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This paper proposes a formalization of the neutral niveau of Rudolph Reti's approach to motivic analysis within our mathematical model based on topological spaces of motives. Reti developed a substantial approach favouring melodic relationships below the musical surface. However, his approach has been much criticized for reasons such as an evident lack of methodology. This paper suggests that, when Reti's terminology is redefined in a precise mathematical setup, his approach can fit a computer-aided motivic analysis, and a topological solution to his problematic identity concept and limitation of transformations can be proposed. Our mathematical model, based on motif, contour, gestalt, and motif similarity, involves neighbourhoods of a motif that include similar motives of different cardinalities. It yields a topological space on the set of all motives of a composition, and in which Reti's concepts of shape, imitation, variation, and transformation are naturally formalized. The 'germinal motif' corresponds to the 'most dense' motif in the space. [ABSTRACT FROM AUTHOR]

A hybrid distributed parameter model of a heating system for domestic hot water is presented in this paper. This heating system comprises a condensing boiler (burner), a counter current heat exchanger, and a so-called stratified storage tank which is the state of the art domestic hot water storage unit. The paper presents the model for the different operational modes of the plant which are described by a finite state automaton representing the discrete-event dynamics and driving the underlying continuous-time dynamics of the storage tank, the heat exchanger, and the burner. These interconnected components are modelled by a system of six coupled, quasi-linear partial differential equations (PDEs) comprising diffusion-, convection-, and source terms. In order to perform numerical simulations, the set of PDEs is spatially discretized using the method of lines. Thereby, the influence of various discretization schemes on the temporal evolution of the traveling temperature profiles in the single components is investigated. A high resolution slope limiter scheme for the stratified storage tank and a higher order up-/downwind scheme for the heat exchanger and the burner are found to be an appropriate choice for the spatial discretization of the model equations in order to adequately cover the plant dynamics. Simulation results fortify the effectiveness of the chosen discretization schemes and show the excellent performance of the suggested model representing the measurement data. [ABSTRACT FROM AUTHOR]

This paper reports how students talk about their aspirations in regard to higher education (HE) and their mathematics, what 'repertoires' they use to mediate this discourse, and how students' predominant 'repertoire style' relates to their cultural background. Our analyses draw on an interview sample (n=40) of students selected because they are 'on the cusp' of participation or non-participation in mathematically demanding programmes in further and higher education. The interviews explored the students' aspirations for their future in general and HE in particular, influences on these choices, and the place of mathematics in these. Thematic analysis revealed four interpretative repertoires commonly in use, which we call 'becoming successful', 'personal satisfaction', 'vocational', and 'idealist' repertoires. Most of the sample was found to use a single, predominant repertoire, which we call their repertoire 'style': what is more, this style is found to be strongly related to background factors independently obtained. The implications for policy and practice are discussed. [ABSTRACT FROM AUTHOR]

This paper characterizes the existence and form of the possible limit error signals in typical parameter-optimal iterative learning control. The set of limit errors has attracting and repelling components and the behaviour of the algorithm in the vicinity of these sets can be associated with the undesirable properties of apparent (but in fact temporary) convergence or permanent slow convergence properties in practice. The avoidance of these behaviours in practice is investigated using novel switching strategies. Deterministic strategies are analysed to prove the feasibility of the concept by proving that each of a number of such strategies is guaranteed to produce global convergence of errors to zero independent of the details of plant dynamics. For practical applications a random switching strategy is proposed to replace these approaches and shown, by example, to produce substantial potential improvements when compared with the non-switching case. The work described in this paper is covered by pending patent applications in the UK and elsewhere. [ABSTRACT FROM AUTHOR]

