This thesis consists of three chapters. Each chapter is on a different subject. However, all three chapters address issues that arise in counting arithmetically interesting objects. Chapter 1 is... Show moreThis thesis consists of three chapters. Each chapter is on a different subject. However, all three chapters address issues that arise in counting arithmetically interesting objects. Chapter 1 is on the unit equation in positive characteristic. Chapter 2 is about the statistical behavior of ray class groups, of fixed integral conductor, in families of imaginary quadratic fields. Chapter 3 concerns the study of the unit group of local fields in the category of filtered groups. Show less
Chapter 1,contains the numerical verification of the Birch and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic curves of genus 2, 3, 4 and 5. Chapter 2 treats the... Show moreChapter 1,contains the numerical verification of the Birch and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic curves of genus 2, 3, 4 and 5. Chapter 2 treats the equivalence of BSD for a certain elliptic curve over Q(∜5), and a pair of hyperelliptic curves over Q. In chapter 3, ordinary Galois covers of smooth curves are constructed from ordinary Galois covers of semi-stable curves. Finally, in chapter 4, the statistics of ordinary reduction for hyperelliptic curves is considered. Show less
If X is a locally compact Hausdorff space, then a representation of the complex C* algebra C_0(X) on a Hilbert space $H$ is given by a spectral measure that takes its values in the orthogonal... Show moreIf X is a locally compact Hausdorff space, then a representation of the complex C* algebra C_0(X) on a Hilbert space $H$ is given by a spectral measure that takes its values in the orthogonal projections on $H$. It is natural to ask whether something similar is true for a positive representation of the ordered Banach algebra C_0(X) on a Banach lattice E. If E is a KB-space or if E is reflexive, then the answer is affirmative: the representation is given by a spectral measure that takes its values in the positive projections on X. The proofs of above results make use of the fact that E is a Banach space, but there should be a purely order-theoretic approach. In chapter 1 of this thesis, we shall explain that this is indeed the case. In chpater 2, we are talking about simultaneous power factorization in Banach algebras. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Show less
This thesis mainly extends the theory of positive operators on Riesz spaces to a setting of pre-Riesz spaces. The theory of pre-Riesz space was established by M. van Haandel in 1993, which yields... Show moreThis thesis mainly extends the theory of positive operators on Riesz spaces to a setting of pre-Riesz spaces. The theory of pre-Riesz space was established by M. van Haandel in 1993, which yields that every directed Archimedean partially ordered vector space (pre-Riesz space) owns a vector lattice cover, that is, it can be embedded order densely into a Riesz space. Then this theory was developed by O. van Gaans and A. Kalauch during 1999-2016. Based on that, we study some properties of operators on pre-Riesz spaces, e.g. disjointness preserving operator, compact operator, disjointness preserving semigroup, local generator, dissipativity etc. on pre-Riesz spaces, which extends the classical operator theories on Riesz spaces and Banach lattices. Show less
During embryonic growth, cells proliferate, differentiate, and collectively migrate to form different tissues at the right position and time in the body. The extracellular matrix, a gel-like... Show moreDuring embryonic growth, cells proliferate, differentiate, and collectively migrate to form different tissues at the right position and time in the body. The extracellular matrix, a gel-like material containing an intricate network of fibres that surrounds cells in tissues affects cell shape and cell migration. Cells can sense the mechanical properties of the matrix, such as its stiffness, by applying forces on the matrix. In response to matrix stiffness, cells change their shape and migratory behaviour. Cells communicate with neighbouring cells by applying forces on the matrix that locally stiffen the matrix. In this thesis, we study how such mechanical cell-cell communication coordinates the patterning of tissues. We have developed multiscale models that describe cell shape and migration, the extracellular matrix and cell-matrix interactions. Our simulations suggest that by communicating via forces on the matrix, cells can form networks. This process resembles dynamics of a cell culture model of blood vessel formation. Furthermore, simulated tissues that actively pull on the matrix can better align to pre-existing stresses in the matrix compared to less contractile tissues. A model that includes the molecular complexes mediating cell-matrix interactions can accurately predict how cell shape and cell motility depends on matrix stiffness Show less
