Academic literature on the topic 'Alphabet arithmetic'

Abstract This fMRI study aimed at unraveling the neural basis of learning alphabet arithmetic facts, as a proxy of the transition from slow and effortful procedural counting-based processing to fast and effortless processing as it occurs in learning addition arithmetic facts. Neural changes were tracked while participants solved alphabet arithmetic problems in a verification task (e.g., F + 4 = J). Problems were repeated across four learning blocks. Two neural networks with opposed learning-related changes were identified. Activity in a network consisting of basal ganglia and parieto-frontal areas decreased with learning, which is in line with a reduction of the involvement of procedure-based processing. Conversely, activity in a network involving the left angular gyrus and, to a lesser extent, the hippocampus gradually increases with learning, evidencing the gradual involvement of retrieval-based processing. Connectivity analyses gave insight in the functional relationship between the two networks. Despite the opposing learning-related trajectories, it was found that both networks become more integrated. Taking alphabet arithmetic as a proxy for learning arithmetic, the present results have implications for current theories of learning arithmetic facts and can give direction to future developments.

Arithmetic coding is a lossless compression algorithm with variable-length source coding. It is more flexible and efficient than the well-known Huffman coding. In this paper we present a non-adaptive FPGA implementation of a multi-alphabet arithmetic coding with separated statistical model of the data source. The alphabet of the data source is a 256-symbol ASCII character set and does not include the special end-of-file symbol. No context switching is used in the proposed design which gives maximal throughput without pipelining. We have synthesized the design for Xilinx FPGA devices and used their built-in hardware resources.

We provide lower bounds for$p$-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the$p$-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the$p$-Lucas property for almost all primes$p$.

Abstract. The present article reports two experiments investigating the influence of natural order of events on the acquisition and use of knowledge about operations, in short mental operators. The principle of use specificity states that task performance depends directly on the similarity between acquisition context and the present situation. In contrast, the principle of natural order proposes that knowledge about operations can always be applied easier (faster) if reasoning follows the natural order of events. In Experiment 1, participants had to apply alphabet-arithmetic operators and LISP functions in a prognosis task (A + 2 = ?) or a retrognosis task (? - 2 = A). In alphabet-arithmetic, an advantage for the first kind of task at the beginning of training decreased with increasing practice. In LISP, however, a preference for this task (corresponding with a prospective knowledge use) emerged with increasing practice. In Experiment 2, arithmetic relations between digit pairs had to be induced. In a causal context condition, relations were described as input and output of electric circuits, in a neutral context the relations were described as arithmetic dependencies. A preference for the prognosis task was found for the causal context condition (corresponding with a prospective knowledge use) but not for the neutral one. The findings suggest that the natural order of events moderates the acquisition and use of mental operators. Further research is required to clarify the bases for this moderation.

The purpose of this study was to evaluate the effects of a training package consisting of response restriction and the reinforcement of self-evaluation on letter reversal errors. Participants were 3 typically developing boys between the age of 5 and 7. The results indicated that the training package was successful in correcting reversals in the absence of a model during training and on application tests. These improvements maintained during subsequent follow-up sessions and generalized across trainers. Fading was not always necessary in correcting reversals, but was effective in correcting reversals that persisted during the overlay training procedures. The advantages to implementing a systematic intervention for reducing letter reversal errors in the classroom, as well as directions for future research, are discussed.

