Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the "lab-on-a-chip" paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results.

Simulations of charge transport in organic compounds

To aid the design of organic semiconductors, we study the charge transport properties of organic liquid crystals, i.e. hexabenzocoronene and carbazole macrocycle, and single crystals, i.e. rubrene, indolocarbazole and benzothiophene derivatives (BTBT, BBBT). The aim is to find structure-property relationships linking the chemical structure as well as the morphology with the bulk charge carrier mobility of the compounds. To this end, molecular dynamics (MD) simulations are performed yielding realistic equilibrated morphologies. Partial charges and molecular orbitals are calculated based on single molecules in vacuum using quantum chemical methods. The molecular orbitals are then mapped onto the molecular positions and orientations, which allows calculation of the transfer integrals between nearest neighbors using the molecular orbital overlap method. Thus we obtain realistic transfer integral distributions and their autocorrelations. In case of organic crystals the differences between two descriptions of charge transport, namely semi-classical dynamics (SCD) in the small polaron limit and kinetic Monte Carlo (KMC) based on Marcus rates, are studied. The liquid crystals are investigated solely in the hopping limit. To simulate the charge dynamics using KMC, the centers of mass of the molecules are mapped onto lattice sites and the transfer integrals are used to compute the hopping rates. In the small polaron limit, where the electronic wave function is spread over a limited number of neighboring molecules, the Schroedinger equation is solved numerically using a semi-classical approach. The results are compared for the different compounds and methods and, where available, with experimental data. The carbazole macrocycles form columnar structures arranged on a hexagonal lattice with side chains facing inwards, so columns can closely approach each other allowing inter-columnar and thus three-dimensional transport. When taking only intra-columnar transport into account, the mobility is orders of magnitude lower than in the three-dimensional case. BTBT is a promising material for solution-processed organic field-effect transistors. We are able to show that, on the time-scales of charge transport, static disorder due to slow side chain motions is the main factor determining the mobility. The resulting broad transfer integral distributions modify the connectivity of the system but sufficiently many fast percolation paths remain for the charges. Rubrene, indolocarbazole and BBBT are examples of crystals without significant static disorder. The high mobility of rubrene is explained by two main features: first, the shifted cofacial alignment of its molecules, and second, the high center of mass vibrational frequency. In comparsion to SCD, only KMC based on Marcus rates is capable of describing neighbors with low coupling and of taking static disorder into account three-dimensionally. Thus it is the method of choice for crystalline systems dominated by static disorder. However, it is inappropriate for the case of strong coupling and underestimates the mobility of well-ordered crystals. SCD, despite its one-dimensionality, is valuable for crystals with strong coupling and little disorder. It also allows correct treatment of dynamical effects, such as intermolecular vibrations of the molecules. Rate equations are incapable of this, because simulations are performed on static snapshots. We have thus shown strengths and weaknesses of two state of the art models used to study charge transport in organic compounds, partially developed a program to compute and visualize transfer integral distributions and other charge transport properties, and found structure-mobility relations for several promising organic semiconductors.

A novel functional renormalization group framework for gauge theories and gravity

In this thesis we develop further the functional renormalization group (RG) approach to quantum field theory (QFT) based on the effective average action (EAA) and on the exact flow equation that it satisfies. The EAA is a generalization of the standard effective action that interpolates smoothly between the bare action for krightarrowinfty and the standard effective action rnfor krightarrow0. In this way, the problem of performing the functional integral is converted into the problem of integrating the exact flow of the EAA from the UV to the IR. The EAA formalism deals naturally with several different aspects of a QFT. One aspect is related to the discovery of non-Gaussian fixed points of the RG flow that can be used to construct continuum limits. In particular, the EAA framework is a useful setting to search for Asymptotically Safe theories, i.e. theories valid up to arbitrarily high energies. A second aspect in which the EAA reveals its usefulness are non-perturbative calculations. In fact, the exact flow that it satisfies is a valuable starting point for devising new approximation schemes. In the first part of this thesis we review and extend the formalism, in particular we derive the exact RG flow equation for the EAA and the related hierarchy of coupled flow equations for the proper-vertices. We show how standard perturbation theory emerges as a particular way to iteratively solve the flow equation, if the starting point is the bare action. Next, we explore both technical and conceptual issues by means of three different applications of the formalism, to QED, to general non-linear sigma models (NLsigmaM) and to matter fields on curved spacetimes. In the main part of this thesis we construct the EAA for non-abelian gauge theories and for quantum Einstein gravity (QEG), using the background field method to implement the coarse-graining procedure in a gauge invariant way. We propose a new truncation scheme where the EAA is expanded in powers of the curvature or field strength. Crucial to the practical use of this expansion is the development of new techniques to manage functional traces such as the algorithm proposed in this thesis. This allows to project the flow of all terms in the EAA which are analytic in the fields. As an application we show how the low energy effective action for quantum gravity emerges as the result of integrating the RG flow. In any treatment of theories with local symmetries that introduces a reference scale, the question of preserving gauge invariance along the flow emerges as predominant. In the EAA framework this problem is dealt with the use of the background field formalism. This comes at the cost of enlarging the theory space where the EAA lives to the space of functionals of both fluctuation and background fields. In this thesis, we study how the identities dictated by the symmetries are modified by the introduction of the cutoff and we study so called bimetric truncations of the EAA that contain both fluctuation and background couplings. In particular, we confirm the existence of a non-Gaussian fixed point for QEG, that is at the heart of the Asymptotic Safety scenario in quantum gravity; in the enlarged bimetric theory space where the running of the cosmological constant and of Newton's constant is influenced by fluctuation couplings.

