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Minimum diameters of plane integral point sets
"... ABSTRACT. Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its ..."
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Cited by 12 (11 self)
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ABSTRACT. Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2, n) of a plane integral point set consisting of n points arises. We give some new exact values and describe stateoftheart algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2, n) achieving the known upper bound nc2 log log n up to a constant in the exponent. A famous question of Erdős asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by ˙ d(2, n). Recently ˙ d(2, 7) = 22 270 could be determined via an exhaustive search. 1.
The Discovery of Simple 7Designs with Automorphism Group ...
, 1995
"... A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly ..."
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Cited by 10 (7 self)
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A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy behind the system is to transform problems into standard problems of e.g. group theory, graph theory, linear algebra, graphics, or databases and then to use highly specialized routines from that field to tackle the problems. The transformations required often follow the same principles especially in the case of generation and isomorphism testing.
MOLecular Structure GENeration with MOLGEN, new features and future developments
 Fresenius J. Anal. Chem
, 1997
"... MOLGEN is a computer program system which is designed for generating molecular graphs fast, redundancy free and exhaustively. In the present paper we describe its basic features, new features of the current release MOLGEN 3.5, and future developments which provide considerable improvements and ex ..."
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Cited by 6 (4 self)
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MOLGEN is a computer program system which is designed for generating molecular graphs fast, redundancy free and exhaustively. In the present paper we describe its basic features, new features of the current release MOLGEN 3.5, and future developments which provide considerable improvements and extensions. 1 Introduction MOLGEN [17] is a generator for molecular graphs (=connectivity isomers or constitutional formulae) allowing to generate all isomers that correspond to a given molecular formula and (optional) further conditions like prescribed and forbidden substructures, ring sizes etc. The input consists of ffl the empirical formula, together with ffl an optional list of macroatoms, which means prescribed substructures that must not overlap, ffl an optional goodlist, that consists of prescribed substructures which may overlap, ffl an optional badlist, containing forbidden substructures, ffl an optional interval for the minimal and maximal size of rings, ffl an optional num...
Molecules in Silico: The Generation of Structural Formulae and its Applications
 J. Comput. Chem. Jpn
"... In information processing, in combinatorial chemistry, in structure elucidation, and in several other fields of chemistry, the computeraided generation of all structures (constitutional formulae) within a defined structure space has become increasingly important. In this brief review the mathematic ..."
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Cited by 6 (5 self)
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In information processing, in combinatorial chemistry, in structure elucidation, and in several other fields of chemistry, the computeraided generation of all structures (constitutional formulae) within a defined structure space has become increasingly important. In this brief review the mathematical foundations of the classical molecular model and thus of the generation process are outlined, and the current state of structure generation as applied in software developed by the Bayreuth group is discussed.
On the characteristic of integral point sets in
 E m , Australas. J. Combin
"... We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to ..."
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Cited by 3 (3 self)
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We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets. Key words: integral distances, minimum diameter 2000 MSC: 52C10*, 11D99, 53C65 1
Discrete Mathematics for Combinatorial Chemistry
, 1998
"... The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry. ..."
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Cited by 2 (1 self)
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The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry.
Some Simple 7Designs
 Combinatorial Designs and Related Structures, Proceedings of the First Pythagorean Conference, volume 245 of London Mathematical Society Lecture Notes
"... Some simple 7designs with small parameters are constructed with the aid of a computer. The smallest parameter set found is 7(24; 8; 4): An automorphism group is prescribed for finding the designs and used for determining the isomorphism types. Further designs are derived from these designs by ..."
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Cited by 1 (1 self)
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Some simple 7designs with small parameters are constructed with the aid of a computer. The smallest parameter set found is 7(24; 8; 4): An automorphism group is prescribed for finding the designs and used for determining the isomorphism types. Further designs are derived from these designs by known construction processes.
History and Progress of the Generation of Structural Formulae in Chemistry and its Applications
"... After a few remarks on the history of molecular modelling we describe certain mathematical aspects of the generation of molecular structural formulae. The focus is on the automatic generation of structural formulae for the purpose of molecular structure elucidation and the examination of molecular l ..."
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Cited by 1 (1 self)
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After a few remarks on the history of molecular modelling we describe certain mathematical aspects of the generation of molecular structural formulae. The focus is on the automatic generation of structural formulae for the purpose of molecular structure elucidation and the examination of molecular libraries. The aim is to give a review and to point to relevant literature. We demonstrate an application in the area of quantitative structureproperty/activity relationships. Then, we give a glance on ongoing research in the generation of 3Dstructures (stereoisomers and conformers), and finally we mention two problems that should be solved in the near future, the possible use of hypergraphs, and the generation of patent libraries.
Mathematical Simulations in Combinatorial Chemistry
, 1996
"... A novel technique for chemical synthesis in drug research is combinatorial chemistry, where usually a set of buildingblock molecules is attached to a core structure in all the combinatorially possible ways. The resulting set of compounds (called a library) can then be systematically screened for a ..."
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A novel technique for chemical synthesis in drug research is combinatorial chemistry, where usually a set of buildingblock molecules is attached to a core structure in all the combinatorially possible ways. The resulting set of compounds (called a library) can then be systematically screened for a desired biological activity. In this paper we discuss ways and limits of a mathematical simulation of this procedure. At first, two methods for selecting the buildingblocks from a given structure pool are presented with the objective to obtain only dissilimar library entries. Next an algorithm is described for the exhaustive and redundancyfree generation of a combinatorial library, illustrated by a singlestep and a multicomponent reaction. Finally equations for the enumeration of the library sizes are derived and the limits of the virtual combinatorial chemistry, i.e. purely in computer and without experiment, are discussed. 1