Résumé : Cette thèse est consacrée à l'étude des équations de Stokes et de Navier-Stokes avec des conditions aux limites de Navier dans un ouvert borné de . Le manuscrit ici est composé de trois chapitres. Dans le premier, nous considérons les équations de Stokes stationnaires avec des conditions aux limites de Navier. Nous démontrons l'existence, l'unicité et la régularité de la solution d'abord dans un cadre hilbertien puis dans le cadre de la théorie . Nous traitons aussi le cas de solutions très faibles. Dans le deuxième chapitre, nous nous intéressons aux équations de Navier-Stokes avec la condition de Navier. Sous certaines hypothèses sur les données, nous démontrons l'existence de solution faible dans , avec en utilisant un théorème du point fixe appliqué à un problème d'Oseen. Nous démontrons examinons ensuite les questions de régularité des solutions en particulier dans . Dans le dernier chapitre, nous étudions le problème d'évolution de Stokes avec la condition de Navier. La résolution de ce problème se fait au moyen de la théorie des semi-groupes analytiques qui jouent un rôle important pour établir l'existence et l'unicité de la solution dans le cas homogène. Nous traitons le cas du problème non homogène par le biais des puissances imaginaires de l'opérateur de Stokes. / This thesis is devoted to the study of the Stokes equations and Navier-Stokes equations with Navier boundary conditions in a bounded domain of . The work contains three chapters: In the first chapter, we consider the stationary Stokes equations with Navier boundary condition. We show the existence, uniqueness and regularity of the solution in the Hilbert case and in the -theory. We prove also the case of very weak solutions. In the second chapter, we focus on the Navier-Stokes equations with the Navier boundary condition. We show the existence of the weak solution in , with by a fixed point theorem over the Oseen equation. We show also the existence of the strong solution in . In chapter three, we study the evolution Stokes problem with Navier boundary condition. For this, we apply the analytic semi-groups theory, which plays a crucial role in the study of existence and uniqueness of solution in the case of the homogeneous evolution problem. We treat the case of non-homogeneous problem through imaginary powers of the Stokes operator.

This thesis deals with modern statistical approaches and their application aimed at robust methods and neural network modelling. Selected methods are analyzed and applied on frequent practical problems in czech industry and technology. Topics and methods are to be benificial in real applications compared to currently used classical methods. Applicability and effectivity of the algorithms is verified and demonstrated on real studies and problems in czech industrial and research bodies. The great and unexploited potential of modern theoretical and computational capacity and the potential of new approaces to statistical modelling and methods. A significant result of this thesis is also an environment for software application development for data analysis with own programming language DARWin (Data Analysis Robot for Windows) for implemenation of effective numerical algorithms for extaction information from data. The thesis should be an incentive for boarder use of robust and computationally intensive methods as neural networks for modelling processes, quality control and generally better understanding of nature.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal. / Parmi la multitude de fonctions que le système visuel effectue, la perception du mouvement est l'une des plus importantes. Il a été clairement démontré qu'il existe des sites cérébraux spécifiques pour la détection, l'analyse et l'intégration du mouvement. De façon classique, les mécanismes neurophysiologiques qui sous-tendent ces processus sont attribués aux aires corticales. Le thalamus quant à lui, est généralement considéré comme un « relais passif », c'est à dire qui transmet l'information sensorielle vers le cortex sans modifier le signal entrant. Le but de ce projet sera de préciser les mécanismes nerveux impliqués dans la perception et l'intégration du mouvement et plus précisément, la contribution des régions cérébrales sous-corticales et corticales intimement liés par des connexions réciproques. Ces régions sont: le complexe LP-pulvinar, situé dans le thalamus, l'aire extra-striée postero-médiane suprasylvienne (PMLS) et le cortex ectosylvien visuel antérieur (AEV); deux régions corticales ayant un rôle spécialisé dans l'analyse du mouvement. Les expériences ont été réalisées sur des chats adultes normaux anesthésiés. Une microélectrode d'enregistrement a été descendue dans ces trois sites afin d'enregistrer l'activité des neurones. Les réponses neuronales à des réseaux sinusoïdaux, des patrons texturés (« bruit visuel ») et des « plaids » ont été caractérisé pour étudier les mécanismes qui sous-tendent l'intégration du mouvement au niveau cellulaire. Afin de caractériser d'avantage ce lien, l'influence des aires corticales sur les propriétés thalamiques a été déterminée par inactivation locale réversible (i.e. micro-injection de l'acide y-aminobutyrique; GABA) ou par inactivation permanente plus vaste (i.e. ablation chirurgicale). Les résultats de cette étude se résument comme suit : 1) les propriétés des réponses neuronales du PMLS au bruit visuel sont similaires à celles du LP-pulvinar. Ce résultat suggère que les processus d'analyse impliquant une boucle cortico-thalamique PMLS-LP sont comparables au niveau cortical et sous-cortical. 