The photosynthetic bacterial reaction center in native and artificial envirnoments: effects on light-induced electron transfer

In this thesis we focussed on the characterization of the reaction center (RC) protein purified from the photosynthetic bacterium Rhodobacter sphaeroides. In particular, we discussed the effects of native and artificial environment on the light-induced electron transfer processes. The native environment consist of the inner antenna LH1 complex that copurifies with the RC forming the so called core complex, and the lipid phase tightly associated with it. In parallel, we analyzed the role of saccharidic glassy matrices on the interplay between electron transfer processes and internal protein dynamics. As a different artificial matrix, we incorporated the RC protein in a layer-by-layer structure with a twofold aim: to check the behaviour of the protein in such an unusual environment and to test the response of the system to herbicides. By examining the RC in its native environment, we found that the light-induced charge separated state P+QB - is markedly stabilized (by about 40 meV) in the core complex as compared to the RC-only system over a physiological pH range. We also verified that, as compared to the average composition of the membrane, the core complex copurifies with a tightly bound lipid complement of about 90 phospholipid molecules per RC, which is strongly enriched in cardiolipin. In parallel, a large ubiquinone pool was found in association with the core complex, giving rise to a quinone concentration about ten times larger than the average one in the membrane. Moreover, this quinone pool is fully functional, i.e. it is promptly available at the QB site during multiple turnover excitation of the RC. The latter two observations suggest important heterogeneities and anisotropies in the native membranes which can in principle account for the stabilization of the charge separated state in the core complex. The thermodynamic and kinetic parameters obtained in the RC-LH1 complex are very close to those measured in intact membranes, indicating that the electron transfer properties of the RC in vivo are essentially determined by its local environment. The studies performed by incorporating the RC into saccharidic matrices evidenced the relevance of solvent-protein interactions and dynamical coupling in determining the kinetics of electron transfer processes. The usual approach when studying the interplay between internal motions and protein function consists in freezing the degrees of freedom of the protein at cryogenic temperature. We proved that the “trehalose approach” offers distinct advantages with respect to this traditional methodology. We showed, in fact, that the RC conformational dynamics, coupled to specific electron transfer processes, can be modulated by varying the hydration level of the trehalose matrix at room temperature, thus allowing to disentangle solvent from temperature effects. The comparison between different saccharidic matrices has revealed that the structural and dynamical protein-matrix coupling depends strongly upon the sugar. The analyses performed in RCs embedded in polyelectrolyte multilayers (PEM) structures have shown that the electron transfer from QA - to QB, a conformationally gated process extremely sensitive to the RC environment, can be strongly modulated by the hydration level of the matrix, confirming analogous results obtained for this electron transfer reaction in sugar matrices. We found that PEM-RCs are a very stable system, particularly suitable to study the thermodynamics and kinetics of herbicide binding to the QB site. These features make PEM-RC structures quite promising in the development of herbicide biosensors. The studies discussed in the present thesis have shown that, although the effects on electron transfer induced by the native and artificial environments tested are markedly different, they can be described on the basis of a common kinetic model which takes into account the static conformational heterogeneity of the RC and the interconversion between conformational substates. Interestingly, the same distribution of rate constants (i.e. a Gamma distribution function) can describe charge recombination processes in solutions of purified RC, in RC-LH1 complexes, in wet and dry RC-PEM structures and in glassy saccharidic matrices over a wide range of hydration levels. In conclusion, the results obtained for RCs in different physico-chemical environments emphasize the relevance of the structure/dynamics solvent/protein coupling in determining the energetics and the kinetics of electron transfer processes in a membrane protein complex.

Modello bayesiano per la riduzione dell'incertezza nella previsione delle piene del fiume Reno

