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Architecture structurale de la ceinture de Gaspé (Canada) : Imagerie sismique intégrée et application à l'évaluation pétrolièreBêche, Martin 08 December 2008 (has links) (PDF)
La péninsule de la Gaspésie (Québec, Canada) présente un potentiel pétrolier dans les roches datées de l'Ordovicien tardif au Dévonien inférieur. Dans l'est de la ceinture de Gaspé, des puits producteurs de gaz prouvent la présence de réservoirs pétroliers. Des nouvelles études structurales et des études de bassin dans la ceinture de Gaspé permettent de mieux évaluer le potentiel pétrolier de cette région. Nous présentons ici une nouvelle méthodologie pour la prospection d'hydrocarbures dans les régions de piémont de type « fold and thrust belt » comme la ceinture de Gaspé. Nous avons développé cette méthodologie en la testant au niveau de la partie centrale de la ceinture de Gaspé. Nous avons intégré les données géologiques et géophysiques disponibles pour l'étape d'imagerie sismique 2D afin de construire une image sismique directement en profondeur, ce qui a permis d'améliorer l'interprétation structurale, notamment la caractérisation des structures profondes et des failles majeures. Ce travail est suivi d'une modélisation de bassin afin d'évaluer le potentiel pétrolier. Cette étude s'effectue en plusieurs étapes : 1) La construction du modèle structural : L'intégration des données géologiques dans l'étape de la migration en profondeur avant sommation permet d'améliorer le rendu des images sismiques. Ces nouvelles images sont plus fiables et, étant migrées directement en profondeur, rendent les interprétations plus proches de la géometrie réelle du sous-sol. Ces informations permettent de construire un modèle géologique plus complexe et de mieux contraindre le modèle structural de la ceinture acadienne. Les nouvelles interprétations ont permis en particulier, de mieux comprendre la relation entre les ceintures acadienne et taconienne. 2) L'évolution du modèle structural : il a été possible de valider la cohérence de la géométrie structurale grâce aux techniques de restauration. Cependant ce procédé n'a été appliqué qu'au niveau du Synclinal du Lac des Huit-Miles sur les successions stratigraphiques siluro-dévoniennes de la ceinture acadienne : les formations cambroordoviciennes ont été déformées par les orogenèses taconiennes et acadiennes ce qui rend impossible leur restauration. Ce scénario cinématique a été utilisé pour comprendre l'évolution géodynamique de la ceinture de Gaspé et ainsi permettre de proposer une nouvelle géométrie plus favorable à la production et à la migration des hydrocarbures. 3) Évaluation du système pétrolier : Suite à l'étape de restauration, la modélisation de bassin avec le logiciel Temis2D® a été appliquée à la succession stratigraphique Silurien-Dévonien du synclinal du Lac des Huit-Milles et à l'anticlinal de Causapscal. Temis2D® a permis de prédire l'évolution de la roche mère et le degré de maturation ainsi que la génération et l'expulsion des hydrocarbures, en utilisant le modèle structural et les données géochimiques des puits de la ceinture de Gaspé.
