Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks

Laser beacon studies final summary report, 30 jun. - 31 oct. 1961

Laser beacon for daylight optical tracking - components description, signal to noise ratio, and signal sensitivit

Hierarchical Graphical Models for Multigroup Shape Analysis using Expectation Maximization with Sampling in Kendall's Shape Space

This paper proposes a novel framework for multi-group shape analysis relying on a hierarchical graphical statistical model on shapes within a population.The framework represents individual shapes as point setsmodulo translation, rotation, and scale, following the notion in Kendall shape space.While individual shapes are derived from their group shape model, each group shape model is derived from a single population shape model. The hierarchical model follows the natural organization of population data and the top level in the hierarchy provides a common frame of reference for multigroup shape analysis, e.g. classification and hypothesis testing. Unlike typical shape-modeling approaches, the proposed model is a generative model that defines a joint distribution of object-boundary data and the shape-model variables. Furthermore, it naturally enforces optimal correspondences during the process of model fitting and thereby subsumes the so-called correspondence problem. The proposed inference scheme employs an expectation maximization (EM) algorithm that treats the individual and group shape variables as hidden random variables and integrates them out before estimating the parameters (population mean and variance and the group variances). The underpinning of the EM algorithm is the sampling of pointsets, in Kendall shape space, from their posterior distribution, for which we exploit a highly-efficient scheme based on Hamiltonian Monte Carlo simulation. Experiments in this paper use the fitted hierarchical model to perform (1) hypothesis testing for comparison between pairs of groups using permutation testing and (2) classification for image retrieval. The paper validates the proposed framework on simulated data and demonstrates results on real data.Comment: 9 pages, 7 figures, International Conference on Machine Learning 201

A novel approach for ANFIS modelling based on Grey system theory for thermal error compensation

The fast and accurate modelling of thermal errors in machining is an important aspect for the implementation of thermal error compensation. This paper presents a novel modelling approach for thermal error compensation on CNC machine tools. The method combines the Adaptive Neuro Fuzzy Inference System (ANFIS) and Grey system theory to predict thermal errors in machining. Instead of following a traditional approach, which utilises original data patterns to construct the ANFIS model, this paper proposes to exploit Accumulation Generation Operation (AGO) to simplify the modelling procedures. AGO, a basis of the Grey system theory, is used to uncover a development tendency so that the features and laws of integration hidden in the chaotic raw data can be sufficiently revealed. AGO properties make it easier for the proposed model to design and predict. According to the simulation results, the proposed model demonstrates stronger prediction power than standard ANFIS model only with minimum number of training samples

The Riemannian Geometry of Deep Generative Models

Deep generative models learn a mapping from a low dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. However, further investigation into this phenomenon is warranted, to identify if there are other architectures or datasets where curvature plays a more prominent role. We believe that exploring the Riemannian geometry of deep generative models, using the tools developed in this paper, will be an important step in understanding the high-dimensional, nonlinear spaces these models learn.Comment: 9 page

A Preliminary Study of Applying Lean Six Sigma Methods to Machine Tool Measurement

Many manufacturers aim to increase their levels of high-quality production in order to improve their market competitiveness. Continuous improvement of maintenance strategies is a key factor to be capable of delivering high quality products and services on-time with minimal operating costs. However, the cost of maintaining quality is often perceived as a non-added-value task. Improving the efficiency and effectiveness of the measurement procedures necessary to guarantee accuracy of production is a more complex task than many other maintenance functions and so deserves particular analysis. This paper investigates the feasibility of producing a concise yet effective framework that will provide a preliminary approach for integrating Lean and Six Sigma philosophies to the specific goal of reducing unnecessary downtime on manufacturing machines while maintaining its ability to machine to the required tolerance. The purpose of this study is to show how a Six Sigma infrastructure is used to investigate the root causes of complication occurring during the machine tool measurement. This work recognises issues of the uncertainty of data, and the measurement procedures in parallel with the main tools of Six Sigma’s Define-Measure-Analyse-Improve-Control (DMAIC). The significance of this work is that machine tool accuracy is critical for high value manufacturing. Over-measuring the machine to ensure accuracy potentially reduces production volume. However, not measuring them or ignoring accuracy aspects possibly lead to production waste. This piece of work aims to present a lean guidance to lessen measurement uncertainties and optimise the machine tool benchmarking procedures, while adopting the DMAIC strategy to reduce unnecessary downtime

Investigation of a new method for improving image resolution for camera tracking applications

Camera based systems have been a preferred choice in many motion tracking applications due to the ease of installation and the ability to work in unprepared environments. The concept of these systems is based on extracting image information (colour and shape properties) to detect the object location. However, the resolution of the image and the camera field-of- view (FOV) are two main factors that can restrict the tracking applications for which these systems can be used. Resolution can be addressed partially by using higher resolution cameras but this may not always be possible or cost effective. This research paper investigates a new method utilising averaging of offset images to improve the effective resolution using a standard camera. The initial results show that the minimum detectable position change of a tracked object could be improved by up to 4 times

Study to minimize hydrogen embrittlement of ultrahigh-strength steels

Hydrogen-stress cracking in high-strength steels is influenced by hydrogen content of the material and its hydrogen absorption tendency. Non-embrittling cleaning, pickling, and electroplating processes are being studied. Protection from this hydrogen embrittlement is important to the aerospace and aircraft industries

Thermal Error Modelling of a CNC Machine Tool Feed Drive System using FEA Method

Recirculating ball screw systems are commonly used in machine tools and are one of the major heat sources which cause considerable thermal drift in CNC machine tools. Finite Element Analysis (FEA) method has been used successfully in the past to model the thermal characteristics of machine tools with promising results. Since FEA predictions are highly dependent on the efficacy of numerical parameters including the surrounding Boundary Conditions (BC), this study emphasises on an efficient modelling method to obtain optimised numerical parameters for acquiring a qualitative response from the feed drive system model. This study was performed on a medium size Vertical Machining Centre (VMC) feed drive system in which two parameter dentification methods have been employed; the general prediction method based on formulae provided by OEMs, and the energy balance method. The parameters obtained from both methods were applied to the FEA model of the machine feed drive system and validated against experimental results. Correlation with which was increased from 70 % to 80 % using the energy balance method