### Refine

#### Year of publication

#### Document Type

- Doctoral Thesis (400)
- Report (49)
- Preprint (23)
- Article (15)
- Working Paper (13)
- Master's Thesis (5)
- Bachelor Thesis (4)
- Conference Proceeding (4)
- Periodical (4)
- Book (2)

#### Language

- English (523) (remove)

#### Keywords

- Meteorologie (27)
- Grid Computing (21)
- Blockcopolymere (20)
- Nanopartikel (14)
- Polyelektrolyt (14)
- Mikrometeorologie (11)
- Polymere (10)
- Klimaänderung (9)
- Waldökosystem (9)
- Atmosphärische Turbulenz (8)

#### Institute

- Chemie (127)
- Geowissenschaften (127)
- Biologie (77)
- Mathematik (56)
- Physik (46)
- Wirtschaftswissenschaften (39)
- Bayreuther Graduiertenschule für Mathematik und Naturwissenschaften (BayNAT) (20)
- Informatik (10)
- BIGSAS (7)
- Facheinheit Ethnologie (6)

- "Smart" Hydrogels based on Trishydrophilic Triblock Terpolymers (2010)
- The work presented in this thesis focuses on the synthesis of double stimuli-responsive, trishydrophilic triblock terpolymers and their utilization for the construction of “smart” hydrogel systems, responding to a variety of external stimuli. The central focus was put on ABC triblock terpolymers composed of a pH-sensitive A block, a water soluble B block and a thermo-sensitive or multi-responsive C block. This concept was used for the construction of hydrogels responding independently to pH, temperature, and UV light. It was further applied to the formation of polymer/nanoparticle hybrid micelles suitable for the formation of magneto-responsive hydrogels (ferrogels). At first, a new route for the synthesis of block copolymers, containing ethylene oxide and glycidol derivatives, was developed. The crucial aspect of this procedure, based on sequential anionic polymerization, was the utilization of the phosphazene base t-BuP4, enabling the anionic polymerization of epoxide monomers in the presence of lithium counterions. It was shown, that ethoxyethyl glycidyl ether polymerizes easily under the established polymerization conditions without unwanted termination. Hence, we were able to synthesize well-defined block copolymers containing vinyl and epoxide monomers in a one-pot reaction, without performing additional intermediate steps. This new synthetic route was then utilized to synthesize a series of poly(2-vinylpyridine)-block-poly(ethylene oxide)-block-poly(glycidyl methyl ether-co-ethyl glycidyl ether) (P2VP-b-PEO-b-P(GME-co-EGE)) triblock terpolymers suitable for pH and temperature dependent hydrogel formation. The reversible gelation for this particular system relies on two distinct mechanisms. Under conditions, where only one outer block is insoluble, core-shell-corona (CSC) micelles are formed, resulting in gelation via close cubic packing of the micelles. On the other hand, the micelles are also able to crosslink through their corona when both outer blocks are insoluble. As a direct consequence, a temperature triggered gel-sol-gel transition occurred at pH = 7, accompanied by a unique gel strengthening. Solubility and gelation studies were performed by DLS, rheology and SANS. The influence of polymer concentrations and block lengths on the gelation behavior and gel properties was studied. In order to derive information about the exact structure of the cubic lattice formed in the low temperature gel phase (simple cubic or body centered cubic), a 19 wt% aqueous solution of a particular P2VP-b-PEO-b-P(GME-co-EGE) triblock terpolymer at pH = 7 was further investigated using SANS under steady shear. By application of shear stress, the irregularly arranged polydomains of the sample oriented macroscopically along a preferred direction, which led to highly defined, strongly anisotropic 2D scattering patterns. The interpretation of these patterns confirmed the presence of a body centered cubic packing. The gel-sol transition upon temperature increase can be explained by a shrinkage of the shell of the CSC micelles. To increase the versatility of the established hydrogel concept, we further synthesized ABC triblock terpolymers with different responsive polymers as C blocks. This required an alternative synthetic route, combining anionic polymerization and ATRP via “click” chemistry. After optimization of each synthetic step, exemplary poly(2-vinylpyridine)-block-poly(ethylene oxide)-block-poly(oligo(ethylene glycol) methacrylate) (P2VP-b-PEO-b-POEGMA) and poly(2-vinylpyridine)-block-poly(ethylene oxide)-block-poly(dimethyl- aminoethyl methacrylate) (P2VP-b-PEO-b-PDMAEMA) triblock terpolymers were synthesized, respectively, and characterized regarding their solubility and gelation behavior. At pH > 5, P2VP-b-PEO-b-PDMAEMA forms CSC micelles with a P2VP core, and a pH- as well as thermo-sensitive PDMAEMA corona. This particular structure represents a hydrogel, whose temperature dependent response can be easily changed from a gel-sol to a sol-gel transition by increasing the pH from 8 to 9. At pH = 7.5 on the other hand, gel formation is induced by the addition of hexacyanocobaltate(III) ions due to electrostatic interactions between the multivalent cobaltate ions and the charged DMAEMA units, causing a physical crosslinking of the CSC micelles. The gel can subsequently be disintegrated by an exposure to UV-light, based on a UV-catalyzed aquation of the trivalent hexacyanocobaltate(III) ions ions to divalent aquapentacyanocobaltate(III)-ions. In the last part, a new approach was developed to create a novel type of magnetic field-responsive hydrogels (ferrogels), in which the nanoparticles are tightly bound to the polymer matrix. The P2VP block of the previously synthesized P2VP-b-PEO-b-P(GME-co-EGE) triblock terpolymers was quaternized to a low extent and complexed with negatively charged, citrate stabilized maghemite (γ-Fe(III)-oxide) nanoparticles. Using different analytical methods it was shown that well-defined CSC hybrid micelles were obtained with cores formed by a complex of P2VP and 3-4 nanoparticles per core. Concentrated solutions of these micelles are able to form gels depending on temperature, as revealed by rheology measurements. Due to the presence of the maghemite particles, it is possible to induce gelation via remote heating using AC magnetic fields, which was demonstrated by high frequency magnetocalorimetry.

