000 05295nam a22006015i 4500
001 978-0-387-45676-8
003 DE-He213
005 20161121231122.0
007 cr nn 008mamaa
008 100301s2007 xxu| s |||| 0|eng d
020 _a9780387456768
_9978-0-387-45676-8
024 7 _a10.1007/978-0-387-45676-8
_2doi
050 4 _aQA402.5-402.6
072 7 _aPBU
_2bicssc
072 7 _aMAT003000
_2bisacsh
082 0 4 _a519.6
_223
100 1 _aSzabó, P. G.
_eauthor.
245 1 0 _aNew Approaches to Circle Packing in a Square
_h[electronic resource] :
_bWith Program Codes /
_cby P. G. Szabó, M. Cs. Markót, T. Csendes, E. Specht, L. G. Casado, I. García.
264 1 _aBoston, MA :
_bSpringer US,
_c2007.
300 _aXIV, 238 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSpringer Optimization and Its Applications,
_x1931-6828 ;
_v6
505 0 _aand Problem History -- Problem Definitions and Formulations -- Bounds for the Optimum Values -- Approximate Circle Packings Using Optimization Methods -- Other Methods for Finding Approximate Circle Packings -- Interval Methods for Validating Optimal Solutions -- The First Fully Interval-based Optimization Method -- The Improved Version of the Interval Optimization Method -- Interval Methods for Verifying Structural Optimality -- Repeated Patterns in Circle Packings -- Minimal Polynomials of Point Arrangements -- About the Codes Used.
520 _aIn one sense, the problem of finding the densest packing of congruent circles in a square is easy to understand: it is a matter of positioning a given number of equal circles in such a way that the circles fit fully in a square without overlapping. But on closer inspection, this problem reveals itself to be an interesting challenge of discrete and computational geometry with all its surprising structural forms and regularities. As the number of circles to be packed increases, solving a circle packing problem rapidly becomes rather difficult. To give an example of the difficulty of some problems, consider that in several cases there even exists a circle in an optimal packing that can be moved slightly while retaining the optimality. Such free circles (or "rattles”) mean that there exist not only a continuum of optimal solutions, but the measure of the set of optimal solutions is positive! This book summarizes results achieved in solving the circle packing problem over the past few years, providing the reader with a comprehensive view of both theoretical and computational achievements. Typically illustrations of problem solutions are shown, elegantly displaying the results obtained. Beyond the theoretically challenging character of the problem, the solution methods developed in the book also have many practical applications. Direct applications include cutting out congruent two-dimensional objects from an expensive material, or locating points within a square in such a way that the shortest distance between them is maximal. Circle packing problems are closely related to the "obnoxious facility location” problems, to the Tammes problem, and less closely related to the Kissing Number Problem. The emerging computational algorithms can also be helpful in other hard-to-solve optimization problems like molecule conformation. The wider scientific community has already been involved in checking the codes and has helped in having the computational proofs accepted. Since the codes can be worked with directly, they will enable the reader to improve on them and solve problem instances that still remain challenging, or to use them as a starting point for solving related application problems. Audience This book will appeal to those interested in discrete geometrical problems and their efficient solution techniques. Operations research and optimization experts will also find it worth reading as a case study of how the utilization of the problem structure and specialities made it possible to find verified solutions of previously hopeless high-dimensional nonlinear optimization problems with nonlinear constraints.
650 0 _aMathematics.
650 0 _aArithmetic and logic units, Computer.
650 0 _aComputer science
_xMathematics.
650 0 _aComputer mathematics.
650 0 _aConvex geometry.
650 0 _aDiscrete geometry.
650 0 _aMathematical optimization.
650 1 4 _aMathematics.
650 2 4 _aOptimization.
650 2 4 _aMath Applications in Computer Science.
650 2 4 _aConvex and Discrete Geometry.
650 2 4 _aArithmetic and Logic Structures.
650 2 4 _aComputational Science and Engineering.
700 1 _aMarkót, M. Cs.
_eauthor.
700 1 _aCsendes, T.
_eauthor.
700 1 _aSpecht, E.
_eauthor.
700 1 _aCasado, L. G.
_eauthor.
700 1 _aGarcía, I.
_eauthor.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9780387456737
830 0 _aSpringer Optimization and Its Applications,
_x1931-6828 ;
_v6
856 4 0 _uhttp://dx.doi.org/10.1007/978-0-387-45676-8
912 _aZDB-2-SMA
950 _aMathematics and Statistics (Springer-11649)
999 _c508867
_d508867