Diss. (sammanfattning) Uppsala : Univ., 2002. / Härtill 4 uppsatser.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 190-199).

AMARAL, Marcelo Dário dos Santos. Funções absolutamente minimizantes. 2013. 85 f. Dissertação(Mestrado em Matemática)- Centro de Ciências, Universidade Federal do Ceará, Programa de Pós-Graduação em Matemática, Fortaleza, 2013. / Submitted by Rocilda Sales (rocilda@ufc.br) on 2014-02-06T11:12:21Z No. of bitstreams: 1 2013_dis_mdsamaral.pdf: 573351 bytes, checksum: 14bf6d0610fcc1d41b238d5524bb8144 (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2014-02-06T13:26:35Z (GMT) No. of bitstreams: 1 2013_dis_mdsamaral.pdf: 573351 bytes, checksum: 14bf6d0610fcc1d41b238d5524bb8144 (MD5) / Made available in DSpace on 2014-02-06T13:26:35Z (GMT). No. of bitstreams: 1 2013_dis_mdsamaral.pdf: 573351 bytes, checksum: 14bf6d0610fcc1d41b238d5524bb8144 (MD5) Previous issue date: 2013 / The objective of this dissertation is to give an exposition of the theory of absolutely minimizing Lipschitz extensions, based on the work of Gunnar Aronsson, Michael G. Crandall and Petri Juutinen in [1], showing various details in a form accessible to readers without any prior knowledge of the subject. In particular, we retrace the improved results on the existence through arguments that are simpler than those that can be found in literature. We present a proof of the known uniqueness result, which is not based on the theory of viscosity solutions. In our approach we will show that the absolutely minimizing functions are the functions that satisfy a geometric condition which we will call to enjoy comparison with cones. This elementary geometric device renders the theory versatile and transparent. Here we will nd a priori continuity estimates, Harnack inequality, Perron's method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a uni ed sum-mary of the existing theory as well as some results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some signi cant analytical tools. / O objetivo desta dissertação e dar uma exposi ção da teoria das extensões Lipschitz absolutamente minimizantes, baseada no trabalho de Gunnar Aronsson, Micha G.Crandall e Petri Juutinen em [1], apresentando v ários detalhes em uma forma acess ível aos leitores sem qualquer conhecimento pr évio do assunto. Em particular, refazemos resultados melhorados relativos a existência atrav és de argumentos que são mais simples do que aqueles que podem ser encontrados na literatura. N ós apresentamos uma prova do conhecido resultado de unicidade, o qual não se baseia na teoria de solu ções de viscosidade. Em nossa nossa abordagem, mostraremos que as fun ções absolutamente minimizantes são as fun ções que satisfazem uma condi ção geom étrica a qual chamaremos de gozar de compara ção com cones. Este elementar dispositivo geom étrico torna a teoria vers átil e transparente. Aqui, encontraremos por exemplo, estimativas de continuidade a priori, desigualdade de Harnack, o m étodo de Perron para comprovar os resultados de existência, questões de unicidade e regularidade, e algumas ferramentas b ásicas da teoria de solu ções de viscosidade. N ós acreditamos que a nossa apresenta ção fornece um resumo uni ficado da teoria uni ficada existente, assim como alguns resultados de interesse para os peritos e pesquisadores e, ao mesmo tempo, uma fonte que possa ser utilizada para introduzir estudantes a algumas signficantes ferramentas analí ticas.

