Arithmetic Operations Research Articles

Provably Secure and Area-Efficient Modular Addition over Boolean Shares

Several cryptographic schemes, including lattice-based cryptography and the SHA-2 family of hash functions, involve both integer arithmetic and Boolean logic. Each of these classes of operations, considered separately, can be efficiently implemented under the masking countermeasure when resistance against vertical attacks is required. However, protecting interleaved arithmetic and logic operations is much more expensive, requiring either additional masking conversions to switch between masking schemes, or implementing arithmetic functions as nonlinear operations over a Boolean masking. Both solutions can be achieved by providing masked arithmetic addition over Boolean shares, which is an operation with relatively long latency and usually high area utilization in hardware. A further complication arises when the arithmetic performed by the scheme is over a prime modulus, which is common in lattice-based cryptography. In this work, we propose a first-order masked implementation of arithmetic addition over Boolean shares occupying a very small area, while still having reasonable latency. Our proposal is specifically tuned for efficient addition and subtraction modulo an arbitrary integer, but it can also be configured at runtime for power-of-two arithmetic. To the best of our knowledge, we propose the first such construction whose security is formally proven in the glitch+transition-robust probing model.

Impact of working memory on arithmetic problem solving in primary school pupils with mathematical learning difficulties

Learning arithmetic operations is fundamental for social and professional emancipation, where counting and solving numerical operations is part of everyday behavior; this reflects the more or less serious difficulties that pupils with mathematics learning difficulties (MLD) may encounter. The interest of this work is to underline the importance of the cognitive deficit at the memory level in these subjects. In this sense we selected 30 pupils aged 7 to 8 years old, educated from 2nd to3rd year of the primary cycle with MLD on whom we applied tests of arithmetic problem solving and working memory (WM). Analysis of the results indicates a significant correlation between the difficulties recorded in arithmetic and WM. These results lead us to a systematic evaluation of WM in subjects with MLD, in any therapeutic diagnostic approach.

Enhancing computational efficiency in 3-D seismic modelling with half-precision floating-point numbers based on the curvilinear grid finite-difference method

SUMMARY Large-scale and high-resolution seismic modelling are very significant to simulating seismic waves, evaluating earthquake hazards and advancing exploration seismology. However, achieving high-resolution seismic modelling requires substantial computing and storage resources, resulting in a considerable computational cost. To enhance computational efficiency and performance, recent heterogeneous computing platforms, such as Nvidia Graphics Processing Units (GPUs), natively support half-precision floating-point numbers (FP16). FP16 operations can provide faster calculation speed, lower storage requirements and greater performance enhancement over single-precision floating-point numbers (FP32), thus providing significant benefits for seismic modelling. Nevertheless, the inherent limitation of fewer 16-bit representations in FP16 may lead to severe numerical overflow, underflow and floating-point errors during computation. In this study, to ensure stable wave equation solutions and minimize the floating-point errors, we use a scaling strategy to adjust the computation of FP16 arithmetic operations. For optimal GPU floating-point performance, we implement a 2-way single instruction multiple data (SIMD) within the floating-point units (FPUs) of CUDA cores. Moreover, we implement an earthquake simulation solver for FP16 operations based on curvilinear grid finite-difference method (CGFDM) and perform several earthquake simulations. Comparing the results of wavefield data with the standard CGFDM using FP32, the errors introduced by FP16 are minimal, demonstrating excellent consistency with the FP32 results. Performance analysis indicates that FP16 seismic modelling exhibits a remarkable improvement in computational efficiency, achieving a speedup of approximately 1.75 and reducing memory usage by half compared to the FP32 version.

Prevalence of Dehydration in Egyptian Primary School Children and its Impact on their Cognitive performance

