35 resultados para Bifurcation Diagrams


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The liver segmentation system, described by Couinaud, is based on the identification of the three hepatic veins and the plane passing by the portal vein bifurcation. Nowadays, Couinaud's description is the most widely used classification since it is better suited for surgery and more accurate for the localisation and monitoring of intra-parenchymal lesions. Knowledge of the anatomy of the portal and venous system is therefore essential, as is knowledge of the variants resulting from changes occurring during the embryological development of the vitelline and umbilical veins. In this paper, the authors propose a straightforward systematisation of the liver in six steps using several additional anatomical points of reference. These points of reference are simple and quickly identifiable in any radiological examination with section imaging, in order to avoid any mistakes in daily practice. In fact, accurate description impacts on many diagnostic and therapeutic applications in interventional radiology and surgery. This description will allow better preparation for biopsy, portal vein embolisation, transjugular intrahepatic portosystemic shunt, tumour resection or partial hepatectomy for transplantation. Such advance planning will reduce intra- and postoperative difficulties and complications.

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Recent studies have indicated that gamma band oscillations participate in the temporal binding needed for the synchronization of cortical networks involved in short-term memory and attentional processes. To date, no study has explored the temporal dynamics of gamma band in the early stages of dementia. At baseline, gamma band analysis was performed in 29 cases with mild cognitive impairment (MCI) during the n-back task. Based on phase diagrams, multiple linear regression models were built to explore the relationship between the cognitive status and gamma oscillation changes over time. Individual measures of phase diagram complexity were made using fractal dimension values. After 1 year, all cases were assessed neuropsychologically using the same battery. A total of 16 MCI patients showed progressive cognitive decline (PMCI) and 13 remained stable (SMCI). When adjusted for gamma values at lag -2, and -3 ms, PMCI cases displayed significantly lower average changes in gamma values than SMCI cases both in detection and 2-back tasks. Gamma fractal dimension of PMCI cases displayed significantly higher gamma fractal dimension values compared to SMCI cases. This variable explained 11.8% of the cognitive variability in this series. Our data indicate that the progression of cognitive decline in MCI is associated with early deficits in temporal binding that occur during the activation of selective attention processes.

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Recent research has examined the factors controlling the geometrical configuration of bifurcations, determined the range of stability conditions for a number of bifurcation types and assessed the impact of perturbations on bifurcation evolution. However, the flow division process and the parameters that influence flow and sediment partitioning are still poorly characterized. To identify and isolate these parameters, three-dimensional velocities were measured at 11 cross-sections in a fixed-walled experimental bifurcation. Water surface gradients were controlled, and systematically varied, using a weir in each distributary. As may be expected, the steepest distributary conveyed the most discharge ( was dominant) while the mildest distributary conveyed the least discharge ( was subordinate). A zone of water surface super-elevation was co-located with the bifurcation in symmetric cases or displaced into the subordinate branch in asymmetric cases. Downstream of a relatively acute-angled bifurcation, primary velocity cores were near to the water surface and against the inner banks, with near-bed zones of lower primary velocity at the outer banks. Downstream of an obtuse-angled bifurcation, velocity cores were initially at the outer banks, with near-bed zones of lower velocities at the inner banks, but patterns soon reverted to match the acute-angled case. A single secondary flow cell was generated in each distributary, with water flowing inwards at the water surface and outwards at the bed. Circulation was relatively enhanced within the subordinate branch, which may help explain why subordinate distributaries remain open, may play a role in determining the size of commonly-observed topographic features, and may thus exert some control on the stability of asymmetric bifurcations. Further, because larger values of circulation result from larger gradient disadvantages, the length of confluence-diffluence units in braided rivers or between diffluences within delta distributary networks may vary depending upon flow structures inherited from upstream and whether, and how, they are fed by dominant or subordinate distributaries. Copyright (C) 2011 John Wiley & Sons, Ltd.

