Destruction of Chemical Agents in a Plasma Reactor with Working Gas of Hydrogen

Exploring Semantic Textual Similarity

[EN]Measuring semantic similarity and relatedness between textual items (words, sentences, paragraphs or even documents) is a very important research area in Natural Language Processing (NLP). In fact, it has many practical applications in other NLP tasks. For instance, Word Sense Disambiguation, Textual Entailment, Paraphrase detection, Machine Translation, Summarization and other related tasks such as Information Retrieval or Question Answering. In this masther thesis we study di erent approaches to compute the semantic similarity between textual items. In the framework of the european PATHS project1, we also evaluate a knowledge-base method on a dataset of cultural item descriptions. Additionaly, we describe the work carried out for the Semantic Textual Similarity (STS) shared task of SemEval-2012. This work has involved supporting the creation of datasets for similarity tasks, as well as the organization of the task itself.

A Convergent Synthetic Route to the Tunicamycin Antiobiotics. Synthesis of (+)-Tunicamycin V

A concise synthetic route to the tunicamycin antibiotics is described, illustrated by the preparation of (+)-tunicamycin-V (1-V). Key features of the synthesis include: (1) the development and application of a silicon-mediated reductive coupling of aldehydes and allylic alcohols to construct the undecose core of the natural product; and (2) the development of an efficient procedure for the synthesis of the trehalose glycosidic bond within the antibiotic. These innovations allow for the coupling of a uridine-derived aldehyde fragment with a preformed trehalose-linked disaccharide allylic alcohol to form the carbohydrate core (1) of the natural product in a highly convergent manner. The resultant amino polyol is a versatile intermediate for the synthesis of any of the homologous tunicamycin antibiotics.

Reactions of small molecules facilitated by metal-acceptor interactions

The role of metal-acceptor interactions arising from M–BR3 and M–PR3 bonding is discussed with respect to reactions between first-row transition metals and N2, H2, and CO. Thermally robust, S = 1/2 (TPB)Co(H2) and (TPB)Co(N2) complexes (TPB = tris(2- (diisopropylphosphino)phenyl)borane) are described and the energetics of N2 and H2 binding are measured. The H2 and N2 ligands are bound more weakly in the (TPB)Co complexes than in related (SiP3)M(L) complexes (SiP3 = tris(2- (diisopropylphosphino)phenyl)silyl). Comparisons within and between these two ligand platforms allow for the factors that affect N2 (and H2) binding and activation to be delineated. The characterization and reactivity of (DPB)Fe complexes (DPB = bis(2- (diisopropylphosphino)phenyl)phenylborane) in the context of N2 functionalization and E–H bond addition (E = H, C, N, Si) are described. This platform allows for the one-pot transformation of free N2 to an Fe hydrazido(-) complex via an Fe aminoimide intermediate. The principles learned from the N2 chemistry using (DPB)Fe are applied to CO reduction on the same system. The preparation of (DPB)Fe(CO)2 is described as well as its reductive functionalization to generate an unprecedented Fe dicarbyne. The bonding in this highly covalent complex is discussed in detail. Initial studies of the reactivity of the Fe dicarbyne reveal that a CO-derived olefin is released upon hydrogenation. Alternative approaches to uncovering unusual reactivity using metal- acceptor interactions are described in Chapters 5 and 6, including initial studies on a new π-accepting tridentate diphosphinosulfinyl ligand and strategies for designing ligands that undergo site-selective metallation to generate heterobimetallic complexes.

Oxidative chemisorption of SO_2 on γ - Al_2O_3 and deposition of H_2-permselective SiO_2 films

The interaction of SO_2 with γ - Al_2O_3 and the deposition of H_2 permselective SiO_2 films have been investigated. The adsorption and oxidative adsorption of SO_2 on γ - Al_2O_3 have been examined at temperatures 500-700°C by Fourier transform infrared spectroscopy (FTIR) and thermogravimetric analysis (TGA). At temperatures above 500°C most of SO_2 adsorbed on the strong sites on alumina. The adsorbed SO_2 species was characterized by an IR band at 1065 cm^(-1). The equilibrium coverage and initial rate of adsorption decreased with temperature suggesting a two-step adsorption. When γ - Al_2O_3 was contacted with a mixture of SO_2 and O_2, adsorption of SO_2 and oxidation of the adsorbed SO_2 to a surface sulfate characterized by broad IR bands at 1070 cm^(-1), 1390 cm^(-1) took place. The results of a series of TGA experiments under different atmospheres strongly suggest that surface SO_2 and surface sulfate involve the same active sites such that SO_2 adsorption is inhibited by already formed sulfate. The results also indicate a broad range of site strengths.