Abstract Background More than 75% of children do not meet the water intake recommendations by the Institute of medicine. In addition, more than 80% of children in many European countries do not achieve the water intake recommended by the guidelines of the European food safety authority. Studies have revealed a correlation between hydration status and cognitive achievements in visual attention, immediate memory span, semantic flexibility, and automatic application of arithmetic operations in school children living in hot climates. Aim of the Work Assessment of the hydration status in Egyptian primary school children and determining the relationship between hydration status and cognitive function. Patients and Methods A cross-sectional questionnaire-based study was conducted on 200 children, the age group was from 9 to 13 years old. Data collection included the governmental, private and international schools in Cairo. The questionnaire was fulfilled during school time. A child-friendly “fluids and liquid food” diary was designed to prospectively record the volume of total fluid intake including water, sugar sweetened beverages and fruit juices in the previous day and during school day. Hydration status was assessed by self-reported urine colour using eight-point colour scale. The students were asked to answer four cognitive function tests that were designed to test the cognitive skills which were known to be affected by dehydration (Visual attention, Visual memory, Visual search, Visuomotor performance). Results 54 children (27%) had “Satisfactory knowledge” i.e. the children who met the criteria in the self- answered questionnaire and 146 children (73%) of them had “Unsatisfactory knowledge of proper total fluid intake. 166 children (83%) reported low total fluid intake and 34 children (17%) had adequate total fluid intake (according to the guidelines of the European food safety authority), 118 children (59%) were dehydrated, and 34 children (17%) were severely dehydrated. There was a statistically significant higher total fluid intake in international school regarding the amount of total fluid intake during school time and the amount of total fluid intake per day. In our study, the most consumed fluids after water were sugar sweetened beverages and fruit juices. There was a highly statistically significant positive correlation between total fluid intake and cognitive function regarding visual attention, visual memory, visual search, visuomotor performance and total score of cognitive functions, with p-value (p < 0.001). Conclusion We concluded that there was a significant correlation between school students’ hydration status and total fluid intake and their cognitive functions regarding their Visual attention, Visual memory, Visual search, Visuomotor performance. That calls for the importance of corroboration of our study results with studies that further explore students’ awareness and perception about their daily need of water and benefits of proper hydration.

Eksplorasi Etnomatematika pada Permainan Tradisional Congklak di Kelurahan Srengseng Sawah

Ethnomathematics contributes greatly to the improvement of mathematics learning, because it relates to students' experiences in everyday life that touches the realm of local cultural arts so that students become more understanding of the mathematical concepts explained. This study aims to explore the concept of mathematics in the traditional game of congklak in Srengseng Sawah Village. This research is a qualitative research with an ethnographic approach. The data collection techniques used are observation, interview and documentation. The instruments in this study were researchers, observation guidelines, interview guidelines and documentation. Based on the research analysis and discussion of the history and philosophy of the traditional game of congklak, mathematical concepts in the game of congklak such as the concept of flat shapes, namely semicircles, circles, rectangles and spatial shapes, namely half a ball; the concept of transformation in this case reflection / mirroring; the concept of arithmetic calculation operations including addition, subtraction, multiplication and division. The game also has cultural and philosophical values.

Analisis Kesulitan Belajar Peserta Didik Terhadap Materi Matematika, Khususnya Dalam Perkalian Berbentuk Cerita Di Kelas II Sekolah Dasar

Students who have difficulty learning mathematics will often make mistakes in learning to count and mistakes in solving word problems in mathematics. The difficulties experienced by students are difficulty in working on story problems due to the inability to understand the meaning of the questions and confusion when determining what arithmetic operations will be used to solve the story problems. The purpose of this research was to analyze the learning difficulties experienced by students as well as the factors that caused students’ learning difficulties in working on multiplication questions in the form of stories in class II elementary school. The method used in this research is descriptive case study. The subjects of this research are students and teachers of class II A at SDN 15 ULU GADUT. The data collection procedures used in the research are test’s, interviews, observation and documentation. The results of the research show that difficulties in learning mathematics include: 1) Difficulty in distinguishing multiplication problems from other material, 2) Student’s writing cannot be read, 3) Difficulty doing calculations in solving multiplication problems. Internal factors include: 1) Lack of enthusiasm for studying, 2) Low numeracy skills and difficulty understanding questions, 3) Not focusing on studying. External factors include: 1) Lack of attention from parents, 2) Often asking students to memorize multiplication, 3) Not using interesting learning media in teaching.

Employing Novel Ranking Function for Solving Fully Fuzzy Fractional Linear Programming Problems

