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Journal articles on the topic 'Mathematical notation'

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Author: Grafiati

Published: 4 June 2021

Last updated: 30 July 2024

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1

Sokolowski, Andrzej. "Developing Covariational Reasoning Among Students Using Contexts of Formulas." Physics Educator 02, no. 04 (December 2020): 2050016. http://dx.doi.org/10.1142/s266133952050016x.

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Multiple studies have been conducted to assess students’ ability to apply covariational reasoning to sketching graphs in physics. This study is supported by research on developing students’ skills in sketching functions in mathematics. It attempts to evaluate physics students’ ability to apply these skills to identify critical algebraic attributes of physics formulas for their potential to be sketched. Rather than seeking formulas’ physical interpretation, this study is posited to challenge students’ skills to merge their mathematical knowledge within physics structures. A group of thirty ([Formula: see text]) first-year college-level physics students were provided with two physically identical equations that described the object’s position. However, one equation was expressed in functional mathematical notation, whereas the other in a standard formula notation. The students were asked to classify the symbols in each formula as variables or parameters and determine these formulas’ potential to be graphed in respective coordinates. The analysis revealed that 93% of these students considered function notation as possessing sketchable potential against 13% who envisioned such potential in the standard formula notation. Further investigations demystified students’ confusion about the classification of the symbols used in the formula notation. These results opened up a gate for discussing the effects of algebraic notations in physics on activating students’ covariational skills gained in mathematics courses. Suggestions for improving physics instructions stemming from this study are discussed.

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Yang, Seong Hyun. "A View on the Diversity of the Word and Mathematical Notation Expression Used in High School Mathematics Textbooks." Korean School Mathematics Society 20, no. 3 (September 30, 2017): 211–37. http://dx.doi.org/10.30807/ksms.2017.20.3.001.

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Depending on the type of textbook, the word and mathematical notation expression used in high school mathematics textbooks varied and there were also some differences on the mathematical definition and the content description methods. Not only the composition of textbooks but also various expressing ways of textbooks have significant impacts on teaching and learning of teacher and student. The diversity of expression had pros and cons like both sides of a coin. There is a positive aspect that we can pursue pedagogical diversity. Simultaneously there is a negative aspect that the possibility of acting as a learning burden exists in the viewpoint of the student and the equality of evaluation may be undermined. In this study, Preferentially we focused on analyzing the actual situation rather than judging what is more appropriate about the diversity of words and notation expressions used in mathematics textbooks which is based on the current curriculum. For this purpose, we analyzed 56 kinds of mathematics textbooks based on the 2009 revised mathematics curriculum, and presented four aspects(terms expressing, notations expression, mathematical definition, content description method) with examples about differences of the various expressions used in textbooks including ‘terms and notations’.

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3

Williams, Travis D. "Mathematical Enargeia: The Rhetoric of Early Modern Mathematical Notation." Rhetorica 34, no. 2 (2016): 163–211. http://dx.doi.org/10.1525/rh.2016.34.2.163.

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This article proposes and explicates a rhetorical model for the function of notational writing in sixteenth- and seventeenth-century European mathematics. Drawing on enargeia's requirement that both author and reader contribute to the full realization of a text, mathematical enargeia enables the transformation of images of mathematical imagination resulting from an encounter with mathematical writing into further written acts of mathematical creation. Mathematical enargeia provides readers with an ability to understand a text as if they created it themselves. Within the period's dominant reading of classical geometry as a synthetic presentation that suppressed, hid, or obscured analytic mathematical reality, notational mathematics found favor as a rhetorically unmediated expression of mathematical truth. Consequently, mathematical enargeia creates an operational and presentational link between mathematics' past and its future.

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Williams, Travis D. "Mathematical Enargeia: The Rhetoric of Early Modern Mathematical Notation." Rhetorica 34, no. 2 (March 2016): 163–211. http://dx.doi.org/10.1353/rht.2016.0017.

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5

Huuskonen, Taneli. "Polish Notation." Formalized Mathematics 23, no. 3 (September 1, 2015): 161–76. http://dx.doi.org/10.1515/forma-2015-0014.

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Abstract This article is the first in a series formalizing some results in my joint work with Prof. Joanna Golinska-Pilarek ([12] and [13]) concerning a logic proposed by Prof. Andrzej Grzegorczyk ([14]). We present some mathematical folklore about representing formulas in “Polish notation”, that is, with operators of fixed arity prepended to their arguments. This notation, which was published by Jan Łukasiewicz in [15], eliminates the need for parentheses and is generally well suited for rigorous reasoning about syntactic properties of formulas.

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DE TOFFOLI, SILVIA. "‘CHASING’ THE DIAGRAM—THE USE OF VISUALIZATIONS IN ALGEBRAIC REASONING." Review of Symbolic Logic 10, no. 1 (October 28, 2016): 158–86. http://dx.doi.org/10.1017/s1755020316000277.

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AbstractThe aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation.

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7

Grabiner, Judith. "“Notation, Notation, Notation” or Book Review : Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers by Joseph Mazur." Journal of Humanistic Mathematics 5, no. 2 (July 2015): 151–60. http://dx.doi.org/10.5642/jhummath.201502.14.