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Albitization is a common process during which hydrothermal fluids convert plagioclase and/or K-feldspar into nearly pure albite; however, its specific mechanism in granitoids is not well understood. The c. 1700 Ma A-type metaluminous ferroan granites in the Khetri complex of Rajasthan, NW India, have been albitized to a large extent by two metasomatic fronts, an initial transformation of oligoclase to nearly pure albite and a subsequent replacement of microcline by albite, with sharp contacts between the microcline-bearing and microcline-free zones. Albitization has bleached the original pinkish grey granite and turned it white. The mineralogical changes include transformation of oligoclase (similar to An(12)) and microcline (similar to Or(95)) to almost pure albite (similar to An(0 center dot 5-2)), amphibole from potassian ferropargasite (X-Fe 0 center dot 84-0 center dot 86) to potassic hastingsite (X-Fe 0 center dot 88-0 center dot 97) and actinolite (X-Fe 0 center dot 32-0 center dot 67), and biotite from annite (X-Fe 0 center dot 71-0 center dot 74) to annite (X-Fe 0 center dot 90-0 center dot 91). Whole-rock isocon diagrams show that, during albitization, the granites experienced major hydration, slight gain in Si and major gain in Na, whereas K, Mg, Fe and Ca were lost along with Rb, Ba, Sr, Zn, light rare earth elements and U. Whole-rock Sm-Nd isotope data plot on an apparent isochron of 1419 +/- 98 Ma and reveal significant disturbance and at least partial resetting of the intrusion age. Severe scatter in the whole-rock Rb-Sr isochron plot reflects the extreme Rb loss in the completely albitized samples, effectively freezing Sr-87/Sr-86 ratios in the albite granites at very high values (0 center dot 725-0 center dot 735). This indicates either infiltration of highly radiogenic Sr from the country rock or, more likely, radiogenic ingrowth during a considerable time lag (estimated to be at least 300 Myr) between original intrusion and albitization. The albitization took place at similar to 350-400 degrees C. It was caused by the infiltration of an ascending hydrothermal fluid that had acquired high Na/K and Na/Ca ratios during migration through metamorphic rocks at even lower temperatures in the periphery of the plutons. Oxygen isotope ratios increase from delta O-18 = 7 parts per thousand in the original granite to values of 9-10 parts per thousand in completely albitized samples, suggesting that the fluid had equilibrated with surrounding metamorphosed crust. A metasomatic model, using chromatographic theory of fluid infiltration, explains the process for generating the observed zonation in terms of a leading metasomatic front where oligoclase of the original granite is converted to albite, and a second, trailing front where microcline is also converted to albite. The temperature gradients driving the fluid infiltration may have been produced by the high heat production of the granites themselves. The confinement of the albitized granites along the NE-SW-trending Khetri lineament and the pervasive nature of the albitization suggest that the albitizing fluids possibly originated during reactivation of the lineament. More generally, steady-state temperature gradients induced by the high internal heat production of A-type granites may provide the driving force for similar metasomatic and ore-forming processes in other highly enriched granitoid bodies.