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Structural Information and Hidden Markov Models for Biological Sequence AnalysisTångrot, Jeanette January 2008 (has links)
Bioinformatics is a fast-developing field, which makes use of computational methods to analyse and structure biological data. An important branch of bioinformatics is structure and function prediction of proteins, which is often based on finding relationships to already characterized proteins. It is known that two proteins with very similar sequences also share the same 3D structure. However, there are many proteins with similar structures that have no clear sequence similarity, which make it difficult to find these relationships. In this thesis, two methods for annotating protein domains are presented, one aiming at assigning the correct domain family or families to a protein sequence, and the other aiming at fold recognition. Both methods use hidden Markov models (HMMs) to find related proteins, and they both exploit the fact that structure is more conserved than sequence, but in two different ways. Most of the research presented in the thesis focuses on the structure-anchored HMMs, saHMMs. For each domain family, an saHMM is constructed from a multiple structure alignment of carefully selected representative domains, the saHMM-members. These saHMM-members are collected in the so called "midnight ASTRAL set", and are chosen so that all saHMM-members within the same family have mutual sequence identities below a threshold of about 20%. In order to construct the midnight ASTRAL set and the saHMMs, a pipe-line of software tools are developed. The saHMMs are shown to be able to detect the correct family relationships at very high accuracy, and perform better than the standard tool Pfam in assigning the correct domain families to new domain sequences. We also introduce the FI-score, which is used to measure the performance of the saHMMs, in order to select the optimal model for each domain family. The saHMMs are made available for searching through the FISH server, and can be used for assigning family relationships to protein sequences. The other approach presented in the thesis is secondary structure HMMs (ssHMMs). These HMMs are designed to use both the sequence and the predicted secondary structure of a query protein when scoring it against the model. A rigorous benchmark is used, which shows that HMMs made from multiple sequences result in better fold recognition than those based on single sequences. Adding secondary structure information to the HMMs improves the ability of fold recognition further, both when using true and predicted secondary structures for the query sequence. / Bioinformatik är ett område där datavetenskapliga och statistiska metoder används för att analysera och strukturera biologiska data. Ett viktigt område inom bioinformatiken försöker förutsäga vilken tredimensionell struktur och funktion ett protein har, utifrån dess aminosyrasekvens och/eller likheter med andra, redan karaktäriserade, proteiner. Det är känt att två proteiner med likande aminosyrasekvenser också har liknande tredimensionella strukturer. Att två proteiner har liknande strukturer behöver dock inte betyda att deras sekvenser är lika, vilket kan göra det svårt att hitta strukturella likheter utifrån ett proteins aminosyrasekvens. Den här avhandlingen beskriver två metoder för att hitta likheter mellan proteiner, den ena med fokus på att bestämma vilken familj av proteindomäner, med känd 3D-struktur, en given sekvens tillhör, medan den andra försöker förutsäga ett proteins veckning, d.v.s. ge en grov bild av proteinets struktur. Båda metoderna använder s.k. dolda Markov modeller (hidden Markov models, HMMer), en statistisk metod som bland annat kan användas för att beskriva proteinfamiljer. Med hjälp en HMM kan man förutsäga om en viss proteinsekvens tillhör den familj modellen representerar. Båda metoderna använder också strukturinformation för att öka modellernas förmåga att känna igen besläktade sekvenser, men på olika sätt. Det mesta av arbetet i avhandlingen handlar om strukturellt förankrade HMMer (structure-anchored HMMs, saHMMer). För att bygga saHMMerna används strukturbaserade sekvensöverlagringar, vilka genereras utifrån hur proteindomänerna kan läggas på varandra i rymden, snarare än utifrån vilka aminosyror som ingår i deras sekvenser. I varje proteinfamilj används bara ett särskilt, representativt urval av domäner. Dessa är valda så att då sekvenserna jämförs parvis, finns det inget par inom familjen med högre sekvensidentitet än ca 20%. Detta urval görs för att få så stor spridning som möjligt på sekvenserna inom familjen. En programvaruserie har utvecklats för att välja ut representanter för varje familj och sedan bygga saHMMer baserade på dessa. Det visar sig att saHMMerna kan hitta rätt familj till en hög andel av de testade sekvenserna, med nästan inga fel. De är också bättre än den ofta använda metoden Pfam på att hitta rätt familj till helt nya proteinsekvenser. saHMMerna finns tillgängliga genom FISH-servern, vilken alla kan använda via Internet för att hitta vilken familj ett intressant protein kan tillhöra. Den andra metoden som presenteras i avhandlingen är sekundärstruktur-HMMer, ssHMMer, vilka är byggda från vanliga multipla sekvensöverlagringar, men också från information om vilka sekundärstrukturer proteinsekvenserna i familjen har. När en proteinsekvens jämförs med ssHMMen används en förutsägelse om sekundärstrukturen, och den beräknade sannolikheten att sekvensen tillhör familjen kommer att baseras både på sekvensen av aminosyror och på sekundärstrukturen. Vid en jämförelse visar det sig att HMMer baserade på flera sekvenser är bättre än sådana baserade på endast en sekvens, när det gäller att hitta rätt veckning för en proteinsekvens. HMMerna blir ännu bättre om man också tar hänsyn till sekundärstrukturen, både då den riktiga sekundärstrukturen används och då man använder en teoretiskt förutsagd. / Jeanette Hargbo.
