Multi-objective mission flight planning in civil unmanned aerial systems

Unmanned Aerial Vehicles (UAVs) are emerging as an ideal platform for a wide range of civil applications such as disaster monitoring, atmospheric observation and outback delivery. However, the operation of UAVs is currently restricted to specially segregated regions of airspace outside of the National Airspace System (NAS). Mission Flight Planning (MFP) is an integral part of UAV operation that addresses some of the requirements (such as safety and the rules of the air) of integrating UAVs in the NAS. Automated MFP is a key enabler for a number of UAV operating scenarios as it aids in increasing the level of onboard autonomy. For example, onboard MFP is required to ensure continued conformance with the NAS integration requirements when there is an outage in the communications link. MFP is a motion planning task concerned with finding a path between a designated start waypoint and goal waypoint. This path is described with a sequence of 4 Dimensional (4D) waypoints (three spatial and one time dimension) or equivalently with a sequence of trajectory segments (or tracks). It is necessary to consider the time dimension as the UAV operates in a dynamic environment. Existing methods for generic motion planning, UAV motion planning and general vehicle motion planning cannot adequately address the requirements of MFP. The flight plan needs to optimise for multiple decision objectives including mission safety objectives, the rules of the air and mission efficiency objectives. Online (in-flight) replanning capability is needed as the UAV operates in a large, dynamic and uncertain outdoor environment. This thesis derives a multi-objective 4D search algorithm entitled Multi- Step A* (MSA*) based on the seminal A* search algorithm. MSA* is proven to find the optimal (least cost) path given a variable successor operator (which enables arbitrary track angle and track velocity resolution). Furthermore, it is shown to be of comparable complexity to multi-objective, vector neighbourhood based A* (Vector A*, an extension of A*). A variable successor operator enables the imposition of a multi-resolution lattice structure on the search space (which results in fewer search nodes). Unlike cell decomposition based methods, soundness is guaranteed with multi-resolution MSA*. MSA* is demonstrated through Monte Carlo simulations to be computationally efficient. It is shown that multi-resolution, lattice based MSA* finds paths of equivalent cost (less than 0.5% difference) to Vector A* (the benchmark) in a third of the computation time (on average). This is the first contribution of the research. The second contribution is the discovery of the additive consistency property for planning with multiple decision objectives. Additive consistency ensures that the planner is not biased (which results in a suboptimal path) by ensuring that the cost of traversing a track using one step equals that of traversing the same track using multiple steps. MSA* mitigates uncertainty through online replanning, Multi-Criteria Decision Making (MCDM) and tolerance. Each trajectory segment is modeled with a cell sequence that completely encloses the trajectory segment. The tolerance, measured as the minimum distance between the track and cell boundaries, is the third major contribution. Even though MSA* is demonstrated for UAV MFP, it is extensible to other 4D vehicle motion planning applications. Finally, the research proposes a self-scheduling replanning architecture for MFP. This architecture replicates the decision strategies of human experts to meet the time constraints of online replanning. Based on a feedback loop, the proposed architecture switches between fast, near-optimal planning and optimal planning to minimise the need for hold manoeuvres. The derived MFP framework is original and shown, through extensive verification and validation, to satisfy the requirements of UAV MFP. As MFP is an enabling factor for operation of UAVs in the NAS, the presented work is both original and significant.

Predictive motion planning for AUVs subject to strong time-varying currents and forecasting uncertainties

This paper presents a novel path planning method for minimizing the energy consumption of an autonomous underwater vehicle subjected to time varying ocean disturbances and forecast model uncertainty. The algorithm determines 4-Dimensional path candidates using Nonlinear Robust Model Predictive Control (NRMPC) and solutions optimised using A*-like algorithms. Vehicle performance limits are incorporated into the algorithm with disturbances represented as spatial and temporally varying ocean currents with a bounded uncertainty in their predictions. The proposed algorithm is demonstrated through simulations using a 4-Dimensional, spatially distributed time-series predictive ocean current model. Results show the combined NRMPC and A* approach is capable of generating energy-efficient paths which are resistant to both dynamic disturbances and ocean model uncertainty.