L'addition, l'une des opérations fondamentales de l'arithmétique, constitue une des premières opérations enseignées aux enfants. Parmi les diverses formes d’additions, celles impliquant deux opérandes à un chiffre, comme 5+3, sont omniprésentes dans la vie quotidienne et requièrent souvent des calculs mentaux rapides. A ce jour, les mécanismes cognitifs sous-jacents à la résolution de ces opérations restent mal compris. Deux grands modèles théoriques s’opposent. Les théories associationnistes (Ashcraft, 1982; Campbell & Graham, 1985; Logan, 1988; Siegler & Shrager, 1984) postulent que l'apprentissage conduit à une récupération en mémoire de la réponse. Au début de l'apprentissage, les enfants utilisent une procédure explicite de comptage (par exemple 6...7...8) qui crée une trace mnésique associant le problème à sa solution. Après de nombreuses répétitions, le résultat peut être récupéré directement en mémoire sans nécessiter de calcul. Plus récemment, une théorie propose que l'apprentissage conduit à l'automatisation du comptage pour les plus petites additions (Barrouillet & Thevenot, 2013; Uittenhove et al., 2016; Thevenot & Barrouillet, 2016). Même après une grande expérience, le résultat ne serait pas récupéré en mémoire, mais calculé grâce à une procédure ultra-rapide et inconsciente qui parcourrait la ligne numérique mentale. L'objectif de cette thèse est de contribuer à ce champ de recherche en explorant les mécanismes cognitifs employés grâce à une approche expérimentale et computationnelle. Le volet expérimental vise à déterminer comment les stratégies de comptage et de récupération opèrent durant l’apprentissage de la résolution des additions. Il s'agit également d'examiner si des facteurs comme la magnitude des opérandes et la structure des problèmes peuvent influencer ces stratégies. Le volet expérimental comprend deux études d’apprentissage basées sur des tâches similaires à celles de l’arithmétique alphabétique et menées avec des participants adultes. La première étude explore l'automatisation des additions en comparant deux conditions d'apprentissage, mémorisation et comptage, à l'aide d’additions construites sur une séquence artificielle et montre que le comptage est toujours utilisé par la majorité des participants tandis que d’autres mémorisent les plus grands problèmes. La seconde étude examine l'influence du matériel d'apprentissage, en comparant des additions construites à partir de séquences contiguës et non contiguës, et montre que la structure des séquences affecte également les stratégies utilisées par les participants. Le volet modélisation computationnelle a pour objectif d’expliquer et reproduire les évolutions stratégiques observées entre le comptage et la récupération en mémoire. Une première version du modèle, fondée uniquement sur l'accélération du comptage, ne suffit pas à expliquer pleinement les données expérimentales. Une nouvelle version du modèle, intégrant un mécanisme de compétition dynamique entre le comptage et la récupération en mémoire, a permis de simuler de manière plus précise la transition entre ces deux stratégies en fonction de la taille des problèmes et de leur structure, comme observé dans les expériences. Les résultats des deux approches montrent qu'aucune stratégie unique ne prévaut à la fin de l'apprentissage. Les résultats sont plus nuancés et révèlent que la taille des problèmes ainsi que la structure du matériel influencent le choix des stratégies. De plus, des différences inter-individuelles ont été observées, certains individus privilégiant la récupération en mémoire, tandis que d'autres continuent d'utiliser des procédures de comptage même après une pratique prolongée. Ces observations soulignent l'importance de proposer un modèle flexible pour comprendre les mécanismes d'automatisation des additions basiques
Addition, one of the fundamental operations in arithmetic, is among the first operations taught to children. Among the various forms of addition, those involving two single-digit operands, such as 5+3, are ubiquitous in daily life and often require fast mental calculations. To date, the cognitive mechanisms underlying the resolution of these operations remain poorly understood. Two major theoretical models are in opposition. Associationist theories (Ashcraft, 1982; Campbell & Graham, 1985; Logan, 1988; Siegler & Shrager, 1984) posit that learning leads to the retrieval of answers from memory. At the beginning of learning, children use an explicit counting procedure (e.g., 6...7...8) that creates a memory trace associating the problem with its solution. After numerous repetitions, the result can be retrieved directly from memory without requiring calculation. More recently, a theory proposes that learning leads to the automatization of counting for smaller additions (Barrouillet & Thevenot, 2013; Uittenhove et al., 2016; Thevenot & Barrouillet, 2016). Even after significant experience, the result is not retrieved from memory but is calculated using an ultra-fast and unconscious procedure that would scroll the mental number line. The objective of this thesis is to contribute to this field of research by exploring the cognitive mechanisms employed through both experimental and computational approaches. The experimental component aims to determine how counting and retrieval strategies operate during the learning of addition resolution. It also seeks to examine whether factors such as operand magnitude and problem structure can influence these strategies. The experimental component comprises two learning studies based on tasks similar to those of alphabet arithmetic and conducted with adults. The first study explores the automatization of additions by comparing two learning conditions, memorization and counting, using additions built from an artificial sequence, and shows that counting is still used by most participants, while others memorize larger problems. The second study examines the influence of learning material by comparing additions built from contiguous and non-contiguous sequences, demonstrating that the structure of the sequences also affects the strategies used by participants. The computational modeling component aims to explain and reproduce the strategic shifts observed between counting and memory retrieval. A first version of the model, based solely on counting acceleration, does not fully explain the experimental data. A new version of the model, incorporating a dynamic competition mechanism between counting and memory retrieval, more precisely simulates the transition between these two strategies depending on problem size and structure, as observed in the experiments. The results from the two approaches show that no single strategy prevails at the end of learning. The results are more nuanced, revealing that problem size and material structure influence the choice of strategies. Additionally, individual differences were observed, with some individuals favoring memory retrieval, while others continue to use counting procedures even after prolonged practice. These findings highlight the importance of proposing a flexible model to understand the mechanisms underlying the automatization of basic additions

