A New Approach to Fuzzy Arithmetic

This work shows an application of a generalized approach for constructing dilation-erosion adjunctions on fuzzy sets. More precisely, operations on fuzzy quantities and fuzzy numbers are considered. By the generalized approach an analogy with the well known interval computations could be drawn and thus we can define outer and inner operations on fuzzy objects. These operations are found to be useful in the control of bioprocesses, ecology and other domains where data uncertainties exist.

On the Arithmetic of Errors

An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.

A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations

2000 Mathematics Subject Classification: 65H10.

Examining Cognitive Processing in Children with an Arithmetic Disability

Currently there is no consensus as to the specific cognitive impairments that characterize mathematical disabilities (MD) or specific subtypes such as an arithmetic disability (AD). The present study sought to address this concern by examining cognitive processes that might undergird AD in children. The present study utilized archival data to conduct two investigations. The first investigation examined the executive functioning and working memory of children with AD. An age-matched achievement-matched design was employed to explore whether children with AD exhibit developmental lags or deficits in these cognitive domains. While children with AD did not exhibit impairments in verbal working memory or colour word inhibition, they did demonstrate impairments in shifting attention, visual-spatial working memory, and quantity inhibition. As children with AD did not perform more poorly than their younger achievement-matched peers on any of these tasks, impairments in specific areas of executive functioning and working memory appeared to reflect a developmental lag rather than a cognitive deficit. The second study examined the phonological processing performance of children with AD compared to children with comorbid disabilities in arithmetic and word recognition (AD/WRD) and to typically achieving (TA) children. Results indicated that, while children with AD did demonstrate impairments on all isolated naming speed tasks, trail making digits, and memory for digits, they did not demonstrate impairments on measures of phonological awareness, nonword repetition, serial processing speed, or serial naming speed. In contrast, children with AD/WRD demonstrated impairments on measures of phonological awareness, phonological short-term memory, isolated naming speed, serial processing speed, and the alphabet a-z task. Overall, results suggested that phonological processing impairments are more prominent in children with a WRD than children with an AD. Together, these studies further our understanding of the nature of the cognitive processes that underlie AD by focusing upon rarely used methods (i.e., age-matched achievement-matched design) and under-examined cognitive domains (i.e., phonological processing).

Differential ability and attainment in language and arithmetic of Dutch primary school pupils.

Background. In pre-school and primary education pupils differ in many abilities and competences (‘giftedness’). Yet mainstream educational practice seems rather homogeneous in providing age-based or grade-class subject matter approaches. Aims. To clarify whether pupils scoring initially at high ability level do develop and attain differently at school with respect to language and arithmetic compared with pupils displaying other initial ability levels. To investigate whether specific individual, family or educational variables co-vary with the attainment of these different types of pupils in school. Samples. Data from the large-scale PRIMA cohort study including a total of 8258 grade 2 and 4 pupils from 438 primary schools in The Netherlands. Methods. Secondary analyses were carried out to construct gain scores for both language and arithmetic proficiency and a number of behavioural, attitudinal, family and educational characteristics. The pupils were grouped into different ability categories (highly able; able; above average; average and below). Further analyses used Pearson correlations and analyses of variance both between and within ability categories. Cross-validation was done by introducing a cohort of younger pupils in pre-school and grouping both cohorts into decile groups based on initial ability in language and arithmetic. Results. Highly able pupils generally decreased in attainment in both language and arithmetic, whereas pupils in average and below average groups improved their language and arithmetic scores. Only with highly able pupils were some educational characteristics correlated with the pupils’ development in achievement, behaviour and attitudes. Conclusions. Pre-school and primary education should better match pupils’ differences in abilities and competences from their start in pre-school to improve their functioning, learning processes and outcomes. Recommendations for educational improvement strategies are presented in closing.

Number comparison and number ordering as predictors of arithmetic performance in adults: Exploring the link between the two skills, and investigating the question of domain-specificity.