An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension >= 3. Suppose that the sectional curvature K satisfies -1-s(r) <= K <= -1, where r denotes distance to a fixed point in M. If lim(r ->infinity) e(2r) s(r) = 0, then (M, g) has to be isometric to H-n.The same proof also yields that if K satisfies -s(r) <= K <= 0 where lim(r ->infinity) r(2) s(r) = 0, then (M, g) is isometric to R-n, a result due to Greene and Wu.Our second result is a local one: Let (M, g) be any Riemannian manifold. For a E R, if K < a on a geodesic ball Bp (R) in M and K = a on partial derivative B-p (R), then K = a on B-p (R).

Lepton masses and flavor violation in the Randall-Sundrum model

Lepton masses and mixing angles via localization of 5-dimensional fields in the bulk are revisited in the context of Randall-Sundrum models. The Higgs is assumed to be localized on the IR brane. Three cases for neutrino masses are considered: (a) The higher-dimensional neutrino mass operator (LH.LH), (b) Dirac masses, and (c) Type I seesaw with bulk Majorana mass terms. Neutrino masses and mixing as well as charged lepton masses are fit in the first two cases using chi(2) minimization for the bulk mass parameters, while varying the O(1) Yukawa couplings between 0.1 and 4. Lepton flavor violation is studied for all the three cases. It is shown that large negative bulk mass parameters are required for the right-handed fields to fit the data in the LH.LH case. This case is characterized by a very large Kaluza-Klein (KK) spectrum and relatively weak flavor-violating constraints at leading order. The zero modes for the charged singlets are composite in this case, and their corresponding effective 4-dimensional Yukawa couplings to the KK modes could be large. For the Dirac case, good fits can be obtained for the bulk mass parameters, c(i), lying between 0 and 1. However, most of the ``best-fit regions'' are ruled out from flavor-violating constraints. In the bulk Majorana terms case, we have solved the profile equations numerically. We give example points for inverted hierarchy and normal hierarchy of neutrino masses. Lepton flavor violating rates are large for these points. We then discuss various minimal flavor violation schemes for Dirac and bulk Majorana cases. In the Dirac case with minimal-flavor-violation hypothesis, it is possible to simultaneously fit leptonic masses and mixing angles and alleviate lepton flavor violating constraints for KK modes with masses of around 3 TeV. Similar examples are also provided in the Majorana case.

On Wilking's criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators , which are nonnegative in a suitable sense, to every invariant subset . In this article we show that if is an invariant subset of such that is closed and denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in also admits a metric with curvature operator in (b) The normalized Ricci flow on any compact Riemannian manifold with curvature operator in converges to a metric of constant positive sectional curvature. We also point out that if is an arbitrary subset, then is contained in the cone of curvature operators with nonnegative isotropic curvature.

Uso de espinores na investigação do limite de Karlhede para ondas pp

Neste trabalho foi feito um estudo do limite de Karlhede para ondas pp. Para este fim, uma revisão rigorosa de Geometria Diferencial foi apresentada numa abordagem independente de sistemas de coordenadas. Além da abordagem usual, a curvatura de uma variedade riemanniana foi reescrita usando os formalismos de referenciais, formas diferenciais e espinores do grupo de Lorentz. O problema de equivalência para geometrias riemannianas foi formulado e as peculiaridades de sua aplicação é a Relatividade Geral são delineadas. O limite teórico de Karlhede para espaços-tempo de vácuo de tipo Petrov N foi apresentado. Esse limite é estudado na prática usando técnicas espinores e as condições para sua existência são resolvidas sem a introdução de sistemas de coordenadas.

Delayed decision-making in bistable models

Switching between two modes of operation is a common property of biological systems. In continuous-time differential equation models, this is often realised by bistability, i.e. the existence of two asymptotically stable steadystates. Several biological models are shown to exhibit delayed switching, with a pronounced transient phase, in particular for near-threshold perturbations. This study shows that this delay in switching from one mode to the other in response to a transient input is reflected in local properties of an unstable saddle point, which has a one dimensional unstable manifold with a significantly slower eigenvalue than the stable ones. Thus, the trajectories first approximatively converge to the saddle point, then linger along the saddle's unstable manifold before quickly approaching one of the stable equilibria. ©2010 IEEE.

