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Análisis del clima local y micro-local de la ciudad de Bahía BlancaFerrelli, Federico 09 June 2016 (has links)
Los estudios climáticos se realizan en distintas escalas de análisis espacio-temporales. Se pueden estudiar procesos globales, regionales, locales y micro-locales. Los dos últimos hacen referencia a los espacios urbanos y generan variaciones climáticas en un territorio pequeño dado que sus características radiativas, térmicas, físicas y aerodinámicas difieren de las de su entorno natural.
El objetivo principal de esta investigación fue estudiar el clima local y micro-local de la ciudad de Bahía Blanca y comparar los resultados con estudios previos. De esta forma se pudieron evaluar los efectos de la variabilidad climática, del aumento de la población, etc. en el clima local.
A escala regional se estudió la variabilidad climática de Bahía Blanca a partir del análisis de 50 años de registros meteorológicos diarios. Se aplicaron distintas escalas temporales de análisis: decádica, interanual y estacional con el objetivo de caracterizar el comportamiento temporal de las variables de temperatura del aire, humedad relativa, precipitaciones y velocidad máxima del viento. La temperatura del aire se incrementó 0,9 ºC, la velocidad máxima del viento se redujo 7,5 km h-1 y la humedad relativa presentó dos ciclos diferenciados, marcando períodos húmedos y secos. El análisis de las precipitaciones diferenció en el período de estudio dos eventos extremadamente secos y dos extremadamente húmedos.
Para estudiar la Isla de Calor Urbana se diseñaron instrumentales que registraron datos en alta frecuencia (2 Hz) basado en electrónica de código abierto. Este instrumental permitió monitorear el ambiente urbano con alta resolución espacial y temporal. Asimismo, posibilitó un aumento de la cobertura espacial y la disminución de los tiempos de medición en relación con los métodos tradicionales. Se comparó la distribución térmica urbana estival e invernal observados en 1985 y 2014. Durante este período, el crecimiento urbano de la ciudad de Bahía Blanca modificó la distribución espacial de la temperatura y la humedad relativa generando un aumento de las condiciones de desconfort en la población.
Se realizó por primera vez la caracterización térmica de la ciudad y su relación con las distintas coberturas del suelo urbano, el estudio de la expansión urbana y su relación con el Índice Normalizado de Vegetación a partir del procesamiento digital de imágenes satelitales. Se calculó una expansión urbana de 110 % desde 1986 hasta 2014. Como consecuencia los valores de Temperatura de Superficie Terrestre (TST) y NDVI disminuyeron 0,6 ºC y 0,32, respectivamente. La periferia y las áreas con escasa construcción fueron más cálidas que el centro de la ciudad (debido a que las imágenes satelitales son tomadas a la hora de la mañana, representando el estudio de la "Isla de frescor" o "Isla fría"). Las coberturas del suelo Periferia y Urbano Disperso fueron las que presentaron los mayores valores de TST y los menores fueron en el Centro y las Áreas Verdes.
Para el análisis a escala micro-local se diseñaron 10 estaciones de monitoreo de temperatura y humedad relativa. Se denominaron Estaciones Portátiles (EP) y registraron información en alta frecuencia (30 minutos) durante 408 días. Se localizaron en distintos sectores de la ciudad y permitieron analizar el comportamiento de las variables meteorológicas teniendo en cuenta las distintas condiciones de sitio. Las diferencias espaciales de temperatura en la ciudad de Bahía Blanca fueron mayores en el invierno. Las áreas más frías y húmedas fueron la costera, la periferia y las que tuvieron mayor densidad de vegetación urbana localizadas en el noreste. Los espacios construidos y con mayor densidad de edificación fueron cálidos y secos durante esta estación del año.