2) les neurones du PMLS et du LP peuvent coder le mouvement relatif entre un objet et son arrièreplan. De plus, l'inactivation réversible du LP perturbe ces réponses au niveau du PMLS. Ces résultats sont essentiels dans l'établissement d'un lien fonctionnel entre ces deux régions quant à l'analyse de certains aspects du mouvement. 3) certains neurones du complexe LP-pulvinar sont capables d'intégrer l'information directionnelle telle que définie par des « plaids ». Ceci constitue la première démonstration de propriétés de haut-niveau en dehors du cortex. De plus, cette découverte suggère que le LP-pulvinar participe de façon parallèle et en coopération avec le cortex dans l'analyse de scènes visuelles complexes via l'exploitation des boucles cortico-thalamiques. Les résultats de cette étude sont importants non seulement pour appuyer des notions théoriques novatrices sur le rôle du thalamus, mais aussi dans le but de réévaluer et de préciser les mécanismes nerveux qui sous-tendent la perception du mouvement et l'intégration sensorielle en général. / Among the multitude of functions the visual system carries out, the perception of motion is one of the most important. It has been clearly demonstrated that the visual system contains numerous specialised areas implicated in the detection, analysis, and integration of motion. Classically, the neurophysiological mechanisms underlying these processes have been uniquely attributed to regions of the cerebral cortex. The thalamus for its part, has generally been regarded as a passive relay transferring information to the cortex without any modification of the sensory signal. The purpose of this study is to investigate the neurophysiological mechanisms implicated in the perception and integration of motion and more specifically, delineate the contribution of cortical and subcortical structures that are intimately related via reciprocal connections. These areas are: the LP-pulvinar complex; located in the thalamus, and the extrastriate posteromedial lateral suprasylvian (PMLS) and anterior ectosylvian visual (AEV) cortical areas; two regions whose role in motion analysis are well established. Experiments were carried out on normal adult anaesthetised cats. A recording microelectrode was descended in one of the aforementioned areas to record neuronal activity. Neuronal responses to drifting sine-wave gratings, moving texture patterns ("visual noise"), and "plaid patterns" were recorded in order to investigate the mechanisms underlying the integration of motion information at the neuronal level. As a continuation of the study, the influence of cortical motion areas on recorded thalannic responses will be determined by local reversible deactivation (i.e. microinjection of y-aminobutyric acid; GABA) or by irreversible deactivation (i.e. surgical ablation). The results of the study are as follows: 1) Response properties of PMLS neurons to moving texture patterns are similar to those found in the LP-pulvinar connplex. These results suggest that motion processing along both components of the PMLS-LP cortico-thalamic loop is carried out within a similar envelope of analysis. 2) Neurons in both PMLS and LP are able to code the relative motion of an object with respect to its background. Furthermore, reversible deactivation of LP disrupts these responses in PMLS. These results are important in establishing that both these areas are functionally linked in the analysis of specific aspects of motion. 3) The fact that pattern-selective responses to moving plaids can be found in the LP-pulvinar complex suggests that this area is capable of carrying out higher-order motion computations. The importance of this later results is two-fold. First, these findings represent the first demonstration that higher-order properties exists outside extrastriate cortical areas. Second, they further suggest that certain thalamic nuclei, via the establishment of cortico-thalamic loops, participate in parallel and in co-operation with the cortex in the analysis of complex visual scenes. The results of this study are important not only to reinforce current and novel theoretical notions regarding the role of the thalamus, but also in the re-evaluation of the neurophysiological mechanisms involved in motion perception and sensory integration as a whole.