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Computer methods for voice analysisGranqvist, Svante January 2003 (has links)
This thesis consists of five articles and a summary. Thethesis deals with methods for measuring properties of thevoice. The methods are all computer-based, but utilisedifferent approaches for measuring different aspects of thevoice. Paper I introduces the Visual Sort and Rate (VSR) method forperceptual rating of voice quality. The method is based on theVisual Analogue Scale (VAS), but simultaneously shows allstimuli as icons along the VAS on the computer screen. As thelistener places similar-sounding stimuli close to each otherduring the rating process, comparing stimuli becomeseasier. Paper II introduces the correlogram. Fundamental frequencyF0 sometimes cannot be strictly defined, particularly forperturbed voice signals. The method displays multipleconsecutive correlation functions in a grey scale image. Thus,the correlogram avoids selecting a single F0 value. Rather itpresents an unbiased image of periodicity, allowing theinvestigator to select among several candidates, ifappropriate. PaperIII introduces a method for detection of phonation tobe utilised in voice accumulators. The method uses twomicrophones attached near the subjects ears. Phase andamplitude relations of the microphone signals are used to forma phonation detector. The output of the method can be used tomeasure phonation time, speaking time and fundamental frequencyof the subject, as well as sound pressure level of both thesubjects voicing and the ambient sounds. Paper IV introduces a method for Fourier analysis ofhigh-speed laryngoscopic imaging. The data from the consecutiveimages are re-arranged to form time-series that reflect thetime-variation of light intensity in each pixel. Each of thesetime series is then analysed by means of Fouriertransformation, such that a spectrum for each pixel isobtained. Several ways of displaying these spectra aredemonstrated. Paper V examines a test set-up for simultaneous recording ofairflow, intra-oral pressure, electro-glottography, audio andhigh-speed imaging. Data are analysed with particular focus onsynchronisation between glottal area and inverse filteredairflow. Several methodological aspects are also examined, suchas the difficulties in synchronising high-speed imaging datawith the other signals. / QC 20100609
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Rééchantillonnage et Sélection de modèlesArlot, Sylvain 13 December 2007 (has links) (PDF)
Cette thèse s'inscrit dans les domaines de la statistique non-paramétrique et de la théorie statistique de l'apprentissage. Son objet est la compréhension fine de certaines méthodes de rééchantillonnage ou de sélection de modèles, du point de vue non-asymptotique. <br /><br />La majeure partie de ce travail de thèse consiste dans la calibration précise de méthodes de sélection de modèles optimales en pratique, pour le problème de la prédiction. Nous étudions la validation croisée V-fold (très couramment utilisée, mais mal comprise en théorie, notamment pour ce qui est de choisir V) et plusieurs méthodes de pénalisation. Nous proposons des méthodes de calibration précise de pénalités, aussi bien pour ce qui est de leur forme générale que des constantes multiplicatives. L'utilisation du rééchantillonnage permet de résoudre des problèmes difficiles, notamment celui de la régression avec un niveau de bruit variable. Nous validons théoriquement ces méthodes du point de vue non-asymptotique, en prouvant des inégalités oracle et des propriétés d'adaptation. Ces résultats reposent entre autres sur des inégalités de concentration.<br /><br />Un second problème que nous abordons est celui des régions de confiance et des tests multiples, lorsque l'on dispose d'observations de grande dimension, présentant des corrélations générales et inconnues. L'utilisation de méthodes de rééchantillonnage permet de s'affranchir du fléau de la dimension, et d'"apprendre" ces corrélations. Nous proposons principalement deux méthodes, et prouvons pour chacune un contrôle non-asymptotique de leur niveau.
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Cenozoic evolution of a fragmented foreland basin, Altiplano plateau, southern PeruFitch, Justin David 14 November 2013 (has links)
Debate persists on the timing, magnitude and style of crustal shortening, uplift and basin evolution in the Andes. Many studies suggest that the central Andes, including the Altiplano plateau, were gradually uplifted as a result of protracted Cenozoic retroarc shortening. However, recent isotopic studies conclude that the Andes instead rose in pulses, with the most significant event occurring at 10-6 Ma. Many researchers attribute these rapid pulses of uplift to lower lithosphere delamination events. A better understanding of the history of Cenozoic crustal shortening is essential for determination of the mechanism(s) of Andean uplift.
The well-exposed Cenozoic San Jerónimo Group was studied in the Ayaviri basin of the northern Altiplano in southern Peru. The 3-5 km-thick succession is situated at 3900-4800 m elevation, between the Western Cordillera magmatic arc and the Eastern Cordillera fold-thrust-belt.
