Is proof dead in the age of computers?

Some knowledge of what it means to construct a proof is an extremely important part of mathematics. All mathematics teachers and students should have some exposure to the ideas of proof and proving. This paper deals with the issue of creating proofs in mathematics problems.

High rate filtration using buoyant medium : experiments and mathematical models

In this study, the high rate filtration system using buoyant medium of polypropylene or polystyrene was introduced with the study on its performance evaluation via: (i) specific surface coverage, (ii) ultimate specific deposit, (iii) blocking effect and (iv) particle detachment. The filter system was arranged in the downflow mode with an in-line flocculation. Experimental results obtained were analyzed using filtration models. This study showed that: (i) the specific surface coverage increased with the increase in the velocity; (ii) the ultimate specific deposit changed slightly at different filtration velocities and different flocculants; (iii) the blocking effect was significant in the transient stage removal of flocs; and (iv) the detachment model parameter was found to decrease with the increase in the adhesive force, Fad that acts on the flocs (above-micron size).

Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation : public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study (LPS) documented the voicing of mathematical ideas in public discussion and in teacher-student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher-student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation are central to the development of any theory of mathematics instruction.

Classification of healthcare data using genetic fuzzy logic system and wavelets

Healthcare plays an important role in promoting the general health and well-being of people around the world. The difficulty in healthcare data classification arises from the uncertainty and the high-dimensional nature of the medical data collected. This paper proposes an integration of fuzzy standard additive model (SAM) with genetic algorithm (GA), called GSAM, to deal with uncertainty and computational challenges. GSAM learning process comprises three continual steps: rule initialization by unsupervised learning using the adaptive vector quantization clustering, evolutionary rule optimization by GA and parameter tuning by the gradient descent supervised learning. Wavelet transformation is employed to extract discriminative features for high-dimensional datasets. GSAM becomes highly capable when deployed with small number of wavelet features as its computational burden is remarkably reduced. The proposed method is evaluated using two frequently-used medical datasets: the Wisconsin breast cancer and Cleveland heart disease from the UCI Repository for machine learning. Experiments are organized with a five-fold cross validation and performance of classification techniques are measured by a number of important metrics: accuracy, F-measure, mutual information and area under the receiver operating characteristic curve. Results demonstrate the superiority of the GSAM compared to other machine learning methods including probabilistic neural network, support vector machine, fuzzy ARTMAP, and adaptive neuro-fuzzy inference system. The proposed approach is thus helpful as a decision support system for medical practitioners in the healthcare practice.

Medical data classification using interval type-2 fuzzy logic system and wavelets

This paper introduces an automated medical data classification method using wavelet transformation (WT) and interval type-2 fuzzy logic system (IT2FLS). Wavelet coefficients, which serve as inputs to the IT2FLS, are a compact form of original data but they exhibits highly discriminative features. The integration between WT and IT2FLS aims to cope with both high-dimensional data challenge and uncertainty. IT2FLS utilizes a hybrid learning process comprising unsupervised structure learning by the fuzzy c-means (FCM) clustering and supervised parameter tuning by genetic algorithm. This learning process is computationally expensive, especially when employed with high-dimensional data. The application of WT therefore reduces computational burden and enhances performance of IT2FLS. Experiments are implemented with two frequently used medical datasets from the UCI Repository for machine learning: the Wisconsin breast cancer and Cleveland heart disease. A number of important metrics are computed to measure the performance of the classification. They consist of accuracy, sensitivity, specificity and area under the receiver operating characteristic curve. Results demonstrate a significant dominance of the wavelet-IT2FLS approach compared to other machine learning methods including probabilistic neural network, support vector machine, fuzzy ARTMAP, and adaptive neuro-fuzzy inference system. The proposed approach is thus useful as a decision support system for clinicians and practitioners in the medical practice. copy; 2015 Elsevier B.V. All rights reserved.

Fuzzy logic controller and cascade inverter for direct torque control of IM

In this paper, a five-level cascaded H-bridge multilevel inverters topology is applied on induction motor control known as direct torque control (DTC) strategy. More inverter states can be generated by a five-level inverter which improves voltage selection capability. This paper also introduces two different control methods to select the appropriate output voltage vector for reducing the torque and flux error to zero. The first is based on the conventional DTC scheme using a pair of hysteresis comparators and look up table to select the output voltage vector for controlling the torque and flux. The second is based on a new fuzzy logic controller using Sugeno as the inference method to select the output voltage vector by replacing the hysteresis comparators and lookup table in the conventional DTC, to which the results show more reduction in torque ripple and feasibility of smooth stator current. By using Matlab/Simulink, it is verified that using five-level inverter in DTC drive can reduce the torque ripple in comparison with conventional DTC, and further torque ripple reduction is obtained by applying fuzzy logic controller. The simulation results have also verified that using a fuzzy controller instead of a hysteresis controller has resulted in reduction in the flux ripples significantly as well as reduces the total harmonic distortion of the stator current to below 4 %.

