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The problem of covering a circle, a square or a regular triangle with \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} $n$ \end{document} congruent circles of minimum diameter (the {\it circle covering} problem) has been investigated by a number of authors and the smallest diameter has been found for several values of \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} $n$ \end{document}. This paper is devoted to the study of an analogous problem, the {\it diameter covering} problem, in which the shape and congruence of the covering pieces is relaxed and -- invariably -- the maximal diameter of the pieces is minimized. All cases are considered when the solution of the first problem is known and in all but one case the diameter covering problem is solved.

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Abstract  

A concept of finite coverings of continua with a linear order of their members is given. A characterization is obtained of hereditarily locally connected continua which have a finite supremum of cardinalities of the considered coverings.

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43 Östergård, P. R. J. And Blass, U., On the size of optimal binary codes of length 9 and covering radius 1, IEEE Trans. Inform. Theory 47 (2001), 2556-2557. MR 2002i :94092

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Lassak, M., Covering a plane convex body by four homothetical copies with the smallest positive ratio, Geom. Dedicata 21 (1986), 157-167. MR 88c :52013 Covering a plane convex body by

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Bertolo, R., Östergård, P. R. J. and Weakley, W. D. , An updated table of binary/ternary mixed covering codes, J. Combin. Des. 12 (2004), 157–176. MR 2005a :94086

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– 18 10.1007/BF01313488 . [5] Guo , W. , Shum , K. P. , Skiba , A. N. 2003 G -covering subgroup systems for the classes of supersoluble and nilpotent groups Israel J. Math. 138 125

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Abstract  

An equilateral triangle T e of sides 1 can be parallel covered with any sequence of squares whose total area is not smaller than 1:5. Moreover, any sequence of squares whose total area does not exceed

\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{3} {4}(2 - \sqrt 3 )$$ \end{document}
(2 − √3) can be parallel packed into T e.

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Abstract  

By using coverings we introduce the concepts of fibrewise covering uniform space and its generalizations (fibrewise generalized uniform space and fibrewise semi-uniform space), and study the fibrewise completions of fibrewise generalized uniform spaces and fibrewise semi-uniform spaces.

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coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87–124. MR 80b :57003 Edmonds A. L. On the construction of branched coverings of low

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