The Economics of Managing Wildlife Disease

The spread of infectious disease among and between wild and domesticated animals has become a major problem worldwide. Upon analyzing the dynamics of wildlife growth and infection when the diseased animals cannot be identified separately from healthy wildlife prior to the kill, we find that harvest-based strategies alone have no impact on disease transmission. Other controls that directly influence disease transmission and/or mortality are required. Next, we analyze the socially optimal management of infectious wildlife. The model is applied to the problem of bovine tuberculosis among Michigan white-tailed deer, with non-selective harvests and supplemental feeding being the control variables. Using a two-state linear control model, we find a two-dimensional singular path is optimal (as opposed to a more conventional bang-bang solution) as part of a cycle that results in the disease remaining endemic in the wildlife. This result follows from non-selective harvesting and intermittent wildlife productivity gains from supplemental feeding.

CHEMICAL AND BIOCHEMICAL PROPERTIES OF ARAUCARIA ANGUSTIFOLIA (BERT.) KTZE. FOREST SOILS IN THE STATE OF SAO PAULO

Araucaria angustifolia, commonly named Araucaria, is a Brazilian native species that is intensively exploited due to its timber quality. Therefore, Araucaria is on the list of species threatened by extinction. Despite the importance of soil for forest production, little is known about the soil properties of the highly fragmented Araucaria forests. This study was designed to investigate the use of chemical and biological properties as indicators of conservation and anthropogenic disturbance of Araucaria forests in different sampling periods. The research was carried out in two State parks of Sao Paulo: Parque Estadual Turistico do Alto do Ribeira and Parque Estadual de Campos de Jordao. The biochemical properties carbon and nitrogen in microbial biomass (MB-C and MB-N), basal respiration (BR), the metabolic quotient (qCO(2)) and the following enzyme activities: beta-glucosidase, urease, and fluorescein diacetate hydrolysis (FDA) were evaluated. The sampling period (dry or rainy season) influenced the results of mainly MB-C, MB-N, BR, and qCO(2). The chemical and biochemical properties, except K content, were sensitive indicators of differences in the conservation and anthropogenic disturbance stages of Araucaria forests. Although these forests differ in biochemical and chemical properties, they are efficient in energy use and conservation, which is shown by their low qCO(2), suggesting an advanced stage of succession.

A NEW METHOD TO QUANTIFY X-RAY SUBSTRUCTURES IN CLUSTERS OF GALAXIES

We present a new method to quantify substructures in clusters of galaxies, based on the analysis of the intensity of structures. This analysis is done in a residual image that is the result of the subtraction of a surface brightness model, obtained by fitting a two-dimensional analytical model (beta-model or Sersic profile) with elliptical symmetry, from the X-ray image. Our method is applied to 34 clusters observed by the Chandra Space Telescope that are in the redshift range z is an element of [0.02, 0.2] and have a signal-to-noise ratio (S/N) greater than 100. We present the calibration of the method and the relations between the substructure level with physical quantities, such as the mass, X-ray luminosity, temperature, and cluster redshift. We use our method to separate the clusters in two sub-samples of high-and low-substructure levels. We conclude, using Monte Carlo simulations, that the method recuperates very well the true amount of substructure for small angular core radii clusters (with respect to the whole image size) and good S/N observations. We find no evidence of correlation between the substructure level and physical properties of the clusters such as gas temperature, X-ray luminosity, and redshift; however, analysis suggest a trend between the substructure level and cluster mass. The scaling relations for the two sub-samples (high-and low-substructure level clusters) are different (they present an offset, i. e., given a fixed mass or temperature, low-substructure clusters tend to be more X-ray luminous), which is an important result for cosmological tests using the mass-luminosity relation to obtain the cluster mass function, since they rely on the assumption that clusters do not present different scaling relations according to their dynamical state.

Finite-size corrections of the entanglement entropy of critical quantum chains

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement alpha-Renyi entropy of a single interval for several critical quantum chains. We considered models with U(1) symmetry such as the spin-1/2 XXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of a, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the alpha-Renyi entropies. We conjecture that the exponent of the leading finite-size correction of the alpha-Renyi entropies is p(alpha) = 2X(epsilon)/alpha for alpha > 1 and p(1) = nu, where X-epsilon denotes the dimensions of the energy operator of the model and nu = 2 for all the models.