New detrital zircon U-Pb geochronological results from four sandstones and one reworked tuff in the San Jerónimo succession show large age populations indicative of syndepositional volcanism between approximately 38 and 27 Ma. A 1600-m-thick magnetostratigraphic section further constrains the depositional timing and accumulation rate of the upper portion of the succession. Sedimentological observations show a rapid transition from cross-stratified braided-fluvial sandstones to proximal channel-fill and alluvial-fan conglomerates at ~30 Ma. Paleocurrent measurements show important temporal and spatial variations in sediment dispersal patterns while conglomerate clast counts show an upsection transition from almost exclusively volcanic input to increasing contributions of clastic, quartzite, and limestone detritus. The corresponding shifts in depositional environment and sediment provenance are attributed to the activation of new thrust structures in close proximity to the basin, namely the Pucapuca-Sorapata fault system, indicating the presence of an eastward advancing fold-thrust belt dating to at least 38 Ma and reaching the Ayaviri basin within the northern Altiplano plateau at ~30 Ma. / text
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Structure of the Patagonian fold-thrust belt in the Magallanes region of Chile, 53° - 55° S Lat.Betka, Paul Michael 18 February 2014 (has links)
The southern Patagonian Andes record the Late Cretaceous closure and inversion of the Late Jurassic – Early Cretaceous Rocas Verdes marginal basin, subsequent development of the Patagonian retroarc fold-thrust belt and the Neogene to present tectonic superposition of a left-lateral strike-slip plate margin defined by the Magallanes- Fagnano fault zone. In this dissertation, I present new geologic maps, cross sections and detailed macro- and microscopic structural analyses that describe the geometry and kinematic evolution of the fold-thrust belt and superposed strike-slip deformation over ~200 km along-strike between 53° and 55° S latitude. Results are discussed in the context of the regional tectonic development of the southernmost Andes and are relevant to the understanding of important tectonic processes including the development of a retroarc fold-thrust belt, the formation of a basal décollement below and toward the hinterland of a fold-thrust belt and the spatial distribution of deformation along a strike-slip plate margin.
New maps and balanced cross-sections of the Patagonian fold-thrust belt show that it developed during two main phases of Late Cretaceous to Paleogene shortening that were partly controlled by the antecedent geology and mechanical stratigraphy of the Rocas Verdes basin. During the Late Cretaceous, a thin-skinned thrust belt developed above a décollement that formed first in relatively weak shale deposits of the Rocas Verdes basin and later deepened to <1 km below the basement-cover contact. Ramps that cut mechanically rigid volcanic rocks of the marginal basin link the two décollements. Basement-involved reverse faults that cut the early décollements and probably reactivate Jurassic normal faults reflect Paleogene shortening. Shortening estimates increase northwest to southeast from 26 to 37% over 100 km along-strike and are consistent with regional models of the fold-thrust belt.
Structural data, kinematic analyses, and microstructural observations from the lower décollement show that it is defined by transposition of several generations of northeast-vergent noncylindrical folds, shear bands, and a quartz stretching lineation that are kinematically compatible with first-generation structures of the fold-thrust belt. Quartz microstructural data from the décollement are consistent with deformation temperatures that decrease from ~500-650° C to ~400-550° C over ~75 km in the transport direction, indicating that the décollement dipped shallowly (~6°) toward the hinterland. The décollement decoupled the underthrust continental margin from the fold- thrust belt and exemplifies the kinematic relationship between shortening that occurs coevally in a retroarc fold thrust-belt and its polydeformed metamorphic ‘basement’.
