Arithmetic Procedures Research Articles

IFM/IFEk/1 Queuing Model with Erlang Service under Heptagonal and Octagonal Intuitionistic Fuzzy Numbers

Objectives: This research identifies the steps involved in determining the membership and non-membership functions of significant State execution proportions in Erlang services and models α-cut operations. Additionally, we applied the DSW Algorithmic Rule to intuitionistic fuzzy heptagonal and octagonal numbers. In this case, the inter-entry rate is Poisson and exponential, both of which have an intuitionistic Erlang nature. The main objective of this research is to evaluate the performance of a single server queuing model in terms of fuzzy and intuitionistic queuing theory. Fuzzy and intuitionistic fuzzies are the characteristics of the entering and exiting rates, respectively. Using a predetermined technique, an approach is carried out to assess the process metrics, taking the fuzzy numbers at face value without converting them into crisp values. Methods: We use the -cut method in IFM/IFEk/1 Single Server Queuing Model in FCFS with Erlang service. By applying increasing and decreasing entrance costs, the Poisson process is used to manage the arrival rates while delivering the input. Intuitionistic fuzzy queuing models are used to extend the queuing model. Findings: We noticed that the model indicated that performance can be enhanced by adding more factors. Comparing this procedure to fuzzification, accuracy is better. Shopkeepers, dealers, and companies will all benefit from the suggested model's assistance in precisely identifying the best performance metrics for the awaiting system. Novelty: Using an intuitionistic fuzzy queue, this research presents a novel method of assessing the M/Ek/1 queuing model in Erlang services. The computation of -cut intervals is an arithmetic procedure for intuitionistic fuzzy numbers. This novel approach assesses the on-server performance of an Erlang queuing model by extending fuzzy into intuitionistic fuzzy. The fuzzy waiting theory modeling evaluation metrics are given as a range of values, in contrast to the intuitionistic model, which offers a wide range of values. Applications: Several emails arrive at a network device in a sequential manner according to a Poisson arrival process, and each email's execution time is random. When mail arrives, it is queued and processed in the order that it arrives if the network server is busy. Using fuzzy queuing theory, the emails are "customers" and the network is the "server". The single server mathematical framework is a simple tool that can be used to restore the fuzzy queuing model. The parameters are all uncertain. Keywords: Queuing Theory, Heptagonal and Octagonal Fuzzy Numbers, Heptagonal and Octagonal Intuitionistic Fuzzy Numbers, Erlang Service, DSW Algorithm, Performance Measures

Chromatic aberration compensation in a digital camera lens for fringe projection profilometry

This work presents a simplified method for compensating the chromatic aberration of a digital camera lens using orthogonal fringe patterns displayed on an LCD monitor. The proposed compensation method drastically reduces the complex mathematical calculations to an arithmetic procedure of the simple rule of three. First, the implemented method is simulated, and then the experimental results confirm the simulated model. The mean squared error and standard deviation of the chromatic aberration compensation are calculated. Additionally, an arrangement is presented to avoid systematic errors when calculating the chromatic aberration of the lens to be compensated without additional digital processing.

Profiles of mathematical deficits in children with dyslexia

Despite a high rate of concurrent mathematical difficulties among children with dyslexia, we still have limited information regarding the prevalence and severity of mathematical deficits in this population. To address this gap, we developed a comprehensive battery of cognitive tests, known as the UCSF Mathematical Cognition Battery (MCB), with the aim of identifying deficits in four distinct mathematical domains: number processing, arithmetical procedures, arithmetic facts retrieval, and geometrical abilities. The mathematical abilities of a cohort of 75 children referred to the UCSF Dyslexia Center with a diagnosis of dyslexia, along with 18 typically developing controls aged 7 to 16, were initially evaluated using a behavioral neurology approach. A team of professional clinicians classified the 75 children with dyslexia into five groups, based on parents’ and teachers’ reported symptoms and clinical history. These groups included children with no mathematical deficits and children with mathematical deficits in number processing, arithmetical procedures, arithmetic facts retrieval, or geometrical abilities. Subsequently, the children underwent evaluation using the MCB to determine concordance with the clinicians’ impressions. Additionally, neuropsychological and cognitive standardized tests were administered. Our study reveals that within a cohort of children with dyslexia, 66% exhibit mathematical deficits, and among those with mathematical deficits, there is heterogeneity in the nature of these deficits. If these findings are confirmed in larger samples, they can potentially pave the way for new diagnostic approaches, consistent subtype classification, and, ultimately personalized interventions.

