Le travail présenté dans ce mémoire est divisé en deux parties. La première partie est consacrée aux calculs des fonctions d'appui de la Jacobienne généralisée de Clarke et de son enveloppe plénière, associées à une fonction localement lipschtizienne à valeurs vectorielles. Clarke avait établi en 1975 que la fonction d'appui du sous-différentiel généralisé était une dérivée directionnelle généralisée. Il est donc satisfaisant de constater que la fonction d'appui de la Jacobienne généralisée est une sorte de "divergence directionnelle généralisée". Dans la seconde partie, nous présentons un certain nombre d'applications de techniques issues de l'Analyse non lisse à la résolution d'équations de Hamilton-Jacobi du premier ordre. Ainsi nous utilisons la dualité convexe et le calcul sous-différentiel pour prouver que les formules dites de Hopf-Lax définissent des solutions explicites des équations de Hamilton-Jacobi associées (avec données initiales semicontinues inférieurement). Nous n'utilisons ni le fameux principe de comparaison de la théorie des solutions de viscosité ni régularisation. Nous traitons successivement le cas de la dimension finie et de la dimension infinie. Ces résultats nous permettent de trouver des estimations des solutions d'équations dont l'hamiltonien est la différence de deux fonctions convexes. Enfin, nous nous attachons à l'étude des solutions sci dans des espaces de Banach dits ``lisses''. Le théorème de la valeur moyenne de Clarke et Ledyaev nous permet de montrer un résultat d'``enveloppe'' : nous construisons une solution sci pour une équation dont l'hamiltonien est le supremum d'une famille d'hamiltoniens. Nous appliquons enfin les mêmes techniques pour prouver l'existence d'une solution sci minimale sous des hypothèses plus faibles que celles que l'on recontre généralement dans la littérature.

I Vänern, Sverige, fångades årligen ca 75 ton lax och öring av yrkesfisket, sportfisket och fritidsfisket under 1990-talet och början av 2000-talet. Fångsterna av lax och öring verkar ha sjunkit under de senaste åren men det råder stor osäkerhet över fångstuppskattningar. För en ökad förståelse över smolts mortalitet släpps varje år ett visst antal Carlin-märkta laxar och öringar ut i Vänern och Klarälven. Syftet med studien var att sammanställa och analysera databasen för Carlin-märkt lax och öring i Vänern under åren 1965 till 2005. Målet var att åskådliggöra långsiktiga tendenser över återfångster av Carlin-märkt lax och öring i både Vänern och Klarälven samt bedöma om återfångster varierar beroende på utsättningsplats (Vänern eller Klarälven). Microsoft Excel användes för att sammanställa och analysera återfångad och inrapporterad lax och öring från 1965 till 2005. För att bedöma trender för återfångster av de fyra stammarna, Gullspångs- och Klarälvs-, lax och öring, användes linjär regression. Totalt 299 165 Carlin-märkta fiskar fördelade över 388 utsättningsgrupper har släppts ut i Vänern med tillflöden under 40 års tid. Sammanlagt återfångades 14 504 fiskar, vilket motsvarar knappt 5 % av antalet Carlin-märkta och utsläppta fiskar. Återfångsterna har varierat genom åren (<1 % - >20 %), högst var återfångsterna under 1970-och 1980-talet, sedan 1990-talet har de minskat betydligt. Dessa tendenser är liknande för alla fyra stammar. De flesta återfångsterna sker i Vänern. Fisk utsläppt i Vänern återfångas som regel i något högre grad i Vänern än fisk utsläppt i Klarälven. Slutligen framkom att ingen lax eller öring utsläppt i Vänern återfångades i Klarälven. / In Lake Vänern, Sweden, commercial, sport, and subsistence fisheries in the 1990s and the beginning of 2000 caught about 75 tons of salmon and trout. Catches of salmon and trout appeared to have declined in recent years, although there is much uncertainty in catch estimates. In order to better understand smolt-adult mortality, a number of Carlin-tagged salmon and trout are released in Vänern and Klarälven each year. The aim of the study was to assemble and analyze the Carlin-tag database for salmon and trout in Vänern during the years 1965 to 2005. My objectives were to identify long-term trends in tag returns rates in both Vänern and Klarälven, and to assess whether return rates varied by release location (Vänern or Klarälven). Microsoft Excel was used to compile and analyze reported recoveries of Carlin-tagged salmon and trout from 1965 to 2005. Linear regression was used to assess trends in return rates of four stocks, Gullspångsälven and Klarälven salmon and trout. Total 299 165 tagged fish in 388 release groups have been released in Vänern over the 40-year period. Total recaptures were 14 504, which equates to just under 5 % of the number of Carlin-tagged and released fish. Recapture rates have varied throughout the years (<1 % - >20 %), the highest return rates were in the 1970s – 1980s, but they have decreased significantly since the 1990s. These trends are similar for all four stocks. Most of the recaptures occur in the lake, return rates of fish released in the lake are most often caught in the lake than are fish released in Klarälven. Finally, revealed that no salmon or trout that were released in the lake were recaptured in the Klarälven.

