Journal articles: 'Introductio arithmetica'

1

Burke, Maxim R., M. Holz, K. Steffens, and E. Weitz. "Introduction to Cardinal Arithmetic." Bulletin of Symbolic Logic 8, no. 4 (December 2002): 524. http://dx.doi.org/10.2307/797958.

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Abbott, Steve, M. Holz, K. Steffens, and E. Weitz. "Introduction to Cardinal Arithmetic." Mathematical Gazette 84, no. 499 (March 2000): 171. http://dx.doi.org/10.2307/3621544.

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Kilmister, C. W., M. Holz, K. Steffens, and E. Weitz. "Introduction to Cardinal Arithmetic." Mathematical Gazette 84, no. 500 (July 2000): 363. http://dx.doi.org/10.2307/3621721.

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Ferguson, Thomas Macaulay, and Graham Priest. "Introduction." Australasian Journal of Logic 18, no. 5 (July 21, 2021): 132–45. http://dx.doi.org/10.26686/ajl.v18i5.6901.

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Ardeshir, M., and B. Hesaam. "An Introduction to Basic Arithmetic." Logic Journal of IGPL 16, no. 1 (July 10, 2007): 1–13. http://dx.doi.org/10.1093/jigpal/jzm013.

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Kim, Minhyong. "Arithmetic gauge theory: A brief introduction." Modern Physics Letters A 33, no. 29 (September 20, 2018): 1830012. http://dx.doi.org/10.1142/s0217732318300124.

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Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

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Bohlender, Gerd, Arnold Kaufmann, and Madan M. Gupta. "Introduction to Fuzzy Arithmetic, Theory and Applications." Mathematics of Computation 47, no. 176 (October 1986): 762. http://dx.doi.org/10.2307/2008199.

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Nannarelli, Alberto, Peter-Michael Seidel, and Ping Tak Peter Tang. "Guest Editors’ Introduction: Special Sectionon Computer Arithmetic." IEEE Transactions on Computers 63, no. 8 (August 1, 2014): 1852–53. http://dx.doi.org/10.1109/tc.2014.2331711.

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Chen, N. X., F. Y. Zhu, and Y. H. Ku. "Introduction to fuzzy arithmetic—theory and applications." Journal of the Franklin Institute 321, no. 3 (March 1986): 189–90. http://dx.doi.org/10.1016/0016-0032(86)90009-8.

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Eastman, Caroline M. "Introduction to fuzzy arithmetic: Theory and applications." International Journal of Approximate Reasoning 1, no. 1 (January 1987): 145–46. http://dx.doi.org/10.1016/0888-613x(87)90010-7.

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Eastman, Caroline M. "Introduction to fuzzy arithmetic: Theory and applications." International Journal of Approximate Reasoning 1, no. 1 (January 1987): 141–43. http://dx.doi.org/10.1016/0888-613x(87)90009-0.

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Koren, I., and P. Kornerup. "Guest editors' introduction - Special issue on computer arithmetic." IEEE Transactions on Computers 49, no. 7 (July 2000): 625–27. http://dx.doi.org/10.1109/tc.2000.863030.

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Kornerup, Peter, Paolo Montuschi, Jean-Michel Muller, and Eric Schwarz. "Guest Editors' Introduction: Special Section on Comuter Arithmetic." IEEE Transactions on Computers 58, no. 2 (February 2009): 145–47. http://dx.doi.org/10.1109/tc.2009.11.

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Bruguera, Javier, Marius Cornea, and Debjit Das Sarma. "Guest Editors' Introduction: Special Section on Computer Arithmetic." IEEE Transactions on Computers 60, no. 2 (February 2011): 145–47. http://dx.doi.org/10.1109/tc.2011.15.

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Antelo, Elisardo, David Hough, and Paolo Ienne. "Guest Editors' Introduction: Special Section on Computer Arithmetic." IEEE Transactions on Computers 61, no. 8 (August 2012): 1057–58. http://dx.doi.org/10.1109/tc.2012.153.

