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Januszewski, J. , A simple method of translative packing triangles in a triangle, Geombinatorics , 12(2) (2002), 61–68. MR 2003h :52020 Januszewski J. A simple method of

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., Translative covering of a convex body with its positive homothetic copies, Proceedings of the International Scientific Conference on Mathematics (Źilina, 1998), 29-34. MR 2000j :52025 Translative covering of a convex body with its

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Dembowski, P. , Finite geometries , Springer, 1968. MR 38 #1597 Hering, Ch. , On shears of translation planes, Abh. Math. Sem. Univ. Hamburg , 37 (1972), 258–268. MR 46 #4353

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Let t be an infinite graph, let p be a double ray in t, and letd anddp denote the distance functions in t and in p, respectively. One calls p anaxis ifd(x,y)=d p (x,y) and aquasi-axis if lim infd(x,y)/d p (x,y)>0 asx, y range over the vertex set of p andd p (x,y)?8. The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if a is a translation whose directionD(a) is a thin end, then there exists an axis inD(a) andD(a-1) invariant under ar for somer not exceeding the maximum number of disjoint rays inD(a).The thinness ofD(a) is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.

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Abstract  

A characterization formula of an orthonormal multiwavelet with di_erent real dilations and translations for L E 2(R) is presented. The result includes the known result on the classical Hardy space H 2(R).

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Abstract  

The fundamentals have been developed for a quantitative theory on the structure and dynamics of scientific networks. These fundamentals were conceived through a new vision of translation, defined mathematically as the derivative or gradient of the quality of the actors as a function of the coordinates for the space in which they perform. If we begin with the existence of a translation barrier, or an obstacle that must be overcome by the actors in order to translate, and if we accept the Maxwell-Boltzmann distribution as representative of the translating capacity of the actors, it becomes possible to demonstrate the known principle of “success breeds success.” We also propose two types of elemental translation: those which are irreverisble and those which are in equilibrium. In addition, we introduce the principle of composition, which enables, from elemental translations, the quantification of more complex ones.

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Abstract  

Every sequence of positive homothetic copies of a planar convex body C whose total area does not exceed a quarter of the area of C can be translatively packed in C.

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Summary  

Every sequence of positive or negative homothetic copies of a triangle~\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} $T$ \end{document} 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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\frac{2}{9}$ \end{document} of the area 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} $T$ \end{document} can be translatively packed into \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} $T$ \end{document}. The bound 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} $\frac{2}{9}$ \end{document} cannot be improved upon here.

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