An Axiomatic Approach to Computer Arithmetic with an Appendix on Interval Hardware

Signal processing had a significant impact on the development of many elements of modern life, including telecommunications, education, healthcare, industry, and security. The semiconductor industry is the primary driver of signal processing innovation, producing ever-more sophisticated electronic devices and circuits in response to global demand. In addition, the central processing unit (CPU) is described as the “brain” of a computer or all electronic devices and signal processing. CPU is a critical electronic device that includes vital components such as memory, multiplier, adder, etc. Also, one of the essential components of the CPU is the arithmetic and logic unit (ALU), which executes the arithmetic and logical operations within all types of CPU operations, such as addition, multiplication, and subtraction. However, delay, occupied areas, and energy consumption are essential parameters in ALU circuits. Since the recent ALU designs experienced problems like high delay, high occupied area, and high energy consumption, implementing electronic circuits based on new technology can significantly boost the performance of entire signal processing devices, including microcontrollers, microprocessors, and printed devices, with high-speed and low occupied space. Quantum dot cellular automata (QCA) is an effective technology for implementing all electronic circuits and signal processing applications to solve these shortcomings. It is a transistor-less nanotechnology being explored as a successor to established technologies like CMOS and VLSI due to its ultra-low power dissipation, high device density, fast operating speed in THz, and reduced circuit complexity. This research proposes a ground-breaking ALU that upgrades electrical devices such as microcontrollers by applying cutting-edge QCA nanotechnology. The primary goal is to offer a novel ALU architecture that fully utilizes the potential of QCA nanotechnology. Using a new and efficient approach, the fundamental gates are skillfully utilized with a coplanar layout based on a single cell not rotated. Furthermore, this work presents an enhanced 1-bit and 2-bit arithmetic logic unit in quantum dot cellular automata. The recommended design includes logic, arithmetic operations, full adder (FA) design, and multiplexers. Using the powerful simulation tools QCADesigner, all proposed designs are evaluated and verified. The simulation outcomes indicates that the suggested ALU has 42.48 and 64.28% improvements concerning cell count and total occupied area in comparison to the best earlier single-layer and multi-layer designs.

Satisfiability Modulo Theories (SMT) solvers have been widely applied in automated software analysis to reason about the queries that encode the essence of program semantics, relieving the heavy burden of manual analysis. Many SMT solving techniques rely on solving Boolean satisfiability problem (SAT), which is an NP-complete problem, so they use heuristic search strategies to seek possible solutions, especially when no known theorem can efficiently reduce the problem. An emerging challenge, named Mixed-Bitwise-Arithmetic (MBA) obfuscation, impedes SMT solving by constructing identity equations with both bitwise operations (and, or, negate) and arithmetic computation (add, minus, multiply). Common math theorems for bitwise or arithmetic computation are inapplicable to simplifying MBA equations, leading to performance bottlenecks in SMT solving. In this paper, we first scrutinize solvers' performance on solving different categories of MBA expressions: linear, polynomial, and non-polynomial. We observe that solvers can handle simple linear MBA expressions, but facing a severe performance slowdown when solving complex linear and non-linear MBA expressions. The root cause is that complex MBA expressions break the reduction laws for pure arithmetic or bitwise computation. To boost solvers' performance, we propose a semantic-preserving transformation to reduce the mixing degree of bitwise and arithmetic operations. We first calculate a signature vector based on the truth table extracted from an MBA expression, which captures the complete MBA semantics. Next, we generate a simpler MBA expression from the signature vector. Our large-scale evaluation on 3000 complex MBA equations shows that our technique significantly boost modern SMT solvers' performance on solving MBA formulas.

Arithmetic and Logic Unit (ALU) is a combination circuit that performs a number of arithmetic and logical operations within a microprocessor.We present guidelines for low- and ultra low-power & high performance 32- bit ALU [1] units and analyze structures for logical operations to determine the most suitable for these regions of operations for low power applications.An ALU design is then modeled in VHDL for further testing with pre-manufactured parts of the 32 bit ALU using XILINX ISE tools. A novel method that chooses better designs of flipflops and latches in different places of the data path, based on the data and clock activities.Our simulation results indicate that, for the 180nm-65nm CMOS technologies [6] it is possible to reduce the ALU total energy by 18%-24%, with minimal delay degradation. In addition, there is up to 22%-32% reduction in leakage power in the standby mode.

