Negations generated by bounded lattices t-norms

From the birth of fuzzy sets theory, several extensions have been proposed changing the possible membership values. Since fuzzy connectives such as t-norms and negations have an important role in theoretical as well as applied fuzzy logics, these connectives have been adapted for these generalized frameworks. Perhaps, an extension of fuzzy logic which generalizes the remaining extensions, proposed by Joseph Goguen in 1967, is to consider arbitrary bounded lattices for the values of the membership degrees. In this paper we extend the usual way of constructing fuzzy negations from t-norms for the bounded lattice t-norms and prove some properties of this construction.

(S,N)-implications on bounded lattices

Since the birth of the fuzzy sets theory several extensions have been proposed. For these extensions, different sets of membership functions were considered. Since fuzzy connectives, such as conjunctions, negations and implications, play an important role in the theory and applications of fuzzy logics, these connectives have also been extended. An extension of fuzzy logic, which generalizes the ones considered up to the present, was proposed by Joseph Goguen in 1967. In this extension, the membership values are drawn from arbitrary bounded lattices. The simplest and best studied class of fuzzy implications is the class of (S,N)-implications, and in this chapter we provide an extension of (S,N)-implications in the context of bounded lattice valued fuzzy logic, and we show that several properties of this class are preserved in this more general framework.

Primary teachers’ algebraic thinking: example from lesson study

Previous studies have reported on primary children’s algebraic thinking and generalising in a range of problem settings but there is little evidence of primary teachers’ knowledge of algebraic thinking. In this paper the development in algebraic thinking of one primary teacher who taught a research lesson in a Japanese Lesson Study project involving teachers from three primary schools is presented. The findings suggest the need for professional learning in algebra and reasoning and indicate the value of Lesson Study.

TensorPack : a Maple-based software package for the manipulation of algebraic expressions of tensors in general relativity

In this paper we: (1) introduce TensorPack, a software package for the algebraic manipulation of tensors in covariant index format in Maple; (2) briefly demonstrate the use of the package with an orthonormal tensor proof of the shearfree conjecture for dust. TensorPack is based on the Riemann and Canon tensor software packages and uses their functions to express tensors in an indexed covariant format. TensorPack uses a string representation as input and provides functions for output in index form. It extends the functionality to basic algebra of tensors, substitution, covariant differentiation, contraction, raising/lowering indices, symmetry functions and other accessory functions. The output can be merged with text in the Maple environment to create a full working document with embedded dynamic functionality. The package offers potential for manipulation of indexed algebraic tensor expressions in a flexible software environment.

GHP: a maple package for performing calculations in the Geroch-Held-Penrose formalism

We present a new symbolic algebra package, written for Maple, for performing computations in the Geroch-Held-Penrose formalism. We demonstrate the essential features and capabilities of our package by investigating Petrov-D vacuum solutions of Einstein's field equations.

Reduced-order output feedback controllers for discrete-time systems

This article considers the stabilization by output feedback controllers for discrete-time systems. The controller can place all of the closed-loop poles within a specified disk D(-α, 1/β), centred at (-α,0) with radius 1/β, where | - α|  + 1/β < 1. The design method involves the decomposition of the system into two portions. The first portion comprises of all of the poles that are lying outside of the specified disk. A reduced-order model is constructed for this portion. The second portion comprises of all of the remaining poles of the system and is characterized by an H_∞-norm bound. The controller design is then accomplished by using H_∞-control theory. It is shown that, subject to the solvability of an algebraic Riccati equation, output feedback controllers can be systematically derived. The order of the controller is low, and can be as low as the number of the open-loop poles that are lying outside of the specified disk. A step-by-step design algorithm is provided. Numerical examples are given to illustrate the attractiveness of the design method.

Constrained pole placement for linear systems using low-order output feedback controllers

The paper presents a simple approach to the problem of designing low-order output feedback controllers for linear continuous systems. The controller can place all of the closed-loop poles within a circle, C(- , 1/ β) , with centre at - and radius of 1/ β in the left half s-plane. The design method is based on transformation of the original system and then applying the bounded-real-lemma to the transformed system. It is shown that subjected to the solvability of an algebraic Riccati equation (ARE), output feedback controllers can then be systematically derived. Furthermore, the order of the controller is low and equals only the number of the open-loop poles lying outside the circle. A step-by-step design algorithm is given. Numerical examples are given to illustrate the design method.