- "The New Chemistry" - Sustainable Catalysis with Alcohols (2013)
- Subject of the thesis are new iridium complexes stabilized by anionic P,N- or P,N,P-ligands. These complexes were used in homogeneous catalysis. Furthermore, mechanistic studies were performed to provide an insight into the catalytic cycles. Synthesis protocols for a multitude of different product classes have been developed. The iridium complex 1, stabilized by a neutral P,N-ligand, reacts under basic conditions with 2-aminopyridines. By elimination of dipyridylamine the new catalyst species 2a was formed, which is more stable than catalyst 1. Based on this finding eight new anionic P,N-ligands and the resulting iridium complexes were synthesized. After optimization of the reaction conditions (solvent, base, temperature and catalyst loading) these catalysts were used in BH (borrowing hydrogen)/HA (hydrogen autotransfer) reactions. The selective monoalkylation of anilines with primary alcohols was investigated. In comparative experiments the superiority of the new class of catalysts versus the original catalyst 1 was clearly shown. Under mild reaction conditions (70 °C) the selectivity profile with respect to the monoalkylation has been preserved. The catalytic protocol was subsequently extended to the alkylation of aromatic diamines. Therefore various diaminobenzenes were used as substrates. Also Dapsone®, an important drug in treatment of leprosy could be used as starting material. We succeeded in both symmetric and unsymmetric monoalkylations of diamines. Due to the selectivity profile of the catalyst regarding aromatic amines, also unprotected amino alcohols could be used as alkylating reagents. By the use of tridentate P,N,P-ligands, a novel class of more stable catalysts compared to complexes 2a-9a, could be developed. Due to sealing the synthesis reactor with a semipermeable membrane, the retransfer of the “borrowed” hydrogen could be prevented and H2 is released. Dehydrogenation and condensation steps are now possible instead of BH/HA. By reacting secondary alcohols with β-amino alcohols, pyrroles were accessible. After adapting the synthesis protocol to this new class of products the tolerance of functional groups was tested. Diversely functionalized alcohols were used. Under mild reaction conditions (90 °C) and very low catalyst loadings (down to 0.03 mol% iridium), a large number of novel pyrroles was accessible. Using this protocol 21 differently substituted α,α-pyrroles, 12 bicyclic pyrroles, symmetrically as well as non-symmetrically substituted oligopyrroles and three β-aminopyrroles were synthesized. The catalyst resting state was identified by NMR experiments and X-ray structure analysis to be an iridium trihydride. This trihydride is formed under catalytic conditions, by treatment of the pre-catalyst with alcohols or in hydrogen atmosphere. In the final part of the work, a catalytic pyridine synthesis was developed. In this so far unknown heterocycle synthesis up to four different substituents could be introduced within a single reaction step. 2,6-, 2,5-, 2,4- and 2,3-substituted pyridines were synthesized selectively by using variously substituted primary or secondary alcohols and γ-amino alcohols. Furthermore, both the synthesis of bicyclic pyridines as well as the synthesis of pyridines that bear chiral substituents is possible.

