Towards a web-based progressive handwriting recognition environment for mathematical problem solving

The emergence of pen-based mobile devices such as PDAs and tablet PCs provides a new way to input mathematical expressions to computer by using handwriting which is much more natural and efficient for entering mathematics. This paper proposes a web-based handwriting mathematics system, called WebMath, for supporting mathematical problem solving. The proposed WebMath system is based on client-server architecture. It comprises four major components: a standard web server, handwriting mathematical expression editor, computation engine and web browser with Ajax-based communicator. The handwriting mathematical expression editor adopts a progressive recognition approach for dynamic recognition of handwritten mathematical expressions. The computation engine supports mathematical functions such as algebraic simplification and factorization, and integration and differentiation. The web browser provides a user-friendly interface for accessing the system using advanced Ajax-based communication. In this paper, we describe the different components of the WebMath system and its performance analysis.

Modelling the dynamics of cutting during turning

This thesis presents an approach to cutting dynamics during turning based upon the mechanism of deformation of work material around the tool nose known as "ploughing". Starting from the shearing process in the cutting zone and accounting for "ploughing", new mathematical models relating turning force components to cutting conditions, tool geometry and tool vibration are developed. These models are developed separately for steady state and for oscillatory turning with new and worn tools. Experimental results are used to determine mathematical functions expressing the parameters introduced by the steady state model in the case of a new tool. The form of these functions are of general validity though their coefficients are dependent on work and tool materials. Good agreement is achieved between experimental and predicted forces. The model is extended on one hand to include different work material by introducing a hardness factor. The model provides good predictions when predicted forces are compared to present and published experimental results. On the other hand, the extension of the ploughing model to taming with a worn edge showed the ability of the model in predicting machining forces during steady state turning with the worn flank of the tool. In the development of the dynamic models, the dynamic turning force equations define the cutting process as being a system for which vibration of the tool tip in the feed direction is the input and measured forces are the output The model takes into account the shear plane oscillation and the cutting configuration variation in response to tool motion. Theoretical expressions of the turning forces are obtained for new and worn cutting edges. The dynamic analysis revealed the interaction between the cutting mechanism and the machine tool structure. The effect of the machine tool and tool post is accounted for by using experimental data of the transfer function of the tool post system. Steady state coefficients are corrected to include the changes in the cutting configuration with tool vibration and are used in the dynamic model. A series of oscillatory cutting tests at various conditions and various tool flank wear levels are carried out and experimental results are compared with model—predicted forces. Good agreement between predictions and experiments were achieved over a wide range of cutting conditions. This research bridges the gap between the analysis of vibration and turning forces in turning. It offers an explicit expression of the dynamic turning force generated during machining and highlights the relationships between tool wear, tool vibration and turning force. Spectral analysis of tool acceleration and turning force components led to define an "Inertance Power Ratio" as a flank wear monitoring factor. A formulation of an on—line flank wear monitoring methodology is presented and shows how the results of the present model can be applied to practical in—process tool wear monitoring in • turning operations.