The desorption of adsorbed SO_2 and the reductive desorption of oxidatively adsorbed SO_2 have been investigated by microreactor experiments and thermogravimetric analysis (TGA). Temperature programmed reduction (TPR) of adsorbed SO_2 showed that SO_2 was desorbed without significant reaction with H_2 when H_2 concentration was low while considerable reaction occurred when 100% H_2 was used. SO_2 adsorbed on the strong sites on alumina was reduced to sulfur and H_2S. The isothermal reduction experiments of oxidatively adsorbed SO_2 reveal that the rate of reduction is very slow below 550°C even with 100% H_2. The reduction product is mainly composed of SO_2. TPR experiments of oxidatively adsorbed SO_2 showed that H_2S arose from a sulfate strongly chemisorbed on the surface.

Films of amorphous SiO_2 were deposited within the walls of porous Vycor tubes by SiH_4 oxidation in an opposing reactants geometry : SiH_4 was passed inside the tube while O_2 was passed outside the tube. The two reactants diffused opposite to each other and reacted within a narrow front inside the tube wall to form a thin SiO_2 film. Once the pores were plugged the reactants could not reach each other and the reaction stopped. At 450°C and 0.1 and 0.33 atm of SiH_4 and O_2, the reaction was complete within 15 minutes. The thickness of the SiO_2 film was estimated to be about 0.1 µm. Measurements of H_2 and N_2 permeation rates showed that the SiO_2 film was highly selective to H_2 permeation. The H_2:N_2 flux at 450°C varied between 2000-3000.

Thin SiO_2 films were heat treated in different gas mixtures to determine their stability in functioning as high-temperature hydrogen-permselective membranes. The films were heat-treated at 450-700°C in dry N_2, dry O_2, N_2-H_2O, and O_2-H_2O mixtures. The permeation rates of H_2 and N_2 changed depending on the original conditions of film formation as well as on the heat treatment. Heating in dry N_2 slowly reduced the permeation rates of both H_2 and N_2. Heating in a N_2-H_2O atmosphere led to a steeper decline of H_2 permeability. But the permeation rate of N_2 increased or decreased according to whether the film deposition had been carried out in the absence or presence of H_2O vapor, respectively. Thermal treatment in O_2 caused rapid decline of the permeation rates of H_2 and N_2 in films that were deposited under dry conditions. The decline was moderate in films deposited under wet conditions.

Mechanistic and reactivity studies of bent metallocene complexes of niobium and tantalum

A variety of olefin hydride complexes of niobium and tantalum has been prepared in order to study their reactivity and to gain insight into organometallic reaction mechanisms. Examination of a series of ethylene and propylene complexes of niobocene (CP_2Nb; Cp = η^5-C_5H_5), permethylniobocene (Cp*_2Nb; Cp* = η^5-C_5(CH_3)_5), tantalocene, and permethyltantalocene has indicated that there are both large electronic and steric effects deriving from the metal (and its ancillary ligands) in the olefin insertion (β-migratory insertion) process. Furthermore, a thermodynamic and kinetic analysis has been completed for a series of substituted styrene complexes of niobocene in order to better understand the important electronic properties of the olefin. The results are in accord with a concerted four-center process with only moderate charge development.

The special case of β-migratory insertion of a hydride ligand into coordinated benzyne has also been studied for the permethyltantalocene system. The coordinatively unsaturated (sixteen electron) phenyl tautomer, which is made accessible by the facile benzyne hydride insertion reaction, readily reacts with a variety of ligands, L, to afford Cp*_2 Ta(C_6H_5)L complexes (L = CO, O_2, NC≡R, :CH_2, H_2, etc.). This family of compounds exhibits interesting reactivity (a-migratory insertion, O_2 activation, and reductive elimination) which is discussed in some detail.