The need to construct optimal drape surfaces arises in airborne geophysical surveys where it is necessary to fly a safe distance above the ground and within the performance limits of the aircraft used, but as close as possible to the surface. The problem is formulated as an LP with constraints at every point of a grid covering the area concerned, yielding a very large problem. The objective of the LP is to create as "good" a surface as possible. This formulation is new, as previous methods did not aim to minimise an objective function. If the desired surface has only slope limitations, the resulting drape surface must be constrained in the first derivative. Such a drape surface is readily constructed using the Lifting Algoritlun. It is shown that the Litling Algorithm is both exact and has great speed advantages. Some numerical results confinning exacmcss and speed are presented, as is the algorithm's analogy to a flow network method. An enhanced lifting method with a better order of complexity is also proposed and tested numerically. In most practical situations a drape surface is required which has both first and second derivatives constrained. If only a cut through such a surface is considered, the problem can be solved with relative ease by exploiting its nctwork~Jike structure. This method fonns the basis of one of the preferred heuristics developed later. It was not possible to generalise this method to a full two~dimensional drape surface. A commercially available LP package fares better in finding the optimal solution. Several heuristic methods were examined. first a general heuristic method based on a lifting approach was developed. This was followed by a method using repeated application of the method used for sections (the Alternating One-dimensional Dual Algorithm ["AODA"]). Three heuristics based on thimbles were also designed. Thimbles are caps whose first and second derivatives are acceptable and which are placed over local infeasibilities in the topography The work ends with a chapter comparing the efficiency of various heuristics and comparing the results obtained using a number of test datasets. It was fOLmd that heuristic methods provide acceptable drape surfaces and that the choice lies between speed and accuracy, with a previously designed smoothing method being the fastesl and the AODA the most accurate and quick enough. / Operations Research / D.Phil.

In this digital age, it is more important than ever to have good methods for processing images. We focus on the removal of blur from a captured image, which is called the image deblurring problem. In particular, we make no assumptions about the blur itself, which is called a blind deconvolution. We approach the problem by miniming an energy functional that utilizes total variation norm and a fidelity constraint. In particular, we extend the work of Chan and Wong to use a reference image in the computation. Using the shock filter as a reference image, we produce a superior result compared to existing methods. We are able to produce good results on non-black background images and images where the blurring function is not centro-symmetric. We consider using a general Lp norm for the fidelity term and compare different values for p. Using an analysis similar to Strong and Chan, we derive an adaptive scale method for the recovery of the blurring function. We also consider two numerical methods in this disseration. The first method is an extension of Picards method for PDEs in the discrete case. We compare the results to the analytical Picard method, showing the only difference is the use of the approximation versus exact derivatives. We relate the method to existing finite difference schemes, including the Lax-Wendroff method. We derive the stability constraints for several linear problems and illustrate the stability region is increasing. We conclude by showing several examples of the method and how the computational savings is substantial. The second method we consider is a black-box implementation of a method for solving the generalized eigenvalue problem. By utilizing the work of Golub and Ye, we implement a routine which is robust against existing methods. We compare this routine against JDQZ and LOBPCG and show this method performs well in numerical testing.

Heterojunctions of CuInSe2 (CIS) nanowires with cadmium sulfide (CdS) were fabricated demonstrating for the first time, vertically aligned nanowires of CIS in the conventional Mo/CIS/CdS stack. These devices were studied for their material and electrical characteristics to provide a better understanding of the transport phenomena governing the operation of heterojunctions involving CIS nanowires. Removal of several key bottlenecks was crucial in achieving this. For example, it was found that to fabricate alumina membranes on molybdenum substrates, a thin interlayer of tungsten had to be inserted. A qualitative model was proposed to explain the difficulty in fabricating anodized aluminum oxide (AAO) membranes directly on Mo. Experimental results were used to corroborate this model. Subsequently, a general procedure to use any material that can be deposited using sputtering or evaporation as a back contact for nanowires grown using AAO templates was developed. Experimental work to demonstrate this by transferring thin AAO templates onto flexible Polyimide (PI) substrates was performed. This pattern transfer approach opens doors for a wide variety of applications on almost any substrate. Any material that can be deposited by physical means can then be used as a back contact. Electron-beam induced deposition using a liquid precursor (LP-EBID) was used to selectively grow preconceived patterns of compound semiconductor (CdS) nanoparticles. Stoichiometric CdS nanoparticle patterns were grown successfully using this method. They were structurally and optically characterized indicating high purity deposits. This approach is promising because it marries the precision of e-beam lithography with the versatility of solution based deposition methods.