By enlarging the spatial and temporal spectral problems within a certain Lie algebra, a hierarchy of integrable couplings of the Kaup-Newell soliton equations is constructed. The recursion operator of the resulting hierarchy of integrable couplings is explicitly computed. The integrability of the new coupling hierarchy is exhibited by showing the existence of infinitely many commuting symmetries.

The purpose of this thesis is to implement Finite Volume Weighted Essentially Non-Oscillatory (FV-WENO) scheme to solution of one and two-dimensional discretised Euler equations with different flux algorithms. The effects of the different fluxes on the solution have been tested and discussed. Beside, the effect of the grid on these fluxes has been investigated. Weighted Essentially Non-Oscillatory (WENO) schemes are high order accurate schemes designed for problems with piecewise smooth solutions that involve discontinuities. WENO schemes have been successfully used in applications, especially for problems containing both shocks and complicated smooth solution structures. Fluxes are used as building blocks in FV-WENO scheme. The efficiency of the scheme is dependent on the fluxes used in scheme The applications tested in this thesis are the 1-D Shock Tube Problem, Double Mach Reflection, Supersonic Channel Flow, and supersonic Staggered Wedge Cascade. The numerical solutions for 1-D Shock Tube Problem and the supersonic channel flow are compared with the analytical solutions. The results for the Double Mach Reflection and the supersonic staggered cascade are compared with results from literature.

Förvaltning av lax är en komplicerad uppgift som kräver stor förståelse för både laxen och de antropogena effekter som påverkar den. Vattenkraftverk har stor påverkan på laxen genom oregelbundna flöden, reducerad vattenföring och barriärer i vandringsvägar. Laxen är beroende av att vandra för att optimera tillväxt, överlevnad och reproduktion. Östersjöns laxbestånd minskade kraftigt mellan 1950- och 1980-talet till följd av dammutbyggnad och överfiske. Redan i början av 1900-talet började det lokala laxbeståndet i älven Ljusnan att sjunka av samma orsaker. Idag finns ingen lax kvar i Ljusnan. För att möjliggöra en återintroduktion av lax krävs möjlighet för vandring och tillgång på reproduktionshabitat. Bollnäsströmmarna bedöms med sina 6,6 km historiskt ha stått för 25 % av hela Ljusnans laxproduktion. Syftet med denna studie var att simulera en återintroduktion av lax i Bollnäsströmmarna och att genom sårbarhetsanalys undersöka om utplantering av romkorn kan resultera i en överlevande laxpopulation. Simuleringarna utfördes med dagens förutsättningar med undantag av att de passagefria kraftverken ersattes med passagelösningar av bästa möjliga teknik. Utöver detta undersöktes effekter av ökad passagedödlighet förbi kraftverken, habitatåterställande åtgärder och utökade supplementära utsättningar av romkorn. Sammanlagt nio sårbarhetsanalyser utfördes i simuleringsprogrammet Vortex. Resultatet av studien visar att utplantering av romkorn kan producera en laxpopulation som överlever under en hundraårsperiod. Efter en initial populationsökning under tio år sjunker dock tillväxten konstant resterande nittio år vilket med största sannolikhet hade resulterat i en utdöd population över en större tidsrymd. Den potentiella laxpopulationen var mycket känslig för ökad passagedödlighet med en hög utdöenderisk efter endast en liten ökning. Både habitatåterställande åtgärder och supplementära utsättningar av rom gav positiva effekter på populationsstorleken men inget scenario lyckades producera ett stabilt och livskraftigt bestånd vilket tyder på att fler åtgärder krävs för ett lyckat återintroduktionsprojekt. / Conservation of salmon is a complicated task that requires understanding both the biology of salmon and the anthropogenic impacts affecting it. Atlantic salmon is dependent on its migration to optimize growth, survival and reproduction. Hydropower plants affect salmon by causing highly irregular flows, reduced discharge and barriers along migration routes. The salmon stocks of the Baltic Sea declined considerably between the 1950s and 1980s as a result of dam development and overfishing. For the same reasons the local salmon stock in the River Ljusnan, Sweden began to decline already in the early 1900s, and today there are no salmon left in the River Ljusnan. To restore the population, actions to mediate migration and access to reproduction habitats and rearing habitats are needed. The 6.6 km long river section, “Bollnäsströmmarna”, is estimated to have accounted for 25 % of the total salmon production in the River Ljusnan. The aim of this study was to simulate a reintroduction of Atlantic salmon in Bollnäsströmmarna, and using a population viability analysis (PVA), investigate if stocking of eggs can result in a viable salmon population. Simulations were made on the basis of today’s conditions, with the condition that best practice passage solutions are implemented. In addition, the effects of increased passage mortality, habitat restoration and increased supplementary stocking of eggs were simulated. A total of nine PVA simulation (n= 100) scenarios were performed in the Vortex simulation program. The results of the study show that stocking of eggs can produce a salmon population that has a minimum extinction risk over a one-hundred year period. However, after an initial population increase over ten years, population growth steadily declines for the remaining ninety years, which most likely would result in extinction if a period longer than 100 years was considered. The salmon population was very sensitive to an increase in passage mortality, with a high extinction risk after only a small increase in mortality. Both habitat restoration and increased supplementary stocking impacted the population size positively, but no scenario produced a stable and viable population, which indicates that more measures are required for a successful long-term reintroduction.