Finally a series of paramagnetic seventeen electron Cp*_2 TaX_2 (X = halide, alkyl, hydride) complexes, and the corresponding cationic and anionic species, have been prepared and studied. The odd electron neutral complexes exhibit surprising thermal stability and undergo very little reactivity. While the chemistry of the anionic compounds is almost completely dominated by their potent reducing power, that of the cations is quite diverse and amenable for study. Therefore the syntheses and reactivity (1 ,2-eliminations, ligand insertions, and deprotonation reactions) of these coordinatively unsaturated sixteen electron species are presented.

The application of trimethylsilyl diazomethane to the synthesis of optically active pyrazolines and related studies

The 1,3-dipolar cycloadditions of trimethylsilyl diazomethane with camphorsultam-derived acrylates are reported as a means for the efficient synthesis of optically active pyrazolines. Trimethylsilyl diazomethane is a safe, commercially available diazoalkane which provides Δ¹-pyrazolines 1n good yield and diastereoselectivity when camphorsultam-derived acrylates are used as the reaction dipolarophiles . These initial cycloadducts are subsequently converted to stable, characterizable Δ²-pyrazolines upon desilylation.

A manifold of reactions that can be applied to these Δ²-pyrazolines has been developed which includes pyrazoline reduction, N-N bond reduction, addition to the pyrazoline C=N by mild carbon nucleophiles, and both solvolytic and reductive chiral auxiliary removal. Additionally, it has been demonstrated that the pyrazoline reduction products can take part in peptide coupling reactions that allow for the pyrazolidines to serve as proline-like molecules. The development of this methodology is a general solution to the problem of highly substituted, functionalized pyrazoline synthesis. Importantly, the pyrazolines thus provided have been demonstrated to be amenable to reactions that add to their value as synthetic intermediates.

A unified strategy to ent-kauranoid natural products : total syntheses of (–)-maoecrystal Z, (–)-trichorabdal A, and (–)-longikaurin E

The diterpenoid constituents of the Isodon plants have attracted reasearchers interested in both their chemical structures and biological properties for more than a half-century. In recent years, the isolations of new members displaying previously unprecedented ring systems and highly selective biological properties have piqued interest from the synthetic community in this class of natural products.

Reported herein is the first total synthesis of such a recently isolated diterpenoid, (–)-maoecrystal Z. The principal transformations implemented in this synthesis include two highly diastereoselective radical cyclization reactions: a Sm^(II)-mediated reductive cascade cyclization, which forms two rings and establishes four new stereocenters in a single step, and a Ti^(III)-mediated reductive epoxide-acrylate coupling that yields a functionalized spirolactone product, which forms a core bicycle of maoecrystal Z.

The preparation of two additional ent-kauranoid natural products, (–)-trichorabdal A and (–)-longikaurin E, is also described from a derivative of this key spirolactone. These syntheses are additionally enabled by the palladium-mediated oxidative cyclization reaction of a silyl ketene acetal precursor that is used to install the bridgehead all-carbon quaternary stereocenter and bicyclo[3.2.1]octane present in each natural product. These studies have established a synthetic relationship among three architecturally distinct ent-kaurane diterpenoids and have forged a path for the preparation of interesting unnatural ent-kauranoid structural analogs for more thorough biological study.