Understanding the correct concepts can avoid misconceptions in basic mathematical material. There is still a mathematical misconception among PGSD students due to the low understanding of the concept. Lecturing the basic mathematical concepts as a subject that must be taken by PGSD students can be done by internalizing Islamic values in it. The concepts in mathematics are full of the values they contain including Islamic values. Therefore, it is necessary to have teaching materials on the basic concepts of mathematics based on the internalization of Islamic values. This study aims at 1) to determine the characteristics of teaching materials on the basic concepts of mathematics based on the internalization of Islamic values, 2) to improve understanding of mathematical concepts through learning with teaching materials on the basic concepts of mathematics based on the internalization of Islamic values. This research is a development research. Analysis of the data used included the validity analysis of teaching materials, analysis of practicality of teaching materials, analysis of the effectiveness of teaching materials, descriptive statistics, and Gain normalization tests. Improving the understanding of student mathematical concepts can be seen in the average score of 61.54 pretest and the post-test average score of 85.41. From the normalized gain test calculation shows an increase of 0.62 in the medium category. From the results of the study it is expected that the teaching materials of the basic concepts of mathematics based on the internalization of Islamic values can be applied in learning through various learning innovations with student centered. Keywords: Teaching materials of mathematics basic concepts in Elementary, Internalization of Islamic values, concept understanding Ability.

An all-optical arithmetic and logic unit is the integral part of optical computing and data processing. In this paper, the author proposes a method to develop an all-optical arithmetic and logic unit by means of which so many binary logic operations such as AND, OR, NAND, NOR, EXOR, data comparator etc., as well as some arithmetic operations, such as half-addition, half-subtraction, full-addition, full-subtraction multiplication, etc., can be performed by properly selecting the frequencies of the control signals and using the frequency encoded input data. For this purpose, the author has exploited the frequency conversion scheme of semiconductor optical amplifier (SOA) using nonlinear polarization rotation character of the probe beam in SOA and, cross-gain modulation character of reflecting semiconductor optical amplifier (RSOA). Frequency-dependent optical beam routing to different channels have done using optical add/drop multiplexer made of SOA with multiquantum well structure. Frequency conversion efficiency of SOA and RSOA are very high as well as switching speed of SOA is very fast with very good ON-OFF contrast ratio.

Design of integrated reversible fault-tolerant arithmetic and logic unit

Leading Zero Count (LZC) plays a crucial role in arithmetic and logic operations (ALU) within digital circuits, both sequential and combinational. LZC refers to the process of determining the number of consecutive leading zeros in a binary number. Both combinational and sequential LZC circuits find extensive usage in digital signal processing tasks, where quick and accurate determination of leading zeros is essential for various operations like normalization, filtering and feature extraction. Furthermore, they are integral components in hardware implementation of algorithms for data compression and encryption, contributing to enhanced efficiency and performance. The design principles, implementation strategies and applications of leading zero count in both combinational and sequential settings, highlighting their significance in modern digital systems. The LZC system is simulated and synthesized using Xilinx ISE tool and implemented on Artix-7 FPGA.

Back to table of contents Previous article Next article Book ReviewNo AccessThe Growth of Basic Mathematical and Scientific Concepts in ChildrenRichard Trautman, M.D.Richard TrautmanNew York, N.Y.Search for more papers by this author, M.D.Published Online:30 Apr 2018https://doi.org/10.1176/appi.psychotherapy.1963.17.2.332aAboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail PDFThis content is only available as a PDF. Access content To read the fulltext, please use one of the options below to sign in or purchase access. Personal login Institutional Login Sign in via OpenAthens Purchase Save for later Item saved, go to cart PPV Articles - APT - American Journal of Psychotherapy $35.00 Add to cart PPV Articles - APT - American Journal of Psychotherapy Checkout Please login/register if you wish to pair your device and check access availability. Not a subscriber? Subscribe Now / Learn More PsychiatryOnline subscription options offer access to the DSM-5 library, books, journals, CME, and patient resources. This all-in-one virtual library provides psychiatrists and mental health professionals with key resources for diagnosis, treatment, research, and professional development. Need more help? PsychiatryOnline Customer Service may be reached by emailing [email protected] or by calling 800-368-5777 (in the U.S.) or 703-907-7322 (outside the U.S.). FiguresReferencesCited byDetailsCited byNone Volume 17Issue 2 1963, pp. 169–362Pages 332-334 Metrics PDF download History Published online 30 April 2018 Published in print 1 April 1963

Arithmetic Logic Unit (ALU) as one of the main parts of any computing hardware plays an important role in digital computers. In quantum computers which can be realized by reversible logics and circuits, reversible ALUs should be designed. In this paper, we proposed three different designs for reversible 1-bit ALUs using our proposed 3×3 and 4×4 reversible gates called MEB3 and MEB4 (Moallem Ehsanpour Bolhasani) gates, respectively. The first proposed reversible ALU consists of six logical operations. The second proposed ALU consists of eight operations, two arithmetic, and six logical operations. And finally, the third proposed ALU consists of sixteen operations, four arithmetic operations, and twelve logical operations. Our proposed ALUs can be used to construct efficient quantum computers in nanotechnology, because the proposed designs are better than the existing designs in terms of quantum cost, constant input, reversible gates used, hardware complexity, and functions generated.