BEZERRA, Kelton Silva. Instabilidade e rigidez de hipersuperfícies e um teorema de unicidade em variedades semi-riemannianas. 2015. 58 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015. / Submitted by Rocilda Sales (rocilda@ufc.br) on 2016-04-04T15:45:05Z No. of bitstreams: 1 2015_tese_ksbezerra.pdf: 883491 bytes, checksum: f9e199361f39af1569cb1321de363c92 (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2016-04-04T15:48:16Z (GMT) No. of bitstreams: 1 2015_tese_ksbezerra.pdf: 883491 bytes, checksum: f9e199361f39af1569cb1321de363c92 (MD5) / Made available in DSpace on 2016-04-04T15:48:16Z (GMT). No. of bitstreams: 1 2015_tese_ksbezerra.pdf: 883491 bytes, checksum: f9e199361f39af1569cb1321de363c92 (MD5) Previous issue date: 2015 / Our aim in this work is threefold. First, we get an extension, to the spherical case, of a theorem due to J. Simons, which concerns unstability of minimal cones constructed over a certain class of minimal submanifolds of the Euclidean sphere. Second, we classify the quasi-Einstein structures of the Riemannian product Hn x R. Third, we get a rigidity theorem for complete hypersurfaces into the De Sitter space, under certain conditions on the mean and scalar curvatures. / Este trabalho aborda três problemas em Geometria Diferencial. Primeiro, obtemos uma extensão, para o caso esférico, de um teorema devido a J. Simons sobre instabilidade de cones mínimos construídos sobre uma certa classe de subvariedades mínimas da esfera Euclidiana. Depois, classificamos as estruturas quasi-Einstein existentes sobre o produto Riemanniano Hn X R. Por fim, obtemos um teorema de rigidez para hipersuperfícies tipo-espaço completas do espaço de De Sitter, sob certas condições sobre as curvaturas média e escalar, além de uma condição de integrabilidade.

L’imagerie super-résolue délivrée par l’optique adaptative permet d’imager la rétine à l’échelle cellulaire et rentre au service du diagnostic et du suivi de pathologie de l’oeil mais aussi à une meilleure connaissance de l’anatomie, des fonctions et des mécanismes de la rétine.Dans un premier temps nous avons défini la méthode permettant de positionner le centre de la fovéa ainsi que de mesurer la densité des cônes avec la meilleure répétabilité. Le centre de l’ellipse de plus fortes densités permet de positionner le centre de la fovéa. La densité des cônes est mesurée dans une fenêtre de 80 par 80 pixels à une excentricité mesurée sur un montage établi à partir de clichés réalisés tous les 2°.Un des aspects fondamental de ce projet est la réalisation d’une base de données cliniques normatives liées à des bio-marqueurs (densité, espacement et morphologie des cônes). En effet il est nécessaire de fournir des mesures quantitatives des structures visualisées. Cette étude a permis de caractériser la structure rétinienne d’une population saine ne présentant aucune maladie rétinienne, de façon à établir un référentiel de normalité. Cette étude nous a aussi permis de répondre aux questions laissées sans réponse par la littérature. Nous avons constaté que la densité des cônes différait nettement entre les méridiens horizontaux et verticaux. Nous avons également observé que la longueur axiale influait sur la densité des cônes exprimée en unité de surface mais pas en terme d'angle visuel, supportant l'hypothèse que la rétine s'étirait avec l'allongement du globe oculaire. Enfin, l'âge ne semble pas avoir d'impact sur la densité des cônes excepté à 2°.Nous nous sommes ensuite intéressé au lien entre l’acuité visuelle périphérique et l’espacement des cônes. Il est connu que les cellules ganglionnaires midgets (mRGC) sont responsables de la résolution visuelle. Mais, au niveau de la fovéa, chaque cône se connecte, via des cellules bipolaires, à une mRGC ON et OFF. Cette particularité anatomique rend la mesure de l’espacement entre les cônes un substitut observable aux cellules ganglionnaires permettant ainsi le lien avec l'acuité visuelle. Nous avons donc déterminé jusqu'à quelle excentricité les cônes ne sont plus un substitut observable des cellules ganglionnaires. Nous avons constaté que l’acuité visuelle est régit par l’espacement des cônes jusqu’à 2° d’excentricité, à la fois en rétine nasale et inférieure, soutenant l'idée que seulement 50% des mRGC fovéale détermine l’acuité visuelle (mRGC ON ou OFF). Au delà, l’espacement des cônes surestime l’acuité visuelle. Watson a développé un modèle permettant de prédire l’acuité visuelle sur la base de l’espacement entre les mRGC lui-même dérivé de l’espacement entre les cônes. Le modèle 50% (mRGC ON ou OFF) prédit également l’acuité visuelle dans le méridien vertical (inférieur) à 4° et 6° d'excentricité. Le long du méridien nasal, le modèle 50% sous-estime l’acuité visuelle. Un chevauchement partiel des champs récepteurs des mRGC ON + OFF pourrait en être la cause. Nous avons ensuite mesuré la densité des cônes dans les 48° centraux le long du méridien horizontal dans deux groupes, l’un présentant une histoire d’évolution myopique (i.e. allongement de l'oeil) et l'autre non. À ce jour, il n'existe qu’une seule étude ayant des mesures in-vivo de la densité des cônes au delà des 15° centraux. Cette étude nous a également permis d'affirmer que lors de l'évolution myopique, la rétine s’étirerait de manière uniforme dans la zone mesurée, excepté proche de la tête du nerf optique, où il semblerait qu'il y ait une adhérence plus forte. / The high-resolved imagery delivered by the adaptive optic is used to form pictures of the retina at the cellular level and to diagnose and follow up on eyes pathologies, but also to improve knowledge of the anatomy, functions and mechanisms of the retina.First we defined the method that allows us to position the center of the fovea and to measure the density of the cones with the best repeatability. The center of the ellipse of higher densities makes it possible to position the center of the fovea. The cone density is measured in a region of interest of 80 x 80 pixels to an eccentricity measured on a montage established from pictures made every 2°.One of the fundamental aspects of this project is the creation of a normative clinical database linked to bio-markers (density, spacing and morphology of cones). Indeed it is necessary to provide quantitative measures of the structures visualized. This study made it possible to characterize the retinal structure of a healthy population with no retinal disease, so as to establish a reference of normality. This study also allowed us to answer questions left unanswered by the literature. The density of the cones differs sharply between the horizontal and vertical meridians. We have also observed that the axial length influences cone density expressed in metric unit but not in terms of visual angle, supporting the hypothesis that the retina is stretched with the lengthening of the eyeball. Finally, age does not seem to have an impact on cone density except at 2°.We then examined the relationship between peripheral visual acuity and cone spacing. It is known that midget ganglion cells are in charge of visual resolution. But, at the fovea, each cone connects, via bipolar cells, to a midget ganglion cell ON and OFF. This anatomical particularity makes the measurement of the cone spacing an observable substitute for ganglion cells thus allowing the link with visual acuity. We have therefore determined to what eccentricity the cones are no longer an observable substitute for ganglion cells. We found that visual acuity is governed by the arrangement of cones up to 2° of eccentricity, both in nasal and inferior retinas, supporting the idea that only 50% of foveal mRGCs determine VA (mRGC ON or OFF). Beyond this, cone spacing over-samples ganglion cells. Watson developed a model to predict visual acuity based on the mRGC spacing, itself derived from the cone spacing. The 50% model (mRGC ON or OFF) also predicts visual acuity in the inferior meridian at 4° and 6° of eccentricity. Along the nasal meridian, the 50% model underestimates visual acuity. Partial overlap of the mRGC ON + OFF receptive fields may be the cause.Finally, we measured the cone density in the central 48° along the horizontal meridian in two study groups, one with a history of myopic evolution (i.e. elongation of the eye) and the other without. To this date, there is only one study with in-vivo measurements of cone density beyond the central 15°. This study also allowed us to assert that during myopic evolution, the retina stretches uniformly in the measured area, excepted near the optic nerve head, where it appears that there is a stronger adhesion.