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Hormigo, Javier, Jean-Michel Muller, Stuart Oberman, Nathalie Revol, Arnaud Tisserand, and Julio Villalba-Moreno. "Introduction to the Special Issue on Computer Arithmetic." IEEE Transactions on Computers 66, no. 12 (December 1, 2017): 1991–93. http://dx.doi.org/10.1109/tc.2017.2761278.

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Bruguera, Javier D., and Florent de Dinechin. "Guest Editors Introduction: Special Section on Computer Arithmetic." IEEE Transactions on Computers 68, no. 7 (July 1, 2019): 951–52. http://dx.doi.org/10.1109/tc.2019.2918447.

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Harin, Alexander. "Introduction to sub-interval analysis. Sub-interval arithmetic." Applied Mathematical Sciences 14, no. 12 (2020): 607–20. http://dx.doi.org/10.12988/ams.2020.914230.

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19

Fennell, Francis (Skip). "By Way of Introduction." Arithmetic Teacher 36, no. 1 (September 1988): 2. http://dx.doi.org/10.5951/at.36.1.0002.

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Did you see it? What? Why, the new subtitle, of course. Beginning with this volume, 36, the Arithrnetic Teacher will be subtitled “Mathematics Education through the Middle Grades.” Our subtitle accurately represents the content and intended audience of those articles published in the journal. The subtitle also attempts to publicize to our readership that a contemporary mathematics curriculum include attention to a variety of topic other than arithmetic.

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20

Brandemberg, João Cláudio. "AN INTRODUCTION TO CHRISTOPHORI CLAVII EPITOME ARITHMETICAE PRACTICAE (1614." Boletim Cearense de Educação e História da Matemática 4, no. 11 (June 1, 2018): 81–92. http://dx.doi.org/10.30938/bocehm.v4i11.33.

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Neste artigo fazemos uma introdução ao trabalho do matemático jesuíta alemão Christoph Clavius (1538-1612) partindo de uma leitura dos quatro primeiros capítulos de seu livro Epitome Arthmeticae Practicae (1614), onde, descrevemos aspectos das operações elementares apresentadas, seu estudo e a importância da aritmética no início do século XVII. Temos em Clavius, um professor, que além de sua contribuição teórica para a matemática, foi um de seus maiores promotores. Provavelmente, nenhum outro intelectual alemão do século XVI fez mais do que ele para a promoção da matemática; principalmente, por sua influência no ensino da Aritmética e da Álgebra e por sua participação na reforma do calendário gregoriano. Com relação à importância de seu trabalho relacionada ao ensino de aritmética, consideramos o início de um novo estágio no desenvolvimento de notações e algoritmos. Trata-se de uma aritmética prática, para ser empregada, inicialmente, nas transações comerciais, uma representação de receitas e despesas por uma lista de números e suas operações de adição e subtração para indicar os acréscimos e as retiradas e esclarecendo o porquê destas circunstancias, principalmente, na prestação de contas, tanto na esfera pública quanto na privada. Uma ferramenta indispensável para o cálculo de impostos, evitando e reconhecendo possíveis fraudes. Para Clavius, com a manipulação dos números e suas operações aritméticas, o homem encontra-se com a mente arejada e pronta a receber outros conhecimentos matemáticos que lhe venham a ser ensinados. Com seu texto ele deseja proporcionar aos leitores as vantagens do conhecimento aritmético. Enfim, seu trabalho se faz importante na sistematização e divulgação do conhecimento matemático produzido em seu tempo.

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21

Hayes, D. R. "Introduction to Chapter 19 of "The Arithmetic of Polynomials"." Finite Fields and Their Applications 1, no. 2 (April 1995): 152–56. http://dx.doi.org/10.1006/ffta.1995.1012.

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22

Casselman, W. "Introduction to the Schwartz Space of T\G." Canadian Journal of Mathematics 41, no. 2 (April 1, 1989): 285–320. http://dx.doi.org/10.4153/cjm-1989-015-6.

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Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G. This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

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(Andy) Reeves, Charles A., and Darcy Webb. "Balloons on the Rise: A Problem-Solving Introduction to Integers." Mathematics Teaching in the Middle School 9, no. 9 (May 2004): 476–82. http://dx.doi.org/10.5951/mtms.9.9.0476.

