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Journal articles on the topic 'Graphical calculus'

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Published: 4 June 2021
Last updated: 26 April 2022

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1

Lanese, Ivan, and Ugo Montanari. "A Graphical Fusion Calculus." Electronic Notes in Theoretical Computer Science 104 (November 2004): 199–215. http://dx.doi.org/10.1016/j.entcs.2004.08.026.

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2

BIAN, Xiaoning, and Quanlong WANG. "Graphical Calculus for Qutrit Systems." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E98.A, no. 1 (2015): 391–99. http://dx.doi.org/10.1587/transfun.e98.a.391.

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3

FIORENZA, DOMENICO, and RICCARDO MURRI. "FEYNMAN DIAGRAMS VIA GRAPHICAL CALCULUS." Journal of Knot Theory and Its Ramifications 11, no. 07 (November 2002): 1095–131. http://dx.doi.org/10.1142/s0218216502002165.

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We use Reshetikhin-Turaev graphical calculus to define Feynman diagrams and prove that asymptotic expansions of Gaussian integrals can be written as a sum over a suitable family of graphs. We discuss how different kinds of interactions give rise to different families of graphs. In particular, we show how symmetric and cyclic interactions lead to "ordinary" and "ribbon" graphs respectively. As an example, the 't Hooft-Kontsevich model for 2D quantum gravity is treated in some detail.
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4

Khovanov, Mikhail. "Heisenberg algebra and a graphical calculus." Fundamenta Mathematicae 225, no. 1 (2014): 169–210. http://dx.doi.org/10.4064/fm225-1-8.

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5

Alves, Sandra, Maribel Fernández, and Ian Mackie. "A new graphical calculus of proofs." Electronic Proceedings in Theoretical Computer Science 48 (February 11, 2011): 69–84. http://dx.doi.org/10.4204/eptcs.48.8.

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6

KAUFFMAN, LOUIS H., and PIERRE VOGEL. "LINK POLYNOMIALS AND A GRAPHICAL CALCULUS." Journal of Knot Theory and Its Ramifications 01, no. 01 (March 1992): 59–104. http://dx.doi.org/10.1142/s0218216592000069.

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This paper constructs invariants of rigid vertex isotopy for graphs embedded in three dimensional space. For the Homfly and Dubrovnik polynomials, the skein formalism for these invariants is as shown below. Homfly. [Formula: see text] Dubrovnik. [Formula: see text]
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7

Gowrisankar, Arulprakash, Alireza Khalili Golmankhaneh, and Cristina Serpa. "Fractal Calculus on Fractal Interpolation Functions." Fractal and Fractional 5, no. 4 (October 8, 2021): 157. http://dx.doi.org/10.3390/fractalfract5040157.

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In this paper, fractal calculus, which is called Fα-calculus, is reviewed. Fractal calculus is implemented on fractal interpolation functions and Weierstrass functions, which may be non-differentiable and non-integrable in the sense of ordinary calculus. Graphical representations of fractal calculus of fractal interpolation functions and Weierstrass functions are presented.
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8

Berry, John S., and Melvin A. Nyman. "Promoting students’ graphical understanding of the calculus." Journal of Mathematical Behavior 22, no. 4 (January 2003): 479–95. http://dx.doi.org/10.1016/j.jmathb.2003.09.006.

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9

Lu, Xuexing, Yu Ye, and Sen Hu. "A Graphical Calculus for Semi-Groupal Categories." Applied Categorical Structures 27, no. 2 (November 19, 2018): 163–97. http://dx.doi.org/10.1007/s10485-018-9549-8.

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10

Kim, Joon-Hwi, Maverick S. H. Oh, and Keun-Young Kim. "Boosting vector calculus with the graphical notation." American Journal of Physics 89, no. 2 (February 2021): 200–209. http://dx.doi.org/10.1119/10.0002142.

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11

MA, MINGHUI, and AHTI-VEIKKO PIETARINEN. "PEIRCE’S CALCULI FOR CLASSICAL PROPOSITIONAL LOGIC." Review of Symbolic Logic 13, no. 3 (October 29, 2018): 509–40. http://dx.doi.org/10.1017/s1755020318000187.

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AbstractThis article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted byPC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to presentPCas a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, inPC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection betweenPCand the alpha system.
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12

Wood, Christopher J., Jacob D. Biamonte, and David G. Cory. "Tensor networks and graphical calculus for open quantum systems." Quantum Information and Computation 15, no. 9&10 (July 2015): 759–811. http://dx.doi.org/10.26421/qic15.9-10-3.

