Isogeny graphs of ordinary abelian varieties

Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called $\mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(\ell, \ell)$-isogenies: those whose kernels are maximal isotropic subgroups of the $\ell$-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure

Zyklische Levelzeichnungen gerichteter Graphen

The Sugiyama framework proposed in the seminal paper of 1981 is one of the most important algorithms in graph drawing and is widely used for visualizing directed graphs. In its common version, it draws graphs hierarchically and, hence, maps the topological direction to a geometric direction. However, such a hierarchical layout is not possible if the graph contains cycles, which have to be destroyed in a preceding step. In certain application and problem settings, e.g., bio sciences or periodic scheduling problems, it is important that the cyclic structure of the input graph is preserved and clearly visible in drawings. Sugiyama et al. also suggested apart from the nowadays standard horizontal algorithm a cyclic version they called recurrent hierarchies. However, this cyclic drawing style has not received much attention since. In this thesis we consider such cyclic drawings and investigate the Sugiyama framework for this new scenario. As our goal is to visualize cycles directly, the first phase of the Sugiyama framework, which is concerned with removing such cycles, can be neglected. The cyclic structure of the graph leads to new problems in the remaining phases, however, for which solutions are proposed in this thesis. The aim is a complete adaption of the Sugiyama framework for cyclic drawings. To complement our adaption of the Sugiyama framework, we also treat the problem of cyclic level planarity and present a linear time cyclic level planarity testing and embedding algorithm for strongly connected graphs

Rewriting Codes for Joint Information Storage in Flash Memories

Memories whose storage cells transit irreversibly between states have been common since the start of the data storage technology. In recent years, flash memories have become a very important family of such memories. A flash memory cell has q states—state 0.1.....q-1 - and can only transit from a lower state to a higher state before the expensive erasure operation takes place. We study rewriting codes that enable the data stored in a group of cells to be rewritten by only shifting the cells to higher states. Since the considered state transitions are irreversible, the number of rewrites is bounded. Our objective is to maximize the number of times the data can be rewritten. We focus on the joint storage of data in flash memories, and study two rewriting codes for two different scenarios. The first code, called floating code, is for the joint storage of multiple variables, where every rewrite changes one variable. The second code, called buffer code, is for remembering the most recent data in a data stream. Many of the codes presented here are either optimal or asymptotically optimal. We also present bounds to the performance of general codes. The results show that rewriting codes can integrate a flash memory’s rewriting capabilities for different variables to a high degree

Feynman Categories

In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads, their colored versions, as well as algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI--modules. The usefulness of this approach is that it allows us to handle all the classical as well as more esoteric structures under a common framework and we can treat all the situations simultaneously. Many of the known constructions simply become Kan extensions. In this common framework, we also derive universal operations, such as those underlying Deligne's conjecture, construct Hopf algebras as well as perform resolutions, (co)bar transforms and Feynman transforms which are related to master equations. For these applications, we construct the relevant model category structures. This produces many new examples.Comment: Expanded version. New introduction, new arrangement of text, more details on several constructions. New figure

Evaluation of frictional forces between brackets of different types at various angulations and an arch wire: With and without pulsating vibration

Objective: The objective of this study was to determine the effect of pulsating vibration on the sliding resistance between orthodontic brackets and stainless steel wires. Brackets were placed at two different angulations (0° and 5°) to simulate leveling of a tipped tooth during tooth movement. Pulsating vibration was delivered via the AcceleDent device. Background: Friction is defined as a force that retards or resists the relative motion of two objects in contact, and its direction is tangential to the common boundary of the two surfaces in contact. This has been of interest to the orthodontist since the mid-20th century. Since the time of Stoner’s paper in 1960, the orthodontic literature has been full of studies done on friction in orthodontics including: friction with different ligation methods, friction among different arch wire materials, friction and different bracket materials, and friction with various slot designs. Understanding friction has led to the emergence of new technologies in orthodontics. One of the most popular is the self-ligating bracket. This popularity arose from claims that they reduce friction during treatment. Other innovations have been introduced in the field of orthodontics to help accelerate tooth movement. Among these innovations is the application of a pulsating vibration during active orthodontic treatment. Such pulsating vibration can be delivered during orthodontic treatment by AcceleDent, which is a hands- free device designed by OrthoAccel Technologies, Inc., Bellaire, TX. The company claims the output force helps accelerate bone turnover. The following study investigated whether it could decrease treatment time via a different mechanism: decreasing frictional resistance to tooth movement along the arch wire. Methods: A paper template was made of a typodont tooth with a bracket window cut out. The bracket cut out was made with the bracket window angulated 0° and 5°. 0.022” x 0.028” standard prescription edgewise brackets (American Orthodontics, Sheboygan, WI) of ceramic, twin and self-ligating design were bonded to 3 maxillary 1st premolar typodont teeth using the template. The teeth were leveled with a 0.019” x 0.025” SS arch wire and placed in a metal scaffold. They were held in place with Aquasil Ultra XLV wash material PVS (DENTSPLY Caulk, Milford, DE.). Only the middle bracket was adjusted for angulation and accuracy was checked with the iPhone 6 level. The AcceleDent Aura device (OrthoAccel Technologies, Inc., Bellaire, TX). was attached to the occlusal surface of the teeth via cable ties. The AcceleDent Aura device provided 30 Hz of pulsating vibration. All tests were performed with a 0.019” x 0.025” SS arch wire pulled through the brackets via a Universal Testing Machine (Instron, Grove City, PA) at a crosshead speed of 2.5mm/min for 30 seconds. Frictional resistance was measured by averaging 6 recordings every 5 seconds. Results: The pulsating vibration provided by the AcceleDent device significantly reduced the resistance to sliding for each bracket type at both 0° and 5° (p\u3c0.05). Ceramic brackets had the highest resistance to sliding of all bracket types. Conclusions: Pulsating vibration via the AcceleDent Aura device reduces the resistance to sliding between a bracket and arch wire in vitro. This may potentially decrease overall treatment time but more in vivo studies need to be done to evaluate this