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8

Wege, Theresa Elise, Sophie Batchelor, Matthew Inglis, Honali Mistry, and Dirk Schlimm. "Iconicity in mathematical notation: Commutativity and symmetry." Journal of Numerical Cognition 6, no. 3 (December 3, 2020): 378–92. http://dx.doi.org/10.5964/jnc.v6i3.314.

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Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects – those which visually resemble in some way the concepts they represent – offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd’s hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance.

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9

Bier, Agnieszka, and Zdzisław Sroczyński. "Rule based intelligent system verbalizing mathematical notation." Multimedia Tools and Applications 78, no. 19 (July 4, 2019): 28089–110. http://dx.doi.org/10.1007/s11042-019-07889-3.

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10

Perkins, Isabel, and Alfinio Flores. "Mathematical Notations and Procedures of Recent Immigrant Students." Mathematics Teaching in the Middle School 7, no. 6 (February 2002): 346–51. http://dx.doi.org/10.5951/mtms.7.6.0346.

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Mathematics is often referred to as a universal language. Compared with the differences in language and culture faced by students who are recent immigrants, the differences in mathematical notation and procedures seem to be minor. Nevertheless, immigrant students confront noticeable differences between the way that mathematical ideas are represented in their countries of origin and the manner that they are represented in the United States. If not addressed, the differences in notation and procedures can add to the difficulties that immigrants face during their first years in a new country.

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11

MAURER, W. DOUGLAS. "The influence of the computer upon mathematical notation." Semiotica 125, no. 1-3 (1999): 165–68. http://dx.doi.org/10.1515/semi.1999.125.1-3.165.

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12

Edwards, Andrew M., and Marie Auger‐Méthé. "Some guidance on using mathematical notation in ecology." Methods in Ecology and Evolution 10, no. 1 (December 16, 2018): 92–99. http://dx.doi.org/10.1111/2041-210x.13105.

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13

Shapiro, Stewart, Eric Snyder, and Richard Samuels. "Computability, Notation, and de re Knowledge of Numbers." Philosophies 7, no. 1 (February 18, 2022): 20. http://dx.doi.org/10.3390/philosophies7010020.

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Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

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HUET, GÉRARD. "Preface." Mathematical Structures in Computer Science 21, no. 4 (July 1, 2011): 671–77. http://dx.doi.org/10.1017/s0960129511000235.

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This special issue of Mathematical Structures in Computer Science is devoted to the theme of ‘Interactive theorem proving and the formalisation of mathematics’.The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements.

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J.G. White, Jason. "The Accessibility of Mathematical Notation on the Web and Beyond." Journal of Science Education for Students with Disabilities 23, no. 1 (October 21, 2020): 1–14. http://dx.doi.org/10.14448/jsesd.12.0013.

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This paper serves two purposes. First, it offers an overview of the role of the Mathematical Markup Language (MathML) in representing mathematical notation on the Web, and its significance for accessibility. To orient the discussion, hypotheses are advanced regarding users’ needs in connection with the accessibility of mathematical notation. Second, current developments in the evolution of MathML are reviewed, noting their consequences for accessibility, and commenting on prospects for future improvement in the concrete experiences of users of assistive technologies. Recommendations are advanced for further research and development activities, emphasizing the cognitive aspects of user interface design.

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Iwaszkiewicz, Jan, and Adam Muc. "State and Space Vectors of the 5-Phase 2-Level VSI." Energies 13, no. 17 (August 25, 2020): 4385. http://dx.doi.org/10.3390/en13174385.

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The paper proposes a general description system of the five-phase two-level inverter. The two base methods are presented and discussed. The first one is based on the standard space vector transformation, while the other uses state vectors which enable the definition of the basic physical quantities of the inverter: current and voltage. The proposed notation system offers a general simplification of vector identification. It comprises a standardized proposal of notation and vector marking, which may be extremely useful for the specification of inverter states. The described notation system makes it possible to reach correlation between state and space vectors. It presents space and state vectors using the same digits. These properties suggest that the proposed notation system is a useful mathematical tool and may be really suitable in designing control algorithms. This mathematical tool was verified during simulation tests performed with the use of the Simulation Platform for Power Electronics Systems—PLECS.

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17

Coghetto, Roland. "Groups – Additive Notation." Formalized Mathematics 23, no. 2 (June 1, 2015): 127–60. http://dx.doi.org/10.1515/forma-2015-0013.

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Abstract We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

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18

Thielemann, Henning. "Mathematical notation and the use of functions (Poster Presentation)." PAMM 7, no. 1 (December 2007): 2170001–2. http://dx.doi.org/10.1002/pamm.200700988.

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19

Conner, Kimberly A. "Constructing and Unpacking Diagrams in Geometry." Mathematics Teacher: Learning and Teaching PK-12 113, no. 6 (June 2020): 516–19. http://dx.doi.org/10.5951/mtlt.2019.0302.

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By having students practice constructing diagrams for geometric theorems, teachers can develop students' understanding of mathematical claims, vocabulary, and notation methods. This practice can also strengthen students' ability to interpret mathematical diagrams and recognize their limitations.

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20

DUNNING, DAVID E. "“ALWAYS MIXED TOGETHER”: NOTATION, LANGUAGE, AND THE PEDAGOGY OF FREGE'S BEGRIFFSSCHRIFT." Modern Intellectual History 17, no. 4 (September 26, 2018): 1099–131. http://dx.doi.org/10.1017/s1479244318000410.