若尔盖湿地3种两栖类的种间竞争及畜牧业对其食性的影响

从空间、时间、食物3个方面研究了若尔盖湿地3种两栖类的种间竞争，运用生态位理论探讨了3种两栖类利用环境资源的能力以及它们之间的共存模式，研究了3种两栖类年周期食性，并考察了畜牧业对3种两栖类食性及食物竞争格局的影响；此外，还通过实验室研究对2种两栖类幼体的种间竞争策略进行了考察。主要结果如下： 1、两栖类的空间资源利用状况：在3种两栖类成体生态位宽度的比较中，岷山蟾蜍（Bufo minshanicus）成体在牧场性质(0.41)、牛粪数量(0.42)、植被盖度(0.20)、地表温度(0.50)、地表湿度(0.51) 5个维度上的生态位宽度最窄；而倭蛙（Nanorana pleskei）成体在牛粪数量(0.81)、植被高度(0.63)、植被盖度(0.47)、小水体距离(0.68) 4个维度上的生态位宽度最宽。在3种两栖类亚成体生态位宽度的比较中，岷山蟾蜍亚成体在牧场性质(0.66)、牛粪数量(0.58)、植被高度(0.64)、小水体距离(0.51)、地表湿度(0.79) 5个维度的上生态位宽度最宽；倭蛙亚成体在牧场性质(0.39)、牛粪数量(0.30)、地表温度(0.18)、地表湿度(0.33) 4个维度上的生态位宽度最窄。高原林蛙（Rana kukunoris）在地表温度（成体：0.62；亚成体：0.56）、地表湿度（成体：0.84；亚成体：0.60）两个维度上具有较大的生态位宽度值，而在小水体距离维度上（成体：0.27；亚成体：0.14）的生态位宽度值则很小。比较3种无尾两栖类在不同生长阶段（成体、亚成体）的生态位宽度，发现高原林蛙和倭蛙的亚成体对栖息环境的要求更高。3种两栖类空间资源利用的相似程度很高，高原林蛙与倭蛙之间的生态重叠度(0.87)较之它与岷山蟾蜍(0.81)的生态位重叠度更大。 2、两栖类的日活动节律：高原林蛙成体、亚成体、岷山蟾蜍亚成体活动的最低气温为0℃、2℃、8℃；岷山蟾蜍和高原林蛙亚成体出现的数量与气温成极显著的正相关（r=0.797, p<0.001；r=0.794, p<0.001），高原林蛙成体出现的数量与气温有一定相关性（r=0.456, p<0.05）；晴天时两栖类的活动性明显高于阴天（p<0.001）；多云转晴天气，高原林蛙和岷山蟾蜍亚成体出现两次日活动高峰，分别为中午12:30左右和下午15:30～16:30之间；多云天气，高原林蛙和岷山蟾蜍亚成体出现两次日活动高峰，分别为9:30～10:30之间和15:30～16:30之间。 3、两栖类的食物资源利用状况：春、秋两季，高原林蛙最主要的食物是蜉金龟科(Aphodiidae)昆虫，相对重要性指数(IRI)最高（春季：35.28%，秋季：28.57%），其次为昆虫的幼虫，以及双翅目的毛蚊科(Bibionidae)、蝇科(Muscidae)、丽蝇科(Calliphoridae)昆虫，秋季，蝗虫是高原林蛙食物组成中的重要部分；岷山蟾蜍最主要的食物是蚂蚁(IRI，春季：85.54%，秋季：49.70%)，其次为蜉金龟科、象甲科(Curculionidae)、步甲科(Carabidae)、粪金龟科(Geotrupidae) 等鞘翅目昆虫；倭蛙春季的最主要食物也是蜉金龟科昆虫(IRI，春季：13.41%)，其次为蚂蚁、毛蚊科昆虫、昆虫的幼虫以及狼蛛科(Lycosidae)。3种两栖类中，倭蛙的食性生态位宽度相对较宽（0.43），而岷山蟾蜍（0.09）和高原林蛙（0.22）的生态位宽度较窄，与春季相比，两栖类在秋季的食谱更宽。以利用食物种类为标准，春季高原林蛙与倭蛙的生态位重叠度(0.40)比它与岷山蟾蜍的生态位重叠度（0.33）更大。 4、畜牧业对两栖类食性及食物竞争格局的影响：以藏牦牛粪为食物或寄居场所的昆虫，如蜉金龟科、粪金龟科、毛蚊科、蝇科、丽蝇科昆虫和某些昆虫幼虫，是3种两栖类食物谱中最主要的组成部分，蜉金龟科昆虫在高原林蛙食谱中的比例更高，高原林蛙可能从畜牧业发展中获得更多的好处，使之在食物竞争方面处于优势地位。与无放牧样地相比，在有放牧样地的中，两栖类食谱中的蜉金龟科昆虫数量更多(有放牧：31.94%；无放牧：21.32%)、出现频率更高(有放牧：76.38%；无放牧：44%)。然而在不同样地上（有放牧/无放牧），两栖类的食物组成无显著性差异(P=0.188)，两栖类的数量(P=0.075)、肥满度(P=0.537)均没有显著差别。 5、两栖类幼体的竞争策略：实验室条件下，通过活动性水平，变态时的体重、增长率和完成变态所需时间考察自然条件下常同水塘分布的中华蟾蜍（Bufo gargarizans）和高原林蛙蝌蚪的竞争策略。结果表明：中华蟾蜍蝌蚪在不同食物资源条件下，所选择的生存策略可能不同，即食物资源充足时，增加活动性获取更多食物，食物资源有限时，降低活动性且提前完成变态；与中华蟾蜍蝌蚪相比，在食物资源有限时高原林蛙蝌蚪获取食物能力可能更强。 This paper presented the study of competition of three amphibians (Rana kukunoris, Nanorana pleskei, Bufo minshanicus) based on spatial, temporal and dietary scales in Zoige wetland. We measured coexistence patterns of three amphibians and analyzed their ability of exploiting resource. Effects of grazing on the diet composition and diet competition of amphibians were analyzed by their diet composition during spring and autumn. Furthermore, we examined the competitive ability of larval common frogs (Rana kukunoris)and common toads(Bufo gargarizans) in a laboratory experiment, and analyzed their competitive strategies respectively. The results were as follows: 1 .The status of using spatial resource Niche breadths of B. minshanicus adults on 5 dimensional axes including character of pasture(0.41), number of yaks dung(0.42), vegetation coverage(0.20), temperature (0.50)and humidity(0.51) of ground surface were narrower than adults of R. kukunoris and N. pleskei. Niche breadths of B. minshanicus subadults were broader than R.kukunoris subadults and N.pleskei subadults on 5 dimensional axes including character of pasture (0.66), number of yaks dung (0.58), vegetation height (0.64), distance to small waterbodies (0.51), humidity of ground surface (0.79). Niche breadths of N. pleskei subadults were the narrowest in three anurans subadults on 4 dimensional axes including character of pasture (0.39), number of yaks dung (0.30), temperature (0.18) and humidity (0.33) of ground surface, niche breadths of N. pleskei adults were the broadest in three anurans adults on 4 dimensional axes including number of yaks dung (0.81), vegetation height (0.63) and coverage(0.47), distance to small waterbodies(0.68).Comparatively, niche breadths of R. kukunoris were broader on the two microclimate factors including temperature(adults：0.62；subadults：0.56) and humidity (adults：0.84；subadults：0.60)of ground surface, but was narrow on distance to small waterbodies(adults：0.27；subadults：0.14). Strategies for using habitat resource of adults and subadults of the three species anuran were different. Generally, subadults of R. kukunoris and N. pleskei needs better habitat condition. It was quite similar that three anurans exploited spatial resource, Niche overlap between R. kukunoris and N. pleskei (0.87) was greater than that between R. kukunoris and B.minshanicus(0.81). 