Many efforts have been devoting since last years to reduce uncertainty in hydrological modeling predictions. The principal sources of uncertainty are provided by input errors, for inaccurate rainfall prediction, and model errors, given by the approximation with which the water flow processes in the soil and river discharges are described. The aim of the present work is to develop a bayesian model in order to reduce the uncertainty in the discharge predictions for the Reno river. The ’a priori’ distribution function is given by an autoregressive model, while the likelihood function is provided by a linear equation which relates observed values of discharge in the past and hydrological TOPKAPI model predictions obtained by the rainfall predictions of the limited-area model COSMO-LAMI. The ’a posteriori’ estimations are provided throw a H∞ filter, because the statistical properties of estimation errors are not known. In this work a stationary and a dual adaptive filter are implemented and compared. Statistical analysis of estimation errors and the description of three case studies of flood events occurred during the fall seasons from 2003 to 2005 are reported. Results have also revealed that errors can be described as a markovian process only at a first approximation. For the same period, an ensemble of ’a posteriori’ estimations is obtained throw the COSMO-LEPS rainfall predictions, but the spread of this ’a posteriori’ ensemble is not enable to encompass observation variability. This fact is related to the building of the meteorological ensemble, whose spread reaches its maximum after 5 days. In the future the use of a new ensemble, COSMO–SREPS, focused on the first 3 days, could be helpful to enlarge the meteorogical and, consequently, the hydrological variability.

A panchromatic study of stellar populations in Galactic Globular Clusters

The aim of this Thesis is to investigate (i) how common the bimodal Blue Straggler Stars (BSS) radial distribution is in stellar clusters and (ii) which are the physical processes that can produce this bimodality. We discuss possible relations between the properties of the BSS radial distribution and the dynamical state of the hosting clusters by making use of dynamical models and simulations. When relevant, we also discuss the possible links with some cluster "anomalies" and the effects of a massive object (like Imtermediate Mass Black Hole) in the cluster center. To this purpose we present the observational multiwavelength studies of the BSS populations and their radial distributions in 5 GGCs: M5, M55, M2, NGC 2419 and NGC 6388.

Monte Carlo simulations of Potts glasses

A complete understanding of the glass transition isstill a challenging problem. Some researchers attributeit to the (hypothetical) occurrence of a static phasetransition, others emphasize the dynamical transitionof mode coupling-theory from an ergodic to a non ergodicstate. A class of disordered spin models has been foundwhich unifies both scenarios. One of these models isthe p-state infinite range Potts glass with p>4, whichexhibits in the thermodynamic limit both a dynamicalphase transition at a temperature T_D, and a static oneat T_0 < T_D. In this model every spins interacts withall the others, irrespective of distance. Interactionsare taken from a Gaussian distribution.In order to understand better its behavior forfinite number N of spins and the approach to thethermodynamic limit, we have performed extensive MonteCarlo simulations of the p=10 Potts glass up to N=2560.The time-dependent spin-autocorrelation function C(t)shows strong finite size effects and it does not showa plateau even for temperatures around the dynamicalcritical temperature T_D. We show that the N-andT-dependence of the relaxation time for T > T_D can beunderstood by means of a dynamical finite size scalingAnsatz.The behavior in the spin glass phase down to atemperature T=0.7 (about 60% of the transitiontemperature) is studied. Well equilibratedconfigurations are obtained with the paralleltempering method, which is also useful for properlyestablishing static properties, such as the orderparameter distribution function P(q). Evidence is givenfor the compatibility with a one step replica symmetrybreaking scenario. The study of the cumulants of theorder parameter does not permit a reliable estimation ofthe static transition temperature. The autocorrelationfunction at low T exhibits a two-step decay, and ascaling behavior typical of supercooled liquids, thetime-temperature superposition principle, is observed. Inthis region the dynamics is governed by Arrheniusrelaxations, with barriers growing like N^{1/2}.We analyzed the single spin dynamics down to temperaturesmuch lower than the dynamical transition temperature. We found strong dynamical heterogeneities, which explainthe non-exponential character of the spin autocorrelationfunction. The spins seem to relax according to dynamicalclusters. The model in three dimensions tends to acquireferromagnetic order for equal concentration of ferro-and antiferromagnetic bonds. The ordering has differentcharacteristics from the pure ferromagnet. The spinglass susceptibility behaves like chi_{SG} proportionalto 1/T in the region where a spin glass is predicted toexist in mean-field. Also the analysis of the cumulantsis consistent with the absence of spin glass orderingat finite temperature. The dynamics shows multi-scalerelaxations if a bimodal distribution of bonds isused. We propose to understand it with a model based onthe local spin configuration. This is consistent with theabsence of plateaus if Gaussian interactions are used.