La información de las EP permitió analizar los cambios de la temperatura y la humedad relativa urbana durante distintos eventos meteorológicos tales como vientos fuertes del norte, del sur, olas de calor, olas de frío, tormentas, heladas y nieblas. Las olas de calor de verano incrementaron 8 ºC la temperatura urbana. Los impactos fueron mayores en Villa Floresta, UOM, Villa Mitre y centro y menores en Patagonia y Noroeste. La ola de calor de invierno generó temperaturas más elevadas (3,6 ºC) en Villa Floresta, CONICET y centro y menores en Patagonia e Ing. White. La ola de frío redujo la temperatura urbana en 4 ºC. Los barrios más afectados fueron Cooperación II, UOM y Villa Floresta. Los vientos fuertes del norte aumentaron la 3,9 ºC la temperatura de la ciudad afectando más intensamente los barrios de Bella Vista y Patagonia. Por otro lado los vientos fuertes del sur redujeron 4,1 ºC la temperatura y tuvieron los mayores impactos sobre CONICET, Bella Vista y Cooperación II. Las heladas se presentaron en los barrios de CONICET, Cooperación II, UOM y Patagonia, mientras que las nieblas también se extendieron sobre Ing. White. Cabe señalar que durante todos los eventos analizados el centro fue más cálido que el resto de la ciudad. El confort climático estival e invernal se analizó también con los datos adquiridos con las EP. Los espacios urbanos con alta densidad de edificaciones presentaron un 40 % de los días de verano con desconfort. Los espacios con mayor vegetación urbana y los cercanos al mar fueron los que tuvieron situaciones óptimas de confort.
Finalmente se aplicó el método DPSIR (por sus siglas en inglés Driver-Pressure-State-Impact-Responses Framework) y se diseñó un índice de estado actual de la ciudad de Bahía Blanca con el objetivo de realizar una propuesta de manejo urbano sostenible. Para llevarla a cabo se consideraron los datos climáticos y se incorporó un análisis la percepción de la población. Los impactos de las variables climáticas y sociales consideradas permitieron identificar 4 regiones urbanas bien diferenciadas entre sí. La participación de la población permitió enmarcar el trabajo en el paradigma de la Percepción Social del Cambio Climático. / Climate studies are analyzed at different spatial and temporal scales and they might be studied in different global, regional, local and micro-local processes where the last two refer to urban spaces. The latter generates climate variations in a small territory due to its radiative, thermal, physical and aerodynamic characteristics which differ from those of their natural environment.
The main objective of this research was to study the local climate and the micro-climate of Bahia Blanca by comparing the results with previous studies. Therefore, the impact of climate variability, the increase of the population and the local climate were assessed.
In a regional scale, climate variability in Bahia Blanca was studied by analyzing 50 years of daily climate records. Different time scales of analysis were applied: decadal, interannual and seasonal in order to characterize the temporal behavior of air temperature, relative humidity, rainfall and maximum wind speed. It was observed that air temperature increased 0.9 ° C, the maximum wind speed was reduced in 7.5 km hˉ¹ and relative humidity presented two different cycles related to wet and dry periods. The analysis of rainfall differed in the study period in two extremely dry and two extremely wet events.
New electronic devices were designed to study the urban heat island which recorded data at high frequency (2 Hz) based on open code electronics. This instrument allowed monitoring the urban environment with high spatial and temporal resolution. It also facilitated the increase in spatial coverage and reduced measurement times which differ from traditional methods. It was compared summer and winter urban heat distribution observed in 1985 and 2014. During this period, the urban growth of the city of Bahia Blanca modified the spatial distribution of temperature and relative humidity causing an increase in disconformity in population.
For the first time, a thermal characterization of the city and its relationship with different coverage of urban land, urban expansion studies and their relationship with the Normalized Difference Vegetation Index (NDVI) from digital satellite image processing was performed. Urban expansion raised 110% from 1986 to 2014. As a result of this expansion values, Land Surface Temperature (LST) and NDVI area decreased 0.6 ° C and 0.32, respectively. The periphery and areas with scarce buildings were warmer than the city center (due to images were registered in the morning). The land cover denominated Periphery and Urban Disperse were those that showed the highest values of LST and the lowest were in the downtown and in the green areas.
For the analysis of a micro-local level, 10 monitoring temperature and relative humidity stations were designed. These stations were called Portable Station (PS) and recorded information during 408 days at high frequency (30 minutes). They were located in different areas of the city and they allowed analyzing the behavior of the meteorological variables by taking into account differences in site conditions. The spatial differences in temperature within the city of Bahia Blanca were higher in the winter. The cooler and wetter areas were the coastal periphery and those that had higher density of urban vegetation located in the northeast. Areas with buildings and greater density of buildings were warm and dry during this season.