- (Na,K) Aluminosilicate Hollandites: Structures, Crystal Chemistry, and High-pressure Behaviour (2007)
- Aluminosilicates with the composition (Na,K)AlSi3O8 and the dense hollandite-type structure, in which all Si and Al are in six-fold coordination, are considered as a possible repository of potassium and sodium in the Earth´s mantle. The aim of this research is to explore the phase relation of the K-Na system at different temperatures and pressures, and to determine the physical-chemical properties and high-pressure behaviour of silicate hollandite-type structures containing K and Na in different concentrations. The (Na,K)AlSi3O8 hollandite solid solution has been synthesised using multi-anvil apparatus in the pressure range between 13 and 26 GPa and temperatures between 1500 and 2200 °C, using (Na0-0.6, K1-0.4)AlSi3O8 glasses, NaAlSi3O8 glass, and Na0.75K0.05Ca0.1AlSi3O8 glass as starting materials. The solubility of Na component into the KAlSi3O8 hollandite end-member increases with increasing pressure and temperature. Homogeneous assemblages with a pure hollandite phase (and maximum 1-2% of stishovite) were synthesized at temperature of 1700 °C and different pressures with up to 50% of NaAlSi3O8 component. No pure NaAlSi3O8 hollandite end-member was succsessfully synthesized. Considering the difference in heat dissipation between the shock events in meteorites and the multi-anvil presses, it appears likely that NaAlSi3O8 hollandite forms as a result of local high pressure and high temperature conditions and really fast quenching under non-equilibrium conditions. All synthesized hollandite samples have tetragonal I4/m symmetry at ambient conditions. The unit-cell volume and lattice parameters of the (Na,K)AlSi3O8 hollandite decreases linearly with increasing Na content. The a cell parameter decreases more rapidly than the c cell parameter, suggesting that changing the cation size in the tunnels of the hollandite structure affects more the a axis than the c axis. Structural refinements of single-crystal data collected for KAlSi3O8 and K0.8Na0.2AlSi3O8 hollandites are consistent with Si and Al disorder among the octahedral sites. The major difference between the KAlSi3O8 hollandite end-member and the K0.8Na0.2AlSi3O8 sample is the presence in the latter of a split site away from the 4th-fold axis. This position, occupied by ~ 75% of the total Na content, is closer to the framework walls and has a very distorted coordination polyhedron with only 5 Na1-O bond distances between 2.4 and 2.6 Angström whereas all other Na1-O bond distances are larger than 3 Angström. The high pressure behaviour of hollandite samples with compositions of KAlSi3O8, K0.8Na0.2AlSi3O8, K0.6Na0.4AlSi3O8, and K0.5Na0.5AlSi3O8 have been studied using diamond anvil cells and different pressure transmitting media, by means of X-ray powder diffraction and Raman spectroscopy. High temperature behaviour of K0.5Na0.5AlSi3O8 hollandite at high pressures has also been explored by means of X-ray powder diffraction. At high pressures, all tetragonal hollandite samples transform to a monoclinic (hollandite II) structure with space group I2/m. The transition pressure decreases with increasing Na component. Na substitution, thus, stabilizes the monoclinic phase, likely because the framework walls are more distorted than in the tetragonal phase and therefore more apt to accommodate the smaller Na atom. Second order Birch- Murnaghan equations of state were calculated for the tetragonal and monoclinic phases. If only experiments using He as pressure transmitting medium are compared, it appears that Na has little effect on the bulk modulus value of the tetragonal aluminosilicate hollandite, but increases the axial anisotropy. Monoclinic hollandites are more compressible, and are stable up to the highest pressures reached during the experiments, suggesting that they may be possible host minerals for Na and K in transition zone and even down to the Earth´s lower mantle. The lattice strains associated with the tetragonal I4/m to monoclinic I2/m transition have been determined. The phase transition is proper ferroelastic with negligible volume strain. The symmetry breaking strains e1-e2=a-b/a0 and e6=a/a0 x cos gamma are proportional to the order parameter Q associated with the transition and their squared values vary linearly with pressure indicating that the transition is second-order in character. The variation with pressure of the symmetry breaking strains is similar in K0.8Na0.2AlSi3O8 and KAlSi3O8 hollandites, suggesting that Na substitution mainly affects the transition pressure but not the transition mechanism. Results from the high pressure experiments show that the tetragonal to monoclinic phase transition is very sensitive to deviatoric stresses present during the experiments due to the different pressure transmitting media. These results might also give an indication of the possible effects arising from stresses on the mineral transitions in the Earth´s mantle.