George Lakoff 's theory of categories renews a global epistemological struggle over who best describes reality. Color perception becomes a prototype solution as Lakoff enlists the aid of anthropology and biopsychology to challenge claims from contemporary psycholinguistics, philosophy, mathematics and psychology. The problem is that the color perception prototype upon which Lakoff bases his metaphor is flawed in the same way and for the same reasons that Lakoff attacks conventional approaches and paradigms. This thesis is tantamount to embracing and extending Lakoff 's approach, but with judicious substitutions. Most important, morphogenesis (the development of the human embryo) is substituted for Lakoff 's biopsychology of the brain. In addition to the radical breakdown of logic, language and mathematics encountered by Lakoff, irreconcilable differences between diverse paradigms and diverse cultures add to the numbers of difficult problems. Substituting Thom's morphogenesis mathematics for conventional mathematics resolves the problems with mathematics but distorts conventional logic, language and paradigms to the point of crashing. Supplementing Lakoff 's adaptation of anthropological prototypes with Maruyama's formulation of mindscapes is the first step toward resolving irreconcilable differences between diverse paradigms and cultures. Combining Maruyama's formulation of a morphogenic paradigm with Lakoff 's approach to categories precludes distortion by incompatible paradigms. Outsourcing logic to Eastern countries is the cost effective strategy for resolving reasoning problems. Finally there is the crucial problem of ecological validity, a problem not covered by Lakoff 's cognitive psychology. Direct observations in natural environments and mathematical models that make natural environment predictions supporting the morphogenesis paradigm resolve the problem of ecological validity. Fashion fabrics and twilight color displays falsify predictions made by the color vision prototype. Falsification of this paper will rest primarily on the paper's prediction of demographic catastrophes that are to occur as a consequence of X chromosome intelligence. [ABSTRACT FROM AUTHOR]

This paper describes the process of designing and administering a sufficiently valid and reliable assessment instrument to measure the progress in attainment of adult learners studying numeracy, and discusses some of the inherent difficulties that were involved. The fieldwork took place during 2003-2005 and involved a sample of 34 teachers and 412 learners aged post-16. The paper details how the lack of a range of suitable instruments, the heterogeneous nature of adult numeracy teaching, the remit and requirements of the research project and other practical factors, made it difficult to produce an instrument capable of measuring learners' progress accurately, and hence to produce robust and reliable conclusions, generalisable across the whole sector. Having said this, and despite all the problems faced, the researchers still believe that the test has a sufficient degree of validity for the purposes of the study. The paper ends with some recommendations that may benefit other researchers when it comes to designing future numeracy assessments. [ABSTRACT FROM AUTHOR]

Two aspects of deep mixing method, the difference relating strength gain in dry jet mixing (DJM, reagent powder introduced into the ground) and cement deep mixing (CDM, reagent slurry introduced into the ground), and prediction of unconfined compressive strength of cement stabilized marine clay, are discussed in this paper. The first part of this paper concentrates on the difference between DJM and CDM on strength gain, and suggests a guideline for DJM and CDM selection. An indicator in terms of water content ratio, which is defined as the ratio of water content to the liquid limit of the soil, is presented by statistical analysis from the laboratory and field test data as a guideline for the selection of DJM or CDM. Based on the laboratory test data, a mathematical model relating strength gain of cement stabilized marine clay to related variables is developed. A new simple index designated as total water-cement ratio, which is defined as the ratio of water weight in the soil-cement to the weight of cement in dry state, is proposed for interpretation of test data of soil-cement. The proposed method is then verified with available test data published by other different researchers. [ABSTRACT FROM AUTHOR]

This paper will consider the range of British participation in mathematics internationally during the nineteenth century through an analysis of British mathematical contributions to scientific journals outside of Britain. Viewing scientific papers contained in journals as significant indicators of research, we consider scientists who authored or read and responded in print to papers in a given area within a given group of journals to constitute a publication community. The extent of publication by British mathematicians in these journals can help characterize the role of foreign publication in nineteenth-century British mathematics. Moreover, the isolation of educational, societal, and personal circumstances which motivated British mathematicians to present their work to foreign journals highlights limited but concentrated groups of mathematicians committed to developing and strengthening international mathematical ties with Britain. [ABSTRACT FROM AUTHOR]

At first the paper investigates the asymptotic behavior of the finite-time ruin probability with constant interest rate and subexponentially tailed claim sizes, which extends the result recently established by Tang[14] for the classical risk model to the delayed renewal risk model; then, within the intersection class of the claim sizes with subexponential tails and the claim sizes with dominatedly varying tails, the paper discusses the finite-time ruin probability with claims arriving according to an arbitrary counting process. [ABSTRACT FROM AUTHOR]

The principles, difficulties and challenges in control education are discussed and compared to the similar problems in the teaching of mathematics and systems science in general. The difficulties of today's students to appreciate the classical teaching of engineering disciplines, which are based on rigorous and scientifically sound grounds, are discussed. The new demands for modern engineering education assume that the mathematical skills of the students in modelling, analysis, synthesis, simulation and validation of control problems improve significantly, which sets a major challenge to the education system. To meet that challenge new activating teaching methods are needed, which together with traditional studies bring the students closer to practical applications. In the current paper two new teaching methods are presented, which are aimed at bringing the control world near the student more easily than the traditional courses do. These example cases are control courses in the net with self-study sections, interactive repetitive questions, problems with complete solutions (both pen and paper and computer problems). Also, new 'interactive laboratory exercises' have been included, where the students prepare their preliminary problems first, and after that test the controller with a real process over the net (from their home or from a computer class). [ABSTRACT FROM AUTHOR]