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Gottlob Frege is considered a founder of analytic philosophy and mathematical logic, but the traditions that claim Frege as a forebear never embraced his Begriffsschrift, or “conceptual notation”—the invention he considered his most important accomplishment. Frege believed that his notation rendered logic visually observable. Rejecting the linearity of written language, he claimed Begriffsschrift exhibited a structure endogenous to logic itself. But Frege struggled to convince others to use his notation, as his frustrated pedagogical efforts at the University of Jena illustrate. Teaching Begriffsschrift meant using words to explain it; rather than replacing spoken language, notation became its obverse in a bifurcated style of argument that separated deduction from commentary. Both registers of this discourse, however, remained within Frege's monologue, imposing a consequential passivity on his students. In keeping with Frege's visual understanding of notation, they learned by silently observing it, though never in isolation: notation and language were always mixed together.

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21

Bonda, Moreno. "Modal Difficulty in Medieval Literature Analysis: the Frame-Notation Correlation in Dante’s Quotations of Fibonacci." Aktuālās problēmas literatūras un kultūras pētniecībā: rakstu krājums, no. 26/2 (March 11, 2021): 106–21. http://dx.doi.org/10.37384/aplkp.2021.26-2.106.

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The investigation of medieval literature poses a number of challenges, even to native speaker researchers. Such difficulties are related to (a) linguistic – syntactical and lexical – obstacles, (b) to the ability to recognise dense networks of interdisciplinary references and, (c) mainly to the cognitive challenges posed by “unfamiliar modes of expression”. The aim of this research is to discuss a methodological approach to deal with these unusual manners of composition, technically known as modal difficulty, in medieval literature. The theoretic setting is represented by Davide Castiglione’s monographic study Difficulty in Poetry (2018) and the specific definition of modal difficulty elaborated by James E. Vincent in the premise of his treatise on American poetry (2003). A study case illustrative of challenges in medieval literature analysis has been chosen to illustrate the speculative reasoning: the references to the celebrated mathematician Leonardo Fibonacci (1170–1242) – known for having introduced the Arabic numbers to the Europeans – in Dante Alighieri’s Divine Comedy. Preliminarily, the author discusses unfamiliar mathematical notations implemented from the 13th to the 18th centuries. Subsequently, adopting cognitive linguistics principles and hermeneutic as methodological tools, several veiled citations of the mathematician’s cogitations – such as the chess comparison in Paradise XXVIII, 91–93 and the quadratic expression in Paradise XXVII, 115–117 – are deciphered and illustrated. The analysis of Dante’s cognitive frame indicates that the recourse to Fibonacci’s formulas is functional to depict the incommensurable multitude of the divine in words. In the conclusions, the case studied is adopted as a model to illustrate how the reflection on unusual forms of expression could be employed to investigate ancient literary texts. A preliminary analysis of the frame-notation relation could help, as an example, to recognise mathematical formulas that were expressed in a verbal and non-symbolic notation.

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22

Wong, Yetta Kwailing, and Isabel Gauthier. "A Multimodal Neural Network Recruited by Expertise with Musical Notation." Journal of Cognitive Neuroscience 22, no. 4 (April 2010): 695–713. http://dx.doi.org/10.1162/jocn.2009.21229.

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Prior neuroimaging work on visual perceptual expertise has focused on changes in the visual system, ignoring possible effects of acquiring expert visual skills in nonvisual areas. We investigated expertise for reading musical notation, a skill likely to be associated with multimodal abilities. We compared brain activity in music-reading experts and novices during perception of musical notation, Roman letters, and mathematical symbols and found selectivity for musical notation for experts in a widespread multimodal network of areas. The activity in several of these areas was correlated with a behavioral measure of perceptual fluency with musical notation, suggesting that activity in nonvisual areas can predict individual differences in visual expertise. The visual selectivity for musical notation is distinct from that for faces, single Roman letters, and letter strings. Implications of the current findings to the study of visual perceptual expertise, music reading, and musical expertise are discussed.

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23

Finney, K. "Mathematical notation in formal specification: too difficult for the masses?" IEEE Transactions on Software Engineering 22, no. 2 (1996): 158–59. http://dx.doi.org/10.1109/32.485225.

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24

Chrisomalis, Stephen. "Constraint, cognition, and written numeration." Pragmatics and Cognition 21, no. 3 (December 31, 2013): 552–72. http://dx.doi.org/10.1075/pc.21.3.08chr.

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The world’s diverse written numeral systems are affected by human cognition; in turn, written numeral systems affect mathematical cognition in social environments. The present study investigates the constraints on graphic numerical notation, treating it neither as a byproduct of lexical numeration, nor a mere adjunct to writing, but as a specific written modality with its own cognitive properties. Constraints do not refute the notion of infinite cultural variability; rather, they recognize the infinity of variability within defined limits, thus transcending the universalist/particularist dichotomy. In place of strictly innatist perspectives on mathematical cognition, a model is proposed that invokes domain-specific and notationally-specific constraints to explain patterns in numerical notations. The analysis of exceptions to cross-cultural generalizations makes the study of near-universals highly productive theoretically. The cross-cultural study of patterns in written numbers thus provides a rich complement to the cognitive analysis of writing systems.

