The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle configurations. We also investigate Hopf bifurcation surfaces of various dynamical systems.

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Soit k un corps algébriquement clos de caractéristique 0 et F une suite de n polynômes en intersection complète sur k[X1,...,Xn]. Le Bezoutien de F fournit une forme dualisante sur k[X]/<F> appelée symbole de Kronecker, qui est un analogue algébrique du résidu. L'objet de ce travail est de construire et calculer le symbole de Kronecker dans le tore (C*)n relativement à une famille f de n polynômes de Laurent en n variables. La famille f possède un nombre fini de zéros et est régulière pour ses polytopes de Newton. La représentation du résidu global dans le tore à l'aide d'un résidu torique, donnée par Cattani et Dickenstein, suggère d'interpréter le symbole de Kronecker dans le tore dans la variété torique projective définie par le polytope P, somme de Minkowski des polytopes de Newton de f.Lorsque P est premier, Roy et Szpirglas ont défini le symbole de Kronecker dans le tore à partir des symboles de Kronecker définis sur les ouverts affines de la variété torique Xp relativement à une famille de n + 1 polynômes homogènes sans zéros communs dans la variété Xp. Nous montrons ici que le cas « P non premier » est réductible au cas précédent en explicitant les morphismes d'éclatement qui traduisent le raffinement de l’éventail de Xp en un éventail simplicial. / Let k be an algebraically closed field with char(k) = 0 and let be polynomials F1,..., Fn such that k[X1,...,Xn]/<F1,..., Fn> is a complete intersection k-algebra. The Bezoutian of F1,..., Fn gives a dualizing form acting on k[X1,...,Xn]/<F1,..., Fn> called Kronecker symbol. It is an algebraic analogue of residue. The aim of this work is to build and calculate the Kronecker symbol in the torus (C*)n for a system f of Laurent polynomials with a a finite set of zeroes and regular for its Newton polytopes. In the same way as Cattani and Dickenstein have done for the global residue in the torus, we consider the projective variety given by the Minkowski sum P of the Newton polytopes of f in order to build the Kronecker symbol in the torus.When P is prime, Roy and Szpirglas have defined the Kronecker symbol in the torus from Kronecker symbols on affine subsets of Xp for a system of n+1 homogeneous polynomials with no common zeroes in XP . We prove that the case "P no prime" can be reduced to the previous case by using simplicial refinements of the fan of Xp and making explicit the associated toric morphisms on the total coordinate spaces.

Syfte: Syftet med denna studie var att undersöka om en sfärisk endagars silikonhydrogel lins (clariti® 1day) vid sporadiska tillfällen kan ordineras istället för en torisk endagars silikonhydrogel lins (clariti® 1day toric) vid -0,75 DC i astigmatism. Metod: I studien undersöktes 26 personer varav 14 kunde delta. Deltagarnas ålder var mellan 18-40 år. Mätningarna genomfördes på 14 vänster ögon med astigmatism på  -0,75 DC. Kontaktlinserna som testades var clariti® 1day och clariti® 1day toric. Ett biomikroskop användes för att mäta inklinationen på linserna, undersöka central och korneal täckning samt passform och rörelse i olika led. Även undersökning med flouresecin och blått ljus utfördes på mätögat för att se om signifikanta stainings fanns(≥ grad 2). En ETDRS logMAR syntavla på 4 meters avstånd användes för att mäta hög (100%)  – och lågkontrast (10%). Resultat: Resultatet av denna studie visar ingen klinisk signifikant skillnad i visus dock blev det visusförändringar vid både hög – och lågkontrast mätningar. En förbättring med 0,08 logMAR (4 optotyper) påträffades med de toriska linserna jämfört med de sfäriska vid högkontrast visusmätningar samt 0,03 logMAR (1,5 optotyp) bättre med de toriska vid lågkontrast visusmätningar. Slutsats: Denna studie visar ingen klinisk signifikant skillnad i visus mellan clariti® 1day och clariti® 1day toric. Dock blev det visusförändringar med de toriska linserna vid både hög – och lågkontrast mätningar med 1,5-4 optotyper mer. / The purpose of this study was to determine whether there is a significant difference in visual acuity or not between a spherical one day silicone hydrogel lenses (clariti® 1day) and a toric one day silicone hydrogel lenses (clariti® 1day toric). The study was conducted with the help of an ETDRS logMAR chart, with a testing distance of 4 meters. Both high and low contrast visual acuity was examined. If no statistical significant difference is showed between these two lenses then clariti® 1day may be prescribed instead of clariti® 1day toric, in some infrequently occasion as short vacations. Toric lenses could be harder to find in stores then the spherical lenses, if needed right away. The toric lenses could also be more expensive than the spherical lenses. If clariti® 1day could be prescribed for some occasions and still be able to maintain a good visual acuity when using them, then it might be possible to prescribe them when needed right away. In this study overall 26 eyes were tested, out of which 12 were excluded in the inclusion criteria due to that the participants had more or less astigmatism than what was needed or they had the astigmatism in the right eye. A total of 14 left eyes were examined. Two of the participants were men and 12 were women. All the participants had a cylinder of -0.75 D in the left eye. The participant were first corrected with a clariti® 1day spherical lens using their spherical eqvivalent power and after that with a torical lens (clariti® 1day toric). Both high and low contrast visual acuity were measured monoculary with both lenses in a photopic lightning. The study showed that the visual acuity in both high and low contrast was 1,5-4 letters better with the torical lenses than with the spherical lenses, even with -0.75 D astigmatism. However in this study there is no statistical significant improvement in visual acuity at a cylinder of -0,75D in either high or low contrast visual acuity measurements. That could be due to a small amount of participants in this study.