Electron flow through cytochrome P450

The cytochromes P450 (P450s) are a remarkable class of heme enzymes that catalyze the metabolism of xenobiotics and the biosynthesis of signaling molecules. Controlled electron flow into the thiolate-ligated heme active site allows P450s to activate molecular oxygen and hydroxylate aliphatic C–H bonds via the formation of high-valent metal-oxo intermediates (compounds I and II). Due to the reactive nature and short lifetimes of these intermediates, many of the fundamental steps in catalysis have not been observed directly. The Gray group and others have developed photochemical methods, known as “flash-quench,” for triggering electron transfer (ET) and generating redox intermediates in proteins in the absence of native ET partners. Photo-triggering affords a high degree of temporal precision for the gating of an ET event; the initial ET and subsequent reactions can be monitored on the nanosecond-to-second timescale using transient absorption (TA) spectroscopies. Chapter 1 catalogues critical aspects of P450 structure and mechanism, including the native pathway for formation of compound I, and outlines the development of photochemical processes that can be used to artificially trigger ET in proteins. Chapters 2 and 3 describe the development of these photochemical methods to establish electronic communication between a photosensitizer and the buried P450 heme. Chapter 2 describes the design and characterization of a Ru-P450-BM3 conjugate containing a ruthenium photosensitizer covalently tethered to the P450 surface, and nanosecond-to-second kinetics of the photo-triggered ET event are presented. By analyzing data at multiple wavelengths, we have identified the formation of multiple ET intermediates, including the catalytically relevant compound II; this intermediate is generated by oxidation of a bound water molecule in the ferric resting state enzyme. The work in Chapter 3 probes the role of a tryptophan residue situated between the photosensitizer and heme in the aforementioned Ru-P450 BM3 conjugate. Replacement of this tryptophan with histidine does not perturb the P450 structure, yet it completely eliminates the ET reactivity described in Chapter 2. The presence of an analogous tryptophan in Ru-P450 CYP119 conjugates also is necessary for observing oxidative ET, but the yield of heme oxidation is lower. Chapter 4 offers a basic description of the theoretical underpinnings required to analyze ET. Single-step ET theory is first presented, followed by extensions to multistep ET: electron “hopping.” The generation of “hopping maps” and use of a hopping map program to analyze the rate advantage of hopping over single-step ET is described, beginning with an established rhenium-tryptophan-azurin hopping system. This ET analysis is then applied to the Ru-tryptophan-P450 systems described in Chapter 2; this strongly supports the presence of hopping in Ru-P450 conjugates. Chapter 5 explores the implementation of flash-quench and other phototriggered methods to examine the native reductive ET and gas binding events that activate molecular oxygen. In particular, TA kinetics that demonstrate heme reduction on the microsecond timescale for four Ru-P450 conjugates are presented. In addition, we implement laser flash-photolysis of P450 ferrous–CO to study the rates of CO rebinding in the thermophilic P450 CYP119 at variable temperature. Chapter 6 describes the development and implementation of air-sensitive potentiometric redox titrations to determine the solution reduction potentials of a series of P450 BM3 mutants, which were designed for non-native cyclopropanation of styrene in vivo. An important conclusion from this work is that substitution of the axial cysteine for serine shifts the wild type reduction potential positive by 130 mV, facilitating reduction by biological redox cofactors in the presence of poorly-bound substrates. While this mutation abolishes oxygenation activity, these mutants are capable of catalyzing the cyclopropanation of styrene, even within the confines of an E. coli cell. Four appendices are also provided, including photochemical heme oxidation in ruthenium-modified nitric oxide synthase (Appendix A), general protocols (Appendix B), Chapter-specific notes (Appendix C) and Matlab scripts used for data analysis (Appendix D).

Fundamental Studies of Carbon Oxygen Bond Activation in Nickel Diphosphine Ether Complexes. And, Metallomacrocycles as Ligands: Synthesis and Characterization of Aluminum-Bridged Bisglyoximato Complexes of Iron and Cobalt

In order to develop better catalysts for the cleavage of aryl-X bonds fundamental studies of the mechanism and individual steps of the mechanism have been investigated in detail. As the described studies are difficult at best in catalytic systems, model systems are frequently used. To study aryl-oxygen bond activation, a terphenyl diphosphine scaffold containing an ether moiety in the central arene was designed. The first three chapters of this dissertation focus on the studies of the nickel complexes supported by this diphosphine backbone and the research efforts in regards to aryl-oxygen bond activation.

Chapter 2 outlines the synthesis of a variety of diphosphine terphenyl ether ligand scaffolds. The metallation of these scaffolds with nickel is described. The reactivity of these nickel(0) systems is also outlined. The systems were found to typically undergo a reductive cleavage of the aryl oxygen bond. The mechanism was found to be a subsequent oxidative addition, β-H elimination, reductive elimination and (or) decarbonylation.

Chapter 3 presents kinetic studies of the aryl oxygen bond in the systems outlined in Chapter 2. Using a series of nickel(0) diphosphine terphenyl ether complexes the kinetics of aryl oxygen bond activation was studied. The activation parameters of oxidative addition for the model systems were determined. Little variation was observed in the rate and activation parameters of oxidative addition with varying electronics in the model system. The cause of the lack of variation is due to the ground state and oxidative addition transition state being affected similarly. Attempts were made to extend this study to catalytic systems.