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Cai, Jinfa, and Patricia Ann Kenney. "Fostering Mathematical Thinking through Multiple Solutions." Mathematics Teaching in the Middle School 5, no. 8 (April 2000): 534–39. http://dx.doi.org/10.5951/mtms.5.8.0534.

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The reform movement in school mathematics advocates communication as a necessary component for learning, doing, and understanding mathematics (Elliott and Kenney 1996). Communication in mathematics means that one is able not only to use its vocabulary, notation, and structure to express ideas and relationships but also to think and reason mathematically. In fact, communication is considered the means by which teachers and students can share the processes of learning, doing, and understanding mathematics. Students should express their thinking and problem-solving processes in both written and oral formats. The clarity and completeness of students' communication can indicate how well they understand the related mathematical concepts.

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Kai Shun, Mr Lam. "An Algorithmic Approach to Solve Continuum Hypothesis." Academic Journal of Applied Mathematical Sciences, no. 71 (November 25, 2020): 36–49. http://dx.doi.org/10.32861/ajams.71.36.49.

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The continuum hypothesis has been unsolved for hundreds of years. In other words, can I answer it completely? By refuting the culturally responsible continuum [1], one can link the problem to the mathematical continuum, and it is possible to disproof the continuum hypothesis [2] . To go ahead a step, one may extend our mathematical system (by employing a more powerful set theory) and solve the continuum problem by three conditional cases. This event is sim-ilar to the status cases in the discriminant of solving a quadratic equation. Hence, my proposed al-gorithmic flowchart can best settle and depict the problem. From the above, one can further con-clude that when people extend mathematics (like set theory — ZFC) into new systems (such as Force Axioms), experts can solve important mathematical problems (CH). Indeed, there are differ-ent types of such mathematical systems, similar to ancient mathematical notation. Hence, different cultures have different ways of representation, which is similar to a Chinese saying: “different vil-lages have different laws.” However, the primary purpose of mathematical notation was initially to remember and communicate. This event indicates that the basic purpose of developing any new mathematical system is to help solve a natural phenomenon in our universe.

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Cardona-Hurtado, Oscar Abel. "Beneficios de la notación de Peirce para los conectivos proposicionales binarios." Respuestas 21, no. 1 (January 1, 2016): 56. http://dx.doi.org/10.22463/0122820x.637.

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Antecedentes: En la notación tradicional para los conectivos proposicionales binarios son tenidos en cuenta solamente algunos de estos. A lo largo del siglo XX fueron propuestas varias notaciones que subsanan esa falencia, dando lugar al planteamiento de interesantes problemas matemáticos. Objetivo: En este escrito se presenta la notación creada por el norteamericano Charles Peirce, se muestran algunas propiedades de las cuales goza esta simbología, y se evidencian sus ventajas con respecto a la tradicional. Método: Se describe la notación propuesta por Peirce, y se verifican algunas propiedades de carácter lógico geométrico y algebraico entre sus conectivos; también se analiza la posible actuación de estas propiedades en la notación usual. Resultados: Además de varias propiedades individuales y de múltiples relaciones entre los conectivos, las simetrías del sistema completo de los conectivos proposicionales binarios se evidencian de manera visual en los signos propuestos por Peirce. Conclusión: Diversas bondades de las cuales goza la notación propuesta por Peirce, permiten afirmar que la notación usual es superada de manera clara por la simbología diseñada por el científico norteamericano.Abstract Background: In traditional binary notation for propositional connectives only some of these ones are taken into account. Throughout the twentieth century several notations were proposed which overcome this flaw, leading to the proposal of interesting mathematical problems. Objective: This paper presents the notation created by the American Charles Peirce, showing some of the properties of this symbols, and evidencing the advantages of these compared to the traditional. Method: the notation proposed by Peirce is described, and some properties of the geometric and algebraic logical character among its connective are verified; also, the possible role of these properties in the traditional notation is analyzed. Results: In addition to several individual properties and multiple relations between the connectives, the symmetries of the full set of binary propositional connective is visually evident in the signs proposed by Peirce. Conclusion: Different benefits of the notation proposed by Peirce, support the conclusion that the usual notation is clearly surpassed by the symbolism designed by the American scientist.Palabras clave: Conectivo proposicional, Charles S. Peirce, operación, simetría, tabla de verdad.

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Javid, Tariq, Muhammad Faris, and Pervez Akhtar. "Integrated representation for discrete Fourier and wavelet transforms using vector notation." Mehran University Research Journal of Engineering and Technology 41, no. 3 (July 1, 2022): 175–84. http://dx.doi.org/10.22581/muet1982.2203.18.

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Many mathematical operations are implemented easily through transform domain operations. Multiple transform domain operations are used independently in large and complex applications. There is a need to develop integrated representations for multiple transform domain operations. This paper presents an integrated mathematical representation for the discrete Fourier transformation and the discrete wavelet transformation. The proposed combined representation utilizes the powerful vector notation. A mathematical operator, called the star operator, is formulated that merges coefficients from different transform domains. The star operator implements both convolution and correlation processes in a weighted fashion to compute the aggregated representation. The application of the proposed mathematical formulation is demonstrated successfully through merging transform domain representations of time-domain and image-domain representations. Heart sound signals and magnetic resonance images are used to describe transform-domain data merging applications. The significance of the proposed technique is demonstrated through merging time-domain and image-domain representations in a single- stage that may be implemented as the primary processing engine inside a typical digital image processing and analysis system.