2.Daily activity rhythm R. kukunoris audlts were active when air temperatures were as low as 0℃, R. kukunoris subadults were active at 2℃, B.minshanicus subaudlts were active at 8℃. Positive correlation was found between activities of amphibians and air temperature, Subadults of R.kukunoris, (r=0.797, p<0.001), Subadults,of,B.minshanicus, (r=0.794, p<0.001), andbadults,of,R.kukunoris(r=0.456, p<0.05).Amphibians were more active during sunny days than cloudy days. In cloudy turning into sunny, R. kukunoris and B.minshanicus subadults had two active peak: at noon about 12:30 and 15:30～16:30 pm; in cloudy, R. kukunoris and B.minshanicus subadult had two active peak too : 9:30～10:30am,15:30～16:30pm. 3.Diet analysis Aphodiidae was the most commonly consumed food item by R. kukunoris based on index of relative importance (IRI) during spring (35.28%) and autumn (28.57%) in Zogie wetland. Besides Aphodiidae, larval insect, dipterans such as Bibionidae, Muscidae, Calliphoridae also were important food item for R. kukunoris, in autumn, locust was one of important food item for R. kukunoris. The most important food item for B.minshanicus during spring (IRI:85.54%) and autumn (IRI:49.70%) was ants, following, was coleopterans, such as Aphodiidae, dung beetle. Aphodiidae (IRI:13.41%) were the most important consumed food item by N. pleskei during spring too, following, was ants and Bibionidae. Dietary breadth of N. pleskei (0.43) were greater than R. kukunoris (0.22) and B. minshanicus (0.09). As a whole, Dietary breadth of amphibians during aurumn were greater than spring. Based on prey item, dietary overlap between R. kukunoris and N. pleskei (0.40) was greater than that between R. kukunoris and B.minshanicus (0.33) during spring. 4.Effects of grazing on the diet composition and diet competition of amphibians Amphibians are an important part of the pasture ecosystems as prey and predator. In Zogie wetland, major diet of amphibians was closely associated with dung of yaks, for example, Aphodiidae, Bibionidae, Muscidae, dung beetle. Dung of yaks was major diet and habitat of these insects. Proportion of Aphodiidae was higher in diet composition of R. kukunoris than N. pleskei and B.minshanicus, with development of pasturage, R. kukunoris may have a diet competitive advantage over N. pleskei and B.minshanicus. Number of Aphodiidae in diet composition of amphibians was higher in samples with grazing (31.94%) than in those without grazing (21.32%). Occurrence Frequency of Aphodiidae in diet composition of amphibians was higher in samples with grazing (76.38%) than in those without grazing (44%). However, There was not significantly different on diet composition (P=0.188), and number (P=0.075) and the relative fatness (P=0.537) of amphibians between grazing samples and without grazing. 5.Competitive strategies of amphibian larvae I examined the competitive ability of larval toads (Bufo gargarizans) and frogs (Rana kukunoris) which co-occur in the nature pond by activity level, the growth rate and mass at metamorphosis and larval period in a laboratory experiment. The results suggest: In laborary, B.gargarizans adapted himself to different food level by changing activity. At high food level, B. gargarizans increased activity to gain more diet. At low food level, B. gargarizans decreased activity and achieved early metamorphosis. When food resource was limit, R. kukunoris could gain more food than B. gargarizans.

