vphantom {c s}} \right. \kern-\nulldelimiterspace} s} \leqslant {{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$ \end{document}
, and let f : E2 → Ed for d ≥ 2 be a function such that for every p, qE 2 if p − q = c, then f(p) − f(q) ≤ c, and if p − q = s, then f(p) − f(q) ≥ s. Then f is a congruence. This result depends on and expands a result of Rdo et. al. [9], where a similar result holds, but for
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } 3}} \right. \kern-\nulldelimiterspace} 3}$$ \end{document}
replacing
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$ \end{document}
. We also present a further extensions where E2 is replaced by En for n > 2 and where the range of c/s is enlarged.

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Abstract  

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{E}^d$$ \end{document}
, which are centers of balls Bd(pi, r i) and Bd(qi, r i) of radius r i, for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{E}^d$$ \end{document}
, where the functions in question are constructed from maximum and minimum operations applied to functions each being radially symmetric monotone decreasing and integrable.

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