8 search hits
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Integral point sets over Z_n^m
(2007)
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Axel Kohnert
Sascha Kurz
- There are many papers studying properties of point sets in the Euclidean space or on integer grids, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of Z_n, and study the properties of the resulting combinatorial structures.
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There are integral heptagons, no three points on a line, no four on a circle
(2007)
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Tobias Kreisel
Sasch Kurz
- We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdös.
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On the minimum diameter of plane integral point sets
(2007)
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Sascha Kurz
Alfred Wassermann
- 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 state-of-the-art 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 n^{c_2loglog n} up to a constant in the exponent.
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Inclusion-maximal integral point sets over finite fields
(2007)
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Michael Kiermaier
Sascha Kurz
- We consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in $GF(q)^2$ where the formally defined Euclidean distance of every pair of points is an element of $GF(q)$. From another point of view we consider point sets over $GF(q)^2$ with few and prescribed directions. So this is related to Redeis work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces. In this article we study the spectrum of integral point sets over $GF(q)^2$ which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.
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Maximal integral point sets over Z^2
(2008)
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Sascha Kurz
Andrey Radoslavov Antonov
- Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P={p(1),...,p(n)} in Z^2 a maximal integral point set over Z^2 if all pairwise distances are integral and every additional point p(n+1) destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.
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A note on Erdös-Diophantine graphs and Diophantine carpets
(2005)
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Axel Kohnert
Sascha Kurz
- A Diophantine figure is a set of points on the integer grid $\mathbb{Z}^{2}$ where all mutual Euclidean distances are integers. We also speak of Diophantine graphs. The vertices are points in $\mathbb{Z}^{2}$ (the coordinates)and the edges are labeled with the distance between the two adjacent vertices, which is integral. In this language a Diophantine figure is a complete Diophantine graph. Two Diophantine graphs are equivalent if they only differ by translation or rotation of vertices. Due to a famous theorem of Erdös and Anning there are complete Diophantine graphs which are not contained in larger ones. We call them Erdös-Diophantine graphs. A special class of Diophantine graphs are Diophantine carpets. These are planar triangulations of a subset of the integer grid. We give an effective construction for Erdös-Diophantine graphs and characterize the chromatic number of Diophantine carpets.
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On the characteristic of integral point sets in $\mathbb{E}^m$
(2005)
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Sascha Kurz
- 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.
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Complex nanostructures in triblock terpolymer thin films
(2004)
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Sabine Ludwigs
- The thin film phase behavior of poly(styrene)-block-poly(2-vinylpyridine)-block-poly(tert-butyl methacrylate) (PS-b-P2VP-b-PtBMA) triblock terpolymers with volume fractions f(PS) : f(P2VP) : f(PtBMA) = 1 : 1.2 : x, with x ranging from 3.05 to 4, is studied with a combinatorial gradient approach. Gradients in film thickness are prepared via thin film flow coating of dilute solutions in chloroform. Upon controlled annealing in nearly saturated solvent vapor the films form terraces of well-defined step height. The dependence between morphology and film thickness is studied with optical microscopy, tapping mode SFM, and SEM. Though showing different morphologies in the bulk, the same sequence of surface structures is found with increasing film thickness for the whole range of compositions: a disordered phase in the thinnest regions, a liquid-like distribution of upright standing cylinders, cylinders oriented parallel to the film, and finally a hexagonally ordered perforated lamella structure (PL) on the first terrace with a thickness of d = (37+3) nm. Higher terraces also exhibit PL as surface structures. Due to the chemical nature of the block components and the particular stoichiometry of the polymer a wetting layer with a PtBMA-rich top layer is formed next to the substrate. By imposing an additional gradient in substrate surface energy, orthogonal to the gradually increasing film thickness, the perforated lamella is shown to be a stable phase, regardless of the chemical nature of the substrate, which makes this structure and methodology robust for application in nanotechnology. The complex phase behavior observed in thin films is supported by mesoscale computer simulations based on dynamic density functional theory. Thin films of the above mentioned triblock terpolymers are modeled as a melt of A3B4C12 Gaussian chains which is confined in a slit with film thickness H. By adjusting the interaction parameters between the polymer components and the surfaces, the experimentally observed sequence of surface structures as function of the film thickness can be successfully modeled. At well-defined film thickness the perforated lamella structure is formed. In analogy with earlier work on a two-component system these structures are identified as surface reconstructions of the bulk structure. In particular, the core-shell PL can be seen as analogue to the PL surface reconstruction of cylinder-forming AB and ABA systems. The influence of film thickness, surface field, and the interaction parameters between the different polymer components on the phase behavior is also explored. A large spectrum of surface structures is observed in analogy to the experiments. Further attention has been given to the perforated lamella structure. This structure can be visualized as P2VP/PS/P2VP lamellae which are perforated by channels of PtBMA interconnecting between two outer layers of PtBMA. A highly ordered PL structure could be prepared with a very small number of defects over an area of about 12 x 4 µm2. Because of the special functionalities of the triblock terpolymer a rather versatile nanostructure was produced. By selective UV-depolymerization of the PtBMA matrix phase, the PL phase might potentially be used for lithographic applications similar to the case of perpendicularly oriented poly(methyl methacrylate) (PMMA) cylinders in PS-b-PMMA block copolymer thin films. Furthermore, a responsive membrane can be created by selective removal of the matrix phase. The remaining PL has a P2VP shell which might be either switched via the pH-value or loaded with metal components. A polymer-analogous reaction of the matrix phase of the PL to poly(methacrylic acid) via acid-catalyzed hydrolysis leads to a pH-responsive nanostructure without altering the overall structure. With SFM in aqueous environment structural changes of the PL phase are studied as function of the pH-value. Upon changing the pH of the surrounding medium a strong swelling of the original film thickness is observed at pH-values > 6 to a maximum degree of 7.5-fold swelling. This swelling is explained with a conformational change of the matrix phase poly(methacrylic acid). The hexagonal arrangement of the pattern is not affected. The first two blocks PS and P2VP act as skeleton of the PL phase which withstands the mechanical forces exerted on the strongly swollen PMAA. In contrast to the PL phase core-shell cylinders oriented parallel to the interfaces cannot withstand these forces and are solubilised at high pH-values.