The paper discusses the interaction between intuitive biases of probabilistic thinking and mathematical knowledge. It would appear that students may answer numerical problems correctly but falter on simple descriptive solutions. Students appear to relinquish formal knowledge for simpler heuristics when attempting to describe the outcome of an event. This paper explores some possible theoretical causes of the dissociation between procedural and estimative abilities. [ABSTRACT FROM AUTHOR]

This paper, the second of a two part article, expands on an idea that appeared in literature in the 1950s to show that by resticting the domain to those complex numbers that map onto real numbers, representations of functions other than the ones in the real plane are obtained. In other words, the well-known curves in the real plane only depict part of a bigger whole. This expanded representation brings new insight into visualising complex roots. The suggestion is that this new approach be introduced to students firstly through relating the path in history and secondly by imparting the visual presentation as exposed in the paper to offer a richer teaching and learning approach to the topic. Furthermore this approach provides a new way of employing technology to visualise concepts and curves that were previously not noticed. [ABSTRACT FROM AUTHOR]

In an earlier paper, the author introduced new upper bounds for free linear and nonlinear vibration systems; to compute the best upper bounds, the differential calculus of norms was applied. In the present paper, this work is continued for the corresponding excited systems. Some new techniques and ideas are involved. The results in the applications cannot be obtained by the methods used so far. [ABSTRACT FROM AUTHOR]

With Simson's 1753 paper as a starting point, the current paper reports investigations of Simson's identity (also known as Cassini's) for the Fibonacci sequence as a means to explore some fundamental ideas about recursion. Simple algebraic operations allow one to reduce the standard linear Fibonacci recursion to the nonlinear Simon's recursion that is equivalent to Simson's identity and then further to a nonlinear recursion dependent only on a single preceding term. This leads to a striking nonrecursive characterization of Fibonacci numbers that is much less well-known than it should be. It is then discovered that Simson's recursion itself implies a family of linear recursions and characterizes a class of generalized Fibonacci sequences. [ABSTRACT FROM AUTHOR]

Nonlinear systems of time dependent partial differential equations of real life phenomena tend to be very complicated to solve numerically. One common approach to solve such problems is by applying operator splitting which introduces subproblems that are easier to handle. The solutions of the subsystems are usually glued together by either the Godunov method (first-order) or the Strang method (second-order). However, the accuracy of such an approach may be very hard to analyse because of the complexity of the equations involved. The purpose of the present paper is to introduce another line of reasoning concerning the accuracy of operator splitting. Let us assume that a fully coupled and implicit discretization of the complete system has been developed. Under appropriate conditions on the continuous problem, such discretizations provide reasonable and convergent approximations. As in the continuous case, operator splitting can be utilized to obtain tractable algebraic subsystems. The problem we address in the present paper is to obtain a bound on the difference between the fully coupled implicit discrete solutions, uh, and the solutions, uh, s, obtained by applying operator splitting to these discrete equations. Suppose we know that uh converges to the analytical solution u as the grid is properly refined and, applying the results from this paper, that uh, s converges toward uh under grid refinement. Then, by the triangle inequality, also the splitting approximation uh, s converges toward the analytical solution u and convergence is thus obtained for an approximation that, from a practical point of view, is easier to compute. [ABSTRACT FROM AUTHOR]

In the literature, rank-constrained Hermitian nonnegative-definite solutions to the matrix equation AXAH=B have been investigated, under the conditions that B is Hermitian and nonnegative-definite, and the matrix equation is consistent. In this paper, we discuss rank-constrained Hermitian nonnegative-definite least squares solutions to this matrix equation, in which the above conditions may not hold. We derive the rank range and expression of these least squares solutions. Therefore, the results obtained in this paper generalize those in the literature. [ABSTRACT FROM AUTHOR]