Strukturfaktoren, Modenkopplungsgleichungen und Glasübergang molekularer Kristalle

In dieser Arbeit wird der Orientierungsglasübergang ungeordneter, molekularer Kristalle untersucht. Die theoretische Behandlung ist durch die Anisotropie der Einteilchen-Verteilungsfunktion und der Paarfunktionen erschwert. Nimmt man ein starres Gitter, wird der reziproke Raum im Gegenzug auf die 1. Brillouin-Zone eingeschränkt. Der Orientierungsglasübergang wird im Rahmen der Modenkopplungsgleichungen studiert, die dazu hergeleitet werden. Als Modell dienen harte Rotationsellipsoide auf einem starren sc Gitter. Zur Berechnung der statischen tensoriellen Strukturfaktoren wird die Ornstein-Zernike(OZ)-Gleichung molekularer Kristalle abgeleitet und selbstkonsistent zusammen mit der von molekularen Flüssigkeiten übernommenen Percus-Yevick(PY)-Näherung gelöst. Parallel dazu werden die Strukturfaktoren durch MC-Simulationen ermittelt. Die OZ-Gleichung molekularer Kristalle ähnelt der von Flüssigkeiten, direkte und totale Korrelationsfunktion kommen jedoch wegen des starren Gitters nur ohne Konstantanteile in den Winkelvariablen vor, im Gegensatz zur PY-Näherung. Die Anisotropie bringt außerdem einen nichttrivialen Zusatzfaktor. OZ/PY-Strukturfaktoren und MC-Ergebnisse stimmen gut überein. Bei den Matrixelementen der Dichte-Dichte-Korrelationsfunktion gibt es drei Hauptverläufe: oszillatorisch, monoton und unregelmäßig abfallend. Oszillationen gehören zu alternierenden Dichtefluktuationen, führen zu Maxima der Strukturfaktoren am Zonenrand und kommen bei oblaten und genügend breiten prolaten, schwächer auch bei dünnen, nicht zu langen prolaten Ellipsoiden vor. Der exponentielle monotone Abfall kommt bei allen Ellipsoiden vor und führt zu Maxima der Strukturfaktoren in der Zonenmitte, was die Tendenz zu nematischer Ordnung zeigt. Die OZ/PY-Theorie ist durch divergierende Maxima der Strukturfaktoren begrenzt. Bei den Modenkopplungsgleichungen molekularer Kristalle zeigt sich eine große Ähnlichkeit mit denen molekularer Flüssigkeiten, jedoch spielen auf einem starrem Gitter nur die Matrixelemente mit l,l' > 0 eine Rolle und es finden Umklapps von reziproken Vektoren statt. Die Anisotropie bringt auch hier nichtkonstante Zusatzfaktoren ins Spiel. Bis auf flache oblate Ellipsoide wird die Modenkopplungs-Glaslinie von der Divergenz der Strukturfaktoren bestimmt. Für sehr lange Ellipsoide müssen die Strukturfaktoren zur Divergenz hin extrapoliert werden. Daher treibt nicht der Orientierungskäfigeffekt den Glasübergang, sondern Fluktuationen an einer Phasengrenze. Nahe der Kugelform ist keine zuverlässige Glasline festlegbar. Die eingefrorenen kritischen Dichte-Dichte-Korrelatoren haben nur in wenigen Fällen die Oszillationen der statischen Korrelatoren. Der monotone Abfall bleibt dagegen für lange Zeiten meist erhalten. Folglich haben die kritischen Modenkopplungs-Nichtergodizitätsparameter abgeschwächte Maxima in der Zonenmitte, während die Maxima am Zonenrand meist verschwunden sind. Die normierten Nichtergodizitätsparameter zeigen eine Fülle von Verläufen, besonders tiefer im Glas.