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Brownell, William A. "AT Classic: The Revolution in Arithmetic." Arithmetic Teacher 34, no. 2 (October 1986): 38–42. http://dx.doi.org/10.5951/at.34.2.0038.

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At the suggestion of Glenn Nelson of Cedar Fulls. Iowa. the Editorial Panel has authorized the occasional republication of an article of exceptional merit from the period 1954-1975 that would interest today's readers. Invitations to submit five nominations were sent to individuals familiar with the journal during this period. This article was one of the most highly rated. Introduction comments have been prepared by James E. Inskeep, Jr., who selected the article.

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25

Ford, Susan. "Thierry of Chartres: The commentary on the De arithmetica of Boethius [Book Review]." Journal of the Australian Early Medieval Association 15 (2019): 150. http://dx.doi.org/10.35253/jaema.2019.1.21.

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Review(s) of: Thierry of Chartres: The commentary on the De arithmetica of Boethius, edited with an introduction by Irene Caiazzo (Toronto, Pontifical Institute of Mediaeval Studies, 2015) hardcover, xii + 262 pages, RRP euro90.00; ISBN: 9780888441911.

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26

Bui, Tuan Anh, and Thi Anh Nguyen. "A brief introduction to Quillen conjecture." Science and Technology Development Journal 22, no. 2 (June 14, 2019): 235–38. http://dx.doi.org/10.32508/stdj.v22i2.1229.

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Introduction: In 1971, Quillen stated a conjecture that on cohomology of arithmetic groups, a certain module structure over the Chern classes of the containing general linear group is free. Over time, many efforts has been dedicated into this conjecture. Some verified its correctness, some disproved it. So, the original Quillens conjecture is not correct. However, this conjecture still has great impacts on the field cohomology of group, especially on cohomology of arithmetic groups. This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result that this conjecture has been verified by the authors. Method: In this work, we investigate the key reasons that makes Quillen conjecture fails. We review two of the reasons: 1) the injectivity of the restriction map; 2) the non-free of the image of the Quillen homomorphism. Those two reasons play important roles in the study of the correctness of Quillen conjecture. Results: In section 4, we present the cohomology of ring which is isomorphic to the free module over . This confirms the Quillen conjecture. Conclusion: The scope of the conjecture is not correct in Quillens original statement. It has been disproved in many examples and also been proved in many cases. Then determining the conjectures correct range of validity still in need. The result in section 4 is one of the confirmation of the validity of the conjecture.

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LAMBOV, BRANIMIR. "RealLib: An efficient implementation of exact real arithmetic." Mathematical Structures in Computer Science 17, no. 1 (February 2007): 81–98. http://dx.doi.org/10.1017/s0960129506005822.

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This paper is an introduction to the RealLib package for exact real number computations. The library provides certified accuracy, but tries to achieve this at performance close to the performance of hardware floating point for problems that do not require higher precision. The paper gives the motivation and features of the design of the library and compares it with other packages for exact real arithmetic.

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Ogunti, Erastus O., Oluwasegun A. Somefun, Benedict T. Terkura, and Gideon E. Enoch. "The Algorithm of Fully Fuzzy Cognitive Map Models." European Journal of Engineering Research and Science 4, no. 6 (June 30, 2019): 145–54. http://dx.doi.org/10.24018/ejers.2019.4.6.1325.

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Precision in the real world is covered by imprecision and arithmetic operations serve as the foundations of computation. Since the introduction of Fuzzy Cognitive mapping, the dynamic model used to establish the fuzzy cognitive map, used conventional arithmetic operations on asymmetric fuzzy sets.Therefore, for a cognitive map, to be completely fuzzy, it should incorporate the use of fuzzy arithmetic and fuzzy numbers in describing the concept nodes and the cause-effect lines defining its structure. It then can be stated that the necessary and sufficient condition for a cognitive map to be fully fuzzy is that its dynamic activity or operation, be achieved only through fuzzy mathematics.This paper presents an introductory analysis into the peculiar design of the fully fuzzy structure of the cognitive map.