Chapter 4 investigates aryl oxygen bond activation in the presence of additives. It was found that the addition of certain metal alkyls to the nickel(0) model system lead to an increase in the rate of aryl oxygen bond activation. The addition of excess Grignard reagent led to an order of magnitude increase in the rate of aryl oxygen bond activation. Similarly the addition of AlMe3 led to a three order of magnitude rate increase. Addition of AlMe₃ at -80 °C led to the formation of an intermediate which was identified by NOESY correlations as a system in which the AlMe3 is coordinated to the ether moiety of the backbone. The rates and activation parameters of aryl oxygen bond activation in the presence of AlMe3 were investigated.

The last two chapters involve the study of metalla-macrocycles as ligands. Chapter 5 details the synthesis of a variety of glyoxime backbones and diphenol precursors and their metallation with aluminum. The coordination chemistry of iron on the aluminum scaffolds was investigated. Varying the electronics of the aluminum macrocycle was found to affect the observed electrochemistry of the iron center.

Chapter 6 extends the studies of chapter 5 to cobalt complexes. The synthesis of cobalt dialuminum glyoxime metal complexes is described. The electrochemistry of the cobalt complexes was investigated. The electrochemistry was compared to the observed electrochemistry of a zinc analog to identify the redox activity of the ligand. In the presence of acid the cobalt complexes were found to electrochemically reduce protons to dihydrogen. The electronics of the ancillary aluminum ligands were found to affect the potential of proton reduction in the cobalt complexes. These potentials were compared to other diglyoximate complexes.

Mechanism and applications of the photochemistry of Bis(η^5-cyclopentadienyl)titanacyclobutanes

The photochemically induced reductive elimination of cyclopropanes from bis(η⁵-cyclopentadienyl)titanacyclobutanes has been examined. Stereochemical labelling studies indicate that the cyclopropane is initially formed in a 6±1:1, ratio favoring retention of stereochemistry. The starting titanacyclobutane is isomerized during the course of the reaction. The isomerization of the starting material results from metal-carbon bond homolysis to yield a 1,4-biradical, which can either close to give the starting material or generate cyclopropane. The 1,4-biradical can be observed through a cyclopropyl carbinyl rearrangement employing 2-bis(η⁵- cyclopentadienyl)titana-5,5-dimethylbicyclo[2.1.0]pentane, to give the titanium alkylidene, 1-bis(η⁵-cyclopentadienyl)titana-3,3-dimethyl-1,4- pentadiene, which can be observed directly by NMR at low temperature.

The oxidation of titanacyclobutanes by chemical and electrochemical methods also yields cyclopropanes. Reduction of the metal center does not yield cyclopropanes. Depending on the oxidant, stereochemically labelled titanacyclobutanes yield cyclopropanes that are between 7:1 and 100:1 retention:isomerization. The fragmentation reaction resembles the photochemically induced reductive elimination. Both result from formal oxidation of a metal-carbon bond, which then results in very rapid formation of cyclopropane.

The titanocene generated photochemically reacts with a variety of substrates even at low temperature. Titanocene can be generated in a glass at 77 K. The titanocene can be trapped in noncoordinating solvents in high yield with bulky internal acetylenes to give monoacetylene adducts of titanocene. Less bulky acetylenes give the titanacyclopentadienes. The titanocene can be trapped with olefins to give less stable adducts, which appear by NMR analysis to be intermediate in structure between a titanacyclopropane and an η² olefin adduct of titanocene. Reaction of titanocene with butadiene gives a stable product, which appears to be the s-trans butadiene adduct of titanocene. It does not isomerize on heating. Titanocene reacts with epoxides to give titanocene-µ-oxo polymer and olefin. Stereochemically labelled epoxides and episulfides yield isomerized olefin upon deoxygenation by titanocene. The observations are rationalized as a result of a 1,4-biradical formed by stepwise insertion of titanocene into a carbon-oxygen bond.

Heterometallic Complexes as Models of Enzymatic Active Sites

This dissertation describes studies on two multinucleating ligand architectures: the first scaffold was designed to support tricopper complexes, while the second platform was developed to support tri- and tetrametallic clusters.