Generalized Metropolis dynamics with a generalized master equation: An approach for time-independent and time-dependent Monte Carlo simulations of generalized spin systems

The extension of Boltzmann-Gibbs thermostatistics, proposed by Tsallis, introduces an additional parameter q to the inverse temperature beta. Here, we show that a previously introduced generalized Metropolis dynamics to evolve spin models is not local and does not obey the detailed energy balance. In this dynamics, locality is only retrieved for q = 1, which corresponds to the standard Metropolis algorithm. Nonlocality implies very time-consuming computer calculations, since the energy of the whole system must be reevaluated when a single spin is flipped. To circumvent this costly calculation, we propose a generalized master equation, which gives rise to a local generalized Metropolis dynamics that obeys the detailed energy balance. To compare the different critical values obtained with other generalized dynamics, we perform Monte Carlo simulations in equilibrium for the Ising model. By using short-time nonequilibrium numerical simulations, we also calculate for this model the critical temperature and the static and dynamical critical exponents as functions of q. Even for q not equal 1, we show that suitable time-evolving power laws can be found for each initial condition. Our numerical experiments corroborate the literature results when we use nonlocal dynamics, showing that short-time parameter determination works also in this case. However, the dynamics governed by the new master equation leads to different results for critical temperatures and also the critical exponents affecting universality classes. We further propose a simple algorithm to optimize modeling the time evolution with a power law, considering in a log-log plot two successive refinements.

Chemical and biochemical properties of Araucaria angustifolia (Bert.) Ktze. forest soils in the state of São Paulo

Araucaria angustifolia, commonly named Araucaria, is a Brazilian native species that is intensively exploited due to its timber quality. Therefore, Araucaria is on the list of species threatened by extinction. Despite the importance of soil for forest production, little is known about the soil properties of the highly fragmented Araucaria forests. This study was designed to investigate the use of chemical and biological properties as indicators of conservation and anthropogenic disturbance of Araucaria forests in different sampling periods. The research was carried out in two State parks of São Paulo: Parque Estadual Turístico do Alto do Ribeira and Parque Estadual de Campos de Jordão. The biochemical properties carbon and nitrogen in microbial biomass (MB-C and MB-N), basal respiration (BR), the metabolic quotient (qCO2) and the following enzyme activities: β-glucosidase, urease, and fluorescein diacetate hydrolysis (FDA) were evaluated. The sampling period (dry or rainy season) influenced the results of mainly MB-C, MB-N, BR, and qCO2. The chemical and biochemical properties, except K content, were sensitive indicators of differences in the conservation and anthropogenic disturbance stages of Araucaria forests. Although these forests differ in biochemical and chemical properties, they are efficient in energy use and conservation, which is shown by their low qCO2, suggesting an advanced stage of succession.

Universal behavior of the Shannon mutual information of critical quantum chains

We consider the Shannon mutual information of subsystems of critical quantum chains in their ground states. Our results indicate a universal leading behavior for large subsystem sizes. Moreover, as happens with the entanglement entropy, its finite-size behavior yields the conformal anomaly c of the underlying conformal field theory governing the long-distance physics of the quantum chain. We study analytically a chain of coupled harmonic oscillators and numerically the Q-state Potts models (Q = 2, 3, and 4), the XXZ quantum chain, and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information is a quantity easily computed, and our results indicate that for relatively small lattice sizes, its finite-size behavior already detects the universality class of quantum critical behavior.

Untersuchung der Filmbildung aus Polymerdispersionen mit Hilfe der forcierten Rayleighstreuung

