Focuses of awareness in the process of learning the fundamental theorem of calculus with digital technologies

This paper brings together three themes: the fundamental theorem of the calculus (FTC), digital learning environments in which the FTC may be taught, and what we term “focuses of awareness.” The latter are derived from Radford’s theory of objectification: they are nodal activities through which students become progressively aware of key mathematical ideas structuring a mathematical concept. The research looked at 13 pairs of 17-year-old students who are not yet familiar with the concept of integration. Students were asked to consider possible connections between multiple-linked representations, including function graphs, accumulation function graphs, and tables of values of the accumulation function. Three rounds of analysis yielded nine focuses in the process of students’ learning the FTC with a digital tool as well as the relationship between them. In addition, the activities performed by the students to become aware of the focuses are described and theoretical and pedagogical implementations are also discussed.     

Problem solving and the use of digital technologies within the Mathematical Working Space framework

ZDM – Mathematics Education - The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological...     

Complete rent dissipation in contest with power technologies

We consider contest success functions (CSFs) with power technologies. We present a necessary and sufficient condition for that in all Nash equilibria, complete rent dissipation occurs (i.e., the sum of contestants' efforts is equal to the value).     

The didactical tetrahedron as a heuristic for analysing the incorporation of digital technologies into classroom practice in support of investigative approaches to teaching mathematics

There have been various proposals to expand the heuristic device of the didactical triangle to form a didactical tetrahedron by adding a fourth vertex to acknowledge the significant role of technology in mediating relations between content, student and teacher. Under such a heuristic, the technology vertex can be interpreted at several levels from that of the material resources present in the classroom to that of the fundamental machinery of schooling itself. At the first level, recent research into teacher thinking and teaching practice involving use of digital technologies indicates that, while many teachers see particular tools and resources as supporting the classroom viability of investigative approaches to mathematics, the practical expressions of this idea in lessons vary in the degree of emphasis they give to a didactic of reconstruction of knowledge, as against reproduction. At the final level, examining key structuring features of teaching practice makes clear the scope and scale of the situational adaptation and professional learning required for teachers to successfully incorporate use of digital tools and resources in support of investigative approaches. These issues are illustrated through examining contrasting cases of classroom use of dynamic geometry in professionally well-regarded mathematics departments in English secondary schools.     

The linear formulation of the ZSG-DEA models with different production technologies

The zero sum gains data envelopment analysis models (ZSG-DEA models) are non-linear. In this paper, we first show that the ZSG-DEA models can be transformed to linear or parametric linear models and discuss the feasible domains of the parameters. Second, we show that the linear formulations of ZSG-DEA models under the equal output reduction strategy and the proportional output reduction strategy in a single output case are equivalent to the output-oriented super-efficiency model under variable returns-to-scale (VRS) assumption. As a matter of course, the models may encounter infeasibility. Third, we propose the linear transformations of ZSG-DEA models under constant returns-to-scale (CRS) assumption and compare them with the VRS models. In the end, we evaluate the participant countries at the Olympic Games by the linear equivalent models with multiple outputs under different weight restrictions. Our results are compared with the efficiencies obtained from the original ZSG-DEA model with an aggregated output under both CRS and VRS assumptions. It is found that the original method with aggregated output tends to underestimate the efficiencies of DMUs.     

Computation of equilibria in an economy with increasing returns to scale technologies

The purpose of this paper is to show that equilibria in an economy with increasing returns to scale technologies are computable.     

The impossibility of convex constant returns-to-scale production technologies with exogenously fixed factors

The extensions to the variable (VRS) and the constant (CRS) returns-to-scale models developed by Banker and Morey are considered among the main approaches to the incorporation of exogenously fixed factors in models of data envelopment analysis (DEA). Recently, Syrjänen showed that the Banker and Morey CRS technology is not convex. Taking into account that its subset VRS technology is explicitly assumed convex, this observation leads to difficulties with explaining the fundamental production assumptions of the CRS extension. Motivated by the example of Syrjänen, the contribution of this paper is twofold. First, we show that the nonconvex Banker and Morey CRS technology is nevertheless a suitable reference technology for the assessment of scale efficiency. Second, we ask if a convex technology could be constructed that would “correct” the nonconvexity of the CRS technology of Banker and Morey. The answer to this is negative: one consequence of assuming both convexity and ray unboundness with fixed exogenous factors is that we can always “mix-and-match” discretionary and nondiscretionary factors taken from different units, arriving at totally unrealistic production plans. This demonstrates that generally there exists no meaningful convex CRS technology with exogenously fixed factors that can be used in its own right, apart from its use as a reference technology in the measurement of scale efficiency.     