Picard-Fuchs equations of dimensionally regulated Feynman integrals

Zusammenfassung In der vorliegenden Arbeit besch¨aftige ich mich mit Differentialgleichungen von Feynman– Integralen. Ein Feynman–Integral h¨angt von einem Dimensionsparameter D ab und kann f¨ur ganzzahlige Dimension als projektives Integral dargestellt werden. Dies ist die sogenannte Feynman–Parameter Darstellung. In Abh¨angigkeit der Dimension kann ein solches Integral divergieren. Als Funktion in D erh¨alt man eine meromorphe Funktion auf ganz C. Ein divergentes Integral kann also durch eine Laurent–Reihe ersetzt werden und dessen Koeffizienten r¨ucken in das Zentrum des Interesses. Diese Vorgehensweise wird als dimensionale Regularisierung bezeichnet. Alle Terme einer solchen Laurent–Reihe eines Feynman–Integrals sind Perioden im Sinne von Kontsevich und Zagier. Ich beschreibe eine neue Methode zur Berechnung von Differentialgleichungen von Feynman– Integralen. ¨ Ublicherweise verwendet man hierzu die sogenannten ”integration by parts” (IBP)– Identit¨aten. Die neue Methode verwendet die Theorie der Picard–Fuchs–Differentialgleichungen. Im Falle projektiver oder quasi–projektiver Variet¨aten basiert die Berechnung einer solchen Differentialgleichung auf der sogenannten Griffiths–Dwork–Reduktion. Zun¨achst beschreibe ich die Methode f¨ur feste, ganzzahlige Dimension. Nach geeigneter Verschiebung der Dimension erh¨alt man direkt eine Periode und somit eine Picard–Fuchs–Differentialgleichung. Diese ist inhomogen, da das Integrationsgebiet einen Rand besitzt und daher nur einen relativen Zykel darstellt. Mit Hilfe von dimensionalen Rekurrenzrelationen, die auf Tarasov zur¨uckgehen, kann in einem zweiten Schritt die L¨osung in der urspr¨unglichen Dimension bestimmt werden. Ich beschreibe außerdem eine Methode, die auf der Griffiths–Dwork–Reduktion basiert, um die Differentialgleichung direkt f¨ur beliebige Dimension zu berechnen. Diese Methode ist allgemein g¨ultig und erspart Dimensionswechsel. Ein Erfolg der Methode h¨angt von der M¨oglichkeit ab, große Systeme von linearen Gleichungen zu l¨osen. Ich gebe Beispiele von Integralen von Graphen mit zwei und drei Schleifen. Tarasov gibt eine Basis von Integralen an, die Graphen mit zwei Schleifen und zwei externen Kanten bestimmen. Ich bestimme Differentialgleichungen der Integrale dieser Basis. Als wichtigstes Beispiel berechne ich die Differentialgleichung des sogenannten Sunrise–Graphen mit zwei Schleifen im allgemeinen Fall beliebiger Massen. Diese ist f¨ur spezielle Werte von D eine inhomogene Picard–Fuchs–Gleichung einer Familie elliptischer Kurven. Der Sunrise–Graph ist besonders interessant, weil eine analytische L¨osung erst mit dieser Methode gefunden werden konnte, und weil dies der einfachste Graph ist, dessen Master–Integrale nicht durch Polylogarithmen gegeben sind. Ich gebe außerdem ein Beispiel eines Graphen mit drei Schleifen. Hier taucht die Picard–Fuchs–Gleichung einer Familie von K3–Fl¨achen auf.