The aim of this study is to discuss the conceptual dimensions of pre-service mathematics teachers' approaches to the basic concepts and operations of mathematics. 25 pre-service mathematics teachers in the first semester of the 2022-2023 academic year, who were educated at a state university, participated in the research. Four open-ended questions about the basic concepts of mathematics were asked to pre-service mathematics teachers and their written opinions were received. Content and descriptive analysis were applied to the responses. As a result of the analyzes, pre-service teachers did not approach these questions in the context of conceptual knowledge, but generally gave the answers in the context of procedural knowledge. Pre-service mathematics teachers used reflective thinking, which they used to compare the procedural knowledge processes they gained in the formal education process with the previous solutions, in these question solutions as well.

The socialization of Visual Basic 6.0 based applications for calculating prime, odd and even numbers aims to introduce efficient tools for studying and implementing basic mathematical concepts interactively. This application is designed with a user-friendly interface to support mathematics teaching, both at secondary school level and in basic programming training activities. The app's key features include: Prime Number Check: Allows users to identify whether a number is a prime number. Odd and Even Classification: Provides a feature that automatically determines whether a certain number is odd or even. Calculations in a Specific Range: Provides calculation results for numbers within a specified range, supporting the exploration of mathematical patterns. The socialization process is carried out through training and interactive demos for potential users, such as teachers, students and the technology community. Evaluation shows that this application is able to improve understanding of basic mathematical concepts and speed up the manual calculation process. By utilizing Visual Basic 6.0 as a development platform, this application offers flexibility and compatibility with older operating systems, making it accessible to users with limited hardware. It is hoped that this application can be a useful educational tool in studying mathematics practically.

Several variants of spiking neural P systems (SNPS) have been presented in the literature to perform arithmetic operations. However, each of these variants was designed only for one specific arithmetic operation. In this paper, a complete arithmetic calculator implemented by SNPS is proposed. An application of the proposed calculator to information fusion is also proposed. The information fusion is implemented by integrating the following three elements: (1) an addition and subtraction SNPS already reported in the literature; (2) a modified multiplication and division SNPS; (3) a novel storage SNPS, i.e. a method based on SNPS is introduced to calculate basic probability assignment of an event. This is the first attempt to apply arithmetic operation SNPS to fuse multiple information. The effectiveness of the presented general arithmetic SNPS calculator is verified by means of several examples.

This chapter and the next two have three objectives. First, to introduce the reader to some basic concepts and formulas that will be needed in later chapters. Second, to serve as an introduction to computation and numerical methods and the use of Excel and Matlab procedures. The present chapter is devoted to mathematics and Chap. 3 is an introduction to computation and Chap. 4 will concentrate on probability theory and statistics. Those who are familiar with the material may want to glance through these chapters and move on. A third function of the chapters is to provide a handy reference for readers who, in reading later chapters, might feel a need to refresh their understanding of a concept or to check a formula.

In this paper, some counter-examples are given to analyze deeply several common easily confused basic concepts and theories in unary function calculus. Through the examples, it can seen that we must truly understand the basic concepts of advanced mathematics. In order to learn higher mathematics better; it needs us to have a deep understanding about the concept in unary function calculus. In this paper, a number of counter-examples are given to intensify these basic concepts and theories, and the authors gives some explanations for the concept of several common mistakes.

Numeracy skills are fundamental abilities that every student must possess to understand mathematical concepts. Indonesia ranks 74th out of 79 countries in terms of numeracy proficiency, indicating the low ability of students to comprehend basic mathematical concepts and solve numeracy-based problems. According to the 2024 National Assessment results, the majority of students in this region fall into the low category for numeracy literacy. This is further supported by data from SMPN 16 Mataram, where field observations reveal that many students struggle with mastering basic arithmetic operations such as addition, subtraction, multiplication, and division, which are foundational for understanding more advanced mathematical concepts. Therefore, this community service aims to help students develop skills in understanding and operating basic algebraic concepts, such as addition, subtraction, multiplication, and division of algebraic expressions. This service is conducted at SMPN 16 Mataram, located in Mataram city, with the participants being 23 students from class VII of SMP in Mataram. The data used in this service activity is sourced from the implementation of the Madrasah Indonesia Competency Assessment (AKM) guidance activity to improve students' literacy and numeracy competencies at SMPN 16 Mataram, conducted by 7th grade mathematics teacher. The evaluation of the activity was done through interviews with both the teacher and the students of grade 7th. The results of this activity show a significant improvement in the numeracy skills of students in algebraic arithmetic operations at SMPN 16 Mataram, class VII. This improvement is measurable, with the average score initially at 40, which then increased to 70 after the implementation of the program or community service activities. This indicates that students who previously had difficulty understanding and mastering algebraic arithmetic operations, such as addition, subtraction, multiplication, and division with variables or algebraic expressions, made significant progress after being given guidance and practice. Therefore, the addition of numeracy-focused programs is effective in helping students solve problems.