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In recent years, elevated arsenic concentrations have been found in waters and soils of many, countries, often resulting in a health threat for the local population. Switzerland is not an exception and this paper deals with the release and subsequent fate of arsenic in a 200-km(2) mountainous watershed, characterized by crystalline silicate rocks (gneisses, schists, amphibolites) that contain abundant As-bearing sulfide ore deposits, some of which have been mined for iron and gold in the past. Using analytical methods common for mineralogical, ground water and soil studies (XRD, XRF, XAS-XANES and -EXAFS, electron microprobe, extraction, ICP, AAS with hydride generator, ion chromatography), seven different field situations and related dispersion processes of natural arsenic have been studied: (1) release by rock weathering, (2) transport and deposition by water and ice; (3) release of As to the ground and surface water due to increasing pH; (4) accumulation in humic soil horizons; (5) remobilization by reduction in water-saturated soils and stagnant ground waters; (6) remobilization by using P-rich fertilizers or dung and (7) oxidation, precipitation and dilution in surface waters. Comparison of the results with experimental adsorption studies and speciation diagrams from the literature allows us to reconstruct and identify the typical behavior of arsenic in a natural environment under temperate climatic conditions. The main parameters identified are: (a) once liberated from the primary minerals, sorption processes on Fe-oxy-hydroxides dominate over Al-phases, such as Al-hydroxides or clay minerals and limit the As concentrations in the spring and well waters between 20 and 300 mug/l. (b) Precipitation as secondary minerals is limited to the weathering domain, where the As concentrations are still high and not yet too diluted by rain and soils waters. (c) Although neutral and alkaline pH conditions clearly increase the mobility of As, the main factor to mobilize As is a low redox potential (Eh close or below 0 mV), which favors the dissolution of the Fe-oxy-hydroxides on which the As is sorbed. (d) X-ray absorption spectroscopy (XAS) of As in water-logged humic forest soils indicates that the reduction to As III only occurs at the solid-water interface and that the solid contains As as As V (e) A and Bh horizons of humic cambisols can effectively capture As when As-rich waters flow through them. Complex spatial and temporal variation of the various parameters in a watershed results in repeated mobilization and immobilization of As, which continuously transports As from the upper to the lower part of a watershed and ultimately to the ocean. (C) 2004 Elsevier B.V. All rights reserved.

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Purpose: Results of some PCI clinical trials indicated more procedure related complications and worse clinical outcomes in women than men, but due to a lower representation of female patients this question still remains open. We aimed toinvestigate characteristics and early and late clinical outcomes in female patients as compared to male when treated in a real life setting with a new generation DES.Methods: Among 3069 consecutive patients treated with Nobori DES, and enrolled in NOBORI 2 study, 675 were female. The primary endpoint of the study is target lesion failure (TLF), a composite of cardiac death, MI and target lesion revascularization (TLR) at 12 months. Data are entered in an electronic database; all adverse events are adjudicated by an independent clinical event committee and independent corelabs analyse all angiograms.Results: Compared to male patients, female patients were significantly older (68±10 vs ± 63±11 years; p<0.001), had higher incidence of diabetes (37% vs 27%; p<0.001) and hypertension (75% vs 66%; p<0.001), but lower frequency of previous MI, PCI/CABG and smoking history. Lesion characteristics were similar in two genders, except for lesions located at bifurcation which were more frequent inmale patients (22% vs 15% inmale and female patients respectively; p<0.001). Majority of QCA assessed parameters were similar with the exception for RVD, post-procedure MLD in-segment and %DS in-stent which were significantly lower in female patients (p<0.05 for all). In table 1 results at 6 months follow-up are presented and at the time of presentation 1 year results will be available. Table 1. Clinical results at 6 months follow-up Male (n=2394) Female (n=675) P value Cardiac Death 12 (0.5%) 5 (0.7%) 0.5550 MI 5 (2.1%) 4 (1.3%) 0.5089 TLR rate 17 (0.7%) 13 (1.9%) 0.0124 TLF rate 44 (1.8%) 17 (2.5%) 0.2745 Stent Thrombosis 12 (0.5%) 6 (0.9%) 0.2548Conclusions: Results indicate that there are differences in the demographics and risk factors in female and male patient's population. The frequency of adverse events at 6 months is low in both populations, showing trend toward slightly higher rate in female patients, particularly for target lesion revascularization.