Given a simple graph G, the corresponding edge ideal IG is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for the projective dimensions of the quotient rings of the edge ideals. In the second half of this thesis, we will consider the class of simple bipartite graphs known as Ferrers graphs. In particular, we will study a class of monomial ideals that arise as initial ideals of the defining ideals of the toric rings associated to Ferrers graphs. The toric rings were studied by Corso and Nagel in 2007, and by studying the initial ideals of the defining ideals of the toric rings we are able to show that in certain cases the toric rings of Ferrers graphs are level.

Let (L,h) → (X,ω) be a compact toric polarized Kahler manifold of complex dimension n. For each k ε N, the fibre-wise Hermitian metric hk on Lk induces a natural inner product on the vector space C∞(X,Lk) of smooth global sections of Lk by integration with respect to the volume form ωn /n! . The orthogonal projection Pk : C∞(X,Lk) → H0(X,Lk) onto the space H0(X,Lk) of global holomorphic sections of Lk is represented by an integral kernel Bk which is called the Bergman kernel (with parameter k ε N). The restriction ρk : X → R of the norm of Bk to the diagonal in X × X is called the density function of Bk. On a dense subset of X, we describe a method for computing the coefficients of the asymptotic expansion of ρk as k → ∞ in this toric setting. We also provide a direct proof of a result which illuminates the off-diagonal decay behaviour of toric Bergman kernels. We fix a parameter l ε N and consider the projection Pl,k from C∞(X,Lk) onto those global holomorphic sections of Lk that vanish to order at least lk along some toric submanifold of X. There exists an associated toric partial Bergman kernel Bl,k giving rise to a toric partial density function ρl,k : X → R. For such toric partial density functions, we determine new asymptotic expansions over certain subsets of X as k → ∞. Euler-Maclaurin sums and Laplace’s method are utilized as important tools for this. We discuss the case of a polarization of CPn in detail and also investigate the non-compact Bargmann-Fock model with imposed vanishing at the origin. We then discuss the relationship between the slope inequality and the asymptotics of Bergman kernels with vanishing and study how a version of Song and Zelditch’s toric localization of sums result generalizes to arbitrary polarized Kahler manifolds. Finally, we construct families of induced metrics on blow-ups of polarized Kahler manifolds. We relate those metrics to partial density functions and study their properties for a specific blow-up of Cn and CPn in more detail.

Background: To evaluate the toric intraocular lens (IOL) calculation considering posterior corneal astigmatism, incision-induced posterior corneal astigmatism, and effective lens position (ELP). Methods Two thousand samples of corneal parameters with keratometric astigmatism >= 1.0 D were obtained using boot-strap methods. The probability distributions for incision-induced keratometric and posterior corneal astigmatisms, as well as ELP were estimated from the literature review. The predicted residual astigmatism error using method D with an IOL add power calculator (IAPC) was compared with those derived using methods A, B, and C through Monte-Carlo simulation. Method A considered the keratometric astigmatism and incision-induced keratometric astigmatism, method B considered posterior corneal astigmatism in addition to the A method, method C considered incision-induced posterior corneal astigmatism in addition to the B method, and method D considered ELP in addition to the C method. To verify the IAPC used in this study, the predicted toric IOL cylinder power and its axis using the IAPC were compared with ray-tracing simulation results. Results The median magnitude of the predicted residual astigmatism error using method D (0.25 diopters [D]) was smaller than that derived using methods A (0.42 D), B (0.38 D), and C (0.28 D) respectively. Linear regression analysis indicated that the predicted toric IOL cylinder power and its axis had excellent goodness-of-fit between the IAPC and ray-tracing simulation. Conclusions The IAPC is a simple but accurate method for predicting the toric IOL cylinder power and its axis considering posterior corneal astigmatism, incision-induced posterior corneal astigmatism, and ELP.

This thesis mainly concerns the cohomology of the moduli spaces ℳ3[2] and ℳ3,1[2] of genus 3 curves with level 2 structure without respectively with a marked point and some of their natural subspaces. A genus 3 curve which is not hyperelliptic can be realized as a plane quartic and the moduli spaces 𝒬[2] and 𝒬1[2] of plane quartics without respectively with a marked point are given special attention. The spaces considered come with a natural action of the symplectic group Sp(6,𝔽2) and their cohomology groups thus become Sp(6,𝔽2)-representations. All computations are therefore Sp(6,𝔽2)-equivariant. We also study the mixed Hodge structures of these cohomology groups. The computations for ℳ3[2] are mainly via point counts over finite fields while the computations for ℳ3,1[2] primarily uses a description due to Looijenga in terms of arrangements associated to root systems. This leads us to the computation of the cohomology of complements of toric arrangements associated to root systems. These varieties come with an action of the corresponding Weyl group and the computations are equivariant with respect to this action.

This thesis constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth toric Fano fourfolds. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. We provide two different methods to show that a collection of line bundles is full, whilst the strong exceptional condition is checked using the package QuiversToricVarieties for the computer algebra system Macaulay2, written by the author. A database of the full strong exceptional collections can also be found in this package.

What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the cohomology of the associated variety directly in terms of the graph G itself.