Alternate treatments of jacobian singularities in polar coordinates within finite-difference schemes

Jacobian singularities of differential operators in curvilinear coordinates occur when the Jacobian determinant of the curvilinear-to-Cartesian mapping vanishes, thus leading to unbounded coefficients in partial differential equations. Within a finite-difference scheme, we treat the singularity at the pole of polar coordinates by setting up complementary equations. Such equations are obtained by either integral or smoothness conditions. They are assessed by application to analytically solvable steady-state heat-conduction problems.

Practising Arithmetic Using Educational Video Games with an Interpersonal Computer

Studies show the positive effects that video games can have on student performance and attitude towards learning. In the past few years, strategies have been generated to optimize the use of technological resources with the aim of facilitating widespread adoption of technology in the classroom. Given its low acquisition and maintenance costs, the interpersonal computer allows individual interaction and simultaneous learning with large groups of students. The purpose of this work was to compare arithmetical knowledge acquired by third-grade students through the use of game-based activities and non-game-based activities using an interpersonal computer, with knowledge acquired through the use of traditional paper-and-pencil activities, and to analyze their impact in various socio-cultural contexts. To do this, a quasi-experimental study was conducted with 271 students in three different countries (Brazil, Chile, and Costa Rica), in both rural and urban schools. A set of educational games for practising arithmetic was developed and tested in six schools within these three countries. Results show that there were no significant differences (ANCOVA) in the learning acquired from game-based vs. non-game-based activities. However, both showed a significant difference when compared with the traditional method. Additionally, both groups using the interpersonal computer showed higher levels of student interest than the traditional method group, and these technological methods were seen to be especially effective in increasing learning among weaker students.

Quaternion orders over quadratic integer rings from arithmetic fuchsian groups

In this paper we show that the quaternion orders OZ[ √ 2] ≃ ( √ 2, −1)Z[ √ 2] and OZ[ √ 3] ≃ (3 + 2√ 3, −1)Z[ √ 3], appearing in problems related to the coding theory [4], [3], are not maximal orders in the quaternion algebras AQ( √ 2) ≃ ( √ 2, −1)Q( √ 2) and AQ( √ 3) ≃ (3 + 2√ 3, −1)Q( √ 3), respectively. Furthermore, we identify the maximal orders containing these orders.

Forcing for hat inductive definitions in arithmetic

By forcing, we give a direct interpretation of inline image into Avigad's inline image. To the best of the author's knowledge, this is one of the simplest applications of forcing to “real problems”.

Spatial biases during mental arithmetic: evidence from eye movements on a blank screen

While the influence of spatial-numerical associations in number categorization tasks has been well established, their role in mental arithmetic is less clear. It has been hypothesized that mental addition leads to rightward and upward shifts of spatial attention (along the “mental number line”), whereas subtraction leads to leftward and downward shifts. We addressed this hypothesis by analyzing spontaneous eye movements during mental arithmetic. Participants solved verbally presented arithmetic problems (e.g., 2 + 7, 8–3) aloud while looking at a blank screen. We found that eye movements reflected spatial biases in the ongoing mental operation: Gaze position shifted more upward when participants solved addition compared to subtraction problems, and the horizontal gaze position was partly determined by the magnitude of the operands. Interestingly, the difference between addition and subtraction trials was driven by the operator (plus vs. minus) but was not influenced by the computational process. Thus, our results do not support the idea of a mental movement toward the solution during arithmetic but indicate a semantic association between operation and space.

Unfolding Feasible Arithmetic and Weak Truth

In this paper we continue Feferman’s unfolding program initiated in (Feferman, vol. 6 of Lecture Notes in Logic, 1996) which uses the concept of the unfolding U(S) of a schematic system S in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried through for a schematic system of non-finitist arithmetic NFA in Feferman and Strahm (Ann Pure Appl Log, 104(1–3):75–96, 2000) and for a system FA (with and without Bar rule) in Feferman and Strahm (Rev Symb Log, 3(4):665–689, 2010). The present contribution elucidates the concept of unfolding for a basic schematic system FEA of feasible arithmetic. Apart from the operational unfolding U0(FEA) of FEA, we study two full unfolding notions, namely the predicate unfolding U(FEA) and a more general truth unfolding UT(FEA) of FEA, the latter making use of a truth predicate added to the language of the operational unfolding. The main results obtained are that the provably convergent functions on binary words for all three unfolding systems are precisely those being computable in polynomial time. The upper bound computations make essential use of a specific theory of truth TPT over combinatory logic, which has recently been introduced in Eberhard and Strahm (Bull Symb Log, 18(3):474–475, 2012) and Eberhard (A feasible theory of truth over combinatory logic, 2014) and whose involved proof-theoretic analysis is due to Eberhard (A feasible theory of truth over combinatory logic, 2014). The results of this paper were first announced in (Eberhard and Strahm, Bull Symb Log 18(3):474–475, 2012).

Grid-based histogram arithmetic for the probabilistic analysis of functions

The selection of predefined analytic grids (partitions of the numeric ranges) to represent input and output functions as histograms has been proposed as a mechanism of approximation in order to control the tradeoff between accuracy and computation times in several áreas ranging from simulation to constraint solving. In particular, the application of interval methods for probabilistic function characterization has been shown to have advantages over other methods based on the simulation of random samples. However, standard interval arithmetic has always been used for the computation steps. In this paper, we introduce an alternative approximate arithmetic aimed at controlling the cost of the interval operations. Its distinctive feature is that grids are taken into account by the operators. We apply the technique in the context of probability density functions in order to improve the accuracy of the probability estimates. Results show that this approach has advantages over existing approaches in some particular situations, although computation times tend to increase significantly when analyzing large functions.

Method to analyze the influence of hysteresis in optical arithmetic units

A new method to analyze the influence of possible hysteresis cycles in devices employed for optical computing architectures is reported. A simple full adder structure is taken as the basis for this method. Single units, called optical programmable logic cells, previously reported by the authors, compose this structure. These cells employ, as basic devices, on-off and SEED-like components. Their hysteresis cycles have been modeled by numerical analysis. The influence of the different characteristic cycles is studied with respect to the obtained possible errors at the output. Two different approaches have been adopted. The first one shows the change in the arithmetic result output with respect to the different values and positions of the hysteresis cycle. The second one offers a similar result, but in a polar diagram where the total behavior of the system is better analyzed.