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This paper presents the predicted flow dynamics from the application of a Reynolds-averaged NavierStokes model to a series of bifurcation geometries with morphologies measured during previous flume experiments. The topography of the bifurcations consists of either plane or bedform-dominated beds which may or may not possess discordance between the two bifurcation distributaries. Numerical predictions are compared with experimental results to assess the ability of the numerical model to reproduce the division of flow into the bifurcation distributaries. The hydrodynamic model predicts: (1) diverting fluxes in the upstream channel which direct water into the distributaries; (2) super-elevation of the free surface induced at the bifurcation edge by pressure differences; and (3) counter-rotating secondary circulation cells which develop upstream of the apex of the bifurcation and move into the downstream channels, with water converging at the surface and diverging at the bed. When bedforms are not present, weak transversal fluxes characterize the upstream channel for almost its entire length, associated with clearly distinguishable secondary circulation cells, although these may be under-estimated by the turbulence model used in the solution. In the bedform dominated case, the same hydrodynamic conditions were not observed, with the bifurcation influence restricted and depth scale secondary circulation cells not forming. The results also demonstrate the dominant effect bed discordance has upon flow division between the two distributaries. Finally, results indicate that in bedform dominated rivers. Consequently, we suggest that sand-bed river bifurcations are more likely to have an influence that extends much further upstream and have a greater impact upon water distribution. This may contribute to observed morphological differences between sand-bedded and gravel-bedded braided river networks. Copyright (C) 2012 John Wiley & Sons, Ltd.

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This paper applies probability and decision theory in the graphical interface of an influence diagram to study the formal requirements of rationality which justify the individualization of a person found through a database search. The decision-theoretic part of the analysis studies the parameters that a rational decision maker would use to individualize the selected person. The modeling part (in the form of an influence diagram) clarifies the relationships between this decision and the ingredients that make up the database search problem, i.e., the results of the database search and the different pairs of propositions describing whether an individual is at the source of the crime stain. These analyses evaluate the desirability associated with the decision of 'individualizing' (and 'not individualizing'). They point out that this decision is a function of (i) the probability that the individual in question is, in fact, at the source of the crime stain (i.e., the state of nature), and (ii) the decision maker's preferences among the possible consequences of the decision (i.e., the decision maker's loss function). We discuss the relevance and argumentative implications of these insights with respect to recent comments in specialized literature, which suggest points of view that are opposed to the results of our study.

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INTRODUCTION: The arteries of bifurcation aneurysms are sometimes so angulated or tortuous that an exchange maneuver is necessary to catheterize them with a balloon or stent delivery catheter. Because of the risk of distal wire perforation associated with exchange maneuvers, we sought to find an alternative technique. METHODS: Our experience shows that a microcatheter tends to preferentially follow a previously placed microcatheter, even if the initial catheterization might be challenging. Accessing an artery with two microcatheters simultaneously may thus be an alternative to an exchange maneuver. Because of this tendency for catheters to behave like sheep following one another, we named this method the sheeping technique (ST). The ST consists of (a) first placing a 1.7 French microcatheter into the division branch requiring balloon or stent protection to straighten the course of the arteries in order to facilitate and (b) positioning in the same artery of a larger and stiffer balloon or stent microcatheter. Once the second balloon or stent microcatheter is in place, the first microcatheter can be pulled back and used to coil the aneurysm. RESULTS: Between January 2009 and December 2012, The ST was successfully used in 208/246 procedures (85 %). Conversion to an exchange maneuver was necessary in 38/246 (15 %). There were no arterial perforations or ischemic events related to the handling of both microcatheters. CONCLUSION: The sheeping technique may improve safety by replacing the need for an exchange maneuver during difficult balloon- or stent-assisted coiling.

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The slow-phase velocity of nystagmus is one of the most sensitive parameters of vestibular function and is currently the standard for evaluating the caloric test. However, the assessment of this parameter requires recording the response by using nystagmography. The aim of this study was to evaluate whether frequency and duration of the caloric nystagmus, as measured by using a clinical test with Frenzel glasses, could predict the result of the recorded test. The retrospective analysis of 222 caloric test results recorded by means of electronystagmography has shown a good association between the 3 parameters for unilateral weakness. The asymmetry observed in the velocity can be predicted by a combination of frequency and duration. On the other hand, no relationship was observed between the parameters for directional preponderance. These results indicate that a clinical caloric test with frequency and duration as parameters can be used to predict the unilateral weakness, which would be obtained by use of nystagmography. We propose an evaluation of the caloric test on the basis of diagrams combining the 3 response parameters.

