A note on bounded arithmetic

Abstract: This study aims to find out how the signs of the Qur’an against arithmetic operations in mathematics education, determine the urgency of arithmetic operations in the Qur’an against mathematics education, and see the relevance of arithmetic operations in the Qur’an to modern mathematics education. The method used in this study is qualitative in library research design, because all that wants to be investigated is sourced from literature and in this study using maudhu’s interpretation. The results of the study of arithmetic operations in the Qur’anic cues to Mathematics Education are: first, the Qur’an implies arithmetic operations in mathematics education implicitly wherein the sum by stating the results and in subtraction, multiplication and division only shows the part of the arithmetic operations, the existence of arithmetic operations in the Qur’an as a means of human thinking in order to think mathematically. Second, the urgency of arithmetic operations in the Qur’an for mathematics education is stated that arithmetic operations in the Qur’an are very important for mathematics education as a means of thinking, cues for the need for innovation in learning, justification of mathematical principles, erasing thought of dichotomy in science, and as a cue for learning mathematics. Third, the relevance of arithmetic operations in the Qur’an to modern mathematics education is in line with the spirit of Islamic integration in science and learning carried out by Islamic education in this modern era. Qur’anic arithmetic operations can be material in integrated mathematics learning. The existence of arithmetic operations in the Qur’an for mathematics education is also very related in learning the 2013 curriculum which emphasizes the scientific approach.

The purpose of this thesis was to further our understanding of the role of distributed cognition (with the use of pen and paper) in defusing the impact of evaluative pressure caused by priming gender-related stereotypes about girls’ maths performance and performance-approach goals on mental arithmetic performance. Interactivity is the transferring of internal cognitive process (e.g., computing simple maths tasks) to the outside world by using different tools (e.g., pen and paper). Some members of social groups (e.g., women) may not perform well in mathematics after negative stereotypes about their academic performance in the mathematical domain, which is known as stereotype threat. Stereotype threat is the risk of confirming a negative stereotype expectation about one’s group. Another decrement to performance may be caused by achievement goals, such as performance-approach goals. Performance-approach goals are linked to normative behaviour where the individual is motivated by outperforming others in academic performance. Negative stereotyping and performance-approach goals can generate anxiety that deplete existing working memory resources. However, some of these working memory limitations can be compensated by off-loading the internal cognitive process to the external environment. We tested whether off-loading could buffer the effects of stereotype threat and performance-approach goals in four experiments. In Studies 1 (16-year-old girls) and 2 (female university students), participants carried out mental arithmetic tasks in the stereotype threat condition or control, crossed with interactivity or no interactivity. There was increased maths performance (accuracies) with interactivity, confirming existing literature. Additionally, the solution latencies were improved when the mental arithmetic tasks were in a known format. However, when the maths tasks were in a novel format, the participants of the second study became slower because of speed-accuracy trade-off. The first two studies found no statistically significant effects of stereotype threat on maths performance. Nevertheless, working memory in participants in Study 1 was depleted in the stereotype threat condition, but it did not affect mental arithmetic performance. Finally, the participants in the interactive conditions in Study 2 had a reduction of their state maths anxiety levels measured at the end of the experiment. Studies 3 (Pilot Study) and 4 focused on achievement goals and their differing effects on working memory. Female university students carried out modular arithmetic tasks in a performance-approach goal or mastery-approach goal condition crossed with interactivity or no interactivity. Performance-approach goal endorsement hampered cognitive performance, as measured by maths accuracy in Study 3, but not in Study 4. These findings were extended in Study 4 where these negative effects were reduced with the help of interactivity. Across both studies, individuals in the mastery-approach goal condition had a performance drop in the interactive condition (Study 3 and 4). Thus, interactivity did not benefit the cognitive performance of these participants. Finally, Study 4 reported higher maths anxiety levels for the individuals in the performance-approach condition. However, the increased maths anxiety levels were not reduced with the help of distributed cognition. Reasons for the findings and future implications will be discussed.