Explaining procedures and interleaving practice in fraction arithmetic

BackgroundExplicit instruction is a pedagogical approach in which students are taught step-by-step procedures for solving problems and practice using the procedures. Despite considerable evidence supporting the use of this general approach, relatively little is known about how varying specific design features impact its effectiveness. AimsThis preregistered study investigated the impact of two design features, explaining rationales for procedures and practicing in interleaved sequence, on the efficacy of explicit instruction in fraction arithmetic. SampleParticipants were 156 sixth to ninth grade children. MethodsChildren received a brief intervention involving explicit instruction in fraction arithmetic procedures. They were randomly assigned to receive or not receive explanations of rationales for the procedures, and to practice different procedures in either interleaved or blocked sequence. ResultsRelative to pretest, accuracy improved on immediate and delayed posttests (Cohen's d = 0.88 and 0.55 respectively). The magnitude of this improvement was not affected by either experimentally-manipulated factor or by both of them together. ConclusionsThe results support the use of explicit instruction in fraction arithmetic, but provide no evidence that explaining rationales for procedures, or interleaving practice thereof, increases the effects of such instruction on procedural knowledge.

Distinct Contributions of the Cerebellum and Basal Ganglia to Arithmetic Procedures.

Humans exhibit complex mathematical skills attributed to the exceptional enlargement of neocortical regions throughout evolution. In the current work, we initiated a novel exploration of the ancient subcortical neural network essential for mathematical cognition. Using a neuropsychological approach, we report that degeneration of two subcortical structures, the cerebellum and basal ganglia, impairs performance in symbolic arithmetic. We identify distinct computational impairments in male and female participants with cerebellar degeneration (CD) or Parkinson's disease (PD). The CD group exhibited a disproportionate cost when the arithmetic sum increased, suggesting that the cerebellum is critical for iterative procedures required for calculations. The PD group showed a disproportionate cost for equations with increasing addends, suggesting that the basal ganglia are critical for chaining multiple operations. In Experiment 2, the two patient groups exhibited intact practice gains for repeated equations at odds with an alternative hypothesis that these impairments were related to memory retrieval. Notably, we discuss how the counting and chaining operations relate to cerebellar and basal ganglia function in other task domains (e.g., motor processes). Overall, we provide a novel perspective on how the cerebellum and basal ganglia contribute to symbolic arithmetic. Our studies demonstrate the constraints on the computational role of two subcortical regions in higher cognition.

Halin’s infinite ray theorems: Complexity and reverse mathematics

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well.

The brain lateralization and development of math functions: progress since Sperry, 1974.

In 1974, Roger Sperry, based on his seminal studies on the split-brain condition, concluded that math was almost exclusively sustained by the language dominant left hemisphere. The right hemisphere could perform additions up to sums less than 20, the only exception to a complete left hemisphere dominance. Studies on lateralized focal lesions came to a similar conclusion, except for written complex calculation, where spatial abilities are needed to display digits in the right location according to the specific requirements of calculation procedures. Fifty years later, the contribution of new theoretical and instrumental tools lead to a much more complex picture, whereby, while left hemisphere dominance for math in the right-handed is confirmed for most functions, several math related tasks seem to be carried out in the right hemisphere. The developmental trajectory in the lateralization of math functions has also been clarified. This corpus of knowledge is reviewed here. The right hemisphere does not simply offer its support when calculation requires generic space processing, but its role can be very specific. For example, the right parietal lobe seems to store the operation-specific spatial layout required for complex arithmetical procedures and areas like the right insula are necessary in parsing complex numbers containing zero. Evidence is found for a complex orchestration between the two hemispheres even for simple tasks: each hemisphere has its specific role, concurring to the correct result. As for development, data point to right dominance for basic numerical processes. The picture that emerges at school age is a bilateral pattern with a significantly greater involvement of the right-hemisphere, particularly in non-symbolic tasks. The intraparietal sulcus shows a left hemisphere preponderance in response to symbolic stimuli at this age.