Vagueness is often present in spatial phenomena. Representing and analysing vague spatial phenomena requires vague objects and operators, whereas current GIS and spatial databases can only handle crisp objects. This paper provides mathematical definitions for vague object types and operators. The object types that we propose are a set of simple types, a set of general types, and vague partitions. The simple types represent identifiable objects of a simple structure, i.e. not divisible into components. They are vague points, vague lines, and vague regions. The general types represent classes of simple type objects. They are vague multipoint, vague multiline, and vague multiregion. General types assure closure under set operators. Simple and general types are defined as fuzzy sets in 2 satisfying specific properties that are expressed in terms of topological notions. These properties assure that set membership values change mostly gradually, allowing stepwise jumps. The type vague partition is a collection of vague multiregions that might intersect each other only at their transition boundaries. It allows for a soft classification of space. All types allow for both a finite and an infinite number of transition levels. They include crisp objects as special cases. We consider a standard set of operators on crisp objects and define them for vague objects. We provide definitions for operators returning spatial types. They are regularized fuzzy set operators: union, intersection, and difference; two operators from topology: boundary and frontier; and two operators on vague partitions: overlay and fusion. Other spatial operators, topological predicates and metric operators, are introduced giving their intuition and example definitions. All these operators include crisp operators as special cases. Types and operators provided in this paper form a model for a spatial data system that can handle vague information. The paper is illustrated with an application of vague objects in coastal erosion. [ABSTRACT FROM AUTHOR]

The article reports on the flaw in Leonhard Euler's theory of polyhedra. It examines the proof laid down by Euler for his polyhedral formula and discusses the questions raised about his proof. It cites the papers "Elementa doctrinae solidorum" and the "Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita" which Euler wrote to discuss his theory. It includes discussions on the proof of the angle sum formula and presents analysis an commentary on Euler's paper.

The convergence of the inexact Uzawa method for stabilized saddle point problems was analysed in a recent paper by Cao, Evans and Qin. We show that this method converges under conditions weaker than those stated in their paper. [ABSTRACT FROM AUTHOR]

Today, with the advances of computer storage and technology, there are huge datasets available, offering an opportunity to extract valuable information. Probabilistic approaches are specially suited to learn from data by representing knowledge as density functions. In this paper, we choose Gaussian mixture models (GMMs) to represent densities, as they possess great flexibility to adequate to a wide class of problems. The classical estimation approach for GMMs corresponds to the iterative algorithm of expectation maximization (EM). This approach, however, does not scale properly to meet the high demanding processing requirements of large databases. In this paper we introduce an EM-based algorithm, that solves the scalability problem. Our approach is based on the concept of data condensation which, in addition to substantially diminishing the computational load, provides sound starting values that allow the algorithm to reach convergence faster. We also focus on the model selection problem. We test our algorithm using synthetic and real databases, and find several advantages, when compared to other standard existing procedures. [ABSTRACT FROM AUTHOR]

The objective of this paper is to find an adaptive control strategy which would enable us to estimate the parameters of the ARMAX model as accurately as possible, along with consuming less controlling energy, while keeping the output of the system below a specified level of variability. Using the self-tuning tracker, this paper establishes global convergence of a stochastic adaptive control algorithm for discrete linear system, and the adaptive controller may converge to the one-step-ahead optimal controller at the same time. [ABSTRACT FROM AUTHOR]

The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s, Cayley, Sylvester, and the Irish mathematician, George Salmon, formulated the basic concepts, developed the key techniques, and set the research agenda for the field. As Cayley and Sylvester continued to extend the theory off and on through the 1880s, first Salmon in 1859 and later Edwin Bailey Elliott in 1895 codified it in high-level textbooks. This paper sketches the development of nineteenth-century invariant theory in British hands against a backdrop of personal, nationalistic, and internationalistic mathematical goals. [ABSTRACT FROM AUTHOR]

In a deliberately provocative first part to this paper, I argue that nineteenth-century British mathematicians had an unduly high opinion of themselves and a striking lack of appreciation for contemporary continental developments. I argue that this failure was rooted in the institutions that supported mathematics, and was only remedied towards the end of the century. In the more sober second half of the paper I ask if historians of mathematics have subscribed to this overestimate, and explore some related questions, among them what are historians doing when they write history: telling it as it was, or righting or defending the record? Historians of mathematics also need to consider British priorities for research in the nineteenth century, and a comparison with other minor players (such as Japan, Portugal, or Poland) might be illuminating. [ABSTRACT FROM AUTHOR]

The goal of this paper is to further the investigation begun in Chalmers and Prophet, Numer. Funct. Anal. Optimiz. 1997; 18:507–520. With the benefit of nearly 10 years of work, we begin by indicating how several proofs from Chalmers and Prophet, Numer. Funct. Anal. Optimiz. 1997; 18:507–520, can be substantially improved. We show that the problem of preserving k-convexity onto Πn is one part of a larger shape-preserving problem (multiconvex preservation) relative to Πn, and we completely solve this expanded problem. And finally, we demonstrate that multiconvex preserving projections constructed in this paper are in fact of minimal operator norm in a large class of Banach spaces. [ABSTRACT FROM AUTHOR]