IEEE 754 floating-point arithmetic is widely used in modern, general-purpose computers. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. Modifying the IEEE arithmetic so that it uses transreal arithmetic has a number of advantages. It removes one redundant binade from IEEE floating-point objects, doubling the numerical precision of the arithmetic. It removes eight redundant, relational, floating-point operations and removes the redundant total order operation. It replaces the non-reflexive, floating-point, equality operator with a reflexive equality operator and it indicates that some of the exceptions may be removed as redundant - subject to issues of backward compatibility and transient future compatibility as programmers migrate to the transreal paradigm.

This thesis examines children's arithmetic strategies and their relation to the concepts of commutativity and associativity. Two complementary methods were used in this research: empirical studies and computational models. Empirical studies were carried out to identify the strategies children used for solving problems like 3 + 4, and 3 + 4 + 7, and the conceptual knowledge associated with them. Their understanding of subtraction problems where the minuend is less than the subtrahend (e.g. 6 - 8) was also considered. A study with 105 subjects revealed a variety of strategies and information about children's knowledge of commutativity and associativity. Four levels of performance of commutativity were also identified. A longitudinal study was carried out with 12 children in order to obtain details of children's changes in strategy, and to double check the results obtained in the main study. The strategies observed to be used by children over a 20 month period parallel those found in previous studies, which show a general transition to more efficient methods. However, the longitudinal study revealed that development of such arithmetic strategies is a slow process. Furthermore, the studies indicated that knowledge of commutativity is a prerequisite for associativity. Models of the observed strategies have been implemented in the form of production rules in a computer program called PALM. The process of implementation highlighted features of children's problem solving that had not been 'detected during the studies. In addition to models that describe the space of strategies, a model of learning has been implemented for the transition from procedural knowledge of commutativity to that of associativity. The model is capable of generalizing its inbuilt knowledge, for instance, its ability to solve 2-term arithmetic expressions, to allow it to solve more complex problems, such as 3-term arithmetic expressions. A further model has been constructed for learning arithmetic strategies that are more efficient than those already represented in the program. It learns specific rules by adding conditions for efficient problem solving to its previous general rules.

This study aims to analyze Antonio Trajano's primary Arithmetics as tools for teaching practice adopted by teachers who taught mathematics in the second half of the 19th century. For that, this research is delimited to the study of classbooks named Primary Arithmetics and Illustrated Elementary Arithmetics, published in 1886 and 1879, respectively. His books represented a change in arithmetics teaching methodology at that time and became an editorial success. To fulfill the analysis, It is crucial to bring themes, such as the author's professional trajectory, his experiences in the educacional field, the project of the Presbiterian Church for education, the influence of the laws related to primary education, the education of the mathematics teacher, the teaching material produced at that time and which methodologies were used in primary education. This study takes into account that the production of a classbook is not limited to the words in its pages and that this type of material has two main readers: the student and the teacher. A communicative structure between the author or the book and the teacher is built on interpretations that involve paratextual and textual elements,, such as the text written to teachers, the organizarion of the teaching text, the structure of problems and solutions, the activities proposed and the illustrations. This work aims to displace researches on the classbook by considering its construction as a teacher's tool and not an analysis from the student's point of view.

Decreasing feature sizes allow additional functionality to be added to future microprocessors to improve the performance of important application domains. As a result of rapid growth in financial, commercial, and Internet-based applications, hardware support for decimal floating-point arithmetic is now being considered by various computer manufacturers and specifications for decimal floating-point arithmetic have been added to the draft revision of the IEEE-754 Standard for Floating-Point Arithmetic (IEEE-754R). This work presents an efficient arithmetic algorithm and hardware design for decimal floating-point division. The design uses an optimized piecewise linear approximation, a modified Newton-Raphson iteration, a specialized rounding technique, and a simplified combined decimal incrementer/decrementer. Synthesis results show that a 64-bit (16-digit) implementation of the decimal divider, which is compliant with IEEE-754R, has an estimated critical path delay of 0.69 ns when implemented using LSI Logic's 0.11 micron gflx-p standard cell library.

