<p>The Torelli group Tg of a closed orientable surface Sg of genus g >1 is the group</p><p>of isotopy classes of orientation-preserving diffeomorphisms of Sg which act trivially</p><p>on its first integral homology. The hyperelliptic Torelli group TDg is the subgroup</p><p>of Tg whose elements commute with a fixed hyperelliptic involution. The finiteness</p><p>properties of Tg and TDg are not well-understood when g > 2. In particular, it is not</p><p>known if T3 is finitely presented or if TD3 is finitely generated. In this thesis, we begin</p><p>a study of the finiteness properties of genus 3 Torelli groups using techniques from</p><p>complex analytic geometry. The Torelli space T3 is the moduli space of non-singular</p><p>genus 3 curves equipped with a symplectic basis for the first integral homology and is</p><p>a model of the classifying space of T. Each component of the hyperelliptic locus T hyp 3</p><p>in T3 is a model of the classifying space for TD3. We will investigate the topology</p><p>of the zero loci of certain theta functions and thetanulls and explain how these are</p><p>related to the topology of T3 and T3 hyp. We show that the zero locus in h 2 x C2 </p><p>of any genus 2 theta function is isomorphic to the universal cover of the universal framed genus 2 curve of compact type and that it is homotopy equivalent to an infinite bouquet of 2-spheres. We also derive a necessary and sufficient condition for the zero locus of any genus 3 even thetanull to be homotopy equivalent to a bouquet of 2-spheres and 3-spheres.</p> / Dissertation

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22971号 / 理博第4648号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 入谷 寛, 教授 吉川 謙一, 教授 加藤 毅 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We describe combinatorial techniques for studying log Calabi-Yau surfaces. These can be viewed as generalizing the techniques for studying toric varieties in terms of their character and cocharacter lattices. These lattices are replaced by certain integral linear manifolds described in [GHK11], and monomials on toric varieties are replaced with the canonical theta functions defined in [GHK11] using ideas from mirror symmetry. We classify deformation classes of log Calabi-Yau surfaces in terms of the geometry of these integral linear manifolds. We then describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher rank cluster varieties. / text

This paper includes some basic ideas about the computation of a function theta(G), the theta number of a graph G, which is known as the Lovasz number of G. theta(G^c) lies between two hard-to-compute graph numbers omega(G), the size of the largest lique in a graph G, and chi(G), the minimum number of colors need to properly color the vertices of G. Lovasz and Grotschel called this the "Sandwich Theorem". Donald E. Knuth gives four additional definitions of theta, theta_1, theta_2, theta_3, theta_4 and proves that they are all equal. First I am going to describe the proof of the equality of theta, theta_1 and theta_2 and then I will show the calculation of the theta function for some specific graphs: K_n, graphs related to K_n, and C_n. This will help us understand the theta function, an important function for graph theory. Some of the results are calculated in different ways. This will benefit students who have a basic knowledge of graph theory and want to learn more about the theta function.

This report studies polynomial equations and how one solves them using only the coefficients of the polynomial. It examines why it is impossible to solve equations of degree greater than four using only radicals and how instead one can solve them using elliptic functions. Although the quintic equation is the main area of our investigation, we also present parts of the history of algebraic equations, Galois theory, and elliptic functions.

No description available.

