arXiv subjects

Gordon Hughes

Publications and source records attributed to Gordon Hughes.

5 recordsLinked to original sources

arXiv2024

Edge Geometry of Regular Polygons -- Part 2

There are multiple mappings that can be used to generate what we call the 'edge geometry' of a regular N-gon, but they are all based on piecewise isometries acting on the extended edges of N to form a 'singularity' set W. This singularity set is also known as the 'web' because it is connected and consists of rays or line segments, with possible accumulation points in the limit. We will use three such maps here, all of which appear to share the same local geometry of W. These mappings are the outer-billiards map Tau, the digital-filter map Df and the 'dual-center' map Dc. In 'Outer-billiards, digital filters and kicked Hamiltonians' (arXiv:1206.5223) we show that the Df and Dc maps are equivalent to a 'shear and rotation' in a toral space and the complex plane respectively, and in 'First Families of Regular Polygons and their Mutations' (arXiv:1612.09295) we show that the Tau-web W can also be reduced to a shear and rotation. This equivalence of maps supports the premise that this web geometry is inherent in the N-gon. Since the topology of W is complex, we hope to make some progress by studying the region local to N. The edges of every regular N-gon are part of a Tau-invariant region local to N. The emphasis here are the S[1] and S[2] First Family tiles adjacent to N, but we will also study their interaction with neighboring tiles. Since all S[k] tiles evolve in a multi-step fashion, it is possible to make predictions about the 'next-generation' tiles which survive in the web. The Edge Conjecture defines just 8 classes of N-gons so there is an 'Eightfold Way' for regular polygons.

math.DS↗

arXiv2016

RemIX: A Distributed Internet Exchange for Remote and Rural Networks

The concept of the IXP, an Ethernet fabric central to the structure of the global Internet, is largely absent from the development of community-driven collaborative network infrastructure. The reasons for this are two-fold. IXPs exist in central, typically urban, environments where strong network infrastructure ensures high levels of connectivity. Between rural and remote regions, where networks are separated by distance and terrain, no such infrastructure exists. In this paper we present RemIX a distributed IXPs architecture designed for the community network environment. We examine this praxis using an implementation in Scotland, with suggestions for future development and research.

cs.NI↗

arXiv2012

Solid State Disk Object-Based Storage with Trim Commands

This paper presents a model of NAND flash SSD utilization and write amplification when the ATA/ATAPI SSD Trim command is incorporated into object-based storage under a variety of user workloads, including a uniform random workload with objects of fixed size and a uniform random workload with objects of varying sizes. We first summarize the existing models for write amplification in SSDs for workloads with and without the Trim command, then propose an alteration of the models that utilizes a framework of object-based storage. The utilization of objects and pages in the SSD is derived, with the analytic results compared to simulation. Finally, the effect of objects on write amplification and its computation is discussed along with a potential application to optimization of SSD usage through object storage metadata servers that allocate object classes of distinct object size.

cs.PF↗

arXiv2012

Analysis of Trim Commands on Overprovisioning and Write Amplification in Solid State Drives

This paper presents a performance model of the ATA/ATAPI SSD Trim command under various types of user workloads, including a uniform random workload, a workload with hot and cold data, and a workload with N temperatures of data. We first examine the Trim-modified uniform random workload to predict utilization, then use this result to compute the resultant level of effective overprovisioning. This allows modification of models previously suggested to predict write amplification of a non-Trim uniform random workload under greedy garbage collection. Finally, we expand the theory to cover a workload consisting of hot and cold data (and also N temperatures of data), providing formulas to predict write amplification in these scenarios.

cs.OH↗

arXiv2012

The Polygons of Albrecht Durer -1525

The early Renaissance artist Albrecht Durer published a book on geometry a few years before he died. This was intended to be a guide for young craftsmen and artists giving them both practical and mathematical tools for their trade. In the second part of that book, Durer gives compass and straightedge constructions for the 'regular' polygons from the triangle to the 16-gon. We will examine each of these constructions using the original 1525 text and diagrams along with a translation. Then we will use Mathematica to carry out the constructions which are only approximate, in order to compare them with the regular case. In Appendix A, we discuss Durer's approximate trisection method for angles, which is surprisingly accurate. Appendix B outlines what is known currently about compass and straightedge constructions and includes two elegant 19th century constructions.

math.HO↗