البرمجة الضبابية مفيدة بشكل خاص في الحالات التي تكون فيها المعاملات غامضة. بسبب هذه الميزة ، في السنوات الأخيرة ، ظهرت العديد من التقنيات لمعالجة عدم اليقين. يقترح هذا البحث تقنية دالة ترتيب جديدة مع متغيرات من نوع الأرقام الضبابية العشرية لحل مشاكل البرمجة الخطية الكسرية الضبابية التامة (FFFLP). تعتمد هذه التقنية على إدخال دالة رتبية جديدة لرقم ضبابي عشري ، واستخدام الطريقة المبسطة الضبابية التامة. بعد تحويل مشكلة FFFLP إلى مشكلة البرمجة الخطية الضبابية التامة (FFLP) باستخدام الطريقة التكميلية ، ثم يتم حلها باستخدام الجداول المبسطة الضبابية التامة ، والتي تكون فيها جميع القيم أرقامًا عشرية ضبابية. بمساعدة العمليات الحسابية للأرقام العشرية ، يتم الوصول إلى التكرار الجديد للجدول البسيط. يتم تكرار الخطوات لحين الوصول إلى الحل الضبابي الأمثل. ولبيان الطريقة المقترحة، يتم تقديم مثال عددي يوضح خطوات ايجاد الحل الضبابي الأمثل للمشكلة.

Development of STEM-PjBL Learning Media to Improve Students' Critical Thinking Skills for Counting Whole Number and Fraction

This study aims to describe the process and results of the development of PjBL-based learning media integrated with STEM to improve students' critical thinking skills on the material of counting operations of whole numbers and fractions. This research is a development research with Plomp development model to develop learning tools in the form of Teaching Modules and Project Worksheets on arithmetic operations of whole numbers and fractions with STEM integrated project learning design. The instruments of this research are project worksheets, teaching modules, student response sheets, and pre-test and post-test sheets. Data collection techniques used questionnaires (validation sheets and student responses) and tests. Data analysis was conducted by testing the validity, practicality, and effectiveness of learning devices. The results showed that the learning devices obtained validity criteria of 87.9% for the Project Worksheet while 84.375% for the validity of the Teaching Module and each was classified as very valid. For the effectiveness criteria, the N-gain was 0.7512 in the limited trial and 0.7091 in the class trial and each obtained high effectiveness criteria. The learning tools obtained practical criteria for implementation with an average student response of 79.3% and 80.45% for project learning design. Thus, the learning tools are declared valid, effective, and practical to use.

Analisis Kesulitan Menyelesaikan Soal Cerita Matematika Pada Siswa Kelas IV SDN Mekarjaya 14

This research aims to analyze the difficulties of fourth grade students at SDN Mekarjaya 14 in solving mathematics story problems. This research uses qualitative research with an inductive method. This research aims to gain knowledge based on correct and reliable data and facts about students' mathematics learning difficulties in the form of story problems. The research instrument used was observation of test questions in the form of stories. The technique for checking the validity of the data is using triangulation. Based on the results of observations, there were still many respondents who made mistakes in working on the questions, including mistakes in changing story questions into mathematical sentences and using arithmetic operations. In general, students have difficulty working on story problems which are presented in the form of story problems. Apart from that, respondents also did not master multiplying results up to 100. In accordance with findings in the field, it can be concluded that there were difficulties experienced by respondents, including difficulties in understanding concepts and material,master and use arithmetic operations, for example multiplication and division. Meanwhile, the factors that caused respondents to experience difficulty learning mixed arithmetic operations included their lack of mastery of the concepts of multiplication and division, and the lack of time provided by respondents to repeat lessons at home.

The Effect of Stress on a Personal Identification System Based on Electroencephalographic Signals.

Personal identification systems based on electroencephalographic (EEG) signals have their own strengths and limitations. The stability of EEG signals strongly affects such systems. The human emotional state is one of the important factors that affects EEG signals' stability. Stress is a major emotional state that affects individuals' capability to perform day-to-day tasks. The main objective of this work is to study the effect of mental and emotional stress on such systems. Two experiments have been performed. In the first, we used hand-crafted features (time domain, frequency domain, and non-linear features), followed by a machine learning classifier. In the second, raw EEG signals were used as an input for the deep learning approaches. Different types of mental and emotional stress have been examined using two datasets, SAM 40 and DEAP. The proposed experiments proved that performing enrollment in a relaxed or calm state and identification in a stressed state have a negative effect on the identification system's performance. The best achieved accuracy for the DEAP dataset was 99.67% in the calm state and 96.67% in the stressed state. For the SAM 40 dataset, the best accuracy was 99.67%, 93.33%, 92.5%, and 91.67% for the relaxed state and stress caused by identifying mirror images, the Stroop color-word test, and solving arithmetic operations, respectively.