- A bijection between the d-dimensional simplices with distances in {1,2} and the partitions of d+1 (2005)
- We give a construction for the d-dimensional simplices with all distances in {1,2} from the set of partitions of d+1.

- A Continuum Electrostatic Approach for Calculating The Binding Energetics of Multiple Ligands (2007)
- Complex biomolecules like proteins or nucleic acids can transiently bind various ligands, e.g., electrons, protons, ions or larger molecules. This property is the key to enzymatic catalysis, regulation and energy transduction in biological systems. Interactions between different ligand binding sites can lead to complex titration behaviors, which can be explained based on a microstate description of the system. Previous approaches to calculate the binding behavior of multiple ligand, only treated sites with one or no ligand bound by using a binary state vector to describe the system. Also only one or two ligand types, i.e., protons or electrons, were used for the calculation. In this thesis, I derive a more general formulation of the theory of ligand binding to biomolecules. For each site any number of charge forms and rotamer forms are allowed as well as any number of ligands and any number of ligand types can be bound. Charge and rotamer forms of sites can be parameterized by measurements on model reactions in solution or by quantum chemical calculations. An energy function is described, consistently combining experimentally determined contributions and those, which can be calculated by continuum electrostatics, molecular mechanics and quantum chemistry. Programs (i.e., QMPB and Perl Molecule) were developed to perform calculations based on the generalized ligand binding theory. The class library Perl Molecule was developed to write powerful Perl programs, which perform the necessary processing steps, e.g., for the conversion of a pdb-file into the input required for energy calculations. The generated input contains all experimentally determined or molecular mechanically and quantum chemically obtained parameters. The energy calculations are performed by the program QMPB, which uses other programs for the continuum electrostatics calculations to solve the linearized Poisson-Boltzmann equation. The computations scale linearly with the total number of sites of the system and can easily be performed in parallel. From the obtained energies, microscopic ligand binding probabilities can be calculated as function of chemical potentials of ligands in solution, e.g., by a Monte Carlo program. Additionally, microscopic and macroscopic equilibrium constants can be computed. The usefullness and correctness of the new approach based on a generalized ligand binding theory is demonstrated by a number of studies on diverse examples. Because various groups used Lysozyme as benchmark system for continuum electrostatics, it is chosen to test if previously obtained results can be reproduced with QMPB. Different quantum chemical approaches are applied to the benzoquinone system for parameterizing a site with several protonation and reduction forms. A complex site is also the CuB center in Cytochrome c oxidase, which is studied to decide if multiple protonation forms of the coordinating histidines are involved in the reaction mechanism. Factors influencing the reduction potential of the electron transfer protein ferredoxin are analyzed using the programs Perl Molecule and QMPB. Here, in particular conformational changes of a peptide bond in the vicinity of the [2Fe-2S] center are of interest. The protonation form of a neighboring glutamate turns out to influence the reduction potential strongly. Protonation and phosphorylation studies on the protein HPr lead to the development of a four-microstate model to explain conformational changes on a histidine, which can be observed by experiment, molecular dynamics simulation and electrostatic calculations. The phosphorylation and protonation state dependent conformational change can be related to the dual role of the protein in regulation and phosphate-transfer. The new microstate description does not only allow to analyze thermodynamic properties but also paved the road for the study of the kinetics of charge transfer.