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Staines, Anthony Spiteri. "Concurrency and Petri Net Models." International Journal of Circuits, Systems and Signal Processing 16 (March 11, 2022): 852–58. http://dx.doi.org/10.46300/9106.2022.16.104.

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Concurrency is a fundamental problem and a solution applicable to different areas of computing. Given the complexities and distribution of computer systems and services, concurrency is a modern area requiring proper attention. Petri nets are formalisms based on process representation both from a mathematical view and from a graphical or drawing like view. Petri nets are used to model concurrent processes. This work deals with understanding and representing low level concurrency in Petri nets, when this is not always visible and properly noted from the graphical structure. In this study an algebraic notation has been devised and is used to represent the Petri net structures. This algebraic notation is used as an alternative and simplified way of representation. The notation is explained and several simple examples are given. The notation presented can be used in conjunction with other Petri net analysis and verification methods. Some results and findings are discussed.

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Lin, Psang Dain, and Jung-Fa Hsieh. "A New Method to Analyze Spatial Binary Mechanisms With Spherical Pairs." Journal of Mechanical Design 129, no. 4 (March 29, 2006): 455–58. http://dx.doi.org/10.1115/1.2437782.

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One of the most popular mathematical tools in the fields of robotics and mechanisms is the Denavit-Hartenberg (DH) notation (Denavit and Hartenberg, 1955, J. Appl. Mech., 77, pp. 215–221). It is valid only for mechanisms containing prismatic, revolute, helical, and cylindrical pairs, but cannot be applied to spherical pairs. This paper presents an extended DH notation that includes spherical pairs, consequently allowing the required independent parameters of any spatial binary mechanism to be listed for purposes of analysis and synthesis. Further, the interference-free region with maximum ball-retention capability of a socket in a spherical pair can be determined analytically. Extended DH notation can systematically model arbitrary binary mechanisms with spherical pairs, simplifying their design and study.

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31

Jones, Rafe. "Enlightening Symbols: A Short History of Mathematical Notation by Joseph Mazur." Mathematical Intelligencer 38, no. 2 (December 30, 2015): 85–86. http://dx.doi.org/10.1007/s00283-015-9589-y.

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32

Norena, P., C. Zapata, and A. Villamizar. "Representing Chemical Events by using Mathematical Notation from Pre-conceptual Schemas." IEEE Latin America Transactions 17, no. 01 (January 2019): 46–53. http://dx.doi.org/10.1109/tla.2019.8826694.

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33

Dunning, David E. "The logic of the nation: Nationalism, formal logic, and interwar Poland." Studia Historiae Scientiarum 17 (December 12, 2018): 207–51. http://dx.doi.org/10.4467/2543702xshs.18.009.9329.

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Between the World Wars, a robust research community emerged in the nascent discipline of mathematical logic in Warsaw. Logic in Warsaw grew out of overlapping imperial legacies, launched mainly by Polish-speaking scholars who had trained in Habsburg universities and had come during the First World War to the University of Warsaw, an institution controlled until recently by Russia and reconstructed as Polish under the auspices of German occupation. The intellectuals who formed the Warsaw School of Logic embraced a patriotic Polish identity. Competitive nationalist attitudes were common among interwar scientists – a stance historians have called “Olympic internationalism,” in which nationalism and internationalism interacted as complementary rather than conflicting impulses. One of the School’s leaders, Jan Łukasiewicz, developed a system of notation that he promoted as a universal tool for logical research and communication. A number of his compatriots embraced it, but few logicians outside Poland did; Łukasiewicz’s notation thus inadvertently served as a distinctively national vehicle for his and his colleagues’ output. What he had intended as his most universally applicable invention became instead a respected but provincialized way of writing. Łukasiewicz’s system later spread in an unanticipated form, when postwar computer scientists found aspects of its design practical for working under the specific constraints of machinery; they developed a modified version for programming called “Reverse Polish Notation” (RPN). RPN attained a measure of international currency that Polish notation in logic never had, enjoying a global career in a different discipline outside its namesake country. The ways in which versions of the notation spread, and remained or did not remain “Polish” as they traveled, depended on how readers (whether in mathematical logic or computer science) chose to read it; the production of a nationalized science was inseparable from its international reception.

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Hetmańczyk, Mariusz Piotr. "The Prediction Oriented Analysis of Mechatronic Machine Structures Recorded by Directed Graphs." Solid State Phenomena 220-221 (January 2015): 429–34. http://dx.doi.org/10.4028/www.scientific.net/ssp.220-221.429.

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The paper presents possibility of usage the formalism of structural reliability notation with an implementation of directed graphs and reliability functions recorded in accordance with the Perfect Disjunctive Normal Form notation (PDNF). The author presents the mathematical basis used for an identification of internal structures of mechatronic machines and reduction it to the series-connected blocks. The presented method is a hybrid combination of a binary analysis of evaluated reliability functions (stored in the form of matrixes that relate to defined graphs) and blocks with binary inputs and outputs.

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Sikora, Ryszard, and Stanislaw Pawłowski. "Fractional derivatives and the laws of electrical engineering." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, no. 4 (July 2, 2018): 1384–91. http://dx.doi.org/10.1108/compel-08-2017-0347.