Single polymers at surfaces: adsorption and detachment

This thesis is concerned with the adsorption and detachment of polymers at planar, rigid surfaces. We have carried out a systematic investigation of adsorption of polymers using analytical techniques as well as Monte Carlo simulations with a coarse grained off-lattice bead spring model. The investigation was carried out in three stages. In the first stage the adsorption of a single multiblock AB copolymer on a solid surface was investigated by means of simulations and scaling analysis. It was shown that the problem could be mapped onto an effective homopolymer problem. Our main result was the phase diagram of regular multiblock copolymers which shows an increase in the critical adsorption potential of the substrate with decreasing size of blocks. We also considered the adsorption of random copolymers which was found to be well described within the annealed disorder approximation. In the next phase, we studied the adsorption kinetics of a single polymer on a flat, structureless surface in the regime of strong physisorption. The idea of a ’stem-flower’ polymer conformation and the mechanism of ’zipping’ during the adsorption process were used to derive a Fokker-Planck equation with reflecting boundary conditions for the time dependent probability distribution function (PDF) of the number of adsorbed monomers. The numerical solution of the time-dependent PDF obtained from a discrete set of coupled differential equations were shown to be in perfect agreement with Monte Carlo simulation results. Finally we studied force induced desorption of a polymer chain adsorbed on an attractive surface. We approached the problem within the framework of two different statistical ensembles; (i) by keeping the pulling force fixed while measuring the position of the polymer chain end, and (ii) by measuring the force necessary to keep the chain end at fixed distance above the adsorbing plane. In the first case we treated the problem within the framework of the Grand Canonical Ensemble approach and derived analytic expressions for the various conformational building blocks, characterizing the structure of an adsorbed linear polymer chain, subject to pulling force of fixed strength. The main result was the phase diagram of a polymer chain under pulling. We demonstrated a novel first order phase transformation which is dichotomic i.e. phase coexistence is not possible. In the second case, we carried out our study in the “fixed height” statistical ensemble where one measures the fluctuating force, exerted by the chain on the last monomer when a chain end is kept fixed at height h over the solid plane at different adsorption strength ε. The phase diagram in the h − ε plane was calculated both analytically and by Monte Carlo simulations. We demonstrated that in the vicinity of the polymer desorption transition a number of properties like fluctuations and probability distribution of various quantities behave differently, if h rather than the force, f, is used as an independent control parameter.

Coarse-graining methods and polymer dynamics

This thesis studies molecular dynamics simulations on two levels of resolution: the detailed level of atomistic simulations, where the motion of explicit atoms in a many-particle system is considered, and the coarse-grained level, where the motion of superatoms composed of up to 10 atoms is modeled. While atomistic models are capable of describing material specific effects on small scales, the time and length scales they can cover are limited due to their computational costs. Polymer systems are typically characterized by effects on a broad range of length and time scales. Therefore it is often impossible to atomistically simulate processes, which determine macroscopic properties in polymer systems. Coarse-grained (CG) simulations extend the range of accessible time and length scales by three to four orders of magnitude. However, no standardized coarse-graining procedure has been established yet. Following the ideas of structure-based coarse-graining, a coarse-grained model for polystyrene is presented. Structure-based methods parameterize CG models to reproduce static properties of atomistic melts such as radial distribution functions between superatoms or other probability distributions for coarse-grained degrees of freedom. Two enhancements of the coarse-graining methodology are suggested. Correlations between local degrees of freedom are implicitly taken into account by additional potentials acting between neighboring superatoms in the polymer chain. This improves the reproduction of local chain conformations and allows the study of different tacticities of polystyrene. It also gives better control of the chain stiffness, which agrees perfectly with the atomistic model, and leads to a reproduction of experimental results for overall chain dimensions, such as the characteristic ratio, for all different tacticities. The second new aspect is the computationally cheap development of nonbonded CG potentials based on the sampling of pairs of oligomers in vacuum. Static properties of polymer melts are obtained as predictions of the CG model in contrast to other structure-based CG models, which are iteratively refined to reproduce reference melt structures. The dynamics of simulations at the two levels of resolution are compared. The time scales of dynamical processes in atomistic and coarse-grained simulations can be connected by a time scaling factor, which depends on several specific system properties as molecular weight, density, temperature, and other components in mixtures. In this thesis the influence of molecular weight in systems of oligomers and the situation in two-component mixtures is studied. For a system of small additives in a melt of long polymer chains the temperature dependence of the additive diffusion is predicted and compared to experiments.