Fermionic zero modes in gauge and gravity backgrounds on T-2

Here we study fermionic zero modes in gauge and gravity backgrounds taking a two-dimensional compact manifold T-2 as extra dimensions. The result is that there exist massless Dirac fermions which have normalizable zero modes under quite general assumptions about these backgrounds on the bulk. Several special cases of gauge background on the torus are discussed and some simple fermionic zero modes axe obtained.

Motion Planning with Six Degrees of Freedom

The motion planning problem is of central importance to the fields of robotics, spatial planning, and automated design. In robotics we are interested in the automatic synthesis of robot motions, given high-level specifications of tasks and geometric models of the robot and obstacles. The Mover's problem is to find a continuous, collision-free path for a moving object through an environment containing obstacles. We present an implemented algorithm for the classical formulation of the three-dimensional Mover's problem: given an arbitrary rigid polyhedral moving object P with three translational and three rotational degrees of freedom, find a continuous, collision-free path taking P from some initial configuration to a desired goal configuration. This thesis describes the first known implementation of a complete algorithm (at a given resolution) for the full six degree of freedom Movers' problem. The algorithm transforms the six degree of freedom planning problem into a point navigation problem in a six-dimensional configuration space (called C-Space). The C-Space obstacles, which characterize the physically unachievable configurations, are directly represented by six-dimensional manifolds whose boundaries are five dimensional C-surfaces. By characterizing these surfaces and their intersections, collision-free paths may be found by the closure of three operators which (i) slide along 5-dimensional intersections of level C-Space obstacles; (ii) slide along 1- to 4-dimensional intersections of level C-surfaces; and (iii) jump between 6 dimensional obstacles. Implementing the point navigation operators requires solving fundamental representational and algorithmic questions: we will derive new structural properties of the C-Space constraints and shoe how to construct and represent C-Surfaces and their intersection manifolds. A definition and new theoretical results are presented for a six-dimensional C-Space extension of the generalized Voronoi diagram, called the C-Voronoi diagram, whose structure we relate to the C-surface intersection manifolds. The representations and algorithms we develop impact many geometric planning problems, and extend to Cartesian manipulators with six degrees of freedom.