Recent evidence has highlighted the important role that number ordering skills play in arithmetic abilities (e.g., Lyons & Beilock, 2011). In fact, Lyons et al. (2014) demonstrated that although at the start of formal mathematics education number comparison skills are the best predictors of arithmetic performance, from around the age of 10, number ordering skills become the strongest numerical predictors of arithmetic abilities. In the current study we demonstrated that number comparison and ordering skills were both significantly related to arithmetic performance in adults, and the effect size was greater in the case of ordering skills. Additionally, we found that the effect of number comparison skills on arithmetic performance was partially mediated by number ordering skills. Moreover, performance on comparison and ordering tasks involving the months of the year was also strongly correlated with arithmetic skills, and participants displayed similar (canonical or reverse) distance effects on the comparison and ordering tasks involving months as when the tasks included numbers. This suggests that the processes responsible for the link between comparison and ordering skills and arithmetic performance are not specific to the domain of numbers. Finally, a factor analysis indicated that performance on comparison and ordering tasks loaded on a factor which included performance on a number line task and self-reported spatial thinking styles. These results substantially extend previous research on the role of order processing abilities in mental arithmetic.

The Inverse Jacobian Control Method Applied to a Four Degree of Freedom Confined Space Robot

Thesis (Master's)--University of Washington, 2016-06

Ordinal Arithmetic: A Case Study for Rippling in a Higher Order Domain

This paper reports a case study in the use of proof planning in the context of higher order syntax. Rippling is a heuristic for guiding rewriting steps in induction that has been used successfully in proof planning inductive proofs using first order representations. Ordinal arithmetic provides a natural set of higher order examples on which transfinite induction may be attempted using rippling. Previously Boyer-Moore style automation could not be applied to such domains. We demonstrate that a higher-order extension of the rippling heuristic is sufficient to plan such proofs automatically. Accordingly, ordinal arithmetic has been implemented in lambda-clam, a higher order proof planning system for induction, and standard undergraduate text book problems have been successfully planned. We show the synthesis of a fixpoint for normal ordinal functions which demonstrates how our automation could be extended to produce more interesting results than the textbook examples tried so far.

Twisted Edwards curves revisited

This paper introduces fast algorithms for performing group operations on twisted Edwards curves, pushing the recent speed limits of Elliptic Curve Cryptography (ECC) forward in a wide range of applications. Notably, the new addition algorithm uses for suitably selected curve constants. In comparison, the fastest point addition algorithms for (twisted) Edwards curves stated in the literature use . It is also shown that the new addition algorithm can be implemented with four processors dropping the effective cost to . This implies an effective speed increase by the full factor of 4 over the sequential case. Our results allow faster implementation of elliptic curve scalar multiplication. In addition, the new point addition algorithm can be used to provide a natural protection from side channel attacks based on simple power analysis (SPA).

Jacobi quartic curves revisited

This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, y 2 = d x 4 + 2 a x 2 + 1. With recent bandwidth-efficient proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.

Faster group operations on elliptic curves

This paper improves implementation techniques of Elliptic Curve Cryptography. We introduce new formulae and algorithms for the group law on Jacobi quartic, Jacobi intersection, Edwards, and Hessian curves. The proposed formulae and algorithms can save time in suitable point representations. To support our claims, a cost comparison is made with classic scalar multiplication algorithms using previous and current operation counts. Most notably, the best speeds are obtained from Jacobi quartic curves which provide the fastest timings for most scalar multiplication strategies benefiting from the proposed 12M + 5S + 1D point doubling and 7M + 3S + 1D point addition algorithms. Furthermore, the new addition algorithm provides an efficient way to protect against side channel attacks which are based on simple power analysis (SPA). Keywords: Efficient elliptic curve arithmetic,unified addition, side channel attack.

Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

Investigating functional thinking in the elementary classroom : foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.

Accurate series solutions for gravity-driven Stokes waves

In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that higher order behaviour of series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Pade approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Pade approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Spherical image-based visual servo and structure estimation

This paper presents a formulation of image-based visual servoing (IBVS) for a spherical camera where coordinates are parameterized in terms of colatitude and longitude: IBVSSph. The image Jacobian is derived and simulation results are presented for canonical rotational, translational as well as general motion. Problems with large rotations that affect the planar perspective form of IBVS are not present on the sphere, whereas the desirable robustness properties of IBVS are shown to be retained. We also describe a structure from motion (SfM) system based on camera-centric spherical coordinates and show how a recursive estimator can be used to recover structure. The spherical formulations for IBVS and SfM are particularly suitable for platforms, such as aerial and underwater robots, that move in SE(3).