The paper examines the roots of Husserlian phenomenology in Weierstrass's approach to analysis. After elaborating on Weierstrass's programme of arithmetization of analysis, the paper examines Husserl's Philosophy of Arithmetic as an attempt to provide foundations to analysis. The Philosophy of Arithmetic consists of two parts; the first discusses authentic arithmetic and the second symbolic arithmetic. Husserl's novelty is to use Brentanian descriptive analysis to clarify the fundamental concepts of arithmetic in the first part. In the second part, he founds the symbolic extension of the authentically given arithmetic with stepwise symbolic operations. In the process of doing so, Husserl comes close to defining the modern concept of computability. The paper concludes with a brief comparison between Husserl and Frege. While Frege chose to subject arithmetic to logical analysis, Husserl wants to clarify arithmetic as it is given to us. Both engage in a kind of analysis, but while Frege analyses within Begriffsschrift, Husserl analyses our experiences. The difference in their methods of analysis is what ultimately grows into two separate schools in philosophy in the 20th century. [ABSTRACT FROM AUTHOR]

Towards the end of his 1930 paper on boundary numbers and domains of sets Zermelo briefly discusses the questions of consistency and of the existence of an unbounded sequence of strongly inaccessible cardinals, deferring a detailed discussion to a later paper which never appeared. In a report to the Emergency Community of German Science from December 1930 about investigations in progress he mentions that some of the intended extensions of these topics had been worked out and were nearly ready for publication. Using manuscripts from his Nachlass, we exhibit essential features of these extensions. [ABSTRACT FROM AUTHOR]

In the classical risk theory, it is often used that different type dimensions can be aggregated into a single-dimensional statistic, as well as the assumption of properties on log-concavity of this aggregation. The extreme-order statistics, minimum and maximum, might be used as aggregate statistics. In this paper, we discuss the log-concavity of the survival function of the minimum and maximum from Gumbel bivariate exponential models, through the log-concavity of generalized mixtures of four or fewer exponential distributions, extending the papers of Baggs and Nagaraja [Baggs, G.E. and Nagaraja, H.N., 1996, Reliability properties of order statistics from bivariate exponential distributions. Communications in Statistics—Stochastic Models , 12 , 611–631] and Franco and Vivo [Franco, M. and Vivo, J.M., 2002, Reliability properties of series and parallel systems from bivariate exponential models. Communications in Statistics—Theory and Methods , 31 , 2349–2360] devote to the log-concavity for generalized mixtures of three or fewer exponential distributions. [ABSTRACT FROM AUTHOR]

The article discusses the importance of the paper "Musical Scales and the Generalized Circle of Fifths," by J. Clough and G. Myerson in the mathematical music theory literature. The paper deals with the way the white keys in the piano, called the diatonic set, are embedded in the chromatic scale and studies the different-sounding progressions with the same structure. The use of chromatic length in defining equivalence relations on the set of note lines is also offered. It corrects the theorem and the corollary stated in the paper.

The goal of this paper is to show some of the most important features of flying insects from a control point of view, describe in details the kinematics of insects during flapping flight, establish the corresponding kinematics equations of flying insects, analyze the force and moment acting on the flying insects, and establish the corresponding aerodynamic equations. Through the kinematics equations and aerodynamic equations of flying insects, the trajectory equations and attitude equations have also been established. A detailed mathematical model of flying insects is presented in this paper, which including 3 d.o.f. of wings kinematics, i.e., flapping, lagging and feathering, and 6 d.o.f. of insects body, i.e., yaw, roll, up–down flight, left–right flight and fore–back flight. All these motions of flying insects are interrelated by the kinematics equations, attitude equations, aerodynamics model and dynamics equations of the insects' centroid. In the findal part of this paper, the mathematical simulation model of flapping-wing insects is set up and the corresponding simulation curve created. [ABSTRACT FROM AUTHOR]

This paper is concerned with identifying and assessing the impacts of various technologically enriched approaches to mathematics learning. Its purpose is to address the usefulness of emerging knowledge to enhance practice, and to contribute to theorizing about technology-based learning. Hence the main drivers are intentions to identify and elaborate on obstacles, errors, and unresolved problems on the one hand, and positives and insights on the other. The paper tells a cautionary tale about expectations of technology-enhanced learning, while simultaneously uncovering a rich base from which to theorize and test new appreciations of what is involved when students, technology, and mathematics connect in learning settings. A summary assessment is that this area still very much represents work in progress, but there is now a heightened realization, at least among those not transfixed by technological blindness, that the search for ultimate answers will require much greater exploration of machine–mathematics–learner relationships. [ABSTRACT FROM AUTHOR]