Towards Better Quantity Representations for Solving Math Word Problems

Solving a math word problem requires selecting quantities in it and performing appropriate arithmetic operations to obtain the answer. For deep learning-based methods, it is vital to obtain good quantity representations, i.e., to selectively and emphatically aggregate information in the context of quantities. However, existing works have not paid much attention to this aspect. Many works simply encode quantities as ordinary tokens, or use some implicit or rule-based methods to select information in their context. This leads to poor results when dealing with linguistic variations and confounding quantities. This article proposes a novel method to identify question-related distinguishing features of quantities by contrasting their context with the question and the context of other quantities, thereby enhancing the representation of quantities. Our method not only considers the contrastive relationship between quantities but also considers multiple relationships jointly. Besides, we propose two auxiliary tasks to further guide the representation learning of quantities: (1) predicting whether a quantity is used in the question and (2) predicting the relations (operators) between quantities given the question. Experimental results show that our method outperforms previous methods on SVAMP and ASDiv-A under similar settings, even some newly released strong baselines. Supplementary experiments further confirm that our method indeed improves the performance of quantity selection by improving the representation of both quantities and questions.

Effectiveness of Peer Tutoring in Enhancing Fundamental Arithmetic Operations

This study utilized the descriptive correlational design to determine the effectiveness of Peer tutoring in enhancing Fundamental Arithmetic Operations of the fifty-two (52) learners in Grade III-Lily and Jasmine of U-og Elementary School, Inabanga, Bohol. Most of the respondents were at the age of 9 and majority were female. A researchers-made survey questionnaire was used as tool in gathering the data. The gathered data were tallied and analyzed using frequency count, simple percentage, weighted mean, and Pearson Product Moment Correlation Coefficient to test the relationship between Peer tutoring and Fundamental Arithmetic Operations. Results showed that the computed r-value 0.413782 was higher than the critical r-value 0.2732 at 0.05 level of significance with 50 degrees of freedom, so the null hypothesis was rejected. Therefore, there was a significant relationship between Peer tutoring and Fundamental Arithmetic Operations which implied that peer tutoring is one of the strategies that may help and assist learners in enhancing fundamental arithmetic operations.

Interval-valued (p,q,r)-spherical fuzzy sets and their applications in MCGDM and MCDM based on TOPSIS method and aggregation operators

The (p,q,r)-Spherical fuzzy set is a useful generalization of other fuzzy structures such as fuzzy, intuitionistic fuzzy, Pythagorean fuzzy, picture fuzzy, q-rung orthopair fuzzy, spherical fuzzy, and T-spherical fuzzy sets. In this paper, we define concept of interval-valued (p,q,r)-spherical fuzzy set (IVpqr-SFS) and Set-theoretical and arithmetic operations of them. Two aggregation operators named interval-valued (p,q,r)-spherical fuzzy weighted arithmetic (IVpqr-SFWAA) and interval-valued (p,q,r)-spherical fuzzy weighted geometric (IVpqr-SFWGA) aggregation operators are introduced for IVpqr-SF numbers (IVpqr-SFN) and their properties were examined. The proposed operators are explained with examples. Furthermore, the score and accuracy functions of an IVpqr-SFN are introduced. We also define distance measures between two same types IVpqr-SFSs based on Hamming, Euclidean, and Hausdorff distance measures. Moreover, a multi-criteria group decision-making (MCGDM) method is developed based on the TOPSIS method by using the proposed distance measures and score function, and a numerical example is given to show the process of the proposed MCGDM method. In addition, a multi-criteria decision-making (MCDM) method is developed using the proposed IVpqr-SFWAA, IVpqr-SFWGA operators, and an illustrative example is given to show the process of the proposed MCDM method. Finally, a comparative analysis is presented between the proposed methods and the existing methods.

Integrasi Islam dan Sains: Telaah Terhadap Konsep Matematika dalam Hadits Keutamaan Membaca Shalawat

National education has the goal of shaping the morals of students who are truehearted and dedicated to God. As part of national education, Islamic-integrated mathematics learning becomes a bridge to achieve this goal. This study aims to explore the concept of mathematics in the hadith with the theme of the virtue of reciting shalawat. This research uses a qualitative approach with literature study methods. Data were obtained from hadith books, hadith applications, articles, and relevant documents. The results showed that there are three mathematical concepts in the hadith related to the virtue of reciting shalawat. The mathematical concepts found include the concept of numbers (natural numbers, integers, positive integers, odd numbers, and even numbers), arithmetic operations (addition operations), and the concept of relations and functions (bijective functions). The results of this study can be used as a reference for learning resources related to Islamic integrated mathematics derived from hadith with the theme of the virtue of reading shalawat.