Fault kinematic data and crosscutting relationships show kinematic and temporal relationships between populations of thrust, strike-slip and normal faults that occur in the study area. Thrust faults form an internally compatible population that shows subhorizontal northeast-trending shortening of the fold-thrust belt and is kinematically distinct from populations of normal and strike-slip faults. Both strike-slip and normal faults crosscut the fold-thrust belt, are localized near segments of the Magallanes- Fagnano fault zone, have mutually compatible kinematic axes and are interpreted to be coeval. Strike-slip faults form Riedel and P-shear geometries that are compatible with left-lateral slip on the Magallanes-Fagnano fault-zone. Strike-slip and normal faults occur in a releasing step-over between two overlapping left-lateral, left-stepping segments of the Magallanes fault zone and record a tectonic event defined by sinistral transtension that probably reflects changing plate dynamics associated with the opening of the Drake Passage during the Early Miocene. / text
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Advanced Statistical Methodologies in Determining the Observation Time to Discriminate Viruses Using FTIRLuo, Shan 13 July 2009 (has links)
Fourier transform infrared (FTIR) spectroscopy, one method of electromagnetic radiation for detecting specific cellular molecular structure, can be used to discriminate different types of cells. The objective is to find the minimum time (choice among 2 hour, 4 hour and 6 hour) to record FTIR readings such that different viruses can be discriminated. A new method is adopted for the datasets. Briefly, inner differences are created as the control group, and Wilcoxon Signed Rank Test is used as the first selecting variable procedure in order to prepare the next stage of discrimination. In the second stage we propose either partial least squares (PLS) method or simply taking significant differences as the discriminator. Finally, k-fold cross-validation method is used to estimate the shrinkages of the goodness measures, such as sensitivity, specificity and area under the ROC curve (AUC). There is no doubt in our mind 6 hour is enough for discriminating mock from Hsv1, and Coxsackie viruses. Adeno virus is an exception.
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Discovery and Characterization of Novel ADP-Ribosylating ToxinsFieldhouse, Robert John 20 December 2011 (has links)
This thesis is an investigation of novel mono-ADP-ribosylating toxins. In the current data-rich era, making the leap from sequence data to knowledge is a task that requires an elegant bioinformatics toolset to pinpoint questions. A strategy to expand important protein-family knowledge is required, particularly in cases in which primary sequence identity is low but structural conservation is high. For example, the mono-ADP-ribosylating toxins fit these criteria and several approaches have been used to accelerate the discovery of new family members. A newly developed tactic for detecting remote members of this family -- in which fold recognition dominates -- reduces reliance on sequence similarity and advances us toward a true structure-based protein-family expansion methodology. Chelt, a cholera-like toxin from Vibrio cholerae, and Certhrax, an anthrax-like toxin from Bacillus cereus, are among six new bacterial protein toxins identified and characterized using in silico and cell-based techniques. Medically relevant toxins from Mycobacterium avium and Enterococcus faecalis were also uncovered. Agriculturally relevant toxins were found in Photorhabdus luminescens and Vibrio splendidus. Computer software was used to build models and analyze each new toxin to understand features including: structure, secretion, cell entry, activation, NAD+ substrate binding, intracellular target binding and the reaction mechanism. Yeast-based activity tests have since confirmed activity. Vibrio cholerae produces cholix – a potent protein toxin of particular interest that has diphthamide-specific ADP-ribosyltransferase activity against eukaryotic elongation factor 2. Here we present a 2.1Å apo X-ray structure as well as a 1.8Å X-ray structure of cholix in complex with its natural substrate, nicotinamide adenine dinucleotide (NAD+). Hallmark catalytic residues were substituted and analyzed both for NAD+ binding and ADP-ribosyltransferase activity using a fluorescence-based assay. These new toxins serve as a reference for ongoing inhibitor development for this important class of virulence factors. In addition to using toxins as targets for antivirulence compounds, they can be used to make vaccines and new cancer therapies. / Natural Sciences and Engineering Research Council (CGS-D), Canadian Institutes of Health Research, Cystic Fibrosis Canada, Human Frontier Science Program, Ontario government (OGSST), University of Guelph (Graduate Research Scholarship)
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Structural evolution of the northern Thor–Odin Culmination, Monashee Complex southern Canadian CordilleraKruse, Stefan January 2007 (has links)
The Monashee Complex is a structural culmination which exposes rocks from the
lowest stratigraphic levels of the Canadian Cordillera. The Monashee Complex is
subdivided into two lesser structural culminations; the Frenchman Cap and Thor–Odin
culminations. The lithostratigraphic succession of the Thor–Odin Culmination is
completely transposed by penetrative isoclinal folds with amplitudes from microscopic
(<1 mm) to regional (10’s km). Lower structural levels are occupied by Proterozoic
gneisses and migmatites of the Monashee basement assemblage. These are infolded with
overlying metasedimentary rocks of the Monashee cover assemblage, which are
Proterozoic to possibly Paleozoic in age. The basement and cover assemblages were
subsequently intruded by Eocene granitic pegmatite, aplite and lamprophyre dykes.