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Purpose This paper aims to evaluate the possibilities of fractional calculus application in electrical circuits and magnetic field theories. Design/methodology/approach The analysis of mathematical notation is used for physical phenomena description. The analysis aims to challenge or prove the correctness of applied notation. Findings Fractional calculus is sometimes applied correctly and sometimes erroneously in electrical engineering. Originality/value This paper provides guidelines regarding correct application of fractional calculus in description of electrical circuits’ phenomena. It can also inspire researchers to find new applications for fractional calculus in the future.

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Eremin, E. A., and K. Yu Polyakov. "HOW CORRECTLY WE TELL PUPILS ABOUT THE NUMBER OF NUMERALS IN DIFFERENT NOTATIONS?" Informatics in school, no. 6 (October 10, 2020): 8–18. http://dx.doi.org/10.32517/2221-1993-2020-19-6-8-18.

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The article proposes to take a fresh look at the traditional for any informatics course material, which describes the fundamental principles of number representation. It considers in details the ideas how different number notations are formed; special attention is paid to the problem related to the amount of numerals used for presenting values in every digit. It is shown that frequent in school informatics definition of radix as an amount of numerals that employed in a system becomes incorrect under careful review. For those who want to discuss this problem at a lesson, two ways are possible: one (stricter) is founded on mathematical formulas, and another is based on a generalization of various notation examples; both methods are presented in the article. Besides for demonstrative narrating of the material to pupils the authors designed their own analogy, which facilitates understanding of the topic. Practical application of notations to computer technique is also discussed.

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LAM, VITUS S. W. "FORMAL ANALYSIS OF BPMN MODELS: A NuSMV-BASED APPROACH." International Journal of Software Engineering and Knowledge Engineering 20, no. 07 (November 2010): 987–1023. http://dx.doi.org/10.1142/s0218194010005079.

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Business Process Modeling Notation (BPMN) plays a significant role in the specification of business processes. To ascertain the validity of BPMN models, a disciplined approach to analyze their behavior is of particular interest to the field of business process management. This paper advocates a semantics-preserving method for transforming BPMN models into New Symbolic Model Verifier (NuSMV) language as a means to verify the models. A subset of BPMN is specified rigorously in the form of a mathematical model. With this foundation in place, the translation for the subset of BPMN notational elements is then driven by a set of formally defined rules. The practicality of our approach is exemplified using an on-line flight reservation service.

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De Gosson, Maurice A. "Quantum Harmonic Analysis of the Density Matrix." Quanta 7, no. 1 (September 26, 2018): 74. http://dx.doi.org/10.12743/quanta.v7i1.74.

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We will study rigorously the notion of mixed states and their density matrices. We will also discuss the quantum-mechanical consequences of possible variations of Planck's constant h. This review has been written having in mind two readerships: mathematical physicists and quantum physicists. The mathematical rigor is maximal, but the language and notation we use throughout should be familiar to physicists.Quanta 2018; 7: 74–110.

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Perry, Jill A., and Sandra L. Atkins. "It's Not Just Notation: Valuing Children's Representations." Teaching Children Mathematics 9, no. 4 (December 2002): 196–201. http://dx.doi.org/10.5951/tcm.9.4.0196.

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Mathematics teachers often say to their students, “Show your work,” when what they really mean is, “Show this to me as I showed it to you” or “Show my work.” During our first few years as mathematics teachers, we spent much of our time trying to make our students think as we thought—to use the symbols that we were using in the ways in which we were using them. We find it uncomfortable to admit that during those years we spent little time trying to find out what our students were thinking. We valued the process and product looking like ours more than the process and product being theirs. We expected our students to use conventional representations before we had given them sufficient time to develop mathematical concepts.

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Impelluso, Thomas J. "The moving frame method in dynamics: Reforming a curriculum and assessment." International Journal of Mechanical Engineering Education 46, no. 2 (August 30, 2017): 158–91. http://dx.doi.org/10.1177/0306419017730633.

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Rigid body dynamics, a gateway course to the mechanical engineering major (and related majors), focuses on a view of motion that is not commensurate with the contemporary age in which mobile devices have on-board inertial firmware. The traditional approach to this topic deploys a mathematical notation, and associated algebra, that inordinately privileges the inertial frames and 2D motion. This limits the study of machines to two-dimensional problems, lends an appearance of whimsy to solutions that obfuscates the theory of motion. We propose a new mathematical approach to dynamics to reinvigorate the discipline and motivate students. The new approach uses modern mathematical tools which have been distilled to tractability: Lie Group Theory, Cartan’s Moving Frames and a new compact notation from Geometrical Physics. The reconstructed course abandons the cross product—a toxic algebraic operation due to its failure to adhere to associativity. We minimize the use of vectors and replace them with rotation matrices. Sophomores learn to solve 3D Dynamics problems with as much ease as solving 2D problems. Typical problems include the precession of tops, gyroscopes, inertial devices to prevent ship roll at sea, and 3D robot and crane kinetics. A critical aspect of this new method is the consistency: the notation is the same for 3D and 2D problems, from advanced robotics to introductory dynamics, students learn the name notational method. The first objective of this paper presents the new mathematical approach to rigid body dynamics—it amounts to an introductory, yet simplified, lecture on a new method. The second objective presents assessment over a three-year period. In the first year, we taught using the old 2D vector-based approach. In the second year, we transitioned to the new method and compared student perceptions in the first two years. In the third year, the course was refined. The goal of this effort is to retain students in mechanical engineering by offering them a new view of the discipline, rather than simple pedagogical course interventions such as e-learning or flipped classrooms. The course content is delivered using the emerging visualization technology: WebGL. WebGL represents the future of the 3D web. It requires no downloads and no plugins. Students are directed to a web site where all images for the lectures are 3D and interactive. The animations run on cell phones, laptops and other mobile devices. It is the contention of this paper that modernizing the math will do more to reduce attrition than learning interventions. This new approach reduces conceptual difficulties that accompany 2D restrictions. It opens many questions on how students perceive 3D space and invites research into how exploiting more modern mathematical math may improve learning.