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Jumarie, Guy. "OBSERVATION WITH INFORMATIONAL INVARIANCE: INTRODUCTION TO A NEW MODELING OF FUZZY ARITHMETICS." Cybernetics and Systems 18, no. 5 (January 1987): 407–24. http://dx.doi.org/10.1080/01969728708902148.

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Fan, Ya Qin, Yu Ding, Mei Lin Liu, and Xin Zhang. "Research on the Wireless Sensor Network Data Fusion Technology." Advanced Materials Research 756-759 (September 2013): 751–55. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.751.

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Wireless sensor network ( WSN ) is a kind of energy constrained network, by using data fusion technology, the elimination of redundant data, can save energy, prolong the network life purpose. Data fusion in wireless sensor network can realize different protocol layers, Based on the introduction of wireless sensor network and data fusion related knowledge, prove that the arithmetic mean method is effective, and use OPNET software tool for network simulation, finally, analysis results and conclude, verify effects of the arithmetic average fusion algorithm for wireless sensor network.

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Lubarsky, Robert S. "An introduction to γ-recursion theory (or what to do in KP – Foundation)." Journal of Symbolic Logic 55, no. 1 (March 1990): 194–206. http://dx.doi.org/10.2307/2274962.

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The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

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Crialesi, Clelia V. "The Excerptiuncula: A Short Introduction to Boethius’s De arithmetica from the Early Middle Ages." Journal of Medieval Latin 31 (January 2021): 265–87. http://dx.doi.org/10.1484/j.jml.5.123666.

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Aliev, R. A., A. V. Alizadeh, and O. H. Huseynov. "An introduction to the arithmetic of Z-numbers by using horizontal membership functions." Procedia Computer Science 120 (2017): 349–56. http://dx.doi.org/10.1016/j.procs.2017.11.249.

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Owen, William B., and Richard Lowry. "The Architecture of Chance: An Introduction to the Logic and Arithmetic of Probability." Journal of the American Statistical Association 87, no. 419 (September 1992): 897. http://dx.doi.org/10.2307/2290232.

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Collins, Pieter. "Computable analysis with applications to dynamic systems." Mathematical Structures in Computer Science 30, no. 2 (February 2020): 173–233. http://dx.doi.org/10.1017/s096012952000002x.

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AbstractNumerical computation is traditionally performed using floating-point arithmetic and truncated forms of infinite series, a methodology which allows for efficient computation at the cost of some accuracy. For most applications, these errors are entirely acceptable and the numerical results are considered trustworthy, but for some operations, we may want to have guarantees that the numerical results are correct, or explicit bounds on the errors. To obtain rigorous calculations, floating-point arithmetic is usually replaced by interval arithmetic and truncation errors are explicitly contained in the result. We may then ask the question of which mathematical operations can be implemented in a way in which the exact result can be approximated to arbitrary known accuracy by a numerical algorithm. This is the subject of computable analysis and forms a theoretical underpinning of rigorous numerical computation. The aim of this article is to provide a straightforward introduction to this subject that is powerful enough to answer questions arising in dynamic system theory.

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Harnik, Victor. "Provably total functions of intuitionistic bounded arithmetic." Journal of Symbolic Logic 57, no. 2 (June 1992): 466–77. http://dx.doi.org/10.2307/2275282.

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This note deals with a proof-theoretic characterisation of certain complexity classes of functions in fragments of intuitionistic bounded arithmetic. In this Introduction we survey the background and state our main result.We follow Buss [B1] and consider a language for arithmetic whose nonlogical symbols are 0, S (the successor operation Sx = x + 1), +, ·, ∣ ∣ (∣x∣ being the number of digits in the binary notation for x), rounded down to the nearest integer), # (x#y = 2∣x∣∣y∣) and ≤. We define 1 = S0, 2 = S1, s0x = 2x and s1x = 2x + 1. In Buss's approach the functions s0 and s1 play a special role. Notice that six is the number obtained from x by suffixing the digit i to its binary representation, and thus the natural numbers are generated from 0 by repeated applications of the operations s0 and s1. This means that they satisfy the induction schemeUsing the fact that is x with its last binary digit deleted, this can be stated more compactly in the following form, called by Buss the polynomial induction or PIND schema:Buss defined a theory S2 consisting of a finite set BASIC of open axioms and the PIND-schema restricted to bounded formulas ϕ. The topic of bounded arithmetic is concerned with S2 and its fragments.