碩士
國立交通大學
電機與控制工程系
89
Data compression played an important role in the data transmission and data storage. Arithmetic coding is an efficient loseless data compression technique and has been proposed in industrial standards. Traditionally, arithmetic coding applies certain specification for different types of data and results in average performance. To obtain the near-optimal performance, this thesis proposes a parameterized solution for adaptive arithmetic coding with a finite weighted history buffer. The thesis develops six probability models by tuning the size of history buffer and scaling weight. Our proposed decoder will employ one of the models for different files and allow users to set the selection of parameters. In addition to the parameterization, the thesis proposes a bit-wise binary searching algorithm to reduce the number of bit-compare operations. The reduction of operations can speed up our decoder significantly. As shown in the thesis, our decoder chip operates at 71.4 MHz clock rate and costs the area of 2.86*2.86 .

AbstractSequence theories are an extension of theories of strings with an infinite alphabet of letters, together with a corresponding alphabet theory (e.g. linear integer arithmetic). Sequences are natural abstractions of extendable arrays, which permit a wealth of operations including append, map, split, and concatenation. In spite of the growing amount of tool support for theories of sequences by leading SMT-solvers, little is known about the decidability of sequence theories, which is in stark contrast to the state of the theories of strings. We show that the decidable theory of strings with concatenation and regular constraints can be extended to the world of sequences over an alphabet theory that forms a Boolean algebra, while preserving decidability. In particular, decidability holds when regular constraints are interpreted as parametric automata (which extend both symbolic automata and variable automata), but fails when interpreted as register automata (even over the alphabet theory of equality). When length constraints are added, the problem is Turing-equivalent to word equations with length (and regular) constraints. Similar investigations are conducted in the presence of symbolic transducers, which naturally model sequence functions like map, split, filter, etc. We have developed a new sequence solver, SeCo, based on parametric automata, and show its efficacy on two classes of benchmarks: (i) invariant checking on array-manipulating programs and parameterized systems, and (ii) benchmarks on symbolic register automata.

AbstractThis paper presents a similar approach for existential first-order characterizations of the languages recognizable by finite automata, by Parikh automata, and by multi-counter machines over the alphabet $$\left\{ 0,1,...,k-1\right\} ^{n}$$ 0 , 1 , . . . , k - 1 n for some $$k\ge 2$$ k ≥ 2 . The set of k-FA-recognizable relations coincides with the set of relations, which are existentially definable in the structure "Image missing" , where "Image missing" corresponds to the bitwise minimum of base k. In order to obtain an existential first-order description of k-Parikh automata languages, we extend this structure with the predicate $$ EqNZB _{k}(x,y)$$ E q N Z B k ( x , y ) which is true if and only if x and y have the same number of non-zero bits in k-ary encoding. Using essentially the same ideas, we encode computations of k-multi-counter machines and thus show that every recursively enumerable relation over the natural numbers is existentially definable in the aforementioned structure supplemented with concatenation $$z=x\smallfrown _{k} y\rightleftharpoons z = x + k^{l_{k}(x)}y$$ z = x ⌢ k y ⇌ z = x + k l k ( x ) y , where $$l_{k}(x)$$ l k ( x ) is the bit-length of x in base k. This result gives us another proof of DPR-theorem.