- A Detailed Treatment of the Measurement of Transport Coefficients in Transient Grating Experiments (2007)
- This thesis treats the measurement of transport coefficients in transient grating experiments and is organized into 3 parts. Part 1 provides a brief review of the thermodynamic-phenomenological theory relevant for a correct description of the Soret effect. It comprises the formulation of the first law in open systems, the calculation of the entropy production, and the derivation of the phenomenological equations. This part is based on the books by de Groot and Mazur and by Haase and contains also some own results. We have explicitely derived a relation between reversible work and dissipation function, if heat and mass are exchanged reversibly and irreversibly between the two homogenous phases of a non-isothermal heterogenous system. Moreover we have discussed in detail, whether Onsager coefficients are invariant against shifts of enthalpy or entropy zero. Furthermore some comments on recent literature work have been made, since thermodynamic principles are not always correctly incorporated. In parts 2 and 3 we have treated the measurement of heat, mass and thermal diffusion in transient grating experiments. In part 2 we have presented a two-dimensional model to account for the role of heat conducting walls in the measurement of heat transport and Soret effect driven mass transport in transient holographic grating experiments. Heat diffusion into the walls leads to non-exponential decay of the temperature grating. Under certain experimental conditions it can be approximated by an exponential function and assigned an apparent thermal diffusivity D_{th,app} <D_{th,s}, where D_{th,s} is the true thermal diffusivity of the sample. The ratio D_{th,app}/D_{th,s} depends on only three dimensionless parameters, d/l_s, k_s/k_w, and D_{th,s}/D_{th,w}. d is the grating period, l_s the sample thickness, k_s and k_w the thermal conductivities of sample and wall, respectively, and D_{th,w} the thermal diffusivity of the wall. If at least two measurements are performed at different d/l_s, both D_{th,s} and k_s can be determined. Instead of costly solving partial differential equations, the unknown parameters can be obtained by finding the zero of an analytic function. For thin samples and large grating periods, heat conduction into the walls plays a predominant role and consequently the concentration grating in binary mixtures is no longer one-dimensional. Nevertheless, the normalized heterodyne diffraction efficiency of the concentration grating remains unaffected and the true mass and thermal diffusion coefficient and the correct Soret coefficient are still obtained from a simple one-dimensional model. All theoretical predictions have been tested by experiments on pure and binary liquids over a wide range of grating periods and sample thicknesses. Excellent agreement has been found in all cases. A new transient grating technique for the measurement of heat, mass and thermal diffusion in liquids has been introduced in part 3. Similar to holographic grating experiments, a temperature grating is created in the sample. Thermal expansion transforms the temperature into a refractive-index grating, which is read by diffraction of a readout laser beam. In a multicomponent mixture an additional concentration grating is formed by thermal diffusion driven by the temperature gradients of the temperature grating. Differently to laser induced dynamic grating experiments we use Joule heating instead of optical heating. For that purpose we have built cuvettes which have a grating of transparent conducting strips on the inner side of one of their windows. If heated by an electric current a temperature grating will build up in the sample. Both, the heat equation and the extended diffusion equation, have been solved in two dimensions to allow for quantitative data analysis. Our apparatus and method of analysis have been validated by measurements of heat, mass and thermal diffusion in pure and binary liquids. Heat diffusion can be correctly determined as was shown for pure toluene, pure dodecane and the symmetric mixture of isobutylbenzene-dodecane. Mass and thermal diffusion was studied in the three symmetric mixtures of dodecane, isobutylbenzene and tetralin. The obtained diffusion and Soret coefficients agree with the literature values within the experimental errors. Uncompensated transient heating effects limit the resolution of the experimental technique.

- A generalized job-shop problem with more than one resource demand per task (2011)
- We study a generalized job-shop problem called the Laser Sharing Problem with fixed tours (LSP-T) where the tasks may need more than one resource simultaneously. This fact will be used to model possible collisions between industrial robots. For three robots we will show that the special case where only one resource is used by more than one robot is already NP-hard. This also implies that one machine scheduling with chained min delay precedence constraints is NP-hard for at least three chains. On the positive side, we present a polynomial algorithm for the two robot case and a pseudo-polynomial algorithm together with an FPTAS for an arbitrary but constant number of robots. This gives a sharp boundary of the complexity status for a constant number of robots.