Cette thèse de doctorat est consacrée à l'étude des problèmes directe et inverse en électrophysiologie cardiaque. Comme les équations qui décrivent l'activité électrique du coeur peuvent être très couteuses en temps de calcul, une attention particulière est apportée aux méthodes d'ordre réduit et à leur applications aux modèles de l'électrophysiologie.Dans un premier temps, nous introduisons les modèles mathématiques et numériques de l'électrophysiologie cardiaque. Ces modèles nous permettent de réaliser des simulations numériques que nous validons à l'aide de plusieurs critères qualitatifs et quantitatifs trouvés dans la littérature médicale. Comme notre modèle prend en compte les oreillettes et les ventricules, nous sommes capables de reproduire des cycles complets d'électrocardiogrammes (ECG) à la fois dans des conditions saines et dans des cas pathologiques.Ensuite, plusieurs méthodes d'ordre réduit sont étudiées pour la résolution des équations de l'électrophysiologie. La méthode Proper Orthogonal Decomposition (POD) est appliquée pour la discrétisation des équations de l'électrophysiologie dans plusieurs configurations, comme par exemple la simulation d'un infarctus du myocarde. De plus, cette méthode est utilisée pour résoudre quelques problèmes d'identification de paramètres comme localiser un infarctus à partir de mesures d'un électrocardiogramme ou simuler une courbe de restitution. Pour contourner les limitations de la POD, une nouvelle méthode basée sur des couples de Lax approchés (Approximated Lax Pairs, ALP) est utilisée. Cette méthode est appliquée aux problèmes directe et inverse. Pour finir, un nouvel algorithme, basé sur les méthodes ALP et l'interpolation empirique discrète, est proposé. Cette nouvelle approche améliore significativement l'efficacité de l'algorithme original ALP et nous permet de considérer des modèles plus complexes utilisés en électrophysiologie cardiaque. / This PhD thesis is dedicated to the investigation of the forward and the inverse problem of cardiac electrophysiology. Since the equations that describe the electrical activity of the heart can be very demanding from a computational point of view, a particular attention is paid to the reduced order methods and to their application to the electrophysiology models. First, we introduce the mathematical and numerical models of electrophysiology and we implement them to provide for simulations that are validated against various qualitative and quantitative criteria found in the medical literature. Since our model takes into account atria and ventricles, we are able to reproduce full cycle Electrocardiograms (ECG) in healthy configurations and also in the case of several pathologies. Then, several reduced order methods are investigated for the resolution of the electrophysiology equations. The Proper orthogonal Decomposition (POD) method is applied for the discretization of the electrophysiology equations in several configurations, as for instance the simulation of a myocardial infarction. Also, the method is used in order to solve some parameters identification problems such as the identification of an infarcted zone using the Electrocardiogram measures and for the efficient simulation of restitution curves. To circumvent some limitations of the POD method, a new reduced order method based on the Approximated Lax Pairs (ALP) is investigated. This method is applied to the forward and inverse problems. Finally, a new reduced order algorithm is proposed, based on the ALP and the Discrete Empirical Interpolation methods. This new approach significantly improves the efficiency of the original ALP algorithm and allow us to consider more complex models used in electrophysiology.