Some research studies, many of which used quantitative methods, have suggested that graphics calculators can be used to effectively enhance the learning of mathematics. More recently research studies have started to explore students’ styles of working as they solve problems with technology. This paper describes the use of a software application that records the keystrokes made by students as they use calculators, in order to enable researchers to gain better insights into students’ working styles. The recordings obtained from this software can be replayed to observe how students have actually used their calculator in tackling a problem. The paper describes three pilot studies from quite different contexts, in which the software reveals how the calculators have been used by the students. In all of these studies the software provides insights into the working that would have been very difficult to obtain without the record of the keystrokes provided by the software. [ABSTRACT FROM AUTHOR]

School transfer acts like a prism, diffracting children’s social and learning trajectories. In this paper I explore this sociological effect through two case studies of children moving from the primary to the secondary school. The impacts of the school, peer and family fields are explored using Bourdieu’s theory of practice, and shifts in the relative power of these fields are traced across the primary–secondary interface. My analysis explains why, decades after Nisbet and Entwistle first linked poor school transition with social class, it is still the case that the economically and culturally more well-endowed make the most of moving school. So it is that through processes of harmonization and dissonance, of compliance and resistance, children navigate, and are steered, from their embodied social and education pasts towards future constrained possibilities. These processes are partially contextualized in the learning of school mathematics. [ABSTRACT FROM AUTHOR]

The article focuses on the a theorem about an infinite series which was communicated by Christian Goldbach to Leonhard Euler and is one of the theorems mentioned in Euler's paper "Variae observationes circa series infinitas." Goldbach and Euler's proof is a common example of what some historians consider a misuse of divergent series because it begins by assigning a value to the harmonic series Σ1/n and is then manipulated by subtractions and replacement of other series until the desired result is attained. The proof of the theorem is hereby reviewed in order to vindicate Goldbach and Euler's work. It is shown how the deficiency in the proof can be amended. However, other results obtained in said Euler's paper are not as easily amended.

The article explores the theory of dynamical systems in mathematics, examining the papers of A. N. Sharkovsky and T.-Y. Li and J. A. Yorke on interval dynamics, among others. It should be noted that neither Sharkovsky's theorem nor Li-Yorke's definition of chaos have a significant role in the core of modern one-dimensional dynamics. Meanwhile, J. Guckenheimer's paper established the foundations for the almost everywhere approach to dynamics, which states that if a map is sufficiently smooth, there may be a fair chance of offering an effective description of the asymptotic behavior of almost every point of the interval.

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [ Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L 2 and H 1 norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established. [ABSTRACT FROM AUTHOR]

In this paper, robust regulation of stable infinite-dimensional plants with transfer functions in the H 8 -algebra is considered. The reference and disturbance signals are allowed to have an infinite number of poles on the imaginary axis, which makes it possible to consider e.g. arbitrary periodic signals. A controller depending on a positive scalar e and design parameters K k is proposed. The design parameters are matrices over H 8 , and by the main result of the paper, if the design parameters satisfy a certain condition, the proposed controller is a robust regulator for sufficiently small values of e. The only knowledge needed to verify this condition is the so-called -stability of the plant and values of the plant transfer function at the poles of the reference and disturbance signals. The analysis in this paper is in the frequency domain and it is based on the fractional representation approach. [ABSTRACT FROM AUTHOR]

Abstract. Assignments were altered to determine if interspersing additional briefer and easier mathematics problems enhanced students' accuracy on longer, more difficult target problems across two types of mathematics assignments. Students (N = 52) from three fifth-grade classes completed six math assignments incorporating two task demands and three interspersal ratios. The interspersal ratios were no interspersal, one interspersed problem per three target problems (1:3), and one interspersal problem per one target problem (1:1). For the written (low-attention) assignments, students completed problems via paper and pencil. For the cognitive (high-attention) assignments, the experimenter read problems aloud and students had to compute mathematics problems in their heads, using paper and pencil only to record answers. Results showed that for the cognitive assignments, target problem accuracy was significantly higher for the 1:3 ratio relative to the other two ratios. On the written assignments, target problem accuracy levels were significantly higher on the 1:1 ratio assignments relative to the no interspersal assignments. Discussion focuses on applied and theoretical implications of the current results. [ABSTRACT FROM AUTHOR]

