Hopf algebras of set systems

Hopf algebras play a major rôle in such diverse mathematical areas as algebraic topology, formal group theory, and theoretical physics, and they are achieving prominence in combinatorics through the influence of G.-C. Rota and his school. Our primary purpose in this article is to build on work of Schmitt [18,19], and establish combinatorial models for several of the Hopf algebras associated with umbral calculus and formal group laws. In so doing, we incorporate and extend certain invariants of simple graphs such as the umbral chromatic polynomial, and Stanley's [21] recently introduced symmetric function. Our fundamental combinatorial components are finite set systems, together with a versatile generalization in which they are equipped with a group of automorphisms. Interactions with the Roman-Rota umbral calculus over graded rings of scalars which may contain torsion are a significant feature of our presentation.     

Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts

In this paper we provide a first overview of the landscape with respect to calculus teaching in European classrooms, an area where research is very limited. In particular through a small expert-based survey and a literature review, we trace the development of calculus teaching at schools in a number of European countries and identify commonalities and differences. In the current curriculum developments, we notice a reduction in the content of calculus and a more informal approach. The use of digital tools has started to be integrated in calculus teaching in most countries. However in some nations, teaching of calculus in the classroom is rather traditional, focusing on procedural aspects of knowledge. Moreover, in cases where more informal and conceptual teaching approaches are used in the classroom, often contradictions seem to exist with other contextual matters such as examination requirements. Finally, we discuss the future of calculus teaching in Europe.     

论实数的基本性质与求解存在性问题

大学微积分的教学内容是在实数集合上展开的．存在性问题贯穿微积分教学的始终．从深入讨论实数的基本性质开始，引入了求解存在性问题的一般思路．希望从微积分教学的一开始，就让学生深刻体会到数学思想方法的无穷魅力．     

How Students Use Physics to Reason About Calculus Tasks

The present research study investigates how undergraduate students in an integrated calculus and physics class use physics to help them solve calculus problems. Using Zandieh's (2000) framework for analyzing student understanding of derivative as a starting point, this study adds detail to her “paradigmatic physical” context and begins to address the need for a theoretical basis for investigating learning and teaching in integrated mathematics and science classrooms. A case study design was used to investigate the different ways students use physics ideas as they worked through calculus tasks. Data were gathered through four individual interviews with each of 8 ICP students, classroom participant‐observation, and triangulation of the data through student homework and exams. The main result of this study is the Physics Use Classification Scheme, a tool consisting of four categories used to characterize students' uses of physics on tasks involving average rate of change, derivative, and integral concepts. Two of the categories from the Physics Use Classification Scheme are elucidated with contrasting student cases in this paper.     

A novel mechanism for provisioning of high-level quality of service information in grid environments

Grids, like other heterogeneous systems, have reached the level of maturity where they are addressing the Quality of Service (QoS) provision aspect as a fundamental part of their architecture. However, due to the multi-layer nature of these environments, the QoS characteristics of the infrastructure are revealed to the service-consumer through a complex procedure, in which each layer maps its own QoS parameters to another. Most of these approaches are starting the delegation of QoS information from the service provider’s side, resulting in delivering low-level terms to the service consumer. In this paper, we define the architecture of a mechanism that is able to provide application-oriented and user-specific QoS information in the internal layers of the Grid middleware. This way, it enables the service consumer and especially the end-user to express his business perspectives through the service level agreement life cycle. We also demonstrate the operation of this mechanism and evaluate its performance using a specific Grid scenario.     

Lambek calculus with a unit and one division

In this paper we present a substitution that reduces the derivability in the Lambek calculus with a unit and one division to the derivability in the Lambek calculus with one division permitting empty antecedents. Using this substitution, we establish the existence of an algorithm checking the derivability in the Lambek calculus with a unit and one division in polynomial time.     

Contributions to the calculus of variations

In this paper, the theory of the calculus of variations for the simplest problem is reviewed. Simpler proofs are given for the classical conditions and a new necessary condition is presented which allows strong, necessary, and sufficient conditions to be stated for the first time.The major part of this research was conducted in the Department of Electrical Engineering, University of the West Indies, St. Augustine, Trinidad, West Indies. The author is indebted to Dr. B. Bhatt for interesting discussions.     

基础模糊命题演算系统BL~*的改进系统

基础模糊命题演算系统BL*是一个和基础命题演算系统BL相对独立的命题演算系统。命题演算系统L*是系统BL*的扩张,但不是系统BL的扩张。通过对系统BL*及其它模糊命题演算系统的研究,本文对BL*系统进行了修正,进一步改进了BL*系统中的公理体系。     

Reallocation of paths of ABR services in a multistage interconnection network (MIN)-type ATM switch network

In this paper, we develop algorithms for reallocating paths of available bit rate (ABR) services in an asynchronous transfer mode (ATM) switch network. ATM traffic control for fair share bandwidth allocation is usually performed under the assumption that paths of all services in a switch network are fixed. However, each connection may have multiple paths from an ingress queue to an egress queue since most ATM switch networks have the structure of the multistage interconnection network of switch elements. Therefore, paths already established for ABR connections may have to be changed to enhance throughput of the switch, if the ATM switch has the capability of adjusting paths of ABR connections while they are being served. We present three algorithms, in which throughput for ABR connections is estimated to decide whether or not paths of the connections should be changed. These algorithms are compared with an existing traffic management algorithm through simulation experiments. Results of the experiments show that the suggested algorithms give higher throughput in terms of the number of transmitted ABR cells without increasing the delay time of ABR services as well as quality of service (QoS) guaranteed services or decreasing the number of transmitted cells of QoS guaranteed services.     