Analytische Ableitungen der Energie für den "Equation-of-Motion-Coupled-Cluster"-Ansatz zur Berechnung von Ionisationspotentialen

In der vorliegenden Arbeit wird die Theorie der analytischen zweiten Ableitungen für die EOMIP-CCSD-Methode formuliert sowie die durchgeführte Implementierung im Quantenchemieprogramm CFOUR beschrieben. Diese Ableitungen sind von Bedeutung bei der Bestimmung statischer Polarisierbarkeiten und harmonischer Schwingungsfrequenzen und in dieser Arbeit wird die Genauigkeit des EOMIP-CCSD-Ansatzes bei der Berechnung dieser Eigenschaften für verschiedene radikalische Systeme untersucht. Des Weiteren können mit Hilfe der ersten und zweiten Ableitungen vibronische Kopplungsparameter berechnet werden, welche zur Simulation von Molekülspektren in Kombination mit dem Köppel-Domcke-Cederbaum (KDC)-Modell - in der Arbeit am Beispiel des Formyloxyl (HCO2)-Radikals demonstriert - benötigt werden.rnrnDer konzeptionell einfache EOMIP-CC-Ansatz wurde gewählt, da hier die Wellenfunktion eines Radikalsystems ausgehend von einem stabilen geschlossenschaligen Zustand durch die Entfernung eines Elektrons gebildet wird und somit die Problematik der Symmetriebrechung umgangen werden kann. Im Rahmen der Implementierung wurden neue Programmteile zur Lösung der erforderlichen Gleichungen für die gestörten EOMIP-CC-Amplituden und die gestörten Lagrange-Multiplikatoren zeta zum Quantenchemieprogramm CFOUR hinzugefügt. Die unter Verwendung des Programms bestimmten Eigenschaften werden hinsichtlich ihrer Leistungsfähigkeit im Vergleich zu etablierten Methoden wie z.B. CCSD(T) untersucht. Bei der Berechnung von Polarisierbarkeiten und harmonischen Schwingungsfrequenzen liefert die EOMIP-CCSD-Theorie meist gute Resultate, welche nur wenig von den CCSD(T)-Ergebnissen abweichen. Einzig bei der Betrachtung von Radikalen, für die die entsprechenden Anionen nicht stabil sind (z.B. NH2⁻ und CH3⁻), liefert der EOMIP-CCSD-Ansatz aufgrund methodischer Nachteile keine aussagekräftige Beschreibung. rnrnDie Ableitungen der EOMIP-CCSD-Energie lassen sich auch zur Simulation vibronischer Kopplungen innerhalb des KDC-Modells einsetzen.rnZur Kopplung verschiedener radikalischer Zustände in einem solchen Modellpotential spielen vor allem die Ableitungen von Übergangsmatrixelementen eine wichtige Rolle. Diese sogenannten Kopplungskonstanten können in der EOMIP-CC-Theorie besonders leicht definiert und berechnet werden. Bei der Betrachtung des Photoelektronenspektrums von HCO2⁻ werden zwei Alternativen untersucht: Die vertikale Bestimmung an der Gleichgewichtsgeometrie des HCO2⁻-Anions und die Ermittlung adiabatischer Kraftkonstanten an den Gleichgewichtsgeometrien des Radikals. Lediglich das adiabatische Modell liefert bei Beschränkung auf harmonische Kraftkonstanten eine qualitativ sinnvolle Beschreibung des Spektrums. Erweitert man beide Modelle um kubische und quartische Kraftkonstanten, so nähern sich diese einander an und ermöglichen eine vollständige Zuordnung des gemessenen Spektrums innerhalb der ersten 1500 cm⁻¹. Die adiabatische Darstellung erreicht dabei nahezu quantitative Genauigkeit.