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Ka, Pratik, S. Suma, and Vishwas B R. "Solving Linear Programming Problems Using AMPL Modeling LanguageSolving Linear Programming Problems Using AMPL Modeling Language." International Journal of Research and Review 9, no. 11 (November 3, 2022): 66–69. http://dx.doi.org/10.52403/ijrr.20221110.

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Optimization problems arise in many contexts. A modelling language like AMPL makes it easier to experiment with formulations and use the right solvers to address the resultant optimization issues. Variables, objectives, constraints, sets of possible parameters, and notations that resemble well-known mathematical notation can all be stated using AMPL. The AMPL command language enables computation and display of data regarding the specifics of a problem and the solutions provided by solvers. It also enables the modification of problem formulations and the resolution of problem chains. Both continuous and discrete optimization issues are addressed by AMPL. In this paper, AMPL is used to solve different optimization problems such as Wyndor Glass problem, Transportation and Assignment problem and Purchase Planning problem. Keywords: Optimization, AMPL, Wyndor Glass Problem, Transportation and Assignment Problem, Purchase Planning Problem

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Popa, Stelian, Alexandru Dorin, Florin Adrian Nicolescu, and Andrei Mario Ivan. "Quaternion-Based Algorithm for Direct Kinematic Model of a Kawasaki FS10E Articulated Arm Robot." Applied Mechanics and Materials 762 (May 2015): 249–54. http://dx.doi.org/10.4028/www.scientific.net/amm.762.249.

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This article follows a detailed description of development and validation for the direct kinematic model of six degrees of freedom articulated arm robot - Kawasaki FS10E model. The development of the kinematic model is based on widely used Denavit-Hartenberg notation, but, after the initial parameter identification, the mathematical algorithm itself follows an approach that uses the quaternion number system, taking advantage of their efficiency in describing spatial rotation - providing a convenient mathematical notation for expressing rotations and orientations of objects in three-dimensional space. The proposed algorithm concludes with two quaternion-based relations that express both the position of robot tool center point (TCP) position and end-effector orientation with respect to robot base coordinate system using Denavit-Hartenberg parameters and joint values as input data. Furthermore, the developed direct kinematic model was validated using the programming and offline simulation software Kawasaki PC Roset.

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Evriani, Emi, Yulis Jamiah, and Hamdani Hamdani. "KEMAMPUAN KOMUNIKASI MATEMATIS SISWA DALAM MATERI HIMPUNAN DI KELAS VII." Jurnal AlphaEuclidEdu 3, no. 2 (December 31, 2022): 212. http://dx.doi.org/10.26418/ja.v3i2.58308.

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This study aims to describe students' mathematical communication skills during the covid-19hpandemic in learning mathematics, especially set operations material. .The method used in this research is the descriptive method.. The research subjects in this study were students of SMP Negeri 2 Pontianak class VII G who had studied the set material. Collecting data in this study through tests,aand, interviews. Based on, the results of the study, it was ,found/ that ,the ability of /students to express mathematical ideas in online learning through writing on set operations material was 42.5% relatively low. Students tend not to be able to write set notation, students are also not able to describe Venn diagrams and students still have many errors in working on the given questions. The ability of students to describe mathematical ideas in online learning on set operations material is 55% low. Students tend to be able to describe Venn diagrams but are less clear and incomplete in describing Venn diagrams. The ability to explain students' mathematics in online learning through oral on set operations material is 65% classified as moderate. Students tend to be able to explain the steps of the answers they have done and give reasons for the answers well. The students' mathematical reading ability in online learning through oral on set operations material by 40% is low. Students havehnot been able to convey what is understood orally and haveknot been able to read mathematical notations anddterms.

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Taranchuk, V. B. "FEATURES OF FUNCTIONAL PROGRAMMING OF INTERACTIVE GRAPHICAL APPLICATIONS." Vestnik of Samara University. Natural Science Series 21, no. 6 (May 17, 2017): 178–89. http://dx.doi.org/10.18287/2541-7525-2015-21-6-178-189.

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In the article methodical and technical solutions which essentially expand capabilities of creation of the electronic intelligent educational resources contain- ing mathematical notation of any level of complexity and graphics illustrations of all types and categories are discussed. Base units of program modules, key constructions of codes, functions and options of language of the system of com- puter algebra Mathematica are explained. Main rules of preparation of freely distributed interactive program applications of CDF format are noted. Exam- ples from practice of preparation of teaching materials of discipline ”Computer Graphics” are given. User interface and results of execution of program modules are illustrated.