On flipping first-semester calculus: a case study

High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are ‘flipping’ (or inverting) their classrooms. By flipping, we mean administering course content outside of the classroom and replacing the traditional in-class lectures with discussion, practice, group work, and other elements of active learning. This paper presents the major results from a three-year study of a flipped, first-semester calculus course at a small, comprehensive, American university with a well-known engineering programme. The data we have collected help quantify the positive and substantial effects of our flipped calculus course on failure rates, scores on the common final exam, student opinion of calculus, teacher impact on measurable outcomes, and success in second-semester calculus. While flipping may not be suitable for every teacher, every student, and in every situation, this report provides some evidence that it may be a viable option for those seeking an alternative to the traditional lecture model.     

Tensor calculus: unlearning vector calculus

Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can serve as a bridge for vector calculus into tensor calculus.     

On Combined Connectives

Combined connectives arise in combined logics. In fibrings, such combined connectives are known as shared connectives and inherit the logical properties of each component. A new way of combining connectives (and other language constructors of propositional nature) is proposed by inheriting only the common logical properties of the components. A sound and complete calculus is provided for reasoning about the latter. The calculus is shown to be a conservative extension of the original calculus. Examples are provided contributing to a better understanding of what are the common properties of any two constructors, say disjunction and conjunction.     

中介命题演算系统$MP^{M}$的代数系统

MP~M系统是在中介逻辑系统的基础上建立起来的,用于处理数据库中不完全信息的三值逻辑命题演算系统．本文通过在MP~M系统上建立一个代数系统,对MP~M系统进行了代数抽象,讨论了MP~M系统的代数性质．本文还研究了该代数系统的次直积,以及与其它一些代数系统之间的关系．     

Basic Calculus on Time Scales and some of its Applications

The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hôspital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.     

Simulation and Analysis of a New Mobility Model for Mobile Communication Networks

The traffic generation models, which describe how the clients use the network services, as well as the mobility models, which describe how clients move within the service area covered by the network, are essential tools for QoS analysis in these environments. In this paper we present the simulation of a new mobility model implemented for the analysis of QoS parameters of a mobile network, such as channel occupation time, handoff and new call blocking probabilities.     

Malliavin calculus with time dependent coefficients applied to a class of stochastic differential equations

In this paper, we consider stochastic differential equations with time dependent coefficients driven by an infinite dimensional Brownian motion. Using the stochastic calculus of variations (Malliavin calculus), we prove, that under a local Hörmander condition, the law of the solution possesses a smooth density with respect to Lebesgue measure.     

Integrating design problems in mathematics curriculum: an architecture college case study

This paper reports an attempt to improve results in the mathematics course in one of the architecture colleges in Israel through practise in applications. The effect of integrating structure design problems in the calculus curriculum on students' achievements and attitudes was examined. The applied topics in the curriculum were connected to calculus topics and studied through problembased learning activities. The integrated curriculum was implemented and the learning results in experimental and control groups were assessed by means of achievement tests, attitude questionnaires and student interviews. The learning achievements in the experimental group proved to be significantly higher than in the control group. The positive impact of learning applications on motivation, understanding, creativity and interest in mathematics is indicated.     

A traffic control algorithm for Clos network type ATM switches

This paper presents a new traffic control algorithm for asynchronous transfer mode (ATM) switches composed of Clos network type switch fabric. In most traffic control algorithms previously proposed, an ATM switch has been considered as a single node, although the switch fabric of an ATM switch is usually of a network type. In this paper, we suggest a new control algorithm that is designed not only for the ATM network but also for the switch fabric and that can maintain high speed even in cases where buffer capacity of the switch fabric is limited. Main modules of the suggested algorithm are a traffic status detection mechanism based on non-parametric statistical tests, a cell-loss detection mechanism, and a cluster-based fair share computation procedure. Results of simulation experiments show that the suggested algorithm satisfactorily adjusts traffic rate of available bit rate services according to changes in traffic rate used by quality of service (QoS) guaranteed services. The results also show that the algorithm prevents cell losses relatively well and keeps the delay time of QoS guaranteed services short enough.     

Promoting students’ graphical understanding of the calculus

Our purpose in this paper is to report on an observational study to show how students think about the links between the graph of a derived function and the original function from which it was formed. The participants were asked to perform the following task: they were presented with four graphs that represented derived functions and from these graphs they were asked to construct the original functions from which they were formed. The students then had to walk these graphs as if they were displacement-time graphs. Their discussions were recorded on audio tape and their walks were captured using data logging equipment and these were analysed together with their pencil and paper notes. From these three sources of data, we were able to construct a picture of the students’ graphical understanding of connections in calculus. The results confirm that at the start of the activity the students demonstrate an algebraic symbolic view of calculus and find it difficult to make connections between the graphs of a derived function and the function itself. By being able to ‘walk’ an associated displacement time graph, we propose that the students are extending their understanding of calculus concepts from symbolic representation to a graphical representation and to what we term a ‘physical feel’.     

Undergraduate mathematics achievement in the emerging ethnic engineers programme

This paper addresses the issue of collegiate mathematics achievement of underrepresented minority students as it investigates the impact of a cooperative learning calculus programme on the first-year calculus experience of non-Asian ethnic minority engineering students. The Emerging Ethnic Engineers Programme in the College of Engineering at the University of Cincinnati is a successful, comprehensive programme that focuses on the recruitment, retention, academic success, professional development, and timely graduation of underrepresented coloured students. The objectives of the programme are accomplished through three interrelated phases: pre-college science and mathematics programmes, first-year collegiate programmes, and upper-division programmes. The underlying principles of the first-year programme include academic achievement and establishing a strong sense of community among the cohort. This report will focus on the cooperative learning calculus programme that has been successful in improving retention and academic success rates for coloured freshmen engineering students.