Progress in satellite remote sensing for studying physical processes at the ocean surface and its borders with the atmosphere and sea-ice

Physical oceanography is the study of physical conditions, processes and variables within the ocean, including temperature-salinity distributions, mixing of the water column, waves, tides, currents, and air-sea interaction processes. Here we provide a critical review of how satellite sensors are being used to study physical oceanography processes at the ocean surface and its borders with the atmosphere and sea-ice. The paper begins by describing the main sensor types that are used to observe the oceans (visible, thermal infrared and microwave) and the specific observations that each of these sensor types can provide. We then present a critical review of how these sensors and observations are being used to study i) ocean surface currents, ii) storm surges, iii) sea-ice, iv) atmosphere-ocean gas exchange and v) surface heat fluxes via phytoplankton. Exciting advances include the use of multiple sensors in synergy to observe temporally varying Arctic sea-ice volume, atmosphere- ocean gas fluxes, and the potential for 4 dimensional water circulation observations. For each of these applications we explain their relevance to society, review recent advances and capability, and provide a forward look at future prospects and opportunities. We then more generally discuss future opportunities for oceanography-focussed remote-sensing, which includes the unique European Union Copernicus programme, the potential of the International Space Station and commercial miniature satellites. The increasing availability of global satellite remote-sensing observations means that we are now entering an exciting period for oceanography. The easy access to these high quality data and the continued development of novel platforms is likely to drive further advances in remote sensing of the ocean and atmospheric systems.

Algorithmen zur automatischen 4dimensionalen Flugbahnführung unter Berücksichtigung der momentanen Windsituation

Für das digitale Flugführungssystem des Versuchsflugzeugs HFB 320 wurde am Institut für Flugführung der DFVLR in Braunschweig eine Betriebsart zum automatischen zeitgenauen Fliegen im Flughafen-Nahbereich entwickelt und im Flugversuch erprobt. Der Bericht enthält die Beschreibung der eingesetzten Algorithmen zur Berechnung 4-dimensionaler Flugbahnen unter Berücksichtigung der momentanen Windsituation. Es werden verschiedene Verfahren zur Messung, Filterung und Vorhersage des Windvektors entwickelt und an Hand von Flugversuchsergebnissen diskutiert.

Unfolded singularities of analytic differential equations

La thèse est composée d’un chapitre de préliminaires et de deux articles sur le sujet du déploiement de singularités d’équations différentielles ordinaires analytiques dans le plan complexe. L’article Analytic classification of families of linear differential systems unfolding a resonant irregular singularity traite le problème de l’équivalence analytique de familles paramétriques de systèmes linéaires en dimension 2 qui déploient une singularité résonante générique de rang de Poincaré 1 dont la matrice principale est composée d’un seul bloc de Jordan. La question: quand deux telles familles sontelles équivalentes au moyen d’un changement analytique de coordonnées au voisinage d’une singularité? est complètement résolue et l’espace des modules des classes d’équivalence analytiques est décrit en termes d’un ensemble d’invariants formels et d’un invariant analytique, obtenu à partir de la trace de la monodromie. Des déploiements universels sont donnés pour toutes ces singularités. Dans l’article Confluence of singularities of non-linear differential equations via Borel–Laplace transformations on cherche des solutions bornées de systèmes paramétriques des équations non-linéaires de la variété centre de dimension 1 d’une singularité col-noeud déployée dans une famille de champs vectoriels complexes. En général, un système d’ÉDO analytiques avec une singularité double possède une unique solution formelle divergente au voisinage de la singularité, à laquelle on peut associer des vraies solutions sur certains secteurs dans le plan complexe en utilisant les transformations de Borel–Laplace. L’article montre comment généraliser cette méthode et déployer les solutions sectorielles. On construit des solutions de systèmes paramétriques, avec deux singularités régulières déployant une singularité irrégulière double, qui sont bornées sur des domaines «spirals» attachés aux deux points singuliers, et qui, à la limite, convergent vers une paire de solutions sectorielles couvrant un voisinage de la singularité confluente. La méthode apporte une description unifiée pour toutes les valeurs du paramètre.

Simple piecewise geodesic interpolation of simple and Jordan curves with applications

We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be included in our construction. Then we present two applications of our main results: the generalized Green’s theorem and the uniqueness of signature for planar Jordan curves with finite p -variation for 1⩽p<2.