Due to the continuous work in the automobile industry to reduce the environmental impact, reduce fuel consumption and increase efficiency, new technologies need to be developed and implemented in vehicles. For spark ignited engines, one technology that has received more attention in recent years is long route Exhaust Gas Recirculation (EGR), which means that exhaust gases after the turbine are transported back to the volume before the compressor in the air intake system of the engine. In this work, the components of the long route EGR system is modeled with mean value engine models in Simulink, and implemented in a existing Simulink engine model. Then different methods for estimating the mass flow over the long route EGR system are compared, and the transport delays for the recirculated exhaust gases in the engines air intake system are modeled. This work is based on measurements done on an engine rig, on which a long route EGR system was installed. Finally, some ideas on how a long route EGR system on a gasoline engine can be controlled are presented based on the results in this thesis work.

Histoire et suivi d'une cohorte multicentrique de 100 nouveau-nés à terme ayant eu un infarctus cérébral artériel (cohorte AVCnn). L’homogénéité de la population, le nombre d’inclus, la somme des données collectées et leur comparaison à des groupes témoins puis désormais la durée du suivi font de la cohorte AVCnn un outil de travail unique dans la littérature. Les premiers résultats ont permis de définir un profil obstétrico-néonatal clinique et biologique à risque (objectif principal de l'étude). Inversement d'autres facteurs de risque suspectés (comme les dysgravidies ou les mutations des facteurs II et V de la coagulation) n'ont pu être confirmés, probablement du fait de la sélectivité des critères d'inclusion; les études antérieures ayant en effet inclus des enfants avec divers types d'AVC périnatals. La cause de l'accident n'a été établie (objectif secondaire) que pour une minorité d'enfants. Il semble exister plusieurs mécanismes pathologiques, dont le poids de naissance et le sexe seraient des biomarqueurs. Des facteurs pronostiques (objectif secondaire) de l'évolution motrice, accessibles dès les premiers jours de vie à l'lRM, ont également été déterminés alors que les symptômes n'apparaîtront que plusieurs mois après. Ceci permet d'envisager qu'une prise en charge précoce et ciblée sur ces nouveau-nés serait d'autant plus efficace que l'enfant n'a pas encore de symptômes. L'évaluation de la qualité de vie et de l'autonomie à l'âge de 3,5 ans est terminée. L'évaluation développementale (profil moteur, neuropsychologique, langagier, comportemental, épileptique et réussite scolaire) et de la réorganisation fonctionnelle cérébrale par IRM à 7 ans viennent de débuter / The AVCnn study is a French multicentre cohort of 100 term newborns with symptomatic arterial ischemic stroke. The homogeneity of the population, the number of patients included, the amount of collected data and their comparison with control groups and duration of follow-up allow to make the cohort AVCnn unique in the literature. Initial data confirm that many obstetrical and perinatal factors are risk determinants (principal objective of the study). Conversely, other suspected risk factors (such as dysgravidia or mutations of coagulation factors II and V) were not confirmed, probably due to the selectivity of the inclusion criteria: previous studies have in fact included children with various types of perinatal stroke. The cause of the accident has been formally established (secondary objective) in only two cases. Nevertheless, birth weight and gender may be biomarkers of two populations of neonates with different pathological mechanisms. Prediction of motor outcomes (secondary objective) can be obtained from early imaging data (<28 days of live), while the symptoms will only appear several months later. This is of paramount importance to families and clinicians, because early intervention may help alleviate the resulting disabilities. The evaluation of quality of life and Functional Independence Measure at age 3.5 years is now complete. The developmental assessment (neuropsychological, language, and behavioural profiles, epilepsy and academic achievement) and the functional reorganization of brain MRI at 7 years have just begun

Le sujet principal de cette thèse de doctorat est l'étude de diverses quantités reliées aux processus de fragmentation. Ces processus sont destinés à modéliser un objet de masse unité se fragmentant au cours du temps.<br />Ce travail comporte quatre chapitres. Le premier chapitre est consacré à l'étude de la dimension de Hausdorff de l'ensemble des points ayant une décroissance exponentielle dans une fragmentation homogène en intervalles. Dans le deuxième chapitre, on construit un processus de Markov auto-similaire qui généralise les fragmentations classiques autorisant en particulier la taille des descendants à être plus grande que celle de leurs parents. On établit ensuite certains théorèmes limites en utilisant la théorie des processus auto-similaires. Dans le troisième chapitre, on s'intéresse à un problème statistique provenant de l'industrie minière avec l'estimation statistique de la mesure de Lévy du subordinateur classiquement associé à la fragmentation. Plus précisément, on observe les fragments seulement à l'instant où ils atteignent une taille inférieure à un seuil fixé. Enfin, dans un quatrième chapitre on étudie le coût énergétique d'une succession de fragmentations.