The current knowledge of the visual pathways in primates includes the patterns of projection from the retina through the dorsal lateral geniculate nucleus (dLGN) to the striate cortex (V1) and the extra-striate projections towards the dorsal and ventral streams. Cells with short wavelength sensitive cone (S-cone) inputs in the dLGN have been studied extensively in New World marmosets but not in Old World macaques. This thesis presents results from studies in the macaque monkey which are more relevant to humans since humans are closer in evolution to Old World than New World monkeys. / The spatial, temporal, chromatic and orientation preferences of neurons in the dLGN of the macaque were investigated by electrophysiological methods. The physiological findings of cells with S-cone inputs were compared to cells with opponent inputs from the long and medium wavelength sensitive cones (L-cones & M-cones, respectively). The cells receiving S-cone inputs (blue-yellow or B-Y cells) preferred lower spatial frequencies than the cells with opponent L-cone and M-cone inputs (red-green or R-G cells). Orthodromic latencies from optic chiasm stimulation were measured where possible to distinguish differences in conduction velocity between the cell groups. Although the B-Y cells usually had longer latencies than R-G cells, there wasconsiderable overlap between the cell groups. / The recorded cells were localised through histological reconstruction of dLGN sections stained for Nissl substance. The distribution of B-Y cells within the dLGN was compared to the distribution of R-G cells. The majority of B-Y cells were located within the intercalated koniocellular layers as well as the koniocellular bridges (extensions of the koniocellular layers into the adjacent parvocellular layers). The B-Y cells were also largely segregated within the middle dLGN layers (K3, P3, K4 & P4). The R-G cells were mainly concentrated within the parvocellular layers (P3, P4, P5 & P6) and were evenly distributed throughout the middle and outer layers of the dLGN. / The study also included recordings from the extra-striate middle temporal area (MT) to determine whether a fast S-cone input exists from the dLGN to area MT which bypasses V1. The pattern of cone inputs to area MT neurons was investigated before and during inactivation of V1. The inactivation was done through reversible cooling with a Peltier thermocouple device or focal inactivation with y-amino butyric acid (GABA) iontophoresis. Precise inactivation of V1 to the topographically matching visual fields of the recording sites in area MT revealed a preservation of all three coneinputs in many cells. The subcortical sources of these preserved inputs are discussed with their relevance to blindsight, which is the limited retention of visual perception after V1 damage. Analysis of the latencies of area MT cells revealed a rough segregation into latencies faster or slower than 70 ms. Cells both with and without a significant change in response during V1 inactivation were present in each group. The findings reported in this thesis indicate that some of the preserved inputs in area MT during V1 inactivation may be carried by a direct input from the dLGN which bypasses V1.

Cette thèse est divisée en deux parties. Dans la première partie nous étudions les variétés 2-Fano. Les variétés 2-Fano, définies par De Jong et Starr, satisfont des generalisations de certaines propriétés des varietes Fano. Nous proposons une définition de variété k-Fano (faible) et conjecturons la polyhédralité du cône de k-cycles pseudo-effectives pour ces variétés en analogie avec le cas k=1. Ensuite, nous calculons quelques nombres de Betti d'une grande classe de variétés k-Fano pour prouver un cas particulier de la conjecture. En particulier, la conjecture est vraie pour toutes les variétés 2-Fano d'indice >n-3, et nous complétons également la classification des variétés faibles 2-Fano répondant aux questions 39 et 41 dans l'article d'Araujo et Castravet.Dans la deuxième partie, nous étudions une application rationnelle particulière. Beauville et Donagi ont prouvé que la variété des droites F(Y) d'une hypersurface lisse, cubique Y de dimension quatre