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The Wnt -Wingless (Wg) in Drosophila- signaling is an evolutionary conserved, fundamental signal transduction pathway in animals, having a crucial role in early developmental processes. In the adult animal the Wnt cascade is mainly shut off; aberrant activation leads to cancer. One physiological exception in the adult animal is the activation of Wnt signaling in the nervous system. In the present work, we investigated Wg signaling in the Drosophila neuromuscular junctions (NMJs). The fly NMJs closely resemble the glutamatergic synapses in the mammalian central nervous system and serves as a model system to investigate the mechanism of synapse formation and stability. We demonstrate that the trimeric G-protein Go has a fundamental role in the presynaptic cell in the NMJ. It is implicated in the presynaptic Wg pathway, acting downstream of the ligand Wg and its receptor Frizzled2 (Fz2). Furthermore, we prove that the presynaptic Wg-Fz2-Gαo pathway is essential for correct NMJ formation. The neuronal protein Ankyrin2 (Ank2) localizes to the NMJ and has so far been considered to be a static player in NMJ formation, linking the plasma membrane to the cytoskeleton. We identify Ank2 as a direct target of Gαo. The physical and genetic interaction of Gαo with Ank2 represents a novel branch of the presynaptic Wg pathway, regulating the microtubule cytoskeleton in NMJ formation, jointly with the previously established Futsch-dependent branch, which controls microtubule stability downstream of the kinase Sgg (the homolog of GSK3ß). We moreover demonstrate that the Gαo-Ankyrin interaction to regulate the cytoskeleton is conserved in mammalian neuronal cells. Our findings therefore provide a novel, universally valid regulation of the cytoskeleton in the nervous system. Aberrant inactivation of the neuronal Wnt pathway is believed to be involved in the pathogenesis of the Aß peptide in Alzheimer's disease (AD). We modeled AD in Drosophila by expressing Aß42 in the nervous system and in the eye. Neuronal expression drastically shortens the life span of the flies. We prove that this effect depends on the expression specifically in glutamatergic neurons. However, Aß42 does not induce any morphological changes in the NMJ; therefore this synapse is not suitable to study the mechanism of Aß42 induced neurotoxicity. We furthermore demonstrate that genetic activation of the Wnt pathway does not rescue the Aß42 induced phenotypes - in opposition to the dominating view in the field. These results advice caution when interpreting data on the potential interaction of Wnt signaling and AD in other models. -- La voie de signalisation Wnt (Wingless (Wg) chez la drosophile) est conservée dans l'évolution et fondamentale pour le développement des animaux. Cette signalisation est normalement inactive chez l'animal adulte; une activation anormale peut provoquer le cancer. Or, ceci n'est pas le cas dans le système nerveux des adultes. La présente thèse avait pour but d'analyser le rôle de la voie de signalisation Wingless dans la plaque motrice de Drosophila melanogaster. En effet, cette plaque ressemble fortement aux synapses glutaminergiques du système nerveux central des mammifères et procure ainsi un bon modèle pour l'étude des mécanismes impliqués dans la formation et la stabilisation des synapses. Nos résultats montrent que la protéine trimérique Go joue un rôle fondamental dans la fonction de la cellule présynaptique de la plaque motrice. Go est en effet impliqué dans la voie de signalisation Wg, opérant en aval du ligand Wg et de son récepteur Frizzled2. Nous avons pu démontrer que cette voie de signalisation Wg-Fz2-Gαo est essentielle pour le bon développement et le fonctionnement de la plaque motrice. Fait intéressant, nous avons montré que la protéine neuronale Ankyrin2 (Ank2), qui est connue pour jouer un rôle statique en liant la membrane plasmique au cytosquelette dans la plaque motrice, est une cible directe de Gαo. L'interaction physique et génétique entre Gαo et Ank2 constitue ainsi une bifurcation de la voie de signalisation présynaptique Wg. Cette voie régule le cytosquelette des microtubules en coopération avec la branche liée à la protéine Futsch. Cette protéine est l'homologue de la protéine liant les microtubules MAP1B des mammifères et contrôle la stabilité des microtubules opérant en aval de la kinase Sgg (l'homologue de GSK3ß). De plus, la régulation du cytosquelette par l'interaction entre Gαo et Ankyrin est conservée chez les mammifères. Dans leur ensemble, nos résultats ont permis d'identifier un nouveau mode de régulation du cytosquelette dans le système nerveux, probablement valable de manière universelle. La voie de signalisation Wnt est soupçonnée d'être impliquée dans la toxicité provoquée par le peptide Aß dans le cadre de la maladie d'Alzheimer. Nous avons tenté de modéliser la maladie chez la drosophile en exprimant Aß42 spécifiquement dans le cerveau. Cette expérience a montré que l'expression neuronale d'Aß42 réduit la durée de vie des mouches de manière significative par un mécanisme impliquant les cellules glutamatergiques. Par contre, aucune modification morphologique n'est provoquée par Aß42 dans les plaques motrices glutamatergiques. Ces résultats montrent que ce modèle de Drosophile n'est pas adéquat pour l'étude de la maladie d'Alzheimer. De plus, l'activation génétique de la voie de signalisation Wg n'a pas réussi à restaurer les phénotypes de survie ou ceux des yeux causés par Aß42. Ces résultats indiquent que l'implication de la voie de signalisation Wg dans la maladie d'Alzheimer doit être considérée avec prudence.