A case of distinct difference between the measured blood Ethanol concentration and the concentration estimated by Widmark's equation

In the last century, several mathematical models have been developed to calculate blood ethanol concentrations (BAC) from the amount of ingested ethanol and vice versa. The most common one in the field of forensic sciences is Widmark's equation. A drinking experiment with 10 voluntary test persons was performed with a target BAC of 1.2 g/kg estimated using Widmark's equation as well as Watson's factor. The ethanol concentrations in the blood were measured using headspace gas chromatography/flame ionization and additionally with an alcohol Dehydrogenase (ADH)-based method. In a healthy 75-year-old man a distinct discrepancy between the intended and the determined blood ethanol concentration was observed. A blood ethanol concentration of 1.83 g/kg was measured and the man showed signs of intoxication. A possible explanation for the discrepancy is a reduction of the total body water content in older people. The incident showed that caution is advised when using the different mathematical models in aged people. When estimating ethanol concentrations, caution is recommended with calculated results due to potential discrepancies between mathematical models and biological systems

An Aquaculture-Based Method for Calibrated Bivalve Isotope Paleothermometry

To quantify species- specific relationships between bivalve carbonate isotope geochemistry ( delta O-18(c)) and water conditions ( temperature and salinity, related to water isotopic composition [delta O-18(w)]), an aquaculture-based methodology was developed and applied to Mytilus edulis ( blue mussel). The four- by- three factorial design consisted of four circulating temperature baths ( 7, 11, 15, and 19 degrees C) and three salinity ranges ( 23, 28, and 32 parts per thousand ( ppt); monitored for delta O-18(w) weekly). In mid- July of 2003, 4800 juvenile mussels were collected in Salt Bay, Damariscotta, Maine, and were placed in each configuration. The size distribution of harvested mussels, based on 105 specimens, ranged from 10.9 mm to 29.5 mm with a mean size of 19.8 mm. The mussels were grown in controlled conditions for up to 8.5 months, and a paleotemperature relationship based on juvenile M. edulis from Maine was developed from animals harvested at months 4, 5, and 8.5. This relationship [ T degrees C = 16.19 (+/- 0.14) - 4.69 (+/- 0.21) {delta O-18(c) VPBD - delta O-18(w) VSMOW} + 0.17 (+/- 0.13) {delta O-18(c) VPBD - delta O-18(w) VSMOW}(2); r(2) = 0.99; N = 105; P < 0.0001] is nearly identical to the Kim and O'Neil ( 1997) abiogenic calcite equation over the entire temperature range ( 7 - 19 degrees C), and it closely resembles the commonly used paleotemperature equations of Epstein et al. ( 1953) and Horibe and Oba ( 1972). Further, the comparison of the M. edulis paleotemperature equation with the Kim and O'Neil ( 1997) equilibrium- based equation indicates that M. edulis specimens used in this study precipitated their shell in isotopic equilibrium with ambient water within the experimental uncertainties of both studies. The aquaculture- based methodology described here allows similar species- specific isotope paleothermometer calibrations to be performed with other bivalve species and thus provides improved quantitative paleoenvironmental reconstructions.

Gibbs point process approximation: Total variation bounds using Stein's method

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

Combinations of dexmedetomidine and alfaxalone with butorphanol in cats: application of an innovative stepwise optimisation method to identify optimal clinical doses for intramuscular anaesthesia

OBJECTIVES The aim of this study was to optimise dexmedetomidine and alfaxalone dosing, for intramuscular administration with butorphanol, to perform minor surgeries in cats. METHODS Initially, cats were assigned to one of five groups, each composed of six animals and receiving, in addition to 0.3 mg/kg butorphanol intramuscularly, one of the following: (A) 0.005 mg/kg dexmedetomidine, 2 mg/kg alfaxalone; (B) 0.008 mg/kg dexmedetomidine, 1.5 mg/kg alfaxalone; (C) 0.012 mg/kg dexmedetomidine, 1 mg/kg alfaxalone; (D) 0.005 mg/kg dexmedetomidine, 1 mg/kg alfaxalone; and (E) 0.012 mg/kg dexmedetomidine, 2 mg/kg alfaxalone. Thereafter, a modified 'direct search' method, conducted in a stepwise manner, was used to optimise drug dosing. The quality of anaesthesia was evaluated on the basis of composite scores (one for anaesthesia and one for recovery), visual analogue scales and the propofol requirement to suppress spontaneous movements. The medians or means of these variables were used to rank the treatments; 'unsatisfactory' and 'promising' combinations were identified to calculate, through the equation first described by Berenbaum in 1990, new dexmedetomidine and alfaxalone doses to be tested in the next step. At each step, five combinations (one new plus the best previous four) were tested. RESULTS None of the tested combinations resulted in adverse effects. Four steps and 120 animals were necessary to identify the optimal drug combination (0.014 mg/kg dexmedetomidine, 2.5 mg/kg alfaxalone and 0.3 mg/kg butorphanol). CONCLUSIONS AND RELEVANCE The investigated drug mixture, at the doses found with the optimisation method, is suitable for cats undergoing minor clinical procedures.