Der Filmbildungsprozess wässriger Polymerdispersionen wurde mit forcierter Rayleighstreuung untersucht.Diffusionskoeffizienten D von Farbstoffsonden sind damit zwischen 10^-21 und 10^-9 m^2s^-1 zugänglich. Das Diffusionsverhalten der hydrophoben FarbstoffsondeAberchrome in feuchten und trockenen Filmen aus Poly(n-butylmethacrylat-co-acrylsäure)-Dispersionen sowie bei der Wiederbefeuchtung trockener Filme wurde untersucht.Die Dynamik von Aberchrome weicht in feuchten Filmen von Fickscher Diffusion ab. Dies äußert sich in Abweichungen vomcharakteristischen q^2-Verhalten der Relaxationsrate tau^-1 (tau^-1 = Dq^2; q:Streuvektor) und im Auftreten gestrecktexponentieller Intensitätsrelaxationskurven. Diese Anomalie wurde mit einem apparenten, längenskalenabhängigen Diffusionskoeffizienten Dapp(Lambda) (Lambda= 2Pi/q) beschrieben, der für Lambda -> 0 Werte annimmt, die einem homogen hydroplastifizierten Polymermaterial entsprechen, während Dapp(Lambda) für Lambda -> Unendlich stark anwächst. Diese Anomalien verschwinden bei Wassergehalten entsprechend der Polymerwasserlöslichkeit. Weiteres Trocknen führt zum Absinken des Fickschen-Diffusionskoeffizienten auf einen Grenzwert für trockene Filme. Die Ergebnisse konnten mit einem Zwei-Zustands-Modell beschrieben werden: Die Sonde diffundiert Ficksch in einer hydrophilen Grenzflächenphase und einer langsamen Polymerphase. Austausch zwischendiesen Phasen ist ohne Einschränkung möglich. Das Modell erlaubt die Quantifizierung des Einflusses des Trocknungsprozesses auf Polymer- und Grenzflächenphaseneigenschaften.Dies wurde durch systematische Veränderungen der Grenzflächeneigenschaften demonstriert. Dies geschah durch Acrylsäuregehaltvariationen in den Poly(n-butylmethacrylat-co-acrylsäure)-Dispersionen und Emulgatorbedeckungsgradvariationen. In beiden Fällen reflektieren sich Dispersionsveränderungen in einer Veränderung der Modellparameter in der Grenzflächenphase.

Structural and elastic properties of DNA and chromatin

We investigate a chain consisting of two coupled worm-like chains withconstant distance between the strands. The effects due todouble-strandedness of the chain are studied. In a previous analyticalstudy of this system an intrinsic twist-stretch coupling and atendency of kinking is predicted. Even though a local twist structureis observed the predicted features are not recovered. A new model for DNA at the base-pair level is presented. Thebase-pairs are treated as flat rigid ellipsoids and thesugar-phosphate backbones are represented as stiff harmonic springs.The base-pair stacking interaction is modeled by a variant of theGay-Berne potential. It is shown by systematic coarse-graininghow the elastic constants of a worm-like chain are related to thelocal fluctuations of the base-pair step parameters. Even though a lotof microscopic details of the base-pair geometry is neglected themodel can be optimized to obtain a B-DNA conformation as ground stateand reasonable elastic properties. Moreover the model allows tosimulate much larger length scales than it is possible with atomisticsimulations due to the simplification of the force-field and inparticular due to the possibility of non-local Monte-Carlo moves. Asa first application the behavior under stretching is investigated. Inagreement with micromanipulation experiments on single DNA moleculesone observes a force-plateau in the force-extension curvescorresponding to an overstretching transition from B-DNA to aso-called S-DNA state. The model suggests a structure for S-DNA withhighly inclined base-pairs in order to enable at least partialbase-pair stacking. Finally a simple model for chromatin is introduced to study itsstructural and elastic properties. The underlying geometry of themodeled fiber is based on a crossed-linker model. The chromatosomesare treated as disk-like objects. Excluded volume and short rangenucleosomal interaction are taken into account by a variant of theGay-Berne potential. It is found that the bending rigidity and thestretching modulus of the fiber increase with more compact fibers. Fora reasonable parameterization of the fiber for physiologicalconditions and sufficiently high attraction between the nucleosomes aforce-extension curve is found similar to stretching experiments onsingle chromatin fibers. For very small stretching forces a kinkedfiber forming a loop is observed. If larger forces are applied theloop formation is stretched out and a decondensation of the fibertakes place.