The GHP II package with applications

We present an advanced version of the Maple package GHP called GHPII. In it we provide a number of additional sophisticated tools to assist with problems formulated in the Geroch-Held-Penrose (ghp) formalism. The first part of this article discusses these new tools while in the second part we shall apply the ghp formalism, using the GHPII routines, to vacuum Petrov type D spacetimes and shear-free perfect fluids. We prove that for all shear-free perfect fluids with a barotropic equation of state, where two of the principal null directions are coplanar with the fluid four-velocity and vorticity then either the expansion or vorticity of the fluid must be zero.

Algebra skills and traps and diagnostic teaching for the future

Explores on some research about teaching and learning algebra and related classroom issues. Diagnostic instruments that may be used by senior secondary teachers in teaching algebra to senior classes; Strategies for remediating algebraic difficulties and misconceptions; Impact of technology on the algebra curriculum; Usefulness of copying algebraic expressions while using Computer Algebra Systems or mathematics processing software in a calculus class.

MMSE precoder for unitary space–time codes in correlated time-varying channels

This letter addresses the problem of the design of a precoder for multiple transmit antenna communication systems with spatially and temporally correlated fading channels. By using the asymptotic (high signal-to-noise ratio) mean-square error of the channel estimates, the letter derives a precoder for unitary space-time codes that can exploit the spatiotemporal correlation in the time-varying fading channels. Simulation results illustrate that significant performance gains can be achieved by using the new precoder.

Evaluation of the robustness of predictive yield models based on catchment characteristics using GIS for reservoir fisheries in Sri Lanka

Land-use patterns in the catchment areas of Sri Lankan reservoirs, which were quantified using Geographical Information Systems (GIS), were used to develop quantitative models for yield prediction. The validity of these models was evaluated through the application to five reservoirs that were not used in the development of the models, and by comparing with the actual fish yield data of these reservoirs collected by an independent body. The robustness of the predictive models developed was tested by principal component analysis (PCA) on limnological characteristics, land-use patterns of the catchments and fish yields. The predicted fish yields in five Sri Lankan reservoirs, using the empirical models based on the ratios of forest cover and/or shrub cover to reservoir capacity or reservoir area were in close agreement with the observed fish yields. The scores of PCA ordination of productivity-related limnological parameters and those of land-use patterns were linearly related to fish yields. The relationship between the PCA scores of limnological characteristics and land-use types had the appropriate algebraic form, which substantiates the influence of the limnological factors and land-use types on reservoir fish yields. It is suggested that the relatively high predictive power of the models developed on the basis of GIS methodologies can be used for more accurate assessment of reservoir fisheries. The study supports the importance and the need for an integrated management strategy for the whole watershed to enhance fish yields.

MRD hashing

We propose two new classes of hash functions which are motivated by Maximum Rank Distance (MRD) codes. We analise the security of these schemes. The system setup phase is computationally expensive for general field extensions. To overcome this limitation we derive an algebraic solution which avoids computations in special extension fields in the intended operational range of the hash functions.

Sir, can x equal x plus two?

Technology in classrooms has brought opportunities to develop new ways of teaching and learning, as well as new ways of thinking and communicating. The author describes a lesson which aimed to have the grade four children in the class construct on the computer screen images of rectangles. The lesson provided the opportunity to learn about some powerful generalisations about rectangles as well to learn about their geometric and algebraic understandings.

A stable MCA learning algorithm

Minor component analysis (MCA) is an important statistical tool for signal processing and data analysis. Neural networks can be used to extract online minor component from input data. Compared with traditional algebraic  approaches, a neural network method has a lower computational complexity. Stability of neural networks learning algorithms is crucial to practical applications. In this paper, we propose a stable MCA neural networks learning algorithm, which has a more satisfactory numerical stability than some existing MCA algorithms. Dynamical behaviors of the proposed algorithm are analyzed via deterministic discrete time (DDT) method and the conditions are obtained to guarantee convergence. Simulations are carried out to illustrate the theoretical results achieved.

Differentiation from first principles using spreadsheets

The article demonstrates the ability of spreadsheets to fit a polynomial to a set of discrete points. It is noted that such an ability enables students to not just evaluate a gradient at a single point, but at a whole family of points, which generates the analytical global gradient function of secants without doing any algebraic manipulations. One of the advantages of such an approach is that it enables the class to focus on the concepts being taught, rather than being hindered by the mechanics of trying to factorise a cubic polynomial.