est une variété hyperKähler. Récemment, C. Lehn, M. Lehn, Sorger et van Straten ont prouvé qu'on peut naturellement associer une variété hyperKähler Z(Y) à la variété compacte des cubiques rationnelles dans Y. Puis, Voisin a défini une application rationnelle de degré 6 entre le produit direct F(Y)xF(Y) et Z(Y). Nous montrerons que le lieu d'indétermination de cette application est le lieu des droites concourantes dans Y. / This thesis is divided in two parts. In the first part we study the 2-Fano varieties. The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher dimensional analogous properties of Fano varieties. We propose a definition of (weak) k-Fano variety and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties in analogy with the case k=1. Then, we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index > n-3, and also we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in Araujo and Castravet’s article.In the second part, we study a particular rational map. Beauville and Donagi proved that the variety of lines F(Y) of a smooth cubic fourfold Y is a hyperKähler variety. Recently, C. Lehn, M.Lehn, Sorger and van Straten proved that one can naturally associate a hyperKähler variety Z(Y) to the variety of twisted cubics on Y. Then, Voisin defined a degree 6 rational map between the direct product F(Y)xF(Y) and Z(Y). We will show that the indeterminacy locus of this map is the locus of intersecting lines. / Questa tesi è divisa in due parti. Nella prima parte studiamo le varietà 2-Fano. Le varietà 2-Fano, definite da De Jong e Starr, soddisfano alcune proprietà analoghe (in dimensionie superiore) alle varietà Fano. Diamo una definizione di varietà k-Fano (debole) e congetturiamo la poliedricità del cono di k-cicli pseudoeffettivi per tali varietà, in analogia al caso k=1. Quindi calcoliamo alcuni numeri Betti di molte varietà k-Fano, per dimostrare alcuni casi particolari della congettura. In particolare, la congettura è vera per tutte le varietà 2-Fano d'indice >n-3, e inoltre completiamo la classificazione delle varietà 2-Fano deboli rispondendo alle domande 39 e 41 nell'articolo di Araujo e Castravet. Nella seconda parte studiamo una particolare mappa razionale. Beauville e Donagi hanno dimostrato che la varietà delle rette F(Y) di una ipersuperfice cubica liscia Y di dimensione 4 è una varietà hyperKähler. Recentemente, C. Lehn, M.Lehn, Sorger e van Straten hanno dimostrato che è possibile associare in modo naturale una varietà hyperKähler Z(Y) alla varietà delle cubiche razionali in Y. Successivamente, Voisin ha definito una mappa razionale di grado 6 tra il prodotto diretto F(Y)xF(Y) e Z(Y). Mostreremo che il luogo di indeterminazione di questa mappa è il luogo delle rette secanti.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Os cones de guta-percha usados em endodontia, compostos de poli (1,4-trans-isopreno), ZnO e/ou BaSO4 e ceras/resinas, foram feitos para durar a vida inteira nos canais radiculares por eles preenchidos. Entretanto, existem evidÃncias de que esses materiais sofrem degradaÃÃo com o decorrer do tempo de obturaÃÃo intensificada quando existem infecÃÃes por microrganismos. A bactÃria Enterococcus faecalis à um desses microrganismos, normalmente associada com infecÃÃes periapicais persistentes. A prevalÃncia dessa bactÃria em casos de re-tratamento ocorre entre 29 e 79% dos casos. O objetivo do trabalho à estudar a degradaÃÃo bacteriana de cones de guta-percha em meio contendo E. faecalis, com o foco na matÃria orgÃnica, o poliisopreno. Cada amostra de cone foi imersa em meio de sais minerais com inÃculo de concentraÃÃo de 5,4x105 UFC/mL por perÃodos de 10 a 100 dias. Os cones foram removidos do contato com o inÃculo em tempos variados e analisados. Foram pesados e caracterizados por TGA, GPC, FTIR, MEV e RMN H1. O crescimento do inoculo foi monitorado atravÃs de medidas de densidade Ãptica, determinaÃÃo da porcentagem de mineralizaÃÃo de CO2, medidas da biomassa seca de cÃlulas, medidas de pH e contagem de cÃlulas. A atividade antimicrobiana dos constituintes inorgÃnicos dos cones foi verificada mediante o teste de difusÃo em poÃo. Foi observada perda gradual de massa dos cones com o tempo de