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We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate fidelity and entanglement fidelity of a channel in terms of each of the different representations of the channel.
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13

Comfort, Cole. "The ZX&-calculus: A complete graphical calculus for classical circuits using spiders." Electronic Proceedings in Theoretical Computer Science 340 (September 6, 2021): 60–90. http://dx.doi.org/10.4204/eptcs.340.4.

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14

Coecke, Bob, and Bill Edwards. "Three qubit entanglement within graphical Z/X-calculus." Electronic Proceedings in Theoretical Computer Science 52 (March 9, 2011): 22–33. http://dx.doi.org/10.4204/eptcs.52.3.

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15

SCHÄFER, GISA. "A GRAPHICAL CALCULUS FOR 2-BLOCK SPALTENSTEIN VARIETIES." Glasgow Mathematical Journal 54, no. 2 (March 29, 2012): 449–77. http://dx.doi.org/10.1017/s0017089512000110.

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AbstractWe generalise statements known about Springer fibres associated to nilpotents with two Jordan blocks to Spaltenstein varieties. We study the geometry of generalised irreducible components (i.e. Bialynicki-Birula cells) and their pairwise intersections. In particular, we develop a graphical calculus that encodes their structure as iterated fibre bundles with ℂℙ1 as base spaces, and compute their cohomology. At the end, we present a connection with coloured cobordisms generalising the construction of Khovanov (M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3) (2000), 359–426) and Stroppel (C. Stroppel, Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Mathematica145(4) (2009), 954–992).
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16

Barrett, John W., R. J. Dowdall, Winston J. Fairbairn, Frank Hellmann, and Roberto Pereira. "Lorentzian spin foam amplitudes: graphical calculus and asymptotics." Classical and Quantum Gravity 27, no. 16 (July 7, 2010): 165009. http://dx.doi.org/10.1088/0264-9381/27/16/165009.

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17

Khovanov, Mikhail, Aaron Lauda, Marco Mackaay, and Marko Stošić. "Extended graphical calculus for categorified quantum sl(2)." Memoirs of the American Mathematical Society 219, no. 1029 (2012): 0. http://dx.doi.org/10.1090/s0065-9266-2012-00665-4.

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18

Coecke, Bob, Quanlong Wang, Baoshan Wang, Yongjun Wang, and Qiye Zhang. "Graphical Calculus for Quantum Key Distribution (Extended Abstract)." Electronic Notes in Theoretical Computer Science 270, no. 2 (February 2011): 231–49. http://dx.doi.org/10.1016/j.entcs.2011.01.034.

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19

Lin, Huimin. "A graphical μ-calculus and local model checking." Journal of Computer Science and Technology 17, no. 6 (November 2002): 665–71. http://dx.doi.org/10.1007/bf02960756.

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20

Hogancamp, Matthew. "Morphisms between categorified spin networks." Journal of Knot Theory and Its Ramifications 29, no. 11 (October 2020): 2050045. http://dx.doi.org/10.1142/s0218216520500455.

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We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones–Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknots.
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21

Perrin, John Robert. "An Intriguing Exponential Inequality." Mathematics Teacher 103, no. 1 (August 2009): 50–55. http://dx.doi.org/10.5951/mt.103.1.0050.

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22

Perrin, John Robert. "An Intriguing Exponential Inequality." Mathematics Teacher 103, no. 1 (August 2009): 50–55. http://dx.doi.org/10.5951/mt.103.1.0050.

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23

GADDUCCI, FABIO. "Graph rewriting for the π-calculus." Mathematical Structures in Computer Science 17, no. 3 (June 2007): 407–37. http://dx.doi.org/10.1017/s096012950700610x.

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We propose a graphical implementation for (possibly recursive) processes of the π-calculus, encoding each process into a graph. Our implementation is sound and complete with respect to the structural congruence for the calculus: two processes are equivalent if and only if they are mapped into graphs with the same normal form. Most importantly, the encoding allows the use of standard graph rewriting mechanisms for modelling the reduction semantics of the calculus.
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24

tom Dieck, Tammo. "Bridges with pillars: a graphical calculus of knot algebra." Topology and its Applications 78, no. 1-2 (July 1997): 21–38. http://dx.doi.org/10.1016/s0166-8641(96)00147-2.

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25

Stošić, Marko. "On extended graphical calculus for categorified quantum sl(n)." Journal of Pure and Applied Algebra 223, no. 2 (February 2019): 691–712. http://dx.doi.org/10.1016/j.jpaa.2018.04.015.

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26

Backens, Miriam, and Ali Nabi Duman