Thermodynamic concepts in adaptive resolution simulations

This thesis work is devoted to the conceptual and technical development of the Adaptive Resolution Scheme (AdResS), a molecular dynamics method that allows the simulation of a system with different levels of resolution simultaneously. The simulation domain is divided into high and low resolution zones and a transition region that links them, through which molecules can freely diffuse.rnThe first issue of this work regards the thermodynamic consistency of the method, which is tested and verified in a model liquid of tetrahedral molecules. The results allow the introduction of the concept of the Thermodynamic Force, an external field able to correct spurious density fluctuations present in the transition region in usual AdResS simulations.rnThe AdResS is also applied to a system where two different representations with the same degree of resolution are confronted. This simple test extends the method from an Adaptive Resolution Scheme to an Adaptive Representation Scheme, providing a way of coupling different force fields based on thermodynamic consistency arguments. The Thermodynamic Force is successfully applied to the example described in this work as well.rnAn alternative approach of deducing the Thermodynamic Force from pressure consistency considerations allows the interpretation of AdResS as a first step towards a molecular dynamics simulation in the Grand Canonical ensemble. Additionally, such a definition leads to a practical way of determining the Thermodynamic Force, tested in the well studied tetrahedral liquid. The effects of AdResS and this correction on the atomistic domain are analyzed by inspecting the local distribution of velocities, radial distribution functions, pressure and particle number fluctuation. Their comparison with analogous results coming from purely atomistic simulations shows good agreement, which is greatly improved under the effect of the external field.rnA further step in the development of AdResS, necessary for several applications in biophysics and material science, consists of its application to multicomponent systems. To this aim, the high-resolution representation of a model binary mixture is confronted with its coarse-grained representation systematically parametrized. The Thermodynamic Force, whose development requires a more delicate treatment, also gives satisfactory results.rnFinally, AdResS is tested in systems including two-body bonded forces, through the simulation of a model polymer allowed to adaptively change its representation. It is shown that the distribution functions that characterize the polymer structure are in practice not affected by the change of resolution.rnThe technical details of the implementation of AdResS in the ESPResSo package conclude this thesis work.

Fault detection, diagnosis and active fault tolerant control for a satellite attitude control system

Modern control systems are becoming more and more complex and control algorithms more and more sophisticated. Consequently, Fault Detection and Diagnosis (FDD) and Fault Tolerant Control (FTC) have gained central importance over the past decades, due to the increasing requirements of availability, cost efficiency, reliability and operating safety. This thesis deals with the FDD and FTC problems in a spacecraft Attitude Determination and Control System (ADCS). Firstly, the detailed nonlinear models of the spacecraft attitude dynamics and kinematics are described, along with the dynamic models of the actuators and main external disturbance sources. The considered ADCS is composed of an array of four redundant reaction wheels. A set of sensors provides satellite angular velocity, attitude and flywheel spin rate information. Then, general overviews of the Fault Detection and Isolation (FDI), Fault Estimation (FE) and Fault Tolerant Control (FTC) problems are presented, and the design and implementation of a novel diagnosis system is described. The system consists of a FDI module composed of properly organized model-based residual filters, exploiting the available input and output information for the detection and localization of an occurred fault. A proper fault mapping procedure and the nonlinear geometric approach are exploited to design residual filters explicitly decoupled from the external aerodynamic disturbance and sensitive to specific sets of faults. The subsequent use of suitable adaptive FE algorithms, based on the exploitation of radial basis function neural networks, allows to obtain accurate fault estimations. Finally, this estimation is actively exploited in a FTC scheme to achieve a suitable fault accommodation and guarantee the desired control performances. A standard sliding mode controller is implemented for attitude stabilization and control. Several simulation results are given to highlight the performances of the overall designed system in case of different types of faults affecting the ADCS actuators and sensors.

Measurement of ttbar production differential cross section at \sqrt{s}=8 TeV at ATLAS

Top quark studies play an important role in the physics program of the Large Hadron Collider (LHC). The energy and luminosity reached allow the acquisition of a large amount of data especially in kinematic regions never studied before. In this thesis is presented the measurement of the ttbar production differential cross section on data collected by ATLAS in 2012 in proton proton collisions at \sqrt{s} = 8 TeV, corresponding to an integrated luminosity of 20.3 fb^{−1}. The measurement is performed for ttbar events in the semileptonic channel where the hadronically decaying top quark has a transverse momentum above 300 GeV. The hadronic top quark decay is reconstructed as a single large radius jet and identified using jet substructure properties. The final differential cross section result has been compared with several theoretical distributions obtaining a discrepancy of about the 25% between data and predictions, depending on the MC generator. Furthermore the kinematic distributions of the ttbar production process are very sensitive to the choice of the parton distribution function (PDF) set used in the simulations and could provide constraints on gluons PDF. In particular in this thesis is performed a systematic study on the PDF of the protons, varying several PDF sets and checking which one better describes the experimental distributions. The boosted techniques applied in this measurement will be fundamental in the next data taking at \sqrt{s}=13 TeV when will be produced a large amount of heavy particles with high momentum.