Polyelectrolytes and their counterions studied by EPR spectroscopy

In this thesis methods of EPR spectroscopy were used to investigate polyion-counterion interactions in polyelectrolyte solutions. The fact that EPR techniques are local methods is exploited and by employing spin-carrying (i.e., EPR-active) probe ions it is possible to examine polyelectrolytes from the counterions’ point of view. It was possible to gain insight into i) the dynamics and local geometry of counterion attachment, ii) conformations and dynamics of local segments of the polyion in an indirect manner, and iii) the spatial distribution of spin probe ions that surround polyions in solution. Analysis of CW EPR spectra of dianion nitroxide spin probe Fremy’s salt (FS, potassium nitrosodisulfonate) in solutions of cationic PDADMAC polyelectrolyte revealed that FS ions and PDADMAC form transient ion pairs with a lifetime of less than 1 ns. This effect was termed as dynamic electrostatic attachment (DEA). By spectral simulation taking into account the rotational dynamics as a uniaxial Brownian reorientation, also the geometry of the attached state could be characterized. By variation of solvent, the effect of solvent viscosity and permittivity were investigated and indirect information of the polyelectrolyte chain motion was obtained. Furthermore, analysis of CW EPR data also indicates that in mixtures of organic solvent/water PDADMAC chains are preferentially solvated by the organic solvent molecules, while in purely aqueous mixtures the PDADMAC chain segments were found in different conformations depending on the concentration ratio R of FS counterions to PDADMAC repeat units.Broadenings in CW EPR spectra of FS ions were assigned to spin-exchange interaction and hence contain information on the local concentrations and distributions of the counterions. From analysis of these broadenings in terms of a modified cylindrical cell approach of polyelectrolyte theory, radial distribution functions for the FS ions in the different solvents were obtained. This approach breaks down in water above a threshold value of R, which again indicates that PDADMAC chain conformations are altered as a function of R. Double electron-electron resonance (DEER) measurements of FS ions were carried out to probe the distribution of attached counterions along polyelectrolyte chains. For a significant fraction of FS spin probes in solution with a rigid-rod model polyelectrolyte containing charged Ru2+-centers, a bimodal distance distribution was found that nicely reproduced the spacings of direct and next-neighbor Ru2+-centers along the polyelectrolyte: 2.35 and 4.7 nm. For the system of FS/PDADMAC, DEER data could be simulated by assuming a two-state distribution of spin probes, one state corresponding to a homogeneous (3-dimensional) distribution of spin probes in the polyelectrolyte bulk and the other to a linear (1-dimensional) distribution of spin probes that are electrostatically condensed along locally extended PDADMAC chain segments. From this analysis it is suggested that the PDADMAC chains form locally elongated structures of a size of at least ~5 nm.

Applicability of Process Mining Techniques in Business Environments

This thesis analyses problems related to the applicability, in business environments, of Process Mining tools and techniques. The first contribution is a presentation of the state of the art of Process Mining and a characterization of companies, in terms of their "process awareness". The work continues identifying circumstance where problems can emerge: data preparation; actual mining; and results interpretation. Other problems are the configuration of parameters by not-expert users and computational complexity. We concentrate on two possible scenarios: "batch" and "on-line" Process Mining. Concerning the batch Process Mining, we first investigated the data preparation problem and we proposed a solution for the identification of the "case-ids" whenever this field is not explicitly indicated. After that, we concentrated on problems at mining time and we propose the generalization of a well-known control-flow discovery algorithm in order to exploit non instantaneous events. The usage of interval-based recording leads to an important improvement of performance. Later on, we report our work on the parameters configuration for not-expert users. We present two approaches to select the "best" parameters configuration: one is completely autonomous; the other requires human interaction to navigate a hierarchy of candidate models. Concerning the data interpretation and results evaluation, we propose two metrics: a model-to-model and a model-to-log. Finally, we present an automatic approach for the extension of a control-flow model with social information, in order to simplify the analysis of these perspectives. The second part of this thesis deals with control-flow discovery algorithms in on-line settings. We propose a formal definition of the problem, and two baseline approaches. The actual mining algorithms proposed are two: the first is the adaptation, to the control-flow discovery problem, of a frequency counting algorithm; the second constitutes a framework of models which can be used for different kinds of streams (stationary versus evolving).