In Chapter 2, the synthesis of yttrium (and lanthanide) complexes supported by a tripodal ligand framework designed to bind three copper centers in close proximity is described. Tricopper complexes were shown to react with dioxygen in a 1:1 [Cu₃]/O₂ stoichiometry to form intermediates in which the O–O bond was fully cleaved, as characterized via UV-Vis spectroscopy and determination of the reaction stoichiometry. Pre-arrangement of the three Cu centers was pivotal to cooperative O₂ activation, as mono-copper complexes reacted differently with dioxgyen. The reactivity of the observed intermediates was studied with various substrates (reductants, O-atom acceptors, H-atom donors, Brønsted acids) to determine their properties. In Chapter 3, the reactivity of the same yttrium-tricopper complex with nitric oxide was explored. Reductive coupling to form a trans-hyponitrite complex (characterized by X-ray crystallography) was observed via cooperative reactivity by an yttrium and a copper center on two distinct tetrametallic units. The hyponitrite complex was observed to release nitrous oxide upon treatment with a Brønsted acid, supporting its viability as an intermediate in nitric oxide reduction to nitrous oxide.

In Chapter 4, a different multinucleating ligand scaffold was employed to synthesize heterometallic triiron clusters containing one oxide and one hydroxide bridges. The effects of the redox-inactive, Lewis acidic heterometals on redox potential was studied by cyclic voltammetry, unveiling a linear correlation between redox potential and heterometal Lewis acidity. Further studies on these complexes showed that the Lewis acidity of the redox-inactive metals also affected the oxygen-atom transfer reactivity of these clusters. Comparisons of this reactivity with manganese systems, collaborative efforts to reassign the structures of related manganese oxo-hydroxo clusters, and synthetic attempts to access related dioxo clusters are also described.

In Appendix A, ongoing efforts to synthesize new clusters supported by the same multinucleating ligand platform are described. Studies of novel approaches towards ligand exchange in tetrametallic clusters and incorporation of new supporting and bridging ligand motifs in trinuclear complexes are presented.

Padronização do Direito em tempos de grande encarceramento: os influxos da nova ordem global

O presente trabalho pretende analisar alguns reflexos da incorporação dos influxos da nova ordem globalizada e do neoliberalismo na ordem jurídica nacional. Será demonstrado que os valores sugeridos ainda que essa sugestão tenha caráter quase cogente pelo Banco Mundial, por meio de sua atividade paranormativa, são incorporados pela ordem legal brasileira através de reformas processuais que priorizam a justiça quantitativa. Essas reformas, somadas a outros fatores tais como o aumento das demandas levadas ao Poder Judiciário, a influência da economia na análise do Direito, a utilização equivocada, irrefletida e mecânica de discursos de fundamentação prévia, a pobreza do ensino jurídico e a submissão dos juízes ao que ditam os tribunais, resultam em um patamar de jurisdição padronizada. Paralelamente a esse processo, a população carcerária brasileira cresce em acelerado ritmo, o que pode ser relacionado com o avanço dos valores neoliberais e da retração do estado do bem estar social.Diante desse quadro, é proposto, como forma de aproximação da teoria jurídica com a prática forense e de maneira a proporcionar abertura do direito penal ao mundo dos fatos, a dogmática funcional redutora, de modo que o Direito Penal atue como dique de contenção do estado de polícia que subjaz a cada estado de direito. Ademais, propõe-se que essa adoção se dê nas salas de aula dos cursos de direito, utilizando-se, tanto quanto possível, de diversos recursos didáticos para tornar mais palpáveis e inteligíveis os conceitos e ideias propostos.

Evolutionary status of Entamoeba

In addition to its medical importance as parasitic pathogen, Entamoeba has aroused people's interest in its evolutionary status for a long time. Lacking mitochondrion and other intracellular organelles common to typical eukaryotes, Entamoeba and several other amitochondrial protozoans have been recognized as ancient pre-mitochondriate eukaryotes and named "archezoa", the most primitive extant eukaryotes. It was suggested that they might be living fossils that remained in a primitive stage of evolution before acquisition of organelles, lying close to the transition between prokaryotes and eukaryotes. However, recent studies revealed that Entamoeba contained an organelle, "crypton" or "mitosome", which was regarded as specialized or reductive mitochondrion. Relative molecular phylogenetic analyses also indicated the existence or the probable existence of mitochondrion in Entamoeba. Our phylogenetic analysis based on DNA topoisomerase II strongly suggested its divergence after some mitchondriate eukaryotes. Here, all these recent researches are reviewed and the evolutionary status of Entamoeba is discussed.

Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute. © 2009 Society for Industrial and Applied Mathematics.