Portables Stations’ information permit to analyze the changes in temperature and relative humidity during different meteorological weather events such as strong winds from the north and south as well as, heat waves, cold waves, storms, frosts and fogs. The summer heat waves increased urban temperature in 8 ° C. The impacts were greater in Villa Floresta, UOM, Villa Mitre, the city center, and lower in Patagonia and Northwest. The winter heat wave generated higher temperatures (3.6 ° C) at Villa Floresta, CONICET, the city center and lower in Patagonia and Ing. White. The cold wave reduced the urban temperature in 4 ° C. The most affected districts were Cooperación II, UOM and Villa Floresta. Strong winds from the north increased in 3.9 ° C the temperature of the city by affecting neighborhoods such as Bella Vista and Patagonia. Conversely, strong winds from the south decreased in 4.1 ºC and had the greatest impacts on CONICET, Bella Vista and Cooperación II. Frost occurred in the districts of CONICET, Cooperación II, UOM and Patagonia, while fogs also spread on Ing. White. It is fundamental to highlight that for all the events analyzed, the center was warmer than the rest of the city.
The summer and winter climate comfort was also studied with the data acquired by the PS. In addition, urban areas with high density of buildings showed 40% of summer days with discomfort. Places with more urban vegetation and those which are closer to the sea had optimal conditions of comfort.
Finally the DPSIR method was applied (Driver-Pressure-State-Impact-Responses Framework) and an index to observe the current state of the city of Bahia Blanca was designed. This allowed creating a sustainable urban management proposal. This proposal was carried out by considering climate data and a perception analysis of the population. The impacts of climate and social variables identified four different urban regions. The participation of people permitted to frame this work in the paradigm of the Social Perception of Climate Change.
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Revendications citoyennes et réalité de la gouvernance urbaine locale : les acteurs associatifs locaux à l'épreuve du projet de requalification urbaine de la rue de la République à Marseille / Citizen claims and reality of local urban governance : the local associations to test the urban requalification project in the street of the Republique in MarseilleMaargab, Rafik 04 December 2012 (has links)
Cette recherche traite du sujet de la gouvernance urbaine locale au delà de son aspect conceptuel ou analytique. Il est question d'appréhender la gouvernance urbaine locale à travers : le projet urbain, les moyens et mécanismes de la gestion de la chose publique aussi bien législatifs que techniques, ainsi qu'à travers les pratiques des acteurs publics et l'ensemble des parties prenantes dont les acteurs de la société civile locale. En s'appuyant sur l'exemple du projet de la requalification urbaine de la rue de la République à Marseille, l'étude s'intéresse également à montrer la capacité et l'apport des associations de quartier dans la défense des intérêts des citoyens et la promotion de la démocratie participative locale. Par ailleurs, la recherche aborde la question de l'impact des mutations scocio-économiques, dues au croisement des politiques publiques, et l'intervention des acteurs économiques aussi bien sur l'espace micro-local, en l'occurrence la rue, que sur le milieu urbain et les citoyens. C'est tout l'intérêt de cette recherche de montrer l'impact des logiques économiques et politiques sur la gestion du projet urbain et la place qu'occupent les citoyens et les acteurs associatifs dans le processus décisionnel. / This research deals with the subject of local urban governance beyond its conceptual or analytical aspect. It is question to understanding the local urban governance through: the urban project, the means and mechanisms for the management of the public affairs as well as techniques and practices of public and all stakeholders, including local civil society actors. Based on the example of urban regeneration project in the street of the République in Marseille, the study also showed the ability and the contribution of neighbourhood associations in the interests of citizens and promotion of local participatory democracy. Furthermore, the research addresses the question of the impact of mutations scocio-economic due to the crossing of public policy and the intervention of economic actors on the micro-local space, namely the street, and on the urban citizens. It's the whole interest of this research to show the impact of economic and political logics on the urban project and to show the place of citizens and civil society actors in decision making.