Atmospheric and climate effects of biomass burning using different injection heights

Inspired by the need for a representation of the biomass burning emissions injection height in the ECHAM/MESSy Atmospheric Chemistry model (EMAC)

Action-based dynamical models of the Fornax dwarf spheroidal galaxy

Il lavoro presentato in questa Tesi si basa sul calcolo di modelli dinamici per Galassie Sferoidali Nane studiando il problema mediante l'utilizzo di funzioni di distribuzione. Si è trattato un tipo di funzioni di distribuzione, "Action-Based distribution functions", le quali sono funzioni delle sole variabili azione. Fornax è stata descritta con un'appropriata funzione di distribuzione e il problema della costruzione di modelli dinamici è stato affrontato assumendo sia un alone di materia oscura con distribuzione di densità costante nelle regioni interne sia un alone con cuspide. Per semplicità è stata assunta simmetria sferica e non è stato calcolato esplicitamente il potenziale gravitazionale della componente stellare (le stelle sono traccianti in un potenziale gravitazionale fissato). Tramite un diretto confronto con alcune osservabili, quali il profilo di densità stellare proiettata e il profilo di dispersione di velocità lungo la linea di vista, sono stati trovati alcuni modelli rappresentativi della dinamica di Fornax. Modelli calcolati tramite funzioni di distribuzione basati su azioni permettono di determinare in maniera autoconsistente profili di anisotropia. Tutti i modelli calcolati sono caratterizzati dal possedere un profilo di anisotropia con forte anisotropia tangenziale. Sono state poi comparate le stime di materia oscura di questi modelli con i più comuni e usati stimatori di massa in letteratura. E stato inoltre stimato il rapporto tra la massa totale del sistema (componente stellare e materia oscura) e la componente stellare di Fornax, entro 1600 pc ed entro i 3 kpc. Come esplorazione preliminare, in questo lavoro abbiamo anche presentato anche alcuni esempi di modelli sferici a due componenti in cui il campo gravitazionale è determinato dall'autogravità delle stelle e da un potenziale esterno che rappresenta l'alone di materia oscura.

Data Quality Assessment of Ungated Flow Cytometry Data in High

Background: The recent development of semi-automated techniques for staining and analyzing flow cytometry samples has presented new challenges. Quality control and quality assessment are critical when developing new high throughput technologies and their associated information services. Our experience suggests that significant bottlenecks remain in the development of high throughput flow cytometry methods for data analysis and display. Especially, data quality control and quality assessment are crucial steps in processing and analyzing high throughput flow cytometry data. Methods: We propose a variety of graphical exploratory data analytic tools for exploring ungated flow cytometry data. We have implemented a number of specialized functions and methods in the Bioconductor package rflowcyt. We demonstrate the use of these approaches by investigating two independent sets of high throughput flow cytometry data. Results: We found that graphical representations can reveal substantial non-biological differences in samples. Empirical Cumulative Distribution Function and summary scatterplots were especially useful in the rapid identification of problems not identified by manual review. Conclusions: Graphical exploratory data analytic tools are quick and useful means of assessing data quality. We propose that the described visualizations should be used as quality assessment tools and where possible, be used for quality control.

Cholesky Residuals for Assessing Normal Errors in a Linear Model with Correlated Outcomes: Technical Report

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed models and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated margional residual vector by the Cholesky decomposition of the inverse of the estimated margional variance matrix. The resulting "rotated" residuals are used to construct an empirical cumulative distribution function and pointwise standard errors. The theoretical framework, including conditions and asymptotic properties, involves technical details that are motivated by Lange and Ryan (1989), Pierce (1982), and Randles (1982). Our method appears to work well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series). Our methods can produce satisfactory results even for models that do not satisfy all of the technical conditions stated in our theory.

Marshall's Lemma for Convex Density Estimation

Marshall's (1970) lemma is an analytical result which implies root-n-consistency of the distribution function corresponding to the Grenander (1956) estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on [0,\infty).