Digital subscriber lines (DSL) have revolutionised the provision of high speed data over the ‘last mile’. Subscribers demand even more bandwidth and the penetration of the service is now nearly universal. While it is feasible to provide improved broadband services on the new very high speed DSL, such as VDSL2/3, one of the greatest challenges to further improvements in speed is the problem of crosstalk. Operating over the unused higher frequencies of the twisted pair network, this technology is subjected to electromagnetic coupling among the wires, limiting the DSL data rate and service reach. Crosstalk suppression methods such as zero-forcing or decision feedback mainly use block processing. However, to cope with the time-varying VDSL environment huge computational costs can be incurred. In contrast, adaptive processing approaches are much simpler and are more beneficial to track such a channel environment. An adaptive canceller uses a training sequence and the convergence speed depends on the number of crosstalk coefficients it has to estimate. In a populated DSL binder, only a few of the crosstalking neighbours to a particular user are significant. With the aim to reduce the computational complexity in such environments, this thesis introduces the concept of detection-guided adaptive crosstalk cancellation for DSL. We propose a least-squares test feature to detect and concentrate the adaptation only on the dominant crosstalking coefficients. In comparison to conventional adaptive cancellers, the cancellers proposed in this thesis demonstrate early convergence. Thus, by incorporating the test feature, these cancellers have to detect only the most significant canceller coefficients and therefore, the length of the training sequence is reduced. Together with enhanced adaptive cancellation with a low run-time complexity and improved convergence, the greatest advantage obtained here is in the bandwidth efficiency. While enhanced adaptive cancellation is a bandwidth-efficient approach, the frequent re-transmission of training sequences may still be required for a rapidly changing VDSL channel. Again, this can be a disadvantage in terms of bandwidth consumption. To overcome this difficulty, we propose fast-converging unsupervised cancellers with an aim to improve the bandwidth efficiency by not transmitting a training sequence. An added advantage obtained here is that this would enable Internet service providers to include multiple or improved broadband services within a single subscription. Certain properties of the DSL channel ensure the communication channel is properly conditioned. This ensures the basis vectors of the channel matrix are near-orthogonal and hence, the linear cancellers, such as zero-forcing perform near-optimally. However, this is not the case with wireless channels. We investigate user detection in wireless channels using the principle of lattice reduction. User detection can also be seen as a search for the closest vector point in the lattice of received symbols. Though a maximum likelihood (ML) detector facilitates optimal user-detection, it has exponential complexity. We identify that the closest vector problem can be cast as a non-linear optimisation problem. Using the periodicity of the maximum likelihood function, we first present a novel algorithm that approximates the ML function using the Taylor series expansion of a suitable cosine function. With the aim of minimising the approximation error, we represent the ML function as a Fourier Series expansion and later, propose another approximation using Jacobi theta functions. We study the performance of these approximations when subjected to a suitable unconstrained optimisation algorithm. Through simulations, we demonstrate that the newly-developed approximations perform better than the conventional cancellers, close to the ML and, importantly, converging in polynomial time.

Digital subscriber lines (DSL) have revolutionised the provision of high speed data over the ‘last mile’. Subscribers demand even more bandwidth and the penetration of the service is now nearly universal. While it is feasible to provide improved broadband services on the new very high speed DSL, such as VDSL2/3, one of the greatest challenges to further improvements in speed is the problem of crosstalk. Operating over the unused higher frequencies of the twisted pair network, this technology is subjected to electromagnetic coupling among the wires, limiting the DSL data rate and service reach. Crosstalk suppression methods such as zero-forcing or decision feedback mainly use block processing. However, to cope with the time-varying VDSL environment huge computational costs can be incurred. In contrast, adaptive processing approaches are much simpler and are more beneficial to track such a channel environment. An adaptive canceller uses a training sequence and the convergence speed depends on the number of crosstalk coefficients it has to estimate. In a populated DSL binder, only a few of the crosstalking neighbours to a particular user are significant. With the aim to reduce the computational complexity in such environments, this thesis introduces the concept of detection-guided adaptive crosstalk cancellation for DSL. We propose a least-squares test feature to detect and concentrate the adaptation only on the dominant crosstalking coefficients. In comparison to conventional adaptive cancellers, the cancellers proposed in this thesis demonstrate early convergence. Thus, by incorporating the test feature, these cancellers have to detect only the most significant canceller coefficients and therefore, the length of the training sequence is reduced. Together with enhanced adaptive cancellation with a low run-time complexity and improved convergence, the greatest advantage obtained here is in the bandwidth efficiency. While enhanced adaptive cancellation is a bandwidth-efficient approach, the frequent re-transmission of training sequences may still be required for a rapidly changing VDSL channel. Again, this can be a disadvantage in terms of bandwidth consumption. To overcome this difficulty, we propose fast-converging unsupervised cancellers with an aim to improve the bandwidth efficiency by not transmitting a training sequence. An added advantage obtained here is that this would enable Internet service providers to include multiple or improved broadband services within a single subscription. Certain properties of the DSL channel ensure the communication channel is properly conditioned. This ensures the basis vectors of the channel matrix are near-orthogonal and hence, the linear cancellers, such as zero-forcing perform near-optimally. However, this is not the case with wireless channels. We investigate user detection in wireless channels using the principle of lattice reduction. User detection can also be seen as a search for the closest vector point in the lattice of received symbols. Though a maximum likelihood (ML) detector facilitates optimal user-detection, it has exponential complexity. We identify that the closest vector problem can be cast as a non-linear optimisation problem. Using the periodicity of the maximum likelihood function, we first present a novel algorithm that approximates the ML function using the Taylor series expansion of a suitable cosine function. With the aim of minimising the approximation error, we represent the ML function as a Fourier Series expansion and later, propose another approximation using Jacobi theta functions. We study the performance of these approximations when subjected to a suitable unconstrained optimisation algorithm. Through simulations, we demonstrate that the newly-developed approximations perform better than the conventional cancellers, close to the ML and, importantly, converging in polynomial time.