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Rabelo, Rafaela Silva. "Apropriações de Dewey na Educação Matemática: Estudo de um Livro de Aritmética para o Ensino Primário." Jornal Internacional de Estudos em Educação Matemática 11, no. 2 (September 11, 2018): 186. http://dx.doi.org/10.17921/2176-5634.2018v11n2p186-193.

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O presente artigo tem como foco o primeiro volume da série The Alexander-Dewey arithmetic, voltado ao ensino primário. Tal série teve Georgia Alexander como autora, John Dewey como editor e foi publicada nos anos 1920 nos EUA. O objetivo foi explorar a tradução do pensamento de Dewey no ensino de aritmética no primário no interior do primeiro volume da referida série. Buscou-se estabelecer diálogo com outras obras de Dewey, principalmente os livros How we think (1910) e The school and society (1899), para tanto mobilizando a noção de apropriação e problematizando questões relacionadas à produção escrita e ao processo editorial, com base em autores como Roger Chartier e Peter Burke. Dentre as constatações, percebe-se que o livro de aritmética analisado estrutura a apresentação dos conteúdos e o desenvolvimento das atividades privilegiando a introdução social. Ainda incentiva o aluno enquanto sujeito de sua aprendizagem. Outros elementos fortemente presentes são o recurso à indução e a situações concretas. Tais características podem ser associadas ao pensamento de Dewey, tendo como elemento reforçador o fato que ele foi editor da série.Palavras-chave: John Dewey. Educação Progressiva. The Alexander-Dewey Arithmetic. Introdução Social.AbstractThe focus of the following article is the first volume of the series The Alexander-Dewey arithmetic, designed to primary school. The series was published in the 1920s in the USA, Georgia Alexander was the author and John Dewey was the editor. The aim of the article is to explore the translation of Dewey’s thought in the arithmetic teaching in primary school according to the first volume of the aforesaid series. In order to establish a dialogue with other Dewey’s works, the main references were the books How we think (1910) and The school and society (1899). The analysis was developed based in the notion of appropriation and inquiring about aspects of writing production and publishing process, relying in such authors as Roger Chartier and Peter Burke. It is possible to say that the analysed arithmetic textbook organizes the insertion of the subjects and the development of activities focusing the social introduction. It also stimulates the pupils to be active in the learning process. Some other elements that stand out are induction based introductions and concrete examples. Such characteristics can be traced to Dewey’s thought, which can be reinforced by the fact he was the editor of the series.Keywords: John Dewey. Progressive Education. The Alexander-Dewey Arithmetic. Social Introduction.

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Schwartz, David I., and Stuart S. Chen. "A constraint-based approach for qualitative matrix structural analysis." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 9, no. 1 (January 1995): 23–36. http://dx.doi.org/10.1017/s0890060400002067.

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AbstractQualitative physics, a subfield of artificial intelligence, adapts intuitive and non-numerical reasoning for descriptive analysis of physical systems. The application of a set-based qualitative algebra to matrix analysis (QMA) allows for the development of a qualitative matrix stiffness methodology for linear elastic structural analysis. The unavoidable introduction of arithmetic ambiguity requires the reinforcement of physical constraints complementary to standard matrix operations. The overall analysis technique incorporates such constraints within the set-based framework with logic programming. Truss, beam, and frame structures demonstrate constraint relationships, which prune spurious solutions resulting from qualitative arithmetic relations. Though QMA is not a panacea for all structural applications, it provides greater insight into new notions of physical analysis.

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Urquhart, Alasdair. "The Complexity of Propositional Proofs." Bulletin of Symbolic Logic 1, no. 4 (December 1995): 425–67. http://dx.doi.org/10.2307/421131.