<p>This thesis treats inequalities of Carlson type, i.e. inequalities of the form</p><p><mml:math><mml:semantics><mml:mrow><mml:mrow><mml:msub><mml:mi>∥f∥</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo mml:stretchy="false">≤</mml:mo><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∏</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msubsup><mml:mi>∥f∥</mml:mi><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mi>i</mml:mi></mml:msub></mml:msubsup></mml:mrow></mml:mrow></mml:semantics></mml:math></p><p>where <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mn>i </mml:mn></mml:msub></mml:mrow><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math> and <i>K</i> is some constant, independent of the function <i>f</i>. <i>X</i> and <mml:math><mml:semantics><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:semantics></mml:math> are normed spaces, embedded in some Hausdorff topological vector space. In most cases, we have <mml:math><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math>, and the spaces involved are weighted Lebesgue spaces on some measure space. For example, the inequality</p><p><mml:math><mml:semantics><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∫</mml:mo><mml:mn>0</mml:mn><mml:mo mml:stretchy="false">∞</mml:mo></mml:munderover><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mml:fontstyle="italic">dx</mml:mi><mml:mo mml:stretchy="false">≤</mml:mo><mml:msqrt><mml:mo mml:stretchy="false">π</mml:mo></mml:msqrt></mml:mrow><mml:msup><mml:mfenced mml:open="(" mml:close=")"><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∫</mml:mo><mml:mn>0</mml:mn><mml:mo mml:stretchy="false">∞</mml:mo></mml:munderover><mml:msup><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow></mml:mfenced><mml:mrow><mml:mn>1</mml:mn><mml:mo mml:stretchy="false">/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mfenced mml:open="(" mml:close=")"><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∫</mml:mo><mml:mn>0</mml:mn><mml:mo mml:stretchy="false">∞</mml:mo></mml:munderover><mml:msup><mml:mi>x</mml:mi><mml:mn>2 </mml:mn></mml:msup></mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mn>2 </mml:mn></mml:msup><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow></mml:mfenced><mml:mrow><mml:mn>1</mml:mn><mml:mo mml:stretchy="false">/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></p><p>first proved by F. Carlson, is the above inequality with <mml:math><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math>, <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mn>1 </mml:mn></mml:msub><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mn>2 </mml:mn></mml:msub></mml:mrow><mml:mo mml:stretchy="false">=</mml:mo><mml:mfrac><mml:mn>1 </mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:semantics></mml:math>, <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mo mml:stretchy="false">ℝ</mml:mo><mml:mrow><mml:mo mml:stretchy="false">+</mml:mo><mml:mn>, </mml:mn></mml:mrow></mml:msub><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mn>, </mml:mn><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>1 </mml:mn></mml:msub><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>2 </mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mo mml:stretchy="false">ℝ</mml:mo><mml:mrow><mml:mo mml:stretchy="false">+</mml:mo><mml:mn>, </mml:mn></mml:mrow></mml:msub><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math> and <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>2 </mml:mn></mml:msub><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>2 </mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mo mml:stretchy="false">ℝ</mml:mo><mml:mrow><mml:mo mml:stretchy="false">+</mml:mo><mml:mn>, </mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>x</mml:mi><mml:mn>2 </mml:mn></mml:msup><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math>. In different situations, suffcient, and sometimes necessary, conditions are given on the weights in order for a Carlson type inequality to hold for some constant <i>K</i>. Carlson type inequalities have applications to e.g. moment problems, Fourier analysis, optimal sampling, and interpolation theory.</p>