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This study reassesses the development of compositional layering during the growth of granitic plutons, with emphasis on fractional crystallization and its interaction with both injection and inflation-related deformation. The Dolbel batholith (SW Niger) consists of 14, kilometre-sized plutons emplaced by pulsed magma inputs. Each pluton has a coarse-grained core and a peripheral layered series. Rocks consist of albite (An(<= 11)), K-feldspar (Or(96 99), Ab(1) (4)), quartz, edenite (X(Mg)=0337-0.55), augite (X(Mg)=0.65-0.72) and accessories (apatite, titanite and Fe-Ti-oxides). Whole-rock compositions are metaluminous, sodic (K(2)O/Na(2)O=0.49-0.62) and iron-rich [FeO(tot)/(FeO(tot)+MgO)=0.65-0.82]. The layering is present as size-graded and modally graded, sub-vertical, rhythmic units. Each unit is composed of three layers, which are, towards the interior: edenite +/- plagioclase (C(a/p)), edenite+plagioclase+augite+quartz (C(q)), and edenite+plagioclase+augite+quartz+K-feldspar (C(k)). All phases except quartz show zoned microstructures consisting of external intercumulus overgrowths, a central section showing oscillatory zoning and, in the case of amphibole and titanite, complexly zoned cores. Ba and Sr contents of feldspars decrease towards the rims. Plagioclase crystal size distributions are similar in all units, suggesting that each unit experienced a similar thermal history. Edenite, characteristic of the basal C(a/p) layer, is the earliest phase to crystallize. Microtextures and phase diagrams suggest that edenite cores may have been brought up with magma batches at the site of emplacement and mechanically segregated along the crystallized wall, whereas outer zones of the same crystals formed in situ. The subsequent C(q) layers correspond to cotectic compositions in the Qz-Ab-Or phase diagram at P(H2O)=5 kbar. Each rhythmic unit may therefore correspond to a magma batch and their repetition to crystallization of recurrent magma recharges. Microtextures and chemical variations in major phases allow four main crystallization stages to be distinguished: (1) open-system crystallization in a stirred magma during magma emplacement, involving dissolution and overgrowth (core of edenite and titanite crystals); (2) in situ fractional crystallization in boundary layers (C(a/p) and C(q) layers); (3) equilibrium `en masse' eutectic crystallization (C(k) layers); (4) compaction and crystallization of the interstitial liquid in a highly crystallized mush (e. g. feldspar intercumulus overgrowths). It is concluded that the formation of the layered series in the Dolbel plutons corresponds principally to in situ differentiation of successive magma batches. The variable thickness of the Ck layers and the microtextures show that crystallization of a rhythmic unit stops and it is compacted when a new magma batch is injected into the chamber. Therefore, assembly of pulsed magma injections and fractional crystallization are independent, but complementary, processes during pluton construction.