The development of stylolites, from small-scale heterogeneities to multi-scale roughness

Stylolites are rough paired surfaces, indicative of localized stress-induced dissolution under a non-hydrostatic state of stress, separated by a clay parting which is believed to be the residuum of the dissolved rock. These structures are the most frequent deformation pattern in monomineralic rocks and thus provide important information about low temperature deformation and mass transfer. The intriguing roughness of stylolites can be used to assess amount of volume loss and paleo-stress directions, and to infer the destabilizing processes during pressure solution. But there is little agreement on how stylolites form and why these localized pressure solution patterns develop their characteristic roughness.rnNatural bedding parallel and vertical stylolites were studied in this work to obtain a quantitative description of the stylolite roughness and understand the governing processes during their formation. Adapting scaling approaches based on fractal principles it is demonstrated that stylolites show two self affine scaling regimes with roughness exponents of 1.1 and 0.5 for small and large length scales separated by a crossover length at the millimeter scale. Analysis of stylolites from various depths proved that this crossover length is a function of the stress field during formation, as analytically predicted. For bedding parallel stylolites the crossover length is a function of the normal stress on the interface, but vertical stylolites show a clear in-plane anisotropy of the crossover length owing to the fact that the in-plane stresses (σ2 and σ3) are dissimilar. Therefore stylolite roughness contains a signature of the stress field during formation.rnTo address the origin of stylolite roughness a combined microstructural (SEM/EBSD) and numerical approach is employed. Microstructural investigations of natural stylolites in limestones reveal that heterogeneities initially present in the host rock (clay particles, quartz grains) are responsible for the formation of the distinctive stylolite roughness. A two-dimensional numerical model, i.e. a discrete linear elastic lattice spring model, is used to investigate the roughness evolving from an initially flat fluid filled interface induced by heterogeneities in the matrix. This model generates rough interfaces with the same scaling properties as natural stylolites. Furthermore two coinciding crossover phenomena in space and in time exist that separate length and timescales for which the roughening is either balanced by surface or elastic energies. The roughness and growth exponents are independent of the size, amount and the dissolution rate of the heterogeneities. This allows to conclude that the location of asperities is determined by a polimict multi-scale quenched noise, while the roughening process is governed by inherent processes i.e. the transition from a surface to an elastic energy dominated regime.rn

High-mass Drell-Yan cross-section and search for new phenomena in multi-electron/positron final states with the ATLAS detector

The Standard Model of particle physics is a very successful theory which describes nearly all known processes of particle physics very precisely. Nevertheless, there are several observations which cannot be explained within the existing theory. In this thesis, two analyses with high energy electrons and positrons using data of the ATLAS detector are presented. One, probing the Standard Model of particle physics and another searching for phenomena beyond the Standard Model.rnThe production of an electron-positron pair via the Drell-Yan process leads to a very clean signature in the detector with low background contributions. This allows for a very precise measurement of the cross-section and can be used as a precision test of perturbative quantum chromodynamics (pQCD) where this process has been calculated at next-to-next-to-leading order (NNLO). The invariant mass spectrum mee is sensitive to parton distribution functions (PFDs), in particular to the poorly known distribution of antiquarks at large momentum fraction (Bjoerken x). The measurementrnof the high-mass Drell-Yan cross-section in proton-proton collisions at a center-of-mass energy of sqrt(s) = 7 TeV is performed on a dataset collected with the ATLAS detector, corresponding to an integrated luminosity of 4.7 fb-1. The differential cross-section of pp -> Z/gamma + X -> e+e- + X is measured as a function of the invariant mass in the range 116 GeV < mee < 1500 GeV. The background is estimated using a data driven method and Monte Carlo simulations. The final cross-section is corrected for detector effects and different levels of final state radiation corrections. A comparison isrnmade to various event generators and to predictions of pQCD calculations at NNLO. A good agreement within the uncertainties between measured cross-sections and Standard Model predictions is observed.rnExamples of observed phenomena which can not be explained by the Standard Model are the amount of dark matter in the universe and neutrino oscillations. To explain these phenomena several extensions of the Standard Model are proposed, some of them leading to new processes with a high multiplicity of electrons and/or positrons in the final state. A model independent search in multi-object final states, with objects defined as electrons and positrons, is performed to search for these phenomenas. Therndataset collected at a center-of-mass energy of sqrt(s) = 8 TeV, corresponding to an integrated luminosity of 20.3 fb-1 is used. The events are separated in different categories using the object multiplicity. The data-driven background method, already used for the cross-section measurement was developed further for up to five objects to get an estimation of the number of events including fake contributions. Within the uncertainties the comparison between data and Standard Model predictions shows no significant deviations.