Efficiency decompositions for heterogeneous technologies

This paper examines firms operating under different technologies but under a common metatechnology and provides a decomposition of their efficiency into input-invariant and output-invariant components. To achieve this, it reviews known definitions of technical and scale efficiency and provides alternative expressions, which incorporate a firm operating at the most productive scale size. By extending this approach, analogous decompositions are derived for the gap between two technologies and the (technical) efficiency difference between two firms operating under two distinct technologies or metatechnologies. Two indicative case-studies are also presented in order to illustrate the applicability of the proposed framework.     

Some aspects of algorithms for digital control with a microcomputer

We study the principles of constructing algorithms for direct digital control for an automatic control system of complicated (step-by-step or multiple-input multiple-output) structure, based on modular software in the controlling microcomputer and taking account of the controller saturation mode, initialization conditions of the elements of the system, reliability of the input information, and the principles of automatic blocking of the system elements when the normal operating mode breaks down. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

An exact construction of digital convex polygons with minimal diameter

Given a natural number n, an exact formula is derived for the minimal possible size MD(n) of a square grid, in which a digital convex n-gon can be inscribed. An exact construction of a digital convex n-gon which can be inscribed into a square grid of size MD(n) is also given.     

Negative features of hyperbolic and directional distance models for technologies with undesirable outputs

In data envelopment analysis for environmental performance measurement the undesirable outputs are taken into account. Ones of the standard approaches for dealing with the undesirable outputs are the hyperbolic and the directional distance measures. They both allow a simultaneous expansion of desirable outputs and a contraction of undesirable outputs by means of a single parameter. To meet environmental requirements, a technology with no disposability of undesirable outputs is often considered and the outputs are assumed to be only weakly disposable. We show that the combination of this type of technology with the hyperbolic measure, (or with its linearization, which is a special type of the directional distance model) may lead to a misleading efficiency score of the unit under evaluation. We derive the dual of the hyperbolic model under the environmental technology and describe some of its properties. Then, we use the hyperbolic and directional distance dual models for developing a second-phase method. This enables to detect the misleading scores of the decision making units. We illustrate the results on a real world data set.     

Transformation technologies in forming composite structures

A design model of axisymmetric transformation of a helically wound shell preform into a given shape is considered. Types of design problems are determined, and examples of their solution are given. Examples of fabrication techniques for structural prototypes by using the transformation method are included.     

Extension of continuous functions in digital spaces with the Khalimsky topology

The digital space Zn equipped with Efim Khalimsky's topology is a connected space. We study continuous functions Zn⊃A→Z, from a subset of Khalimsky n-space to the Khalimsky line. We give necessary and sufficient condition for such a function to be extendable to a continuous function Zn→Z. We classify the subsets A of the digital plane such that every continuous function A→Z can be extended to a continuous function on the whole plane.     

A Popov-theory-based survey on digital control of infinite-dimensional systems with unboundedness

This paper represents a Popov-theory-based assessment of thecurrent status of the digital control of infinite–dimensionalsystems with certain unboundedness in control and/or observation,summarizing some results and indicating which, in the authors'opinion, would be the promising directions for future research.     

Robust digital controllers for uncertain chaotic systems: A digital redesign approach

In this paper, a new and systematic method for designing robust digital controllers for uncertain nonlinear systems with structured uncertainties is presented. In the proposed method, a controller is designed in terms of the optimal linear model representation of the nominal system around each operating point of the trajectory, while the uncertainties are decomposed such that the uncertain nonlinear system can be rewritten as a set of local linear models with disturbed inputs. Applying conventional robust control techniques, continuous-time robust controllers are first designed to eliminate the effects of the uncertainties on the underlying system. Then, a robust digital controller is obtained as the result of a digital redesign of the designed continuous-time robust controller using the state-matching technique. The effectiveness of the proposed controller design method is illustrated through some numerical examples on complex nonlinear systems––chaotic systems.     

Minimum decomposition of a digital surface into digital plane segments is NP-hard

This paper deals with the complexity of the decomposition of a digital surface into digital plane segments (DPSs for short). We prove that the decision problem (does there exist a decomposition with less than λ DPSs?) is NP-complete, and thus that the optimization problem (finding the minimum number of DPSs) is NP-hard. The proof is based on a polynomial reduction of any instance of the well-known 3-SAT problem to an instance of the digital surface decomposition problem. A geometric model for the 3-SAT problem is proposed.     

Mathematical digital escape rooms

Escape rooms have been receiving more attention for use in education. Escape rooms are games in which teams solve multiple puzzles and challenges using clues, hints, and strategy to determine how to escape from a locked room. The implementation of escape rooms has the potential for increasing students' mathematical understanding and engagement. They also enable students to develop teamwork and communication skills which are essential for work, everyday life, and being a productive citizen. The importance of technology for use in education has also received more attention of late. Technology can be utilized with escape rooms for the development of digital escape rooms. This article describes a mathematical digital escape room that was implemented with a class of middle school students. Important principles for developing escape rooms are described and how they were integrated with the digital escape room entitled, Blanca's Building.