degradaÃÃo. O aumento no teor de resÃduos em 800ÂC, mostra que houve uma perda progressiva de matÃria orgÃnica e um aumento na proporÃÃo de constituintes inorgÃnicos. ApÃs o contato com E. faecalis, o poliisopreno isolado dos cones das marcas Odous e Tanari foi analisado e comparado com o poliisopreno in natura por GPC e FTIR. Os resultados mostram que nos trÃs casos ocorre uma pequena e gradual perda de massa molar do poliisopreno constituintes do material, mais evidenciada no poliisopreno in natura. Os espectros de infravermelho mostram que a degradaÃÃo tambÃm ocorre com modificaÃÃes estruturais do polÃmero, especialmente no surgimento de bandas de deformaÃÃo de C=O e OH/OOH; e variaÃÃes no padrÃo de cristalinidade das bandas de C=C-H. As imagens de MEV mostram que podem ocorrer mudanÃas na morfologia da superfÃcie dos cones biodegradados, resultado do ataque seletivo ao componente orgÃnico. Os espectros de RMN de H1 do poliisopreno in natura mostram que o material sofreu pequenas mudanÃas estruturais nas molÃculas de poliisopreno durante o bioensaio. Todos os bioensaios de monitoramento indicaram que o inÃculo cresce continuamente durante o perÃodo de contato com os cones ou com o poliisopreno in natura. A bactÃria degrada a matÃria orgÃnica dos cones usando-os como fonte de substrato e, consequentemente, causando vazios nos cones que podem ocasionar falhas no tratamanto endodÃntico. / The gutta-percha cones used in endodontic therapy, consisting of poly (1,4-trans-isoprene), ZnO and/or BaSO4 and wax/resin, were meant to last a lifetime in root canals filled by them. However, there is evidence that these materials suffer degradation with the passage of time when there shutter intensified infections by microorganisms. The bacterium Enterococcus faecalis is one of those microorganisms commonly associated with persistent periapical infections. The prevalence of this bacterium in cases of re-treatment is between 29% and 79% of cases. The objective is to study the bacterial degradation of gutta-percha in medium containing E. faecalis, with the focus on organic matter, polyisoprene. Each sample cone was immersed in mineral salts medium with inoculum concentration of 5.4 x 105 CFU/mL for periods of 10 to 100 days. The cones were removed from contact with the inoculum at various times and analized. Were weighed and characterized by TGA, GPC, FTIR, NMR H1 and SEM. The growth of the inoculum was monitored by optical density measurements, determining of percentage of mineralization of CO2, measured dry biomass cells, measurements of pH and cell count. The antimicrobial activity of inorganic constituents of cones was verified by diffusion test pit. Was observed gradual loss of mass of the cones with the degradation time. The increase in the ash content at 800 ÂC, shows that there was a progressive loss of organic matter and an increased proportion of inorganic constituents. After contact with E. faecalis, polyisoprene isolated cones marks Odous and Tanari was analized and compared and the fresh polyisoprene by GPC and FTIR. The results show that in all three cases there is small and gradual loss of the molar mass of the polyisoprene constituents of material, the more evidente in polyisoprene in nature. Infrared spectra show that degradation also ocurrs with strutural modifications of the polymer, especially in the occurrence of deformation bands of C=O and OH/OOH, and variations in the pattern of cristalinity of the bands of C=C-H. The SEM images show that changes may ocurr in the surface morphology of the cones biodegraded, resulting the from selective attack to the organic component. The NMR H1 spectra of polyisoprene fresh show that the material has undergone minor structural changes in the molecules of polyisoprene during bioassay. All bioassays monitoring indicated that the inoculum grows continuously during the period of contact with the cones or polyisoprene in nature. The bacteria degrades the organic matter of the cones using them as a source of substrate and thereby causing voids cones that can cause failure in endodontic treatment.

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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