STRATEGIES OF REDUCTION OF ABSTRACTION IN ABSTRACT ALGEBRA

This article is based on the study, which tries to unpack strategies of reduction of abstraction in learning abstract algebra from learners’ perspective. Ethnography was used to collect the required information. The study found the strategies of reduction of abstraction in abstract algebra are: making sense and meaning through previous experiences and existing knowledge an analogical creation of mental image, using first person language in course of doing mathematics by students as teachers do in the classroom for logical arguments, focusing on “symbol” or some mathematical entity to manage abstraction for their idiosyncratic understandings of abstract mathematical structure rather than the reflective thinking, using students own idiosyncratic figures to reduce the degrees of complexity of mathematical concepts. This study can lead teachers of abstract algebra to a new awareness of their teaching strategies and their practices.

Screw theory based method to formulate local Jacobian’ magnitude estimator contours for 6 DoF robots

Singularity prediction of Jacobian is inherently related to the design of any manipulator. Optimization of Jacobian is known to improve the joint rates and joint torques for the same set of end-effector twists and wrenches. Any mathematical entity which is an estimator of Jacobian can be used as a manifestation of the stated two schemes. It is required that online computation of Jacobian be implemented so that singular poses can be avoided and performance is maximized. We present an algorithm that relates to the determinant of the Jacobian for robotic manipulators. The algorithm developed is quicker to compute, and a strategy is developed to synthesize the local estimator contour. The contours change dynamically, henceforth optimal motor torques (in case of presence of interaction fields or system requiring force control) and optimal joint rates can be defined, depending on the control algorithm. The algorithm developed is independent of the structure of the matrix and is generalized to any six DoF manipulator structure used.

The Utterly Prosaic Connection between Physics and Mathematics

Eugene Wigner famously argued for the “unreasonable effectiveness of mathematics” as applied to describing physics and other natural sciences in his 1960 essay. That essay has now led to some 58 years of (sometimes anguished) philosophical soul searching—responses range from “So what? Why do you think we developed mathematics in the first place?”, through to extremely speculative ruminations on the existence of the universe (multiverse) as a purely mathematical entity—the Mathematical Universe Hypothesis. In the current essay I will steer an utterly prosaic middle course: Much of the mathematics we develop is informed by physics questions we are trying to solve; and those physics questions for which the most utilitarian mathematics has successfully been developed are typically those where the best physics progress has been made.

Mechanical Homogeneous Continuous Dynamical Systems Holor Algebra - Steady-State Alternating Velocity Analysis

In this article, a new mathematical representation of the sinusoidal alternating velocity, force and power by means of some complex quantities, termed ‘holors’ is proposed. The word holor is a term to describe a mathematical entity that is made up of one or more independent quantities, and includes complex numbers, scalars, vectors, matrices, tensors and other hypernumbers. Holors, thus defined, have been known for centuries but each has been developed more or less independently, accompanied by separate nomenclature and theory.

Posthuman Reconstruction of the World as a Simulation in Charles Stross’ Accelerando

In this paper my aim is to analyze the theme of simulation in Charles Stross’ hard science fiction novel titled ‘Accelerando’ (2005). The discussion shall start by pointing out the enormous significance of the ideas of simulation both in the world of pure science as well as in the fields of philosophy, arts and literature. We shall show by adopting an informational approach and by using the theoretical framework of Baudrillard’s ‘Simulacra and Simulation’ that science fiction novels like Charles Stross’ ‘Accelerando’ treat simulation not just as an abstract mathematical entity existing independently of human beings; rather they treat simulation as an indispensable step for mankind in their journey of ascension to the Posthuman plane of existence. This paper will also try to prove that Stross’ ‘Accelerando’ not only relies upon the description of the effects of use of many incomprehensibly powerful simulation technologies in various forms to produce the effect of estrangement upon the readers but it also implies that the very fabric of reality which deem to be real is composed of binary oppositions between different strands of contradistinctive signs and symbols which in a posthuman world will crumble together to give way to a Posthuman world replete with endless possibilities. This novel like many other contemporary hard science fiction novels implies that the reality is essentially indistinguishable from the simulation once the influence of human perception or consciousness is deducted from the equation and then the entire Universe can be interpreted as one gigantic computational entity consisting of myriads of quantum informational field operators through the manipulation of which we can derive a semblance of power indistinguishable from the actual one.Keywords: Science Fiction, Posthumanism, Simulacra and Simulation, Quantum Nature of Creation, Portrayal of Future

Mission Design of Mobile Manipulators in Cluttered Environments for Service Applications

The purpose of this paper is to present a mission design approach for a service mobile manipulator which is moving and manipulating objects in partly known indoor environments. The mobile manipulator is requested to pick up and place objects on predefined places (stations). The proposed approach is based on the Bump-Surface concept to represent robot's environment through a single mathematical entity. The solution of the mission design problem is searched on a higher dimension Bump-Surface in such a way that its inverse image into the actual robot environment satisfies the given objectives and constraints. The problem's objectives consist of determining the best feasible paths for both the mobile platform and for the manipulator's end-effector so that all the stations are served at the lowest possible cost. Simulation examples are presented to show the effectiveness of the presented approach.

Mission Design of a Team of Service Robots

The purpose of this chapter is to present an integrated approach for Mission Design of a team of Service Robots that is operating in partially known indoor environments such as libraries, hospitals, or warehouses. The robots are requested to serve a number of service stations while taking into account movement safety and other kinematical constraints. The Bump-Surface concept is used to represent the robots' environment through a single mathematical entity and an optimization problem is formulated representing an aggregation of paths length and movement constraints. Then a modified Genetic Algorithm with parallel populations is used for solving the problem of mission design of a team of service robots on the constructed Bump-Surface. Three simulation examples are presented to show the effectiveness of the presented approach.

Mission Planning of Mobile Robots and Manipulators for Service Applications

The purpose of this chapter is to present a mission planning approach for a service robot, which is moving and manipulating objects in semi-structured and partly known indoor environments such as stores, hospitals, and libraries. The recent advances and trends in motion planning and scheduling of mobile robots carrying manipulators are presented. This chapter adds to the existing body of knowledge of motion planning for Service Robots (SRs), an approach that is based on the Bump-Surface concept. The Bump-Surface concept is used to represent the entire robot’s environment through a single mathematical entity. Criteria and constraints for the mission planning are adapted to the service robots. Simulation examples are presented to show the effectiveness of the presented approach.

Mission Planning of Mobile Robots and Manipulators for Service Applications

The purpose of this chapter is to present a mission planning approach for a service robot, which is moving and manipulating objects in semi-structured and partly known indoor environments such as stores, hospitals, and libraries. The recent advances and trends in motion planning and scheduling of mobile robots carrying manipulators are presented. This chapter adds to the existing body of knowledge of motion planning for Service Robots (SRs), an approach that is based on the Bump-Surface concept. The Bump-Surface concept is used to represent the entire robot’s environment through a single mathematical entity. Criteria and constraints for the mission planning are adapted to the service robots. Simulation examples are presented to show the effectiveness of the presented approach.