This thesis has three main parts. The first part gives an algorithm to compute Hilbert modular polynomials for ordinary abelian varieties with maximal real multiplication. Hilbert modular... Show moreThis thesis has three main parts. The first part gives an algorithm to compute Hilbert modular polynomials for ordinary abelian varieties with maximal real multiplication. Hilbert modular polynomials of a given level b give a way of finding all of the abelian varieties that are b-isogeneous to any given abelian varieties satisfying the right conditions. The second part is the proof of a theorem giving the structure of an isogeny graph of simple ordinary abelian varieties with maximal real multiplication. The third part gives a new polynomial time algorithm to count points on genus 2 curves with maximal real multiplication. This algorithm is the fastest known for curves satisfying the right properties. Show less
The main object of study in this thesis is an Arakelov inequalitywhich bounds the degree of an invertible subsheaf of the direct image ofthe pluricanonical relative sheaf of a semistable family of... Show moreThe main object of study in this thesis is an Arakelov inequalitywhich bounds the degree of an invertible subsheaf of the direct image ofthe pluricanonical relative sheaf of a semistable family of curves. A naturalproblem that arises is the characterization of those families for which the equalityis satisfied in that Arakelov inequality, i.e. the case of Arakelov equality.Few examples of such families are known. In this thesis we provide some examplesby proving that the direct image of the bicanonical relative sheaf ofa semistable family of curves uniformized by the unit ball, all whose singularfibers are totally geodesic, contains an invertible subsheaf which satisfiesArakelov equality. Show less
In the first part, we introduce a new condition, called toric-additivity, on a family of abelian varieties degenerating to a semi-abelian scheme over a normal crossing divisor. The condition... Show moreIn the first part, we introduce a new condition, called toric-additivity, on a family of abelian varieties degenerating to a semi-abelian scheme over a normal crossing divisor. The condition depends only on the l-adic Tate module of the generic fibre, for a prime l invertible on S. We show that toric-additivity is strictly related to the property of existence of Neron models. In the second part, we consider the case of a family of nodal curves over a discrete valuation ring, having split singularities. We say that such a family is semi-factorial if every line bundle on the generic bre extends to a line bundle on the total space. We give a necessary and sufficient condition for semifactoriality, in terms of combinatorics of the dual graph of the special fibre. Show less
Assuming everywhere good reduction we generalize the class number formula of Taelman to Drinfeld modules over arbitrary coefficient rings. In order to prove this formula we develop a theory of... Show moreAssuming everywhere good reduction we generalize the class number formula of Taelman to Drinfeld modules over arbitrary coefficient rings. In order to prove this formula we develop a theory of shtukas and their cohomology. Show less
In this thesis we study products of linear error correcting codes. Error correcting codes are used to correct the errors introduced by some noisy communication channel and are essential in all... Show moreIn this thesis we study products of linear error correcting codes. Error correcting codes are used to correct the errors introduced by some noisy communication channel and are essential in all communications that, due to economic or practical constraints, do not allow data retransmission: for instance deep space communications, broadcasting and mass storage. Their products, throughout the last forty years, have appeared in many different fields, such as cryptography, complexity theory, additive combinatorics and cryptanalysis. We study such products and discuss applications to cryptography. First, we prove that typically the product operation generates trivial codes; then, we investigate and characterize some class of codes whose products are non trivial and satisfy interesting properties. Our methods are algebraic-combinatorial in nature, though sometimes probabilistic techniques will be involved. Show less
