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.

Time-Precision Flexible Arithmetic Unit

Paper submitted to the XVIII Conference on Design of Circuits and Integrated Systems (DCIS), Ciudad Real, España, 2003.

Calculation Methodology for Flexible Arithmetic Processing

Paper submitted to the IFIP International Conference on Very Large Scale Integration (VLSI-SOC), Darmstadt, Germany, 2003.

Hough Transform recursive evaluation using Distributed Arithmetic

Paper submitted to the IFIP International Conference on Very Large Scale Integration (VLSI-SOC), Darmstadt, Germany, 2003.

Proposal for a treatise on arithmetic, 1803

Handwritten proposal for the printing of a treatise on arithmetic by Croswell.

Arithmetic copybook : manuscript, [17--]

Arithmetic copybook containing mathematical rules, problems, proofs, and charts of weights and measures.

The harsh reality of Ukraine's fiscal arithmetic. Bruegel Policy Contribution ISSUE 2015/07, June 2015

Ukraine is struggling with both external aggression and the dramatically poor shape of its economy. The pace of political and institutional change has so far been too slow to prevent the deepening of the fiscal and balance-of-payments crises, while business confidence continues to be undermined. • Unfortunately, the 2015 International Monetary Fund Extended Fund Facility programme repeats many weaknesses of the 2014 IMF Stand-by Arrangement: slow pace of fiscal adjustment especially in the two key areas of energy prices and pension entitlements, lack of a comprehensive structural and institutional reform vision, and insufficient external financing to close the expected balance-of-payments gap and allow Ukraine to return to debt sustainability in the long term. • The reform process in Ukraine must be accelerated and better managed. A frontloaded fiscal adjustment is necessary to stabilise public finances and the balance-of-payments, and to bring inflation down. The international community, especially the European Union, should offer sufficient financial aid backed by strong conditionality, technical assistance and support to Ukraine’s independence and territorial integrity.