In the last chapter, we dealt with mathematical languages in considerable generality. We shall now turn to some particular mathematical languages. One of our goals is to reach Gödel’s incompleteness theorem for the particular system known as Peano Arithmetic. We shall give several proofs of this important result; the simplest one is based partly on Tarski’s theorem, to which we first turn. The first concrete language that we will study is the language of first order arithmetic based on addition, multiplication and exponentiation. [We also take as primitive the successor function and the less than or equal to relation, but these are inessential.] We shall formulate the language using only a finite alphabet (mainly for purposes of a convenient Gödel numbering); specifically we use the following 13 symbols. . . . 0’ ( ) f, υ ∽ ⊃ ∀ = ≤ # . . . The expressions 0, 0′, 0″, 0‴, · · · are called numerals and will serve as formal names of the respective natural numbers 0, 1, 2, 3, · · ·. The accent symbol (also called the prime) is serving as a name of the successor function. We also need names for the operations of addition, multiplication and exponentiation; we use the expressions f′, f″, f‴ as respective names of these three functions. We abbreviate f′ by the familiar “+”; we abbreviate f’’ by the familiar dot and f‴ by the symbol “E”. The symbols ~ and ⊃ are the familiar symbols from prepositional logic, standing for negation and material implication, respectively. [For any reader not familiar with the use of the horseshoe symbol, for any propositions p and q, the propositions p ⊃ q is intended to mean nothing more nor less than that either p is false, or p and q are both true.] The symbol ∀ is the universal quantifier and means “for all.” We will be quantifying only over natural numbers not over sets or relations on the natural numbers. [Technically, we are working in first-order arithmetic, not second-order arithmetic.] The symbol “=” is used, as usual, to denote the identity relation, and “≤” is used, as usual, to denote the “less than or equal to” relation.

Abstract Apart from separate individual objects, which we shall also call INDIVIDUALS for short, logic is concerned with CLASSES of objects; in everyday life as well as in mathematics, classes are more often referred to as SETS. Arithmetic, for instance, frequently deals with sets of numbers, and in geometry our interest lies not so much in single points as in sets of points (that is, in geometrical configurations). Now, classes of individuals are called CLASSES OF THE FIRST ORDER. Relatively more rarely in our investigations we come upon CLASSES OF THE SECOND ORDER, that is, upon classes which consist, not of individuals, but of classes of the first order. Sometimes even CLASSES OF THE THIRD, FOURTH, and HIGHER ORDERS have to be dealt with. Here we shall be concerned almost exclusively with classes of the first order, and only exceptionally-as in Section 26-we shall have to deal with classes of the second order; however, our considerations can be applied with practically no changes to classes of any order. In order to distinguish between individuals and classes (and also among classes of different orders), we employ as variables letters of different shape and belonging to different alphabets. It is customary to designate individual objects such as numbers, and classes of such objects, by lower-case and capital letters of the Latin alphabet, respectively. In elementary geometry the opposite notation is the accepted one, capital letters designating points and lower-case letters (of the Latin or the Greek alphabet) designating sets of points.

This chapter discusses the evolution of symbolic algebra that began in the first half of the sixteenth century. Algebra was not always called algebra. In the mid-fifteenth century some Italian and Latin writers called it Regula rei e census. The twentieth-century mathematician and science fiction author Eric Temple Bell allegedly remarked that in the mid-seventeenth century, mathematicians were able to introduce negative and rational exponents because symbolic manipulation liberated their thinking from the wilderness of words. The chapter considers the contributions of the Arab algebraist al-Qalasādi, who used letters of the Arabic alphabet to denote arithmetic operations and whose notation was clearly an attempt at symbolizing algebra through abbreviations, a first approximation to what we would consider true symbols. It also examines how Italy cultivated the seeds of algebra, citing in particular Gerolamo Cardano's Ars Magna.

The rapid development in technology has facilitated human beings in many ways such as automated home appliances, smart vehicles, smart mobile phones, and tablet computers. The uses of these tools and techniques are increasing in our daily lives to facilitate day to day work. The new trends in technology have focused on finding approaches towards improved learning techniques. Various tools are being used to integrate Information and Communication Technology in education. Tablet Personal Computers (PCs) are one of the new and innovative tools used in education for enhancing learning skills. This research has been conducted in five primary schools, where students of class nursery to class three were taught basic lessons using Tablet PC. In this research an application has been developed on android platform with easy to use interface, where the students were able to perform simple arithmetic calculations and learned alphabet of Sindhi and English languages in visual form. During the experiment, it was observed that with visual aids students understood lessons more clearly and easily.