Describing and communicating uncertainty within the semantic web

The Semantic Web relies on carefully structured, well defined, data to allow machines to communicate and understand one another. In many domains (e.g. geospatial) the data being described contains some uncertainty, often due to incomplete knowledge; meaningful processing of this data requires these uncertainties to be carefully analysed and integrated into the process chain. Currently, within the SemanticWeb there is no standard mechanism for interoperable description and exchange of uncertain information, which renders the automated processing of such information implausible, particularly where error must be considered and captured as it propagates through a processing sequence. In particular we adopt a Bayesian perspective and focus on the case where the inputs / outputs are naturally treated as random variables. This paper discusses a solution to the problem in the form of the Uncertainty Markup Language (UncertML). UncertML is a conceptual model, realised as an XML schema, that allows uncertainty to be quantified in a variety of ways i.e. realisations, statistics and probability distributions. UncertML is based upon a soft-typed XML schema design that provides a generic framework from which any statistic or distribution may be created. Making extensive use of Geography Markup Language (GML) dictionaries, UncertML provides a collection of definitions for common uncertainty types. Containing both written descriptions and mathematical functions, encoded as MathML, the definitions within these dictionaries provide a robust mechanism for defining any statistic or distribution and can be easily extended. Universal Resource Identifiers (URIs) are used to introduce semantics to the soft-typed elements by linking to these dictionary definitions. The INTAMAP (INTeroperability and Automated MAPping) project provides a use case for UncertML. This paper demonstrates how observation errors can be quantified using UncertML and wrapped within an Observations & Measurements (O&M) Observation. The interpolation service uses the information within these observations to influence the prediction outcome. The output uncertainties may be encoded in a variety of UncertML types, e.g. a series of marginal Gaussian distributions, a set of statistics, such as the first three marginal moments, or a set of realisations from a Monte Carlo treatment. Quantifying and propagating uncertainty in this way allows such interpolation results to be consumed by other services. This could form part of a risk management chain or a decision support system, and ultimately paves the way for complex data processing chains in the Semantic Web.

Mathematical and numerical modelling of dispersion-managed solitons, autosolitons and self-similar optical pulses

This thesis presents theoretical investigation of three topics concerned with nonlinear optical pulse propagation in optical fibres. The techniques used are mathematical analysis and numerical modelling. Firstly, dispersion-managed (DM) solitons in fibre lines employing a weak dispersion map are analysed by means of a perturbation approach. In the case of small dispersion map strengths the average pulse dynamics is described by a perturbation approach (NLS) equation. Applying a perturbation theory, based on the Inverse Scattering Transform method, an analytic expression for the envelope of the DM soliton is derived. This expression correctly predicts the power enhancement arising from the dispersion management.Secondly, autosoliton transmission in DM fibre systems with periodical in-line deployment of nonlinear optical loop mirrors (NOLMs) is investigated. The use of in-line NOLMs is addressed as a general technique for all-optical passive 2R regeneration of return-to-zero data in high speed transmission system with strong dispersion management. By system optimisation, the feasibility of ultra-long single-channel and wavelength-division multiplexed data transmission at bit-rates ³ 40 Gbit s-1 in standard fibre-based systems is demonstrated. The tolerance limits of the results are defined.Thirdly, solutions of the NLS equation with gain and normal dispersion, that describes optical pulse propagation in an amplifying medium, are examined. A self-similar parabolic solution in the energy-containing core of the pulse is matched through Painlevé functions to the linear low-amplitude tails. The analysis provides a full description of the features of high-power pulses generated in an amplifying medium.

Mathematical methods in power system stability studies

In this thesis various mathematical methods of studying the transient and dynamic stabiIity of practical power systems are presented. Certain long established methods are reviewed and refinements of some proposed. New methods are presented which remove some of the difficulties encountered in applying the powerful stability theories based on the concepts of Liapunov. Chapter 1 is concerned with numerical solution of the transient stability problem. Following a review and comparison of synchronous machine models the superiority of a particular model from the point of view of combined computing time and accuracy is demonstrated. A digital computer program incorporating all the synchronous machine models discussed, and an induction machine model, is described and results of a practical multi-machine transient stability study are presented. Chapter 2 reviews certain concepts and theorems due to Liapunov. In Chapter 3 transient stability regions of single, two and multi~machine systems are investigated through the use of energy type Liapunov functions. The treatment removes several mathematical difficulties encountered in earlier applications of the method. In Chapter 4 a simple criterion for the steady state stability of a multi-machine system is developed and compared with established criteria and a state space approach. In Chapters 5, 6 and 7 dynamic stability and small signal dynamic response are studied through a state space representation of the system. In Chapter 5 the state space equations are derived for single machine systems. An example is provided in which the dynamic stability limit curves are plotted for various synchronous machine representations. In Chapter 6 the state space approach is extended to multi~machine systems. To draw conclusions concerning dynamic stability or dynamic response the system eigenvalues must be properly interpreted, and a discussion concerning correct interpretation is included. Chapter 7 presents a discussion of the optimisation of power system small sjgnal performance through the use of Liapunov functions.