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Contrôle, stabilisation et propagation des singularités pour des EDP dispersives / Control, Stabilization and Propagation of Singularities for dispersive PDEsZhu, Hui 27 March 2019 (has links)
Dans cette thèse, nous étudions les théories étroitement liées du contrôle, de la stabilisation et de la propagation des singularités, pour des équations aux dérivées partielles dispersives linéaires et non-linéaires. Les résultats principaux proviennent des travaux de l’auteur:[1] Zhu, H., 2016. Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM : Control, Optimisation and Calculus of Variations (ESAIM: COCV), à paraître.[2] Zhu, H., 2017. Control of three dimensional water waves. arXiv preprint arXiv:1712.06130.[3] Zhu, H., 2018. Propagation of singularities for gravity-capillary water waves. arXiv preprint arXiv:1810.09339.Dans [1], nous avons étudié la stabilisation des ondes amorties sur les surfaces de révolution de Zoll. Nous avons donné un exemple où la région d’amortissement est à la limite de la condition du contrôle géométrique, alors que les ondes amorties présentent une décroissance exponentielle uniforme de l’énergie. Cet exemple généralise un résultat de Lebeau. Dans [2], nous avons étudié la contrôlabilité du système des ondes de surface avec tension superficielle. Nous avons démontré, en dimensions arbitraires, la contrôlabilité exacte pour des petites données spatialement périodiques à condition du contrôle géométriques. Ce résultat généralise le travail de Alazard, Baldi et Han-Kwan en dimension deux. Dans [3], nous avons étudié la propagation des singularités pour des ondes de surface avec tension superficielle. Nous avons défini le front d’onde quasi-homogène, généralisant le front d’onde de Hörmander et le front d’onde homogène de Nakamura et démontré des résultats de propagation des fronts d’onde quasi-homogènes par le système des ondes de surface avec tension superficielle. Comme corollaires, nous avons obtenu des effets régularisants locaux et micro-locaux pour les données initiales présentant une décroissance spatiale suffisante. / In this thesis, we study the closely related theories of control, stabilization and propagation of singularities for some linear and nonlinear dispersive partial differential equations. Main results come from the author’s works:[1] Zhu, H., 2016. Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), to appear.[2] Zhu, H., 2017. Control of three dimensional water waves. arXiv preprint arXiv:1712.06130.[3] Zhu, H., 2018. Propagation of singularities for gravity-capillary water waves. arXiv preprint arXiv:1810.09339.In [1] we studied the stabilization of the damped wave equation on Zoll surfaces of revolution. We gave an example where the region of damping is at the borderline of the geometric control condition, yet the damped waves exhibit a uniform exponential decay of energy, generalizing an example of Lebeau.In [2] we studied the controllability of the gravity-capillary water wave equation. Under the geometric control condition, we proved in arbitrary spatial dimension the exact controllability for spatially periodic small data. This generalizes a result of Alazard, Baldi and Han-Kwan for the 2D gravity-capillary water wave equation.In [3] we studied the propagation of singularities for the gravity-capillary water wave equation. We defined the quasi-homogeneous wavefront set, generalizing the wavefront set of H¨ ormander and the homogeneous wavefront set of Nakamura, and proved propagation results for quasi-homogeneous wavefront sets by the gravity-capillary water wave equation. As corollaries, we obtained local and microlocal smoothing effects for gravity-capillary water waves with sufficient spatial decay.
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Propagation of singularities for pseudo-differential operators and generalized Schrödinger propagatorsJohansson, Karoline January 2010 (has links)
<p>In this thesis we discuss different types of regularity for distributions which appear in the theory of pseudo-differential operators and partial differential equations. Partial differential equations often appear in science and technology. For example the Schrödinger equation can be used to describe the change in time of quantum states of physical systems. Pseudo-differential operators can be used to solve partial differential equations. They are also appropriate to use when modeling different types of problems within physics and engineering. For example, there is a natural connection between pseudo-differential operators and stationary and non-stationary filters in signal processing. Furthermore, the correspondence between symbols and operators when passing from classical mechanics to quantum mechanics essentially agrees with symbols and operators in the Weyl calculus of pseudo-differential operators.