Continued Fraction Solutions to Hermite's, Legendre's and Laguerre's Differential Equation

The continued fraction method for solving differential equations is illustrated using three famous differential equations used in quantum chemistry.

Means and method for soil testing

An inexpensive device which is easily operated to accurately measure the Coulomb parameters of the soil. The Coulomb parameters are used in Coulomb's equation to calculate the shearing stresses along a failure surface of the soil. The device includes an instrument to test soil shear strength to which several weights have been added. To obtain the Coulomb parameters, the instrument is placed on the soil to be tested and weights are incrementally added to it. The instrument is rotated at each weight increment and the shearing stresses are read from its calibrated dial. The stresses are plotted on a graph from which the Coulomb parameters are determined. The shearing stress of the soil with any known force applied to it can then be determined.

Q method factor analysis of stakeholder social perspectives of the effectiveness of Environmental Impact Assessment public participation in the Western Cape, South Africa

Public participation is an integral part of Environmental Impact Assessment (EIA), and as such, has been incorporated into regulatory norms. Assessment of the effectiveness of public participation has remained elusive however. This is partly due to the difficulty in identifying appropriate effectiveness criteria. This research uses Q methodology to discover and analyze stakeholder's social perspectives of the effectiveness of EIAs in the Western Cape, South Africa. It considers two case studies (Main Road and Saldanha Bay EIAs) for contextual participant perspectives of the effectiveness based on their experience. It further considers the more general opinion of provincial consent regulator staff at the Department of Environmental Affairs and the Department of Planning (DEA&DP). Two main themes of investigation are drawn from the South African National Environmental Management Act imperative for effectiveness: firstly, the participation procedure, and secondly, the stakeholder capabilities necessary for effective participation. Four theoretical frameworks drawn from planning, politics and EIA theory are adapted to public participation and used to triangulate the analysis and discussion of the revealed social perspectives. They consider citizen power in deliberation, Habermas' preconditions for the Ideal Speech Situation (ISS), a Foucauldian perspective of knowledge, power and politics, and a Capabilities Approach to public participation effectiveness. The empirical evidence from this research shows that the capacity and contextual constraints faced by participants demand the legislative imperatives for effective participation set out in the NEMA. The implementation of effective public participation has been shown to be a complex, dynamic and sometimes nebulous practice. The functional level of participant understanding of the process was found to be significantly wide-ranging with consequences of unequal and dissatisfied stakeholder engagements. Furthermore, the considerable variance of stakeholder capabilities in the South African social context, resulted in inequalities in deliberation. The social perspectives revealed significant differences in participant experience in terms of citizen power in deliberation. The ISS preconditions are highly contested in both the Saldanha EIA case study and the DEA&DP social perspectives. Only one Main Road EIA case study social perspective considered Foucault's notion of governmentality as a reality in EIA public participation. The freedom of control of ones environment, based on a Capabilities approach, is a highly contested notion. Although agreed with in principle, all of the social perspectives indicate that contextual and capacity realities constrain its realisation. This research has shown that Q method can be applied to EIA public participation in South Africa and, with the appropriate research or monitoring applications it could serve as a useful feedback tool to inform best practice public participation.

Evapotranspiration and crop coefficients of rice (Oryza sativa L.) under sprinkler irrigation in a semiarid climate determined by the surface renewal method

The evapotranspiration (ETc) of sprinkler-irrigated rice was determined for the semiarid conditions of NE Spain during 2001, 2002 and 2003. The surface renewal method, after calibration against the eddy covariance method, was used to obtain values of sensible heat flux (H) from high-frequency temperature readings. Latent heat flux values were obtained by solving the energy balance equation. Finally, lysimeter measurements were used to validate the evapotranspiration values obtained with the surface renewal method. Seasonal rice evapotranspiration was about 750–800 mm. Average daily ETc for mid-season (from 90 to 130 days after sowing) was 5.1, 4.5 and 6.1 mm day−1 for 2001, 2002 and 2003, respectively. The experimental weekly crop coefficients fluctuated in the range of 0.83–1.20 for 2001, 0.81–1.03 for 2002 and 0.84–1.15 for 2003. The total growing season was about 150–160 days. In average, the crop coefficients for the initial (Kcini), mid-season (Kcmid) and late-season stages (Kcend) were 0.92, 1.06 and 1.03, respectively, the length of these stages being about 55, 45 and 25 days, respectively.