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Forero, Manuel G., and Carlos A. Jacanamejoy. "Unified Mathematical Formulation of Monogenic Phase Congruency." Mathematics 9, no. 23 (November 30, 2021): 3080. http://dx.doi.org/10.3390/math9233080.

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Phase congruency is a technique that has been used for edge, corner and symmetry detection. Its implementation through the use of monogenic filters has improved its computational cost. For this purpose, different methods of implementation have been published, but they do not use a common notation, which makes it difficult to understand. Therefore, this paper presents a unified mathematical formulation that allows a general understanding of the Monogenic phase congruency concepts and establishes criteria for its use. A new protocol for parameter tuning is also described, allowing better practical results to be obtained with this technique. Some examples are presented allowing one to observe the changes produced in the parameter tuning, evidencing the validity of the proposed criteria.

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Atagi, Natsuki, Melissa DeWolf, James W. Stigler, and Scott P. Johnson. "The role of visual representations in college students’ understanding of mathematical notation." Journal of Experimental Psychology: Applied 22, no. 3 (September 2016): 295–304. http://dx.doi.org/10.1037/xap0000090.

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Dyke, Phil. "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers." Leonardo 48, no. 2 (April 2015): 202–3. http://dx.doi.org/10.1162/leon_r_00988.

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Rábová, Ivana. "Using UML and Petri nets for visualization of business document flow." Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis 60, no. 2 (2012): 299–306. http://dx.doi.org/10.11118/actaun201260020299.

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The article deals with two principles of business workflow modeling, Petri nets and UML notation, that are the acceptable approaches to business modeling and can be used also for business documents workflow. The special type of Petri nets, WF-nets and UML activity diagrams are used in this article and both modeling ways are presented on the concrete business workflow and then there are presented and specified their advantage and disadvantage for business documents flows. At beginning it is explained the word workflow in context business documents, its features, principles and using in business environment. After that it is clarified that the UML is OMG’s most-used specification, and the way the world models not only application structure, behavior, and architecture, but also business process, workflows and data structure. Activity diagram UML is good way to show how different workflows in the business are managed, how they start, go and stop. Diagrams also show many different decision paths that can be taken from start to finish. State charts can be used as a detail the transitions or changes of states when documents can go through in the business. They show how a documents moves from one state to another and the rules that govern that change. Petri-nets offer a graphical notation for stepwise processes that include choice, iteration, and concurrent execution. Unlike UML Petri nets have an exact mathematical definition of their execution semantics, with a well-developed mathematical theory for process analysis. In the article there are modeled a special type of Petri nets, the WF-nets. The practical part of article incorporates two models of concrete business documents workflows presented in these notations, their comparison and recommendation for using these diagrams in business process management.

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Haryanti, Sri. "ANALISIS HASIL ASSESMEN DIAGNOSTIK KEMAMPUAN KOMUNIKASI MATEMATIKA SISWA PADA PEMBELAJARAN DIMENSI TIGA TIPE TPS KELAS X." Jurnal Karya Pendidikan Matematika 5, no. 2 (October 19, 2018): 76. http://dx.doi.org/10.26714/jkpm.5.2.2018.76-91.

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This study aims to find out: the initial ability of students' mathematical communication; TPS type learning process accompanied by diagnostic analysis; analysis of student difficulties; effectiveness of TPS-type learning accompanied by diagnostic analysis; and students' mathematical communication skills after TPS type learning is accompanied by diagnostic analysis. The scope of the learning material in this study is the Three Dimensions for class X students. This type of research is mixed methods with concurrent embedded research designs. Qualitative methods as primary methods, while quantitative methods as secondary methods. Samples in qualitative research were taken by purposive sampling technique. Samples in quantitative research were taken by cluster random sampling technique. The research data was collected through questionnaires and written tests. Questionnaire data were analyzed descriptively qualitatively and test data were analyzed by proportion test, Uni Anova and Gain test. The results showed that 1) The initial ability of mathematical communication of students is still below the minimum completeness criteria. Student responses are low; 2) TPS type learning process accompanied by diagnostic analysis is carried out through three stages of think-pair-share; 3) TPS type learning accompanied by effective diagnostic analysis is evident from mathematical communication that achieves completeness, increases and is significantly better than the control group; 4) The difficulty of most students is not paying attention to the notation and not being careful in calculating; 5) Students' mathematical communication skills after TPS type learning accompanied by diagnostic analysis in the upper, middle and lower groups have been able to express mathematical ideas, have been able to draw according to the problems given and write mathematical notations are still lacking.

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Cuka, Klea, and Ergi Bufasi. "Hyper Video for Improving Students’ Math Performance." European Journal of Education and Pedagogy 3, no. 5 (September 14, 2022): 13–15. http://dx.doi.org/10.24018/ejedu.2022.3.5.430.

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Mathematical abilities involve both verbal (numerical knowledge, arithmetic, and reasoning) and nonverbal components (math notation, time, space thinking, and computing). Teachers' ability to adopt and implement a unique perspective to mathematics instruction in the classroom may prove crucial to the improvement of their students' mathematical abilities. The aim of this study was to analyze the effectiveness of a hypervideo-based intervention meant to improve students' mathematical performance. The effects of the hypervideo content were compared between an experimental and a control group using a quasi-experimental design. According to the findings, the utilization of hypermedia can lead to improvements in student performance.

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