Exact quantum Monte Carlo methods for correlated electron systems

Thema dieser Arbeit ist die Entwicklung und Kombination verschiedener numerischer Methoden, sowie deren Anwendung auf Probleme stark korrelierter Elektronensysteme. Solche Materialien zeigen viele interessante physikalische Eigenschaften, wie z.B. Supraleitung und magnetische Ordnung und spielen eine bedeutende Rolle in technischen Anwendungen. Es werden zwei verschiedene Modelle behandelt: das Hubbard-Modell und das Kondo-Gitter-Modell (KLM). In den letzten Jahrzehnten konnten bereits viele Erkenntnisse durch die numerische Lösung dieser Modelle gewonnen werden. Dennoch bleibt der physikalische Ursprung vieler Effekte verborgen. Grund dafür ist die Beschränkung aktueller Methoden auf bestimmte Parameterbereiche. Eine der stärksten Einschränkungen ist das Fehlen effizienter Algorithmen für tiefe Temperaturen.rnrnBasierend auf dem Blankenbecler-Scalapino-Sugar Quanten-Monte-Carlo (BSS-QMC) Algorithmus präsentieren wir eine numerisch exakte Methode, die das Hubbard-Modell und das KLM effizient bei sehr tiefen Temperaturen löst. Diese Methode wird auf den Mott-Übergang im zweidimensionalen Hubbard-Modell angewendet. Im Gegensatz zu früheren Studien können wir einen Mott-Übergang bei endlichen Temperaturen und endlichen Wechselwirkungen klar ausschließen.rnrnAuf der Basis dieses exakten BSS-QMC Algorithmus, haben wir einen Störstellenlöser für die dynamische Molekularfeld Theorie (DMFT) sowie ihre Cluster Erweiterungen (CDMFT) entwickelt. Die DMFT ist die vorherrschende Theorie stark korrelierter Systeme, bei denen übliche Bandstrukturrechnungen versagen. Eine Hauptlimitation ist dabei die Verfügbarkeit effizienter Störstellenlöser für das intrinsische Quantenproblem. Der in dieser Arbeit entwickelte Algorithmus hat das gleiche überlegene Skalierungsverhalten mit der inversen Temperatur wie BSS-QMC. Wir untersuchen den Mott-Übergang im Rahmen der DMFT und analysieren den Einfluss von systematischen Fehlern auf diesen Übergang.rnrnEin weiteres prominentes Thema ist die Vernachlässigung von nicht-lokalen Wechselwirkungen in der DMFT. Hierzu kombinieren wir direkte BSS-QMC Gitterrechnungen mit CDMFT für das halb gefüllte zweidimensionale anisotrope Hubbard Modell, das dotierte Hubbard Modell und das KLM. Die Ergebnisse für die verschiedenen Modelle unterscheiden sich stark: während nicht-lokale Korrelationen eine wichtige Rolle im zweidimensionalen (anisotropen) Modell spielen, ist in der paramagnetischen Phase die Impulsabhängigkeit der Selbstenergie für stark dotierte Systeme und für das KLM deutlich schwächer. Eine bemerkenswerte Erkenntnis ist, dass die Selbstenergie sich durch die nicht-wechselwirkende Dispersion parametrisieren lässt. Die spezielle Struktur der Selbstenergie im Impulsraum kann sehr nützlich für die Klassifizierung von elektronischen Korrelationseffekten sein und öffnet den Weg für die Entwicklung neuer Schemata über die Grenzen der DMFT hinaus.

STUDY OF SPARK IGNITION ENGINE COMBUSTION MODEL FOR THE ANALYSIS OF CYCLIC VARIATION AND COMBUSTION STABILITY AT LEAN OPERATING CONDITIONS

A fundamental combustion model for spark-ignition engine is studied in this report. The model is implemented in SIMULINK to simulate engine outputs (mass fraction burn and in-cylinder pressure) under various engine operation conditions. The combustion model includes a turbulent propagation and eddy burning processes based on literature [1]. The turbulence propagation and eddy burning processes are simulated by zero-dimensional method and the flame is assumed as sphere. To predict pressure, temperature and other in-cylinder variables, a two-zone thermodynamic model is used. The predicted results of this model match well with the engine test data under various engine speeds, loads, spark ignition timings and air fuel mass ratios. The developed model is used to study cyclic variation and combustion stability at lean (or diluted) combustion conditions. Several variation sources are introduced into the combustion model to simulate engine performance observed in experimental data. The relations between combustion stability and the introduced variation amount are analyzed at various lean combustion levels.