Group decision making based on dice similarity measure and 2-dimension linguistic fuzzy weighted arithmetic aggregation operator for 2-dimension linguistic fuzzy variables

In the process of decision-making, it usually applies 2-dimension linguistic fuzzy variable (2DLFV) to describe the uncertainty caused by the decision objects and the decision makers. This proposal aims to design group decision making method based on dice similarity measure (DSM) and 2-dimension linguistic fuzzy weighted arithmetic aggregation (2DLFWAA) operator for 2-dimension linguistic fuzzy framework. Based on the psychological theory and prospect theory, 2-dimension linguistic scale function (2DLSF) about 2DLFV is firstly proposed, and new operational rules of 2DLFV are introduced. Two forms of DSM are proposed to measure the closeness between two 2-dimension linguistic fuzzy sets (2DLFSs). A definition is given for 2DLFWAA operator and some of its properties are studied. Based on the DSM and the 2DLFWAA operator for 2DLFSs, the technique for order of preference by similarity to ideal solution (TOPSIS) method is extended to handle the multiple attribute group decision making (MAGDM) problems under 2-dimension linguistic fuzzy environments. Finally, an illustrative example is provided to show the concrete process of the proposed method. Comparative analysis and sensitivity analysis are conducted to test the rationality and robust of the proposed method. The results suggest that the proposed method offers a flexible and reliable approach in 2-dimension linguistic fuzzy scenarios.

Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras

This article considers arrowhead and diagonal-plus-rank-one matrices in Fn×n where F∈{R,C,H} and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O(n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language.

Joint application of the Monte Carlo method and computational probabilistic analysis in problems of numerical modeling with data uncertainties

Abstract In this paper, we suggest joint application of computational probabilistic analysis and the Monte Carlo method for numerical stochastic modeling problems. We use all the capabilities of computational probabilistic analysis while maintaining all the advantages of the Monte Carlo method. Our approach allows us to efficiently implement a computational hybrid scheme. In this way, we reduce the computation time and present the results in the form of distributions. The crucial new points of our method are arithmetic operations on probability density functions and procedures for constructing on the probabilistic extensions. Relying on specific numerical examples of solving systems of linear algebraic equations with random coefficients, we present the advantages of our approach.

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

In this paper, we derive novel families of inclusion sets for the spectra and pseudospectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687–704].

High Efficiency Carry Save Adder using Modified–gate Diffusion Input Technique

Addition is a fundamental function in the design of a digital system, necessary for applications such as signal processing, arithmetic operations, multiplexers, and control systems. Hence, the digital system’s performance is considerably reliant on the efficiency of the adders. Therefore, designing a 4-bit carry save adder (CSA) that consumes less power, occupies a smaller area, and operates at a higher speed is proposed using the modified–gate diffusion input (MOD–GDI) technique. The primary focus is to reduce the area occupied by decreasing the transistor count as compared with other logic styles (i.e., conventional, and Boolean simplification) for CSA through Cadence Virtuoso simulation based on SilTerra 180 nm technology. Notably, the number of transistors is reduced from 42 in the conventional full adder to 11 in the proposed MOD–GDI design. As a result, the proposed 4-bit CSA with MOD–GDI technique is efficient in improving the speed of addition by reducing the area and power consumption.

Novel hybrid arithmetic optimization algorithm-recursive least square approach for power system harmonic estimation

Abstract Accurate harmonic estimation is essential for effective power quality assessment, designing appropriate harmonic filters, and ensuring the reliable operation of electrical equipment. This article proposes a novel hybrid harmonic estimation technique combining recursive least square (RLS) and arithmetic optimization algorithm (AOA) for accurate estimation of harmonics, inter-harmonics and sub-harmonics. AOA is a new meta-heuristic method based on distribution behaviour of main arithmetic operators such as addition, subtraction, multiplication and division. RLS is used for estimation of amplitude of harmonics, whereas phase is estimated by AOA. The performance of AOA–RLS is investigated in detail for estimation of power system signals using two set of test signals buried with noise. The proposed AOA–RLS is proved to be efficient for estimating both phase and amplitude parameters under different signal to noise ratio (SNR) conditions with an estimation of error of E−3. The efficacy of AOA–RLS technique is demonstrated by comparing with competitive existing techniques. The performance of AOA–RLS also verified in experimental studies.