An optimal high-speed and low-power VLSI architecture requires an efficient arithmetic processing unit that is optimized for speed and power consumption. Adders are one of the widely used in digital integrated circuit and system design. High speed adder is the necessary component in a data path, e.g. Microprocessors and a Digital signal processor. The present paper proposes a novel high-speed adder by combining the advantages of Carry Look Ahead Adder (CLAA) and Carry Select Adder (CSA), devising a hybrid CSA. In the proposed adder, CSA uses CLAA technology to generate the carry bits for each sum bit, which are then used to select the respective multiplexer (MUX) which adds the carry bit to the sum accordingly. The proposed adder has been synthesized with bulk 40 nm standard CMOS library on Synopsys Design Compiler. Analysis has indicated the superiority of proposed adder over CLAA and CSA. As compared to CSA and CLAA, the proposed Carry Select Ahead Adder (CSAA) provides shorter average path and a simpler hardware. This has led to faster processing speed by increasing the complexity of circuit on the chip. The proposed adder finds its applications in various Arithmetic and Logic Units (ALU) of CPUs for faster arithmetic results. Keywords—Carry Look Ahead Adder (CLAA), Carry Select Adder (CSA), Multiplexer (MUX), Carry Select Ahead Adder (CSAA), Arithmetic and Logic Unit (ALU).

The Arithmetization of Analysis

Objective To explore the mathematics cognitive function of epileptic children with mathematical Event-related potential(ERP). Methods ERPs from 15 epileptic children and 15 controls were measured and analyzed when they were instructed to perform a mental calculation task which included addition, subtraction and multiplication operations.Results (1)Compared with the control group, the amplitudes of P3 and N2 decreased obviously and the latencies of P3 and N2 prolonged in epileptic children group( P <0.05).(2)The response time was longer[as follow:(addition(949.0±167.2)ms,subtraction(1007.3±166.5)ms,multiplication(985.0±153.6)ms);addition(769.5±149.5)ms,subtraction(799.5±176.7)ms,multiplication(794.6±167.3)ms,P <0.05] in the all three arithmetic operations and there were more errors [as follow:(addition(18.8±13.0)%,subtraction(33.1±21.1)%,(addition(6.4±4.4)%,subtraction(7.5±7.4)%,P <0.05] in addition and subtraction in epileptic children group compared to the controls.Conclusion There is the dysfunction of mental arithmetic in epileptic children,such as addition,subtraction.The ERP may be used as a quantitative index to monitor arithmetic cognitive impairment in epileptic patients. Key words: Epileptic children; Arithemtic; Event-related potential

This study compares the performance of students with Non-Verbal Learning Disabilities (NVLD) and students with Asperger Syndrome (AS) in Solving Arithmetic Problems (SAP). The participants were 15 students with NVLD and 12 students with AS, who were asked to solve 15 word problems of various arithmetic operations. The results showed that both groups faced significant difficulties in SAP. Difficulties have appeared in all phases of the SAP. More intense difficulties appeared in the phase of forming the necessary coherent mental representation of the problem, which is the prerequisite for the activation of cognitive schemata and the choice of the appropriate arithmetic operation. The two groups showed more similarities than differences. Results are discussed in terms of the methodological choices for their instructional support. Article visualizations:

We propose a novel integration of interval constraint propagation (ICP) with SMT solvers for linear real arithmetic (LRA) to decide nonlinear real arithmetic problems. We use ICP to search for interval solutions of the nonlinear constraints, and use the LRA solver to either validate the solutions or provide constraints to incrementally refine the search space for ICP. This serves the goal of separating the linear and nonlinear solving stages, and we show that the proposed methods preserve the correctness guarantees of ICP. Experimental results show that such separation is useful for enhancing efficiency.