Mathematical modelling and control of agitated extraction columns

Liquid-liquid extraction has long been known as a unit operation that plays an important role in industry. This process is well known for its complexity and sensitivity to operation conditions. This thesis presents an attempt to explore the dynamics and control of this process using a systematic approach and state of the art control system design techniques. The process was studied first experimentally under carefully selected. operation conditions, which resembles the ranges employed practically under stable and efficient conditions. Data were collected at steady state conditions using adequate sampling techniques for the dispersed and continuous phases as well as during the transients of the column with the aid of a computer-based online data logging system and online concentration analysis. A stagewise single stage backflow model was improved to mimic the dynamic operation of the column. The developed model accounts for the variation in hydrodynamics, mass transfer, and physical properties throughout the length of the column. End effects were treated by addition of stages at the column entrances. Two parameters were incorporated in the model namely; mass transfer weight factor to correct for the assumption of no mass transfer in the. settling zones at each stage and the backmixing coefficients to handle the axial dispersion phenomena encountered in the course of column operation. The parameters were estimated by minimizing the differences between the experimental and the model predicted concentration profiles at steady state conditions using non-linear optimisation technique. The estimated values were then correlated as functions of operating parameters and were incorporated in·the model equations. The model equations comprise a stiff differential~algebraic system. This system was solved using the GEAR ODE solver. The calculated concentration profiles were compared to those experimentally measured. A very good agreement of the two profiles was achieved within a percent relative error of ±2.S%. The developed rigorous dynamic model of the extraction column was used to derive linear time-invariant reduced-order models that relate the input variables (agitator speed, solvent feed flowrate and concentration, feed concentration and flowrate) to the output variables (raffinate concentration and extract concentration) using the asymptotic method of system identification. The reduced-order models were shown to be accurate in capturing the dynamic behaviour of the process with a maximum modelling prediction error of I %. The simplicity and accuracy of the derived reduced-order models allow for control system design and analysis of such complicated processes. The extraction column is a typical multivariable process with agitator speed and solvent feed flowrate considered as manipulative variables; raffinate concentration and extract concentration as controlled variables and the feeds concentration and feed flowrate as disturbance variables. The control system design of the extraction process was tackled as multi-loop decentralised SISO (Single Input Single Output) as well as centralised MIMO (Multi-Input Multi-Output) system using both conventional and model-based control techniques such as IMC (Internal Model Control) and MPC (Model Predictive Control). Control performance of each control scheme was. studied in terms of stability, speed of response, sensitivity to modelling errors (robustness), setpoint tracking capabilities and load rejection. For decentralised control, multiple loops were assigned to pair.each manipulated variable with each controlled variable according to the interaction analysis and other pairing criteria such as relative gain array (RGA), singular value analysis (SVD). Loops namely Rotor speed-Raffinate concentration and Solvent flowrate Extract concentration showed weak interaction. Multivariable MPC has shown more effective performance compared to other conventional techniques since it accounts for loops interaction, time delays, and input-output variables constraints.

Calculating the derivative of piecewise functions

Exercises involving the calculation of the derivative of piecewise defined functions are common in calculus, with the aim of consolidating beginners’ knowledge of applying the definition of the derivative. In such exercises, the piecewise function is commonly made up of two smooth pieces joined together at one point. A strategy which avoids using the definition of the derivative is to find the derivative function of each smooth piece and check whether these functions agree at the chosen point. Showing that this strategy works together with investigating discontinuities of the derivative is usually beyond a calculus course. However, we shall show that elementary arguments can be used to clarify the calculation and behaviour of the derivative for piecewise functions.