Scrutinizing the performance of GIS-based analytical Hierarchical process approach and frequency ratio model in flood prediction – Case study of Kakegawa, Japan

Floods are one of the most common catastrophes in the world. This study generates the flood susceptibility maps (FSM) using AHP and FR in Kakegawa, Japan. A set of 100 flood points were presented in an ArcGIS environment where 70 points were chosen at random as a training dataset while 30 ones were used for validation. Eleven flood causative factors were calculated and utilized to generate the flood vulnerability maps. For the validation 30% data sub-sample set, FSM was completed by creating the receiver operating characteristic curve and the area under the curve (AUC). The results indicate that the two methods show sensible accuracy since the AUC for FR and AHP are 67% and 85.5% respectively. AHP showed higher accuracy due to the expert opinion that being shared while FR achieved lower precision because of its simple arithmetic procedures. The results help decision-makers in determining the locations vulnerable to flooding.

Penguatan Konten Matematika bagi Guru Sekolah Dasar di Kecamatan Takari Kabupaten Kupang

Mathematics material for some elementary school teachers in Takari District, Kupang Regency, NTT is difficult material. So that learning mathematics tends to only train technical skills in counting while understanding and thinking skills are ignored. In general, there are three main problems, namely: 1) many teachers have misconceptions about elementary mathematics material, 2) many teachers have difficulty solving elementary math problems, 3) many teachers are not able to plan mathematics lessons according to the cognitive level of elementary students. So that mathematics is seen as an arithmetic procedure which causes more drill methods to be given to students, there is anxiety in learning mathematics, and learning activities are not innovative. The objectives of this PKM are 1) increasing understanding of elementary mathematics content, 2) developing teacher skills in solving problems and designing elementary mathematics learning, 3) cultivating teacher quality commitment to self-development related to their professional duties. The Strengthening Mathematics Content Activity for Elementary School Teachers was held in the SDI Bokong 1 Hall on June 21-23 2022. The participants in this activity were elementary school teachers in Takari District from 34 elementary schools totaling 50 people. The material presented is numbers and geometry as well as learning. The results obtained by the category of teachers' understanding of elementary mathematics material tended to be good, namely 38% of teachers were in the good category and 62% of the teachers were in the very good category.

A simple arithmetic procedure for combining units

A procedure for determining the overall units of a combination of quantities based on simple additions and subtractions is described. This procedure could be programmed into online-quiz software for student exercises.

A symbolic-arithmetic for teaching double-black node removal in red-black trees

A red-black (RB) tree is a data structure with red and black nodes coloration. The red and black color of nodes make up the principal component for balancing a RB tree. A balanced tree has an equal number of black nodes on any simple path. But when a black leaf node is deleted, a double-black (DB) node is formed, thus, causing a reduction in black heights and the tree becomes unbalanced. Rebalancing a RB tree with a DB node is a fairly complex process. Teaching and learning the removal of DB nodes is also challenging. This paper introduces a simplified novel method which is a symbolic-algebraic arithmetic procedure for the removal of DB nodes and the rebalancing of black heights in RB trees. This simplified approach has enhanced student learning of the DB node removal in RB trees. Feedback from students showed the learnability, workability and acceptance of the symbolic-algebraic method in balancing RB trees after a delete operation.

Using mathematics game-based intervention on children with special educational needs: Preliminary findings

Many video games incorporate positive learning principles, stimulating students' cognitive functioning and promoting problem-solving and spatial abilities. The high levels of engagement and involvement that some students can achieve with video games are notable. Hence, video games for children with a long history of school failure, such as children with special educational needs (SEN). Moreover, it may give educators immediate and ongoing assessment of students' progress. When complemented with human tutoring, video games as a game-based intervention may improve mathematics performance since instruction is more effective when adapted to students' learning needs and pace.
 As part of the research project "GBl4deaf – Game-based Learning for Deaf Students (PTDC/COM-CSS/32022/2017), the video game "Space adventure: Defend the planet!" was designed to stimulate arithmetic competencies in deaf and hearing children. The player must use elementary arithmetical and spatial concepts to rebuild an abandoned space station. Each challenge has three difficulty levels. In the game challenge used in this study, the player must add or remove particles to collect resources. The current research focuses on two questions: The study follows two research questions: Q1: Did the students make any progress in mathematics achievement after playing the video game?; Q2: Is the gameplay of Space adventure: Defend the planet! an engaging experience for players? A pre-and post-game mathematical test was applied to measure mathematics achievement and an observational grid to gather information about arithmetical procedures. Ten fourth- to ninth-graders participated in the study - four girls and six boys, aged between 9 and 16, three deaf and seven hearing students with different special needs (dyscalculia, cognitive deficits, autism spectrum disorder, deafness and Asperger syndrome). Due to the COVID-19 pandemic lockdown, children played the video game using Zoom video conference software in 8-12 sessions (50 minutes, two a three times/week). The results show 4% to 19% of mathematics progression after children played the video game and indicate that they maintain the use of counting-based procedures throughout the game sessions. For instance, they kept counting both addends starting from 1 or counting by 1, 2, 5 or 10 using a number line. The current data suggest that the videogame "Space adventure: Defend the planet!" allows educators to gather immediate information about students' difficulties and progression.