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§1. Introduction. The classical propositional calculus has an undeserved reputation among logicians as being essentially trivial. I hope to convince the reader that it presents some of the most challenging and intriguing problems in modern logic. Although the problem of the complexity of propositional proofs is very natural, it has been investigated systematically only since the late 1960s. Interest in the problem arose from two fields connected with computers, automated theorem proving and computational complexity theory. The earliest paper in the subject is a ground-breaking article by Tseitin [62], the published version of a talk given in 1966 at a Leningrad seminar. In the three decades since that talk, substantial progress has been made in determining the relative complexity of proof systems, and in proving strong lower bounds for some restricted proof systems. However, major problems remain to challenge researchers. The present paper provides a survey of the field, and of some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs. A major area only touched upon here is the proof theory of bounded arithmetic and its relation to the complexity of propositional proofs. The reader is referred to the book by Buss [10] for background in bounded arithmetic. The forthcoming book by Krajíček [40] also gives a good introduction to bounded arithmetic, as well as covering most of the basic results in complexity of propositional proofs.

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Grimson, Roger C. "The Architecture of Change. An Introduction to the Logic and Arithmetic of Probability.Richard Lowry." Quarterly Review of Biology 65, no. 3 (September 1990): 349. http://dx.doi.org/10.1086/416852.

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Li, Li Qing, Hai Lu, and Xu Dong Li. "The Study on Software Fault Tolerance." Applied Mechanics and Materials 268-270 (December 2012): 1790–93. http://dx.doi.org/10.4028/www.scientific.net/amm.268-270.1790.

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Software Fault -tolerance is an effective and reliable design technique and Recovery Block Scheme is an important Software Fault-tolerant measure. Here is the brief introduction of the implementation of Software Fault-tolerant technique and the design pattern by employing software redundancy and then proposes the implementation of Recovery Block Scheme. Firstly, based on the program block’s fault captured by exception-progressing mechanism, it applies the Command Pattern and Active Objective Pattern to manage and schedule arithmetic unit to achieve rollback, clears the data generated by fault operation block and restores to the state before the operation. The design pattern provides a widely available recovery block schemes design pattern, simplifies the implementation of arithmetic unit and gives the core algorithm through Java.

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Eriksson, Kenneth, Don Estep, Peter Hansbo, and Claes Johnson. "Introduction to Adaptive Methods for Differential Equations." Acta Numerica 4 (January 1995): 105–58. http://dx.doi.org/10.1017/s0962492900002531.

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Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719).When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results…. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used…. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).

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Sheldahl, Terry K. "AMERICA'S EARLIEST RECORDED TEXT IN ACCOUNTING; SARJEANT'S 1789 BOOK." Accounting Historians Journal 12, no. 2 (September 1, 1985): 1–42. http://dx.doi.org/10.2308/0148-4184.12.2.1.

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In 1789, seven years before the text developed by “pioneer American [accounting] author” William Mitchell appeared, Thomas Sarjeant of Philadelphia published An Introduction to the Counting House. It was a concise and able expression of a long mercantile bookkeeping tradition destined to result in later American texts. A mathematics teacher in England and a Philadelphia “academy,” Sarjeant also contributed works on commercial arithmetic. There is significant bibliographical evidence that An Introduction to the Counting House, which is readily available within a remarkable historical microform series, was the first text on accounting to be produced by an American writer.

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Rager, David L., Jo Ebergen, Austin Lee, Dmitry Nadezhin, Ben Selfridge, and Cuong K. Chau. "A Brief Introduction to Oracle's Use of ACL2 in Verifying Floating-point and Integer Arithmetic." Electronic Proceedings in Theoretical Computer Science 192 (September 18, 2015): 1–2. http://dx.doi.org/10.4204/eptcs.192.1.

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Allen, Nancy L. "Review of The Architecture of Chance: An Introduction to the Logic and Arithmetic of Probability." Contemporary Psychology: A Journal of Reviews 35, no. 6 (June 1990): 616. http://dx.doi.org/10.1037/028746.

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Jech, Thomas. "Singular Cardinals and the PCF Theory." Bulletin of Symbolic Logic 1, no. 4 (December 1995): 408–24. http://dx.doi.org/10.2307/421130.

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§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory (“pcf” stands for “possible cofinalities”). The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4. In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory. §2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P(κ), the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ.