Loss of connectivity in riverine systems due to construction of hydropower dams has resulted in a worldwide decline of anadromous salmonid species such as Atlantic salmon (Salmo salar L.) and sea trout (Salmo trutta L.). The future of these species depend on the presence of available spawning habitat in freshwater river systems. Modern research and mitigation efforts mainly focus on ensuring a successful upstream passage past dams e.g. fish ladders. Atlantic salmon and sea trout are iteroparous, and are thus able to spawn repeatedly during their lifetime. Individuals surviving upstream migration and spawning generally face a hazardous journey back to their marine feeding grounds. In this large scale natural field study I evaluate the possibility of using a bubble barrier as a non-physical structure to guide downstream migrating kelt past the turbines at a large hydropower station in northern Sweden. Results from this study clearly show that kelt effectively can be diverted using a bubble barrier in daylight conditions with a mean water velocity of 1.1 m s-1 (p=0,01). From a fishway managers perspective, increasing survival of salmonid kelt is a substantial step towards achieving a viable population with increased numbers of repeat spawners and large individuals. This study presents new results in a sparsely explored subject; the diversion of post-spawn salmonid migrants using non-physical barriers.

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In this manuscript, we provide foundations of properties of homogeneous polynomials such as the half-plane property, determinantal representability, being weakly determinantal, and having a spectrahedral hyperbolicity cone. One of the motivations for studying those properties comes from the ``generalized Lax conjecture'' stating that every hyperbolicity cone is spectrahedral. The conjecture has particular importance in convex optimization and has curious connections to other areas. We take a combinatorial approach, contemplating the properties on matroids with a particular focus on operations that preserve these properties. We show that the spectrahedral representability of hyperbolicity cones and being weakly determinantal are minor-closed properties. In addition, they are preserved under passing to the faces of the Newton polytopes of homogeneous polynomials. We present a proved-to-be computationally feasible algorithm to test the half-plane property of matroids and another one for testing being weakly determinantal. Using the computer algebra system Macaulay2 and Julia, we implement these algorithms and conduct tests. We classify matroids on at most 8 elements with respect to the half-plane property and provide our test results on matroids with 9 elements. We provide 14 matroids on 8 elements of rank 4, including the Vámos matroid, that are potential candidates for the search of a counterexample for the conjecture.:1 Background 1 1.1 Some Properties of Homogeneous Polynomials . . . . . . . . . . 1 Hyperbolic Polynomials . . . . . . . . . . . . . . . . . . . . . . 1 The Half-Plane Property and Stability . . . . . . . . . . . . . . 8 Determinantal Representability . . . . . . . . . . . . . . . . . . 15 Spectrahedral Representability . . . . . . . . . . . . . . . . . . 19 1.2 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Some Operations on Matroids . . . . . . . . . . . . . . . . . . . 29 The Half-Plane Property of Matroids . . . . . . . . . . . . . . . 36 2 Some Operations 43 2.1 Determinantal Representability of Matroids . . . . . . . . . . . 43 A Criterion for Determinantal Representability . . . . . . . . . 46 2.2 Spectrahedral Representability of Matroids . . . . . . . . . . . 50 2.3 Matroid Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 54 Newton Polytopes of Stable Polynomials . . . . . . . . . . . . . 59 3 Testing the Properties: an Algorithm 61 The Half-Plane Property . . . . . . . . . . . . . . . . . . . . . . 61 Being SOS-Rayleigh and Weak Determinantal Representability 65 4 Test Results on Matroids on 8 and 9 Elements 71 4.1 Matroids on 8 Elements . . . . . . . . . . . . . . . . . . . . . . 71 SOS-Rayleigh and Weakly Determinantal Matroids . . . . . . . 76 4.2 Matroids on 9 Elements . . . . . . . . . . . . . . . . . . . . . . 80 5 Conclusion and Future Perspectives 85 5.1 Spectrahedral Matroids . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Non-negative Non-SOS Polynomials . . . . . . . . . . . . . . . 88 5.3 Completing the Classification of Matroids on 9 Elements and More 89 Bibliography 91

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