Theta Band Transcranial Alternating Current Stimulation Enhances Arithmetic Learning: A Systematic Comparison of Different Direct and Alternating Current Stimulations

Over the last decades, interest in transcranial electrical stimulation (tES) has grown, as it might allow for causal investigations of the associations between cortical activity and cognition as well as to directly influence cognitive performance. The main objectives of the present work were to assess whether tES can enhance the acquisition and application of arithmetic abilities, and whether it enables a better assessment of underlying neurophysiological processes. To this end, the present, double-blind, sham-controlled study assessed the effects of six active stimulations (three tES protocols: anodal transcranial direct current stimulation (tDCS), alpha band transcranial alternating current stimulation (tACS), and theta band tACS; targeting the left dorsolateral prefrontal cortex or the left posterior parietal cortex) on the acquisition of an arithmetic procedure, arithmetic facts, and event-related synchronization/desynchronization (ERS/ERD) patterns. 137 healthy adults were randomly assigned to one of seven groups, each receiving one of the tES-protocols during learning. Results showed that frontal theta band tACS reduced the repetitions needed to learn novel facts and both, frontal and parietal theta band tACS accelerated the decrease in calculation times in fact learning problems. The beneficial effect of frontal theta band tACS may reflect enhanced executive functions, allowing for better control and inhibition processes and hence, a faster acquisition and integration of novel fact knowledge. However, there were no significant effects of the stimulations on procedural learning or ERS/ERD patterns. Overall, theta band tACS appears promising as a support for arithmetic fact training, but effects on procedural calculations and neurophysiological processes remain ambiguous.

Surface waves propagation in a homogeneous liquid layer overlying a monoclinic half-space

This article includes an analytical investigation of the surface waves propagation in a uniform liquid layer overlying a homogeneous anisotropic (monoclinic) half-space. The waves are allowed to propagate through the interface, i.e., between the layer and the medium. Basic arithmetic procedures are employed to derive the dispersion equation for surface waves propagation. A comprising study is accomplished through numerical estimation to study the behavior of surface waves. Further analysis of surface waves in the absence of a liquid layer manifests that the phase velocity equation loses its dispersity like waves propagation in an isotropic medium. Some particular cases are extracted from the dispersion equation by taking correspondence with the results. Graphical software has been emerged to establish a criterion for the usefulness of several parameters discussed.

The association of basic numerical abilities and math achievement: The mediating role of visuospatial and arithmetical abilities

Basic numerical abilities such as number line estimation have been observed repeatedly to be associated with mathematical achievement. Recently, it was argued that the association between basic numerical abilities and mathematical achievement is fully mediated by visuospatial abilities. However, arithmetical abilities have not yet been considered as influencing this association, even though solution strategies in number line estimation as well as mathematical achievement often involve arithmetical procedures. Therefore, we investigated the mediating role of arithmetical and visuospatial abilities on the association between number line estimation and mathematical achievement in a sample of n = 599 German elementary school students. The results indicated that arithmetical abilities as well as visuospatial abilities mediated the association between number line estimation and mathematical achievement. However, neither visuospatial nor arithmetical abilities fully mediated the association between number line estimation and mathematical achievement when considered in isolation. This substantiates the relevance of the intertwined development of visuospatial and arithmetical abilities as well as basic numerical abilities such as number line estimation (i.e., the combination of domain-specific numerical and domain-general abilities) driving mathematical achievement.