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Cárdenas, Sonia Y., Juan Silva-Pereyra, Belén Prieto-Corona, Susana A. Castro-Chavira, and Thalía Fernández. "Arithmetic processing in children with dyscalculia: an event-related potential study." PeerJ 9 (January 27, 2021): e10489. http://dx.doi.org/10.7717/peerj.10489.

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Introduction Dyscalculia is a specific learning disorder affecting the ability to learn certain math processes, such as arithmetic data recovery. The group of children with dyscalculia is very heterogeneous, in part due to variability in their working memory (WM) deficits. To assess the brain response to arithmetic data recovery, we applied an arithmetic verification task during an event-related potential (ERP) recording. Two effects have been reported: the N400 effect (higher negative amplitude for incongruent than for congruent condition), associated with arithmetic incongruency and caused by the arithmetic priming effect, and the LPC effect (higher positive amplitude for the incongruent compared to the congruent condition), associated with a reevaluation process and modulated by the plausibility of the presented condition. This study aimed to (a) compare arithmetic processing between children with dyscalculia and children with good academic performance (GAP) using ERPs during an addition verification task and (b) explore, among children with dyscalculia, the relationship between WM and ERP effects. Materials and Methods EEGs of 22 children with dyscalculia (DYS group) and 22 children with GAP (GAP group) were recorded during the performance of an addition verification task. ERPs synchronized with the probe stimulus were computed separately for the congruent and incongruent probes, and included only epochs with correct answers. Mixed 2-way ANOVAs for response times and correct answers were conducted. Comparisons between groups and correlation analyses using ERP amplitude data were carried out through multivariate nonparametric permutation tests. Results The GAP group obtained more correct answers than the DYS group. An arithmetic N400 effect was observed in the GAP group but not in the DYS group. Both groups displayed an LPC effect. The larger the LPC amplitude was, the higher the WM index. Two subgroups were found within the DYS group: one with an average WM index and the other with a lower than average WM index. These subgroups displayed different ERPs patterns. Discussion The results indicated that the group of children with dyscalculia was very heterogeneous and therefore failed to show a robust LPC effect. Some of these children had WM deficits. When WM deficits were considered together with dyscalculia, an atypical ERP pattern that reflected their processing difficulties emerged. Their lack of the arithmetic N400 effect suggested that the processing in this step was not useful enough to produce an answer; thus, it was necessary to reevaluate the arithmetic-calculation process (LPC) in order to deliver a correct answer. Conclusion Given that dyscalculia is a very heterogeneous deficit, studies examining dyscalculia should consider exploring deficits in WM because the whole group of children with dyscalculia seems to contain at least two subpopulations that differ in their calculation process.

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Lattmann, Claas. "Jamblique, In Nicomachi Arithmeticam. Introduction, texte critique, traduction française et notes de commentaire par Nicolas Vinel." Gnomon 90, no. 6 (2018): 497–502. http://dx.doi.org/10.17104/0017-1417-2018-6-497.

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Hackbusch, W. "A Sparse Matrix Arithmetic Based on $\Cal H$ -Matrices. Part I: Introduction to ${\Cal H}$ -Matrices." Computing 62, no. 2 (April 1, 1999): 89–108. http://dx.doi.org/10.1007/s006070050015.

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Győry, K., L. Hajdu, and Á. Pintér. "Perfect powers from products of consecutive terms in arithmetic progression." Compositio Mathematica 145, no. 4 (July 2009): 845–64. http://dx.doi.org/10.1112/s0010437x09004114.

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AbstractWe prove that for any positive integers x,d and k with gcd (x,d)=1 and 3<k<35, the product x(x+d)⋯(x+(k−1)d) cannot be a perfect power. This yields a considerable extension of previous results of Győry et al. and Bennett et al., which covered the cases where k≤11. We also establish more general theorems for the case where x can also be a negative integer and where the product yields an almost perfect power. As in the proofs of the earlier theorems, for fixed k we reduce the problem to systems of ternary equations. However, our results do not follow as a mere computational sharpening of the approach utilized previously; instead, they require the introduction of fundamentally new ideas. For k>11, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

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Journal articles on the topic 'Introductio arithmetica'

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