Digital subscriber lines (DSL) have revolutionised the provision of high speed data over the ‘last mile’. Subscribers demand even more bandwidth and the penetration of the service is now nearly universal. While it is feasible to provide improved broadband services on the new very high speed DSL, such as VDSL2/3, one of the greatest challenges to further improvements in speed is the problem of crosstalk. Operating over the unused higher frequencies of the twisted pair network, this technology is subjected to electromagnetic coupling among the wires, limiting the DSL data rate and service reach. Crosstalk suppression methods such as zero-forcing or decision feedback mainly use block processing. However, to cope with the time-varying VDSL environment huge computational costs can be incurred. In contrast, adaptive processing approaches are much simpler and are more beneficial to track such a channel environment. An adaptive canceller uses a training sequence and the convergence speed depends on the number of crosstalk coefficients it has to estimate. In a populated DSL binder, only a few of the crosstalking neighbours to a particular user are significant. With the aim to reduce the computational complexity in such environments, this thesis introduces the concept of detection-guided adaptive crosstalk cancellation for DSL. We propose a least-squares test feature to detect and concentrate the adaptation only on the dominant crosstalking coefficients. In comparison to conventional adaptive cancellers, the cancellers proposed in this thesis demonstrate early convergence. Thus, by incorporating the test feature, these cancellers have to detect only the most significant canceller coefficients and therefore, the length of the training sequence is reduced. Together with enhanced adaptive cancellation with a low run-time complexity and improved convergence, the greatest advantage obtained here is in the bandwidth efficiency. While enhanced adaptive cancellation is a bandwidth-efficient approach, the frequent re-transmission of training sequences may still be required for a rapidly changing VDSL channel. Again, this can be a disadvantage in terms of bandwidth consumption. To overcome this difficulty, we propose fast-converging unsupervised cancellers with an aim to improve the bandwidth efficiency by not transmitting a training sequence. An added advantage obtained here is that this would enable Internet service providers to include multiple or improved broadband services within a single subscription. Certain properties of the DSL channel ensure the communication channel is properly conditioned. This ensures the basis vectors of the channel matrix are near-orthogonal and hence, the linear cancellers, such as zero-forcing perform near-optimally. However, this is not the case with wireless channels. We investigate user detection in wireless channels using the principle of lattice reduction. User detection can also be seen as a search for the closest vector point in the lattice of received symbols. Though a maximum likelihood (ML) detector facilitates optimal user-detection, it has exponential complexity. We identify that the closest vector problem can be cast as a non-linear optimisation problem. Using the periodicity of the maximum likelihood function, we first present a novel algorithm that approximates the ML function using the Taylor series expansion of a suitable cosine function. With the aim of minimising the approximation error, we represent the ML function as a Fourier Series expansion and later, propose another approximation using Jacobi theta functions. We study the performance of these approximations when subjected to a suitable unconstrained optimisation algorithm. Through simulations, we demonstrate that the newly-developed approximations perform better than the conventional cancellers, close to the ML and, importantly, converging in polynomial time.

Nous considérons certaines séries de Fourier liées à la théorie des formes modulaires. Nous étudions leurs propriétés analytiques : la dérivabilité, le module de continuité et l'exposant de Hölder. Nous utilisons deux méthodes différentes. La première revient à trouver et itérer une équation fonctionnelle de la fonction étudiée (méthode d'Itatsu) et la deuxième provient de l'analyse en ondelettes (méthode de Jaffard). L'étape essentielle de chacune dépend de la modularité sous-jacente. Nous trouvons que les propriétés analytiques de ces séries aux points irrationnels sont liées aux propriétés diophantiennes de ces points. Ce travail a été motivé par l'étude de la fonction de Riemann. / We consider certain Fourier series which arise from modular or automorphicforms. We study their analytic properties: differentiability, modulus of continuity and theH¨older regularity exponent. We use two different methods. One is based on finding anditerating a functional equation for the function studied (Itatsu’s method), the second onecomes from wavelet analysis (Jaffard’s method). The crucial steps in both of them arebased on the underlined modularity. We find that the analytic properties of these seriesat an irrational x are related to the fine diophantine properties of x, in a very precise way.The work was motivated by the study of the Riemann series.