This dissertation presents new cryptographic protocols, which can be divided into two families. Protocols in the first family achieve unilateral security: this means that they protect legitimate... Show moreThis dissertation presents new cryptographic protocols, which can be divided into two families. Protocols in the first family achieve unilateral security: this means that they protect legitimate users against an external attacker. Concretely, we assume that two users wish to communicate securely over a given communication system, where an external attacker eavesdrops and tampers with some of the wires of the system. We contribute to the topic by presenting protocols with improved efficiency and a simpler definition compared to previous work, and we design interactive protocols that achieve security against a stronger attacker.Protocols of the second type achieve multilateral security, meaning that they protect users against each other. This is the case for multi-party computation or MPC, where several users wish to compute a function on private inputs while keeping inputs private and without appealing to a trusted third party; we contribute to this topic by adding a cheater-detection functionality to a well-established MPC protocol.A key component that underlies these scenarios is secret sharing; we investigate this topic by casting in particular a new light on its connections with coding theory. This allows us to better harness the features of recent code constructions to obtain improved secret-sharing schemes. Show less
This thesis deals with properties of Jacobians of genus two curves that cover elliptic curves. If E is an elliptic curve and C is a curve of genus two that covers it n-to-1 so that the covering... Show moreThis thesis deals with properties of Jacobians of genus two curves that cover elliptic curves. If E is an elliptic curve and C is a curve of genus two that covers it n-to-1 so that the covering does not factor through an isogeny, then C also covers another elliptic curve n-to-1 in such a way and the Jacobian of C is isogenous to the product of the two elliptic curves. The Jacobian is said to be (n,n)-split and the elliptic curves are said to be glued along their n-torsion. The first chapter deals with the geometric aspects of this setup. We describe two approaches to constructing (n,n)-split Jacobians and we investigate which curves can appear in the setup. The second chapter deals with the arithmetic aspects, focusing on height functions and the Lang-Silverman conjecture in particular. We show that this conjecture holds for families of (n,n)-split Jacobians if and only if it holds for the corresponding families of elliptic curves that can be glued along their n-torsion. Show less
This thesis concerns the relation bettween the good reduction of Shimura varieties and the associated loop groups. To be precise, by studying the Breuil-Kisin modules of p-divisible groups, we... Show moreThis thesis concerns the relation bettween the good reduction of Shimura varieties and the associated loop groups. To be precise, by studying the Breuil-Kisin modules of p-divisible groups, we construct a direct morphism from the special fibre of a Shimura varieties of Hodge type to an fpqc subquotient of the associated loop group, and show that the geometric fibres of the morphism gives back the Ekedahl-Oort strata of the Shimura varieties in question. This can be seen as an alternative definition of the Ekedahl-Oort stratification of Shimura varieties, which gives a conceptual explanation of E. Viehmann's new invariants of "truncation of level 1" of loop groups. Show less
In this thesis we classify the pairs (p,G), where p is a prime number and G is a finite p-group possessing an intense automorphism, i.e. an automorphism that sends each subgroup of G to a G... Show moreIn this thesis we classify the pairs (p,G), where p is a prime number and G is a finite p-group possessing an intense automorphism, i.e. an automorphism that sends each subgroup of G to a G-conjugate, that is non-trivial and whose order is coprime to p. Show less
In this thesis we develop a theory of relative displays and give hence a description of stratifications of Hilbert modular varieties with parahoric level structure in the case of bad... Show moreIn this thesis we develop a theory of relative displays and give hence a description of stratifications of Hilbert modular varieties with parahoric level structure in the case of bad reduction. Show less
Computed Tomography (CT) is an imaging technique that is used to calculate the interior of an object using X-rays under multiple projection angles. A well-known application is medical imaging with... Show moreComputed Tomography (CT) is an imaging technique that is used to calculate the interior of an object using X-rays under multiple projection angles. A well-known application is medical imaging with a CT-scanner. The reconstruction methods can roughly be divided into two categories: analytical reconstruction methods and algebraic reconstruction methods (ARMs). An example of an algorithm from the first category is Filtered Backprojection (FBP). This method has a high computational efficiency and it performs well in cases with many equiangularly distributed projection angles and high signal-to-noise ratio. ARMs require in general more computation time. They are more robust with respect to noise and can handle few projection angles or a limited angular range better. In this dissertation, the new algorithm Algebraic filter – Filtered Backprojection (AF-FBP) is introduced, which uses an ARM to create filters that can be used in FBP. The reconstruction quality of AF-FBP approximates that of the corresponding (locally) linear ARM, while the reconstructions are obtained with the computational efficiency of FBP. In cases with a small number of different scanning geometries, using AF-FBP enables the reconstruction of images of relatively high quality for few projection angles, limited angular range, or low signal-to-noise ratio. Show less