Objective To discuss the arithmetic of the body residual nitrogen tension with fast buoyancy ascent escape from the DISSUB,following complicated hyperbaric exposure.Methods The processes of fast buoyancy ascent escape were analyzed,the theoretical arithmetic of the residual nitrogen tension in the body at the break of water surface was established with infinitesimal calculus,and the resulting residual nitrogen tension value was converted into corresponding saturation depth.Results The established formula could be used to calculate the residual nitrogen tension in the body at different escape depths,corresponding to saturation depths following saturation at 7 m and 12 m.Conclusions The established theoretical arithmetic could provide reference for the forecasting of DSC and could also provide theoretical evidence for the safety of fast buoyancy ascent escape following complicated hyperbaric exposure. Key words: Submarine; Fast buoyancy ascent escape; Escape depth curve; Nitrogen tension; Decompression sickness

Most high-capacity, 3D data-hiding schemes introduce irreversible changes into the host model. For artistic or technical models, not modifying the original mesh models is sometimes very important. This work describes a novel, multi-layer, reversible data-hiding scheme for 3D mesh models. The proposed scheme starts with finding an independent set of vertices for embedding data. The prediction residues of the vertices and the secret bit string are then encoded using the arithmetic coding method. The embedding procedure is done by substituting part of the mantissa of these vertices with the coded arithmetic decimal numbers. A maximum distortion constraint is set for each embedding vertex, such that the distortion on the cover model is controlled to be imperceptible. Users can construct a multi-layer embedding scheme by repeatedly applying the single-layer embedding procedure until the secret bit string is completely embedded. By integrating the arithmetic encoder and the substitution operation, the proposed method achieves 4-8 bits/vertex embedding capacity while reserving the ability of rebuilding the original model. Experimental results show that the proposed scheme can successfully reconstruct the original model while extracting the secret bit string.

In recent years there has been a flurry of activity proving lower bounds for homogeneous depth-4 arithmetic circuits, which has brought us very close to statements that are known to imply VP != VNP. It is a big question to go beyond homogeneity, and in this paper we make progress towards this by considering depth-4 circuits of low algebraic rank, which are a natural extension of homogeneous depth-4 arithmetic circuits. A depth-4 circuit is a representation of an N-variate, degree n polynomial P as P = sum_{i=1}^T Q_{i1} * Q_{i2} * ... * Q_{it} where the Q_{ij} are given by their monomial expansion. Homogeneity adds the constraint that for every i in [T], sum_{j} degree(Q_{ij}) = n. We study an extension where, for every i in [T], the algebraic rank of the set of polynomials {Q_{i1}, Q_{i2}, ... ,Q_{it}} is at most some parameter k. We call this the class of spnew circuits. Already for k=n, these circuits are a strong generalization of the class of homogeneous depth-4 circuits, where in particular t<=n (and hence k<=n). We study lower bounds and polynomial identity tests for such circuits and prove the following results. 1. Lower bounds: We give an explicit family of polynomials {P_n} of degree n in N = n^{O(1)} variables in VNP, such that any spnewn circuit computing P_n has size at least exp{(Omega(sqrt(n)*log(N)))}. This strengthens and unifies two lines of work: it generalizes the recent exponential lower bounds for homogeneous depth-4 circuits [KLSS14, KS-full] as well as the Jacobian based lower bounds of Agrawal et al. which worked for spnew circuits in the restricted setting where T * k <= n. 2. Hitting sets: Let spnewbounded be the class of spnew circuits with bottom fan-in at most d. We show that if d and k are at most poly(log(N)), then there is an explicit hitting set for spnewbounded circuits of size quasipolynomial in N and the size of the circuit. This strengthens a result of Forbes which showed such quasipolynomial sized hitting sets in the setting where d and t are at most poly(log(N)). A key technical ingredient of the proofs is a result which states that over any field of characteristic zero (or sufficiently large characteristic), upto a translation, every polynomial in a set of algebraically dependent polynomials can be written as a function of the polynomials in the transcendence basis. We believe this may be of independent interest. We combine this with shifted partial derivative based methods to obtain our final results.

This study investigates the pricing of the arithmetic average reset option. The option price is formulated as the solution of the Black-Scholes equation. In addition, the valuation is derived from a series of initial value problems based on the Green function through integration. Finally, the reset option price is numerically calculated. Throughout the numerical method, we can derive reset option prices of both arithmetic average and geometric average reset options. This study also presents the numerical examples for comparison.