</p><p>In this thesis we concentrate on investigating how regularity properties for solutions of partial differential equations are affected under the mapping of pseudo-differential operators, and in particular of the free time-dependent Schrödinger operators.</p><p>The solution of the free time-dependent Schrödinger equation can be expressed as a pseudo-differential operator, with non-smooth symbol, acting on the initial condition. We generalize a result about non-tangential convergence, which was obtained by Sjögren and Sjölin (1989) for the free time-dependent Schrödinger equation.</p><p>Another way to describe regularity for a distribution is to use wave-front sets. They do not only describe where the singularities are, but also the directions in which these singularities appear. The first types of wave-front sets (analytical wave-front sets) were introduced by Sato (1969, 1970). Later on Hörmander introduced ``classical'' wave-front sets (with respect to smoothness) and showed results in the context of pseudo-differential operators with smooth symbols, cf. Hörmander (1985).</p><p>In this thesis we consider wave-front sets with respect to Fourier Banach function spaces. Roughly speaking, we take <em>B</em> as a Banach space, which is invariant under translations and embedded between the space of Schwartz functions and the space of temperated distributions. Then we say that the wave-front set of a distribution contains all points (x<sub>0</sub>, ξ<sub>0</sub>) such that no localization of the distribution at x<sub>0</sub>, belongs to <em>FB</em> in the direction ξ<sub>0</sub>. We prove that pseudo-differential operators with smooth symbols shrink the wave-front set and we obtain opposite embeddings by using sets of characteristic points of the operator symbols.</p> / <p>I denna avhandling diskuterar vi olika typer av regularitet för distributioner som uppkommer i teorin för pseudodifferentialoperatorer och partiella differentialekvationer. Partiella differentialekvationer förekommer inom naturvetenskap och teknik. Exempelvis kan Schrödingerekvationen användas för att beskriva förändringen med tiden av kvanttillstånd i fysikaliska system. Pseudodifferentialoperatorer kan användas för att lösa partiella differential\-ekvationer. De användas också för att modellera olika typer av problem inom fysik och teknik. Det finns till exempel en naturlig koppling mellan pseudodifferentialoperatorer och stationära och icke-stationära filter i signalbehandling. Vidare gäller att relationen mellan symboler och operatorer vid övergången från klassisk mekanik till kvantmekanik i huvudsak överensstämmer med symboler och operatorer inom Weylkalkylen för pseudodifferentialoperatorer.</p><p>I den här avhandlingen koncentrerar vi oss på att undersöka hur regularitetsegenskaper för lösningar till partiella differentialekvationer påverkas under verkan av pseudodifferentialoperatorer, och speciellt för de fria tidsberoende Schrödingeroperatorerna.</p><p>Lösningen av den fria tidsberoende Schrödingerekvationen kan uttryckas som en pseudodifferentialoperator, med icke-slät symbol, verkande på begynnelsevillkoret. Vi generaliserar ett resultat om icke-tangentiell konvergens av Sjögren och Sjölin (1989) för den fria tidsberoende Schrödingerekvationen.</p><p>Ett annat sätt att beskriva regularitet hos en distribution är med hjälp av vågfrontsmängder. De beskriver inte bara var singulariteterna finns, utan också i vilka riktningar dessa singulariteter förekommer. De första typerna av vågfrontsmängder (analytiska vågfrontsmängder) introducerades av Sato (1969, 1970). Senare introducerade Hörmander ''klassiska'' vågfrontsmängder (med avseende på släthet) och visade resultat för verkan av pseudodifferentialoperatorer med släta symboler, se Hörmander (1985).</p><p>I denna avhandling betraktar vi vågfrontsmängder med avseende på Fourier Banach funktionsrum. Detta kan ses som att vi låter <em>B</em> vara ett Banachrum, som är invariant under translationer och är inbäddat mellan rummet av Schwartzfunktioner och rummet av tempererade distributioner. Vågfrontsmängden av en distribution innehåller alla punkter (x<sub>0</sub>, ξ<sub>0</sub>) så att ingen lokalisering av distributionen kring x<sub>0</sub>, tillhör <em>FB</em> i riktningen ξ<sub>0</sub>. Vi visar att pseudodifferentialoperatorer med släta symboler krymper vågfrontsmängden och vi får motsatta inbäddningar med hjälp mängder av karakteristiska punkter till operatorernas symboler.</p>
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Propagation of singularities for pseudo-differential operators and generalized Schrödinger propagatorsJohansson, Karoline January 2010 (has links)