The paper reports some new solutions for heat transfer through extended surfaces or fins using the symbolic algebra package Maple 8, which is widely available. The four specific problems chosen for the present study are: (a) a rectangular convection fin with the heat transfer coefficient varying either linearly or exponentially with the distance from the base, (b) a truncated conical spine with convection at both ends, (c) a heat-generating annular fin with a constant base heat flux and an adiabatic tip, and (d) a convection fin array made of a rectangular fin and two triangular fins. Each problem is formulated in a manner that makes its solution novel and distinct from what is available in the literature. Solutions are provided in symbolic forms. Using the numerical and graphical capabilities of Maple, the results are presented in the form of numerical data as well as graphical displays. The paper demonstrates that Maple provides an effective and convenient tool for the analysis of extended surface heat transfer problems that otherwise demand tedious algebraic manipulations. [ABSTRACT FROM AUTHOR]

This paper reports on a practitioner study of a first-year undergraduate service course that aligns a web-based, student-lecturer communication system with the mathematical curriculum. The report presents and analyses data from students and from the lecturer and outlines the nature of the technical interface. The paper indicates how communication of the students' affective learning needs had a positive influence on the professional development of the lecturer himself. Furthermore, it is claimed that the success of mathematics/statistics teaching that integrates emotional responses from learners is intimately related to the knowledge, skills, beliefs and values of the lecturer. [ABSTRACT FROM AUTHOR]

For many years, educators have been discussing benefits of educational practices such as the use of real-world examples, visualisation, interactivity, constructivism, self-paced learning and self-paced testing. Macromedia Flash MX has been used to develop online modules for the course Differential Equations offered at the University of South Australia that utilize these practices. This paper demonstrates the following innovative features of the modules that provide an effective online learning environment: • graphs which allow the user to interactively vary parameters are accompanied by ‘walkthroughs’ that break mathematical content into manageable chunks; • a clearly labelled navigational structure that implies a linear order for each module while allowing users to jump forwards or backwards; • animations such as an oscillating spring which depict real-world examples. Online multiple-choice quizzes with randomized numeric values is a unique form of assessment that will be discussed. The integration of a wide range of effective learning principles into online modules will be the focus of this paper. [ABSTRACT FROM AUTHOR]

Papers presented at earlier Delta conferences reported on the use of collaborative tutorials and peer-tutors in undergraduate mathematics courses at The University of Auckland. This paper reports on significant developments within the Department's programmes, including the extension of the collaborative small-class tutorials to cover all first-year undergraduate courses in the department. It provides evidence of the effectiveness of the tutorial programme, as gauged from a survey of students' perceptions. Many of the students enrolled in the Tutoring in Mathematics course that initially trains and provides the peer-group tutors continue on as tutors in other mathematics courses within the Department. Some subsequently enrol in the University's Graduate Diploma in Teaching (Secondary). The effects of the tutoring course on tutors continuing to tutor at higher levels, and their performances as novice teachers in the Diploma course are examined. [ABSTRACT FROM AUTHOR]

Comparing the large number of research papers on the ordered weighted averaging (OWA) operator, the researches on relative quantifier are relatively rare so far. In the present paper, based on the quantifier guided aggregation method with OWA operator which was proposed by Yager [“Quantifier guided aggregation using OWA operators”, Int. J. Intell. Syst. , 11, pp. 49–73, 1996], a generating function representation method for regular increasing monotone (RIM) quantifiers is proposed. We extend the the properties of OWA operator to the RIM quantifier which is represented with a monotone function instead of the OWA weighting vector. A class of parameterized equidifferent RIM quantifier which has minimum variance generating function is proposed and its properties are also analyzed. The equidifferent RIM quantifier is consistent with its orness level for any aggregated elements, which can be used to represent the decision maker's preference. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of part one, in which two macroscopic limits of the Boltzmann–Enskog kinetic equation with three small parameters—the Knudsen number, Mach number, and diameter of particles—were considered at the formal level, and conditional theorems were stated. In this paper, the proof of a rigorous result corresponding to one of the two cases considered in part one is given. Under some suitable assumptions on the relations between the small parameters, it is proved that the Boltzmann–Enskog equation results in the Navier–Stokes equation for incompressible fluids together with the Boussinesq relation and the temperature fluctuations equation. The time and velocity averaging is exploited.Dedicated to Professor Janusz Mika on his 70th birthday. [ABSTRACT FROM AUTHOR]