In classical cryptography, the Hill cipher, invented by Lester Hill in 1929, was the first polygraphic substitution cipher based on linear algebra, which practically processed more than three symbols at once. In the classical Hill cipher, each letter is represented by a number modulo 26. To encrypt a message, each block of n letters (considered as an n-component vector) is multiplied by an invertible matrix of size n×n, again modulus 26. In order to decrypt the message, each block is multiplied by the inverse matrix of the matrix used for encryption. In the paper a modified algorithm for encryption and decryption using the Hill cipher on the basis of 2×2-matrix and 64-symbol alphabet is proposed. Here each symbol (Latin letters, decimal digits and punctuation signs) is represented by a number modulo 64 and all arithmetic operations are done by modulo 64 (the number of letters in the alphabet instead of modulo 26). The paper describes MS EXCEL-based application implementing the processes of encryption and decryption of English texts using the Hill cipher based on the matrix of size 2×2 and 64-symbol alphabet. The processes of encryption and decryption of 6-character words in English are illustrated. Numerous examples obtained by the application developed are presented in the paper. The material is used in the educational process in the course “Telecommunication Security” by students of the specialty “Telecommunication Systems” for the Bachelor degree at the University of Ruse but it can be applied in other disciplines involving issues of cryptographic information protection.

Character recognition by a trinary associative memory (TAM) neural network model is proposed. All the twenty-six letters of the English alphabet are stored in the trinary memory. The proposed scheme will then be able to recognize a character from its partial input. The dot product of the partial input with all the stored patterns is calculated as a measure of discrepancy from the desired pattern. Zero thresholding and arithmetic mean thresholding and some other statistical thresholding methods are then applied to select the desired output. The convergence of the recall procedure of the TAM network depends upon the storage representation and thresholding mode. So from the simulation run, an optimum threshold measure is found that establishes efficient character recognition. Shift invariance is also examined by simulation.

Anxiety and visual task difficulty may interact, impose cognitive demands, and affect task performance. Researchers have found that when comparing task performance across people with different levels of anxiety vulnerability, people with somewhat elevated anxiety vulnerability perform worse than people with lower anxiety susceptibility. Furthermore, vulnerability to anxiety is associated with a decreased ability to control the allocation of attention that is reflected while performing tasks. We aim to see the role of visual task complexity as potential mediators in the relation between different levels of anxiety and eye movements and to see whether increasing task complexity has an effect on visualization behaviour. We use eye-movement analysis to observe and analyze how people view and subsequently process visual information. After the screening, the study recruited 31 students (F= 11; M=20) aged between 21 to 35 years (M=28.02, SD=0.62). Exclusion criteria included individuals with visual impairments, with substance disorders and history of neurological disorders. All participants provided consent before participation. STAI-Y2 was used to categorize participants into low, moderate, and high (N=9, N=10, and N= 12). Visual task involved arithmetic problems and determining the accuracy of the answer displayed (true or false). Each level was divided into two subsets, each containing six equations with a combination of numbers, alphabet characters, and spatial symbols (>, <). To heighten task complexity, one subset required participants to respond synchronously using their right index finger for true and left index finger for false, while the other subset required asynchronous responses (left index finger for true, right index finger for false). Relevant information to solve the problems consisted of numbers and spatial symbols, while alphabet characters were considered irrelevant or noise. The First Fixation Duration (FFD), Time to First Fixation (TTFF), Total Visit Duration (TVD), Total Fixation Duration (TFD) were computed after ascertaining the Area of Interests (AOIs). The findings suggest that individuals with high trait anxiety tend to explore or scan irrelevant information more compared to low and moderate trait anxiety levels while performing the task, particularly during more complex visual tasks. Individuals in the low and moderate anxiety groups showed better ability to fixate on relevant information during the same tasks. The study's findings suggest that eye metrics, such as fixation duration and visit duration on relevant and irrelevant information, could serve as objective markers indicating anxiety related attentional differences during task performance. The results are in line with previous research supporting the correlation between anxiety and eye metrics. Our study corroborates with existing literature, which posits that highly anxious individuals may experience impaired attentional control during task performance. The study provides valuable insights into the impact of anxiety on attention and information processing during visual tasks of varying complexity.