In this thesis we discuss different types of regularity for distributions which appear in the theory of pseudo-differential operators and partial differential equations. Partial differential equations often appear in science and technology. For example the Schrödinger equation can be used to describe the change in time of quantum states of physical systems. Pseudo-differential operators can be used to solve partial differential equations. They are also appropriate to use when modeling different types of problems within physics and engineering. For example, there is a natural connection between pseudo-differential operators and stationary and non-stationary filters in signal processing. Furthermore, the correspondence between symbols and operators when passing from classical mechanics to quantum mechanics essentially agrees with symbols and operators in the Weyl calculus of pseudo-differential operators. In this thesis we concentrate on investigating how regularity properties for solutions of partial differential equations are affected under the mapping of pseudo-differential operators, and in particular of the free time-dependent Schrödinger operators. The solution of the free time-dependent Schrödinger equation can be expressed as a pseudo-differential operator, with non-smooth symbol, acting on the initial condition. We generalize a result about non-tangential convergence, which was obtained by Sjögren and Sjölin (1989) for the free time-dependent Schrödinger equation. Another way to describe regularity for a distribution is to use wave-front sets. They do not only describe where the singularities are, but also the directions in which these singularities appear. The first types of wave-front sets (analytical wave-front sets) were introduced by Sato (1969, 1970). Later on Hörmander introduced ``classical'' wave-front sets (with respect to smoothness) and showed results in the context of pseudo-differential operators with smooth symbols, cf. Hörmander (1985). In this thesis we consider wave-front sets with respect to Fourier Banach function spaces. Roughly speaking, we take B as a Banach space, which is invariant under translations and embedded between the space of Schwartz functions and the space of temperated distributions. Then we say that the wave-front set of a distribution contains all points (x0, ξ0) such that no localization of the distribution at x0, belongs to FB in the direction ξ0. We prove that pseudo-differential operators with smooth symbols shrink the wave-front set and we obtain opposite embeddings by using sets of characteristic points of the operator symbols. / I denna avhandling diskuterar vi olika typer av regularitet för distributioner som uppkommer i teorin för pseudodifferentialoperatorer och partiella differentialekvationer. Partiella differentialekvationer förekommer inom naturvetenskap och teknik. Exempelvis kan Schrödingerekvationen användas för att beskriva förändringen med tiden av kvanttillstånd i fysikaliska system. Pseudodifferentialoperatorer kan användas för att lösa partiella differential\-ekvationer. De användas också för att modellera olika typer av problem inom fysik och teknik. Det finns till exempel en naturlig koppling mellan pseudodifferentialoperatorer och stationära och icke-stationära filter i signalbehandling. Vidare gäller att relationen mellan symboler och operatorer vid övergången från klassisk mekanik till kvantmekanik i huvudsak överensstämmer med symboler och operatorer inom Weylkalkylen för pseudodifferentialoperatorer. I den här avhandlingen koncentrerar vi oss på att undersöka hur regularitetsegenskaper för lösningar till partiella differentialekvationer påverkas under verkan av pseudodifferentialoperatorer, och speciellt för de fria tidsberoende Schrödingeroperatorerna. Lösningen av den fria tidsberoende Schrödingerekvationen kan uttryckas som en pseudodifferentialoperator, med icke-slät symbol, verkande på begynnelsevillkoret. Vi generaliserar ett resultat om icke-tangentiell konvergens av Sjögren och Sjölin (1989) för den fria tidsberoende Schrödingerekvationen. Ett annat sätt att beskriva regularitet hos en distribution är med hjälp av vågfrontsmängder. De beskriver inte bara var singulariteterna finns, utan också i vilka riktningar dessa singulariteter förekommer. De första typerna av vågfrontsmängder (analytiska vågfrontsmängder) introducerades av Sato (1969, 1970). Senare introducerade Hörmander ''klassiska'' vågfrontsmängder (med avseende på släthet) och visade resultat för verkan av pseudodifferentialoperatorer med släta symboler, se Hörmander (1985). I denna avhandling betraktar vi vågfrontsmängder med avseende på Fourier Banach funktionsrum. Detta kan ses som att vi låter B vara ett Banachrum, som är invariant under translationer och är inbäddat mellan rummet av Schwartzfunktioner och rummet av tempererade distributioner. Vågfrontsmängden av en distribution innehåller alla punkter (x0, ξ0) så att ingen lokalisering av distributionen kring x0, tillhör FB i riktningen ξ0. Vi visar att pseudodifferentialoperatorer med släta symboler krymper vågfrontsmängden och vi får motsatta inbäddningar med hjälp mängder av karakteristiska punkter till operatorernas symboler.
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