This paper studies the behavior under iteration of the maps Tab(x, y) = (Fab(x) - y, x) of the plane R², in which Fab(x) = ax if x ≥ 0 and bx if x « 0. The orbits under iteration correspond to solutions of the nonlinear difference equation xn+2 = 1/2(a - b)[xn+1] + 1/2(a + b)xn+1 - xn. This family of piecewise-linear maps has the parameter space (a, b] ϵ R². These maps are area-preserving homeomorphisms of R² that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map Sab: S¹ ↵ S¹ of the circle, which has a well-defined rotation number This paper characterizes the possible dynamics under iteration of Tab when the auxiliary map Sab has rational rotation number, It characterizes cases where the map Tab is a periodic map. [ABSTRACT FROM AUTHOR]

The paper starts by pointing to the lack of any theoretical analysis for the concepts of ‘the commons’ and ‘information’ in the general reference to the ‘enclosure of the information commons’. The author suggests, instead, the concept of ‘enclosure’ as a key to understanding of ‘the information commons’, although it is underdeveloped and often equated with privatization or commodification. This paper attempts to establish a systematic theory of enclosure that enables examination of possible contradictions that may lie deep within the ‘information commons’. The information commons is an idea whose time has come. In part this is a result of pressures that face the commons. Issues related to intellectual property law in particular are leading to what many are now calling the enclosure of the information commons—a process that separates people from ideas. This process is analogous to the fencing off of the English commons, an act that separated people from the material resources they needed for their survival. (Info-commons.org, 2002)Within a repressive society, even progressive movements threaten to turn into their [declared] opposite to the degree to which they accept the rules of the game. (Herbert Marcuse, 1965)Unless we demonstrate greater caution with analytical models that are products, as they all are, of particular histories, we are in danger not only of solving but also of producing tragedies and dilemmas in the commons and elsewhere. (Pauline E. Peters, 1987) [ABSTRACT FROM AUTHOR]

Consider the regression model . In a variety of situations, an estimate of VAR ( Y | X )?=?? ( X ) is needed. The paper compares the small-sample accuracy of five estimators of ? ( X ). The results suggest that the optimal estimator is a somewhat complex function of the underlying distributions. In terms of mean squared error, one of the estimators, which is based in part on a non-robust version of Cleveland's smoother, performed about as well as a bagged version of the so-called running interval smoother, but the running interval smoother was found to be preferable in terms of bias. A modification of Cleveland's smoother, stemming from Ruppert et al. (1997), achieves its intended goal of reducing bias when the error term is homoscedastic, but under heteroscedasticity, bias can be high, and in terms of mean squared error it does not compete well with the kernel method considered in the paper. When ? has a heavy-tailed distribution, a robust version of Cleveland's smoother performed particularly well except in some situations where X has a heavy-tailed distribution as well. A negative feature of using Cleveland's robust smoother is relatively high bias, and when there is heteroscedasticity and X has a heavy-tailed distribution, a kernel-type method and the running interval smoother give superior results in terms of both mean squared error and bias. [ABSTRACT FROM AUTHOR]

Consider f ? to be a probability density function with parameter ?. A set of k populations can now be defined such that the i th population ? i is the set of density functions . This paper proposes a test, based on the f- dissimilarity , of the hypothesis that a new individual from a population ? 0 with a density function , belongs to the i th population. The probabilities of misclassification of the minimum f-dissimilarity classification rule are also obtained. In this paper, it is assumed that the parameters and may be ? 0 are unknown and must be estimated from a set of training samples. Explicit expressions for the hypothesis test and the probabilities of misclassification are derived for the case where the populations ? i consist of homoscedastic normal, as well as for gamma distributions. [ABSTRACT FROM AUTHOR]

All groups considered in this paper are finite soluble groups. Let t p ( G ) = max H??G { j|p j ||| H G : H |} and t ( G ) = max p ?p( G ) t p ( G ). In this paper, we study the influence of t ( G ) on the structure of a finite soluble group G . In particular, the bounds of the nilpotent length (derived length, p -length) of a soluble group G with t ( G ) = 2 are found. [ABSTRACT FROM AUTHOR]

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are only a few results on the numerical solutions for SDEwMSs. By this manuscript we continue our series of papers on numerical solutions of SDEwMSs. The main aim of this paper is to investigate the stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching (EMSDEwMSs) and discuss their convergence. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p -dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat. , 5 , 151-157], the matrix- T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model. [ABSTRACT FROM AUTHOR]