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The molecular and isotopic chemistry of organic residues from archaeological potsherds was used to obtain further insight into the dietary trends and economies at the Constance lake-shore Neolithic settlements. The archaeological organic residues from the Early Late Neolithic (3922-3902 BC) site Hornstaad-Hornle IA/Germany are, at present, the oldest archaeological samples analysed at the Institute of Mineralogy and Geochemistry of the University of Lausanne. The approach includes 13C/12C and 15N/14N ratios of the bulk organic residues, fatty acids distribution and 13C/12C ratios of individual fatty acids. The results are compared with those obtained from the over 500 years younger Neolithic (3384-3370 BC) settlement of Arbon Bleiche 3/Switzerland and with samples of modern vegetable oils and fat of animals that have been fed exclusively on C3 forage grasses. The overall fatty acid composition (C9 to C24 range, maximizing at C14 and C16), the bulk 13C/12C and 15N/14N ratios (delta13C, delta15N) and the 13C/12C ratios of palmitic (C16:0), stearic (C18:0) and oleic acids (C18:1) of the organic residues indicate that most of the studied samples (25 from 47 samples and 5 from 41 in the delta13C18:0 vs. delta13C16:0 and delta13C18:0 vs. delta13C18:1 diagrams, respectively) from Hornstaad-Hornle IA and Arbon Bleiche 3 sherds contain fat residues of pre-industrial ruminant milk, and young suckling calf/lamb adipose. These data provide direct proof of milk and meat (mainly from young suckling calves) consumption and farming practices for a sustainable dairying in Neolithic villages in central Europe around 4000 BC.dagger

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In this paper we de. ne a partial ordering of knots and links using a special property derived from their minimal diagrams. A link K' is called a predecessor of a link K if Cr(K') < Cr(K) and a diagram of K' can be obtained from a minimal diagram D of K by a single crossing change. In such a case, we say that K' < K. We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

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BACKGROUND: So far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate. MODELING: In the present paper, Murray's law which is applicable to an arterial bifurcation, is generalized to a non-Newtonian blood flow model (power-law model). When the vessel size reaches the capillary limitation, blood can be modeled using a non-Newtonian constitutive equation. It is assumed two different constraints in addition to the pumping power: the volume constraint or the surface constraint (related to the internal surface of the vessel). For a seek of generality, the relationships are given for an arbitrary number of daughter vessels. It is shown that for a cost function including the volume constraint, classical Murray's law remains valid (i.e. SigmaR(c) = cste with c = 3 is verified and is independent of n, the dimensionless index in the viscosity equation; R being the radius of the vessel). On the contrary, for a cost function including the surface constraint, different values of c may be calculated depending on the value of n. RESULTS: We find that c varies for blood from 2.42 to 3 depending on the constraint and the fluid properties. For the Newtonian model, the surface constraint leads to c = 2.5. The cost function (based on the surface constraint) can be related to entropy generation, by dividing it by the temperature. CONCLUSION: It is demonstrated that the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, it is shown that the difference of entropy generation between the parent and daughter vessels is smaller for a non-Newtonian fluid than for a Newtonian fluid.