Application of stoichiometric hydrogen atoms for balancing organic combustion reactions

Abstract Combustion is a common redox reaction, and organic combustion is one of the basic contents in chemistry curriculum. The transferred H-atom is commonly used as a redox indicator in organic chemistry and biochemistry. Nevertheless, the relationship between the number of transferred H-atoms and the number of transferred electrons has not been fully revealed. Oxidation number (ON) is an electron-counting concept. Without knowing the ONs, the number of transferred electrons cannot be counted and therefore, the redox reactions cannot be classified, defined, and balanced. This paper explores the new H-atom method for counting the number of transferred H-atoms. It provides a half-reaction approach to balance the overall organic combustion reactions. Only simple arithmetic procedures are needed to determine the number of transferred H-atoms and consequently the number of transferred electrons. According to this method, the mathematical formulas for assigning the number of transferred H-atoms can be deducted by balancing the general chemical formulas of organic compounds in half and overall organic combustions. Furthermore, the number of transferred electrons and their stoichiometric categories can be determined conveniently by any given organic chemical formula in organic combustion reactions.

Exploration of self-regulated learning: Mathematical problem solving

Self-regulated learning is needed to regulate and direct itself, adjust, and control self-learning. This study aims to get an overview of students’ self-regulated learning in solving junior high school mathematics problems. This research is a qualitative descriptive study with three research subjects. The data collection techniques used math ability tests, problem-solving tests, self-regulated learning questionnaires, and interview guidelines. The results showed the profile of self-regulated learning raised by junior high school students when solving math problems, especially problems related to SPLDV, namely: 1) Planning, thinking and activation stages: (understanding the problem is expressed in using one’s own language or the language of the problem and there is an estimate of SPLDV completion through arithmetic procedures, logic and elimination substitution methods; 2) Monitoring Stage: monitoring is carried out on the correctness of the variables with or without involving the whole conversation; 3) Control stage: answer checking with reverse technique; and 4) Reaction and Reflection Stage: Explore the difficulties faced related to routine and non-routine problems.

Modified Krill Herd Algorithm for constrained economic load dispatch problem

ABSTRACT Meta-heuristic techniques are gaining attractiveness to solve constrained economic load dispatch problems due to higher global solution capacity, flexibility and derivative-free structure. In this paper, Modified Krill Herd Algorithm (MKHA) is used to solve economic load dispatch problem having non-linear characteristics of generators and considering transmission losses. The proposed MKHA is a nature-inspired technique that incorporates crossover and mutation process designed in differential evolution which causes the krill to execute local search within definite section. Constraints manipulation trial is also made functional in the direction of approaching equality constraints to change total power generation depending upon the requirement of power by consumers. The proposed algorithm is testified on three systems and assessment is prepared on the basis of minimisation of fuel cost and arithmetic procedures. Relative examination of obtained outcome illustrates the efficacy of the proposed algorithm, with regard to the quality of solutions and higher efficiency.

L’astronomie et Bessarion : tradition et modernité

This paper gives a quick survey of Byzantine training in astronomy at the time of Bessarion. It appears that Bessarion, in spite of his opening to modern astronomical studies under the influence of Regiomontanus, remained attached to the Byzantine tradition. For example, his letter De errore Paschatis to Pope Paul II is directly inspired by the treatise on Easter of Barlaam. The Introduction to the Almagest written in Greek by George of Trebizond was severely criticized by Bessarion and Regiomontanus, but there are no modern studies concerning this work. A quick analysis shows some interesting novelties introduced by George in the arithmetical procedures and terminology, but his criticism of Theon was not accepted by his opponents. A serious study of George’s writings would be highly recommended if one wishes to give a correct estimation of George ’s work.

Decoding chancery records from the 1240s

The English records from the 1240s contain many references to the purchase of gold on behalf of the king, Henry III. For example, the Liberate Rolls give explicit numerical information about the amounts of gold purchased and the price paid for it. These records also contain implicit information, and this can sometimes be extracted by analysing the arithmetical procedures that were used by the king's officials. We shall see how the purity of the gold was assessed, and how the price varied in consequence. The records occasionally mention gold in the form of coins, and in such cases our method can be used to identify the type of coin involved. Since there is little evidence about the methods used to perform the calculations, we shall consider the extent to which the Hindu-Arabic methods popularized by Leonard of Pisa (Fibonacci) were being adopted in England in the thirteenth century.