- A homotopy argument and its applications to the transformation rule for bi-Lipschitz mappings, the Brouwer fixed point theorem and the Brouwer degree (2005)
- The main purpose of the paper is to present an elementary self-contained proof of the change of variables formula for injective, locally bi-Lipschitz mappings. The proof is based on a homotopy argument. Various properties of bi-Lipschitz mappings are studied. As a by-product Lipschitz variants of the classical implicit function theorem and the local diffeomorphism theorem are proved. With the help of the homotopy argument a simple proof is given of Brouwer’s fixed point theorem and the main properties of Brouwer’s degree of mapping.

- A macroscopic and microscopic investigation of the magnesite – otavite solid soluition (2004)
- The magnesite (MgCO3) – calcite (CaCO3) system is the most geologically important of the carbonate minerals. At intermediate compositions a distinct ordered phase exists, dolomite [MgCa(CO3)2], which has symmetry. A complete, disordered ( c symmetry) solid solution exists, but at temperatures in excess of 1400°C. Unmixing and ordering processes that occur at high temperatures in the magnesite – calcite system make experimental work difficult. However, the use of an analogue system allows syntheses under more accessible experimental conditions. The effects of cation substitution and ordering in the magnesite (MgCO3) – otavite (CdCO3) solid solution have, therefore, been investigated using X-ray powder diffraction and Hard Mode Infrared Spectroscopy (HMIS). The results from Rietveld refinements of the X-ray powder diffraction data show that the variation of the a-axis is linear as a function of composition and is not affected by different degrees of order at intermediate compositions. However, the c-axis shows a positive deviation from linearity as a function of composition for the 800°C series. The observed deviation decreases with increasing degree of order for samples of composition Mg0.5Cd0.5CO3. Oxygen-carbon-oxygen bond angles for samples with symmetry showed a decrease in bond angle from 120° (constrained by symmetry in samples with the c structure) with increase in degree of order. Site occupancies were used to determine the long-range order parameter, Q, for samples with symmetry. The square of the order parameter varies linearly as a function of temperature, suggesting a second-order phase transition, and a critical transition temperature of Tc = 719°C. Intensity ratios between reflections present only in the low symmetry phase and reflections present in both the low and high symmetry phases can be used to determine a value proportional to the square of the long-range order parameter, Q. Intensity ratios, between the (101) and (202) reflections for samples of composition Mg0.5Cd0.5CO3, were determined from X-ray powder diffraction patterns and show a linear variation as a function of temperature, with a critical transition temperature, Tc = 716°C. The local mixing and ordering behaviour of the magnesite – otavite solid soluition has been characterised using infrared powder absorption spectroscopy. Phonon bands due to cadmium-oxygen translations, and magnesium-oxygen translations were both observed in the IR spectra for samples of intermediate composition. Band positions for cadmium-oxygen translations remained constant as a function of composition, whereas band positions for magnesium-oxygen translations showed a decrease in wavenumber as a function of composition. Vibrational bands in other spectral regions (libration and translation of carbonate groups and doubly degenetrate in-plane bending and out-of-plane bending of carbonate groups) vary linearly as a function of composition and no effect of order on band position is observed. At high frequencies the samples with c symmetry (800°C series) show a linear shift in band position as a function of composition, whereas, samples in the cadmium dolomite stability field, with symmetry, show a marked increase in frequency compared to their c symmetry counterparts. Cation substitution, or disordering causes broadening of IR vibrational bands for a given material. In this study an alternative method was used, which makes use of the autocorrelation function to establish average line widths for six spectral regions. The autocorrelation results, delta corr, can be interpreted in terms of local strain fields present in the structure due to cation substitution and disordering. In the low frequency region, the delta corr values show a positive deviation from linearity as a function of composition. For samples of composition Mg0.5Cd0.5CO3 the delta corr values are virtually all the same, independent of the degree of order. Over all the spectral regions, the addition of 10mol% MgCO3 (1 Mg atom every 2 unit cells) to the otavite results in a large increase in line width. It is, therefore, suggested that the strain fields surrounding the substituting magnesium atoms into the otavite structure are on the order of 30Å. In the low MIR and at a composition of Mg0.5Cd0.5CO3, an effect of order is observed as a reduction in delta corr values with respect to the sample with c symmetry. The difference in delta corr values between the ordered and disordered samples was used to determine the local-order parameter, q, which scales linearly with the long-range order parameter, Q.

- A note on Erdös-Diophantine graphs and Diophantine carpets (2005)
- 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.