Regional metamorphism of the basement and cover assemblages reached upper
amphibolite to lower granulite facies.
The northeastern portion of the Thor–Odin Culmination of the Monashee
Complex contains a suite of structures and fabrics, which are classified into four sets,
based on their interpreted kinematic significance. These are: 1) transposition related
structures (DT); 2) open, upright folds (DO); 3) exhumation related structures (DE); and 4)
brittle faults (DB). Each successive set of structures exerted a control on the geometry of
the next set. The large-scale geometry of the culmination is an interference structure
between DT folds, a DE arch and high-strain zones, and a DB brittle horst.
Early, DT fold style varies from intrafolial isoclinal “mature” style folds to
upright or inclined asymmetric “immature” folds. This continuum of fold styles, along
with evidence of anticlockwise rotation (looking down a vertical axis toward the shear plane) of fold axes and lineations is interpreted as being a result of penetrative triclinic
non-coaxial flow. DO upright, symmetrical folds overprint early structures and fabrics,
but are only preserved at low structural levels in the culmination where the DE coaxial
stretching overprint is weak. DE normal shear bands and boudins overprint all earlier
structures. A complex high-strain zone, the Thor–Odin High-Strain Zone, outcrops at
high structural levels and along the margins of the culmination. The Thor–Odin High-
Strain Zone developed as a result of material moving away from the crest of the
culmination, outwards toward the flanks.
Eocene brittle faults (DB) and fractures within the Thor–Odin Culmination of the
Monashee Complex are divisible into three distinct sets. Initial 340–010º trending strikeslip
faults (Set 1) were locally overprinted and reactivated by normal faults with a 325–
020º trend (Set 2). A third set of 255–275º trending fractures (Set 3) are interpreted as
conjugates to Set 1, reactivated as transfer faults to the Set 2 normal faults. Large
regional faults weather recessively forming topographic lineaments that transect the
Monashee Complex. The Victor Creek Fault defines one such lineament. Detailed
mapping within the northern Thor–Odin Culmination, reveals piercement points (fold
hinges) on the east side of the fault, which are not readily matched on the west side. The
minimum displacement required on the Victor Creek Fault to down-drop the fold hinge
below the level of exposure on the west side is 1370 m, assuming normal down-to-the
west displacement. However, the geometry of the fault is consistent with a Set 1 dextral
strike-slip fault. Matching the piercement points in the study area with possible
equivalents to the north indicates 55–60 kms of dextral strike-slip displacement.
The Monashee Reflection (MR) is a major crustal-scale, cross-cutting reflection
appearing on two mutually perpendicular Lithoprobe seismic profiles in the southern
Omineca Belt of the Canadian Cordillera. It has previously been interpreted as the downplunge
extension of an arched regional ductile thrust fault, the Monashee Décollement,
and is described as separating the Monashee Complex from the overlying Selkirk
Allochthon. Recent mapping demonstrates that this boundary is not a discrete ductile
thrust, but rather transposed and gradational. Overprinting the transition zone is a
complex, outward-dipping, normal, structure; the Thor−Odin High-Strain Zone.
Three alternative 3-D geometric models have been developed for the MR in order
to project the reflection to the surface. The favoured model correlates the surface trace of
the Thor−Odin High-Strain Zone with MR.
Normal shear sense kinematics are interpreted for the MR based on: 1) the
overall geometry and asymptotic relationship between the MR and reflections in the
hanging wall and footwall; 2) offset of metamorphic and geochronological gradients,
consistent with an extensional zone, rather than with thrust fault interpretation and 3) the
cross-cutting nature of the MR is consistent with normal structures throughout the region.
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Triangle Loop in Scalar Decay and Cutting RulesGhaderi, Hazhar January 2013 (has links)
In this report we will calculate the amplitude for a scalar-to-scalars (φ3 <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Crightarrow" /> φ2φ2) decay which involves a triangle loop. We compute the real and imaginary part of the amplitude separately and will argue that this is much more straightforward and practical in this case rather than having to deal with or worry about branch cuts of logarithms. We will derive simple cutting rules closely related to the imaginary part of the amplitude. In doing this, we derive a formula that deals with expressions of the form δ[f(x,y)]δ[g(x,y)], containing two Dirac delta functions.
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