This thesis is composed of papers on four topics: Bayesian theory for the sparse normal means problem, specifically for the horseshoe prior (Chapters 1-3), Bayesian theory for community detection ... Show moreThis thesis is composed of papers on four topics: Bayesian theory for the sparse normal means problem, specifically for the horseshoe prior (Chapters 1-3), Bayesian theory for community detection (Chapter 4), nested model selection (Chapter 5), and the application of competing risk methods in the presence of time-dependent clustering (Chapter 6). Show less
In this dissertation, a primitive recursive algorithm is given for the computation of the étale Euler-Poincaré characteristic (which is the alternating sum of the étale cohomology groups in... Show moreIn this dissertation, a primitive recursive algorithm is given for the computation of the étale Euler-Poincaré characteristic (which is the alternating sum of the étale cohomology groups in the Grothendieck group of Galois modules) with finite coefficients, and on arbitrary varieties over a field. For smooth curves, a primitive recursive algorithm is given for the computation of the étale cohomology groups themselves, using a geometric interpretation of the elements of the first etale cohomology. The general case is then reduced to the case of smooth curves by making the standard dévissage techniques explicit. Show less
This thesis covers different problems concerning the evaluation of DNA evidence. It is mainly divided into two parts: the first regards the DIP-STR genotyping techniques. It addresses the... Show moreThis thesis covers different problems concerning the evaluation of DNA evidence. It is mainly divided into two parts: the first regards the DIP-STR genotyping techniques. It addresses the imperative need of developing a model to assign the likelihood ratio for DIP-STR results, and compares, from a statistical and forensic perspective, the advantages of this novel marker system compared to traditional marker systems, such as STR and Y-STR. The second part deals with several more general statistical aspects involved in the evaluation of DNA evidence. It aims at defining the differences between full Bayesian methods and ad hoc plug-in approximations, and at solving the rare type match problem for Y-STR data. The issues of the different reductions of data and of the levels of uncertainty involved in frequentist solutions are also discussed. These two parts are connected in the final project, by developing a Bayesian solution for the rare type match problem for DIP-STR marker system. Moreover, the initial model for DIP-STR data is improved in the light of the statistical discussion of the second part: any ad hoc solution is avoided to obtain a full Bayesian approach. Show less
The presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which... Show moreThe presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which vanishes on the spectrum of the linearization about the pattern. For certain 'slowly linear' prototype models it has been shown, via geometric arguments, that the Evans function factorizes in accordance with the scale separation. This leads to asymptotic control over the spectrum through simpler, lower-dimensional eigenvalue problems. Recently, the geometric factorization procedure has been generalized to homoclinic pulse solutions in slowly nonlinear reaction-di ffusion systems. In this thesis we study periodic pulse solutions in the slowly nonlinear regime. This seems a straightforward extension. However, the geometric factorization method fails and due to translational invariance there is a curve of spectrum attached to the origin, whereas for homoclinic pulses there is only a simple eigenvalue residing at 0. We develop an alternative, analytic factorization method that works for periodic structures in the slowly nonlinear setting. We derive explicit formulas for the factors of the Evans function, which yields asymptotic spectral control. Moreover, we obtain a leading-order expression for the critical spectral curve attached to origin. Together these approximation results lead to explicit stability criteria. Show less