Boron carbide reinforced aluminium matrix composite : physical, mechanical characterization and mathematical modelling

This paper investigates the manufacturing of aluminium-boron carbide composites using the stir casting method. Mechanical and physical properties tests to obtain hardness, ultimate tensile strength (UTS) and density are performed after solidification of specimens. The results show that hardness and tensile strength of aluminium based composite are higher than monolithic metal. Increasing the volume fraction of B4C, enhances the tensile strength and hardness of the composite; however over-loading of B4C caused particle agglomeration, rejection from molten metal and migration to slag. This phenomenon decreases the tensile strength and hardness of the aluminium based composite samples cast at 800 °C. For Al-15 vol% B4C samples, the ultimate tensile strength and Vickers hardness of the samples that were cast at 1000 °C, are the highest among all composites. To predict the mechanical properties of aluminium matrix composites, two key prediction modelling methods including Neural Network learned by Levenberg-Marquardt Algorithm (NN-LMA) and Thin Plate Spline (TPS) models are constructed based on experimental data. Although the results revealed that both mathematical models of mechanical properties of Al-B4C are reliable with a high level of accuracy, the TPS models predict the hardness and tensile strength values with less error compared to NN-LMA models.

Mathematics as a second language : looking for a bridge between mathematical language and the language of thought

Our thoughts are in one language, and mathematical results are expressed in a language foreign to the way we think. Mathematics is a unique foreign language with all the components of a language; it has its own grammar, vocabulary, conventions, synonyms, sentence structure, and paragraph structure. Students need to learn these components to partake in a thorough discussion of how to read, write, speak and think mathematics. Beginning with the students natural language and expanding that language to include symbolism and logic is the key. Providing lessons in concrete, pictorial, written and verbal terms allows the instructor to create a translation bridge between the grammar of the mother language and the grammar of mathematics. This papers presents methods to create the translation bridge for students so that they become articulate members of the mathematics community. The students "mother" language, expanded to include the symbols of mathematics and logic, is the the key to both the learning of mathematics and its effective application to problem situations. The use of appropriate language is the key to making mathematics understandable.

Catholic mothers and daughters: resilient conversations

In this paper, conversations between a group of white, middle class, adult Australian mothers and daughters are examined to disclose the symbolic dimensions that mark girlhood, womanhood, motherhood and the mother– daughter dyad within Catholic discourse. In their conversations,women unravel and produce understandings of themselves as women, and as mothers and daughters that operate at both a symbolic and real material level. Thewomen in their intergenerational dialogue depict a visceral account of Catholic desire, guilt, pleasure, piety and anger. They offer insights into the curious patriarchal religious rituals; logic and superstition that shaped their Catholic upbringings and that still permeate their adult lives. For it becomes evident that they are still affected by, and living out, the intensity of this religious force. By constructing traces of the Catholic mother–daughter nexus which relates back to women’s conversations, the intention is not to construct a closed off space but, rather, to construct a place in which women actively talk, listen and read, and by so doing come to a better understanding of their own social gendered embodied selves.

Appropriate choice of aggregation operators in fuzzy decision support systems

Fuzzy logic provides a mathematical formalism for a unified treatment of vagueness and imprecision that are ever present in decision support and expert systems in many areas. The choice of aggregation operators is crucial to the behavior of the system that is intended to mimic human decision making. This paper discusses how aggregation operators can be selected and adjusted to fit empirical data—a series of test cases. Both parametric and nonparametric regression are considered and compared. A practical application of the proposed methods to electronic implementation of clinical guidelines is presented

Understanding Concepts of Efficiency and Effectiveness in Architectural Facilities Space Planning and Design

Although contemporary literature treats efficiency and effectiveness extensively, virtually no objective understanding of both is found in architectural facilities space planning and design. It is imperative that both concepts are definitively explained as they apply to these disciplines. Hence, the contemporary views of both concepts are reviewed and attempts made to interpret them within more amenable contexts. The objectives are (1) to objectively elucidate the factual differences between efficiency and effectiveness; and (2) to contribute to their better understanding in architectural facilities space planning and design. Diagrammatic and mathematical means are engaged to demonstrate efficiency and effectiveness in these fields.

Internationalisation of Singapore television : Singaporean regional and global perspectives and contexts

In this study l investigate the Singaporean characteristics of broadcast media internationalisation. I ask the question "e; Does Internationalisation lead to homogenisation and commercialisation of the television culture in Singapore or does it give way to more diversity, thus stimulating cultural differentiation?"e; . I articulate the constraints and/or tensions of supranational regulation, foreign policy, regional and intraregional alliances upon communication and the cultural and social effects as they impact on and respond to production, programming, scheduling and output in Singapore. I explain how Singaporean Television media culture takes part in the processes of globalisation, and how it challenges existing cultures and creates new and alternative symbolic and cultural communities, within the context of regional communication. In this thesis 1 conclude that whilst Singapore definitely does not have equity in information, wealth or resource flows it is attempting to liberalise. To do so, the government recognises that serious inadequacies and imbalances must be addressed and that the path to greater political and economic growth is through an actively informed public. Despite regulatory restrictions on data flow and technical and service ownership, Singapore is encouraging regional alliances, depoliticising cultural differences and concentrating on economic imperatives to build mutual knowledge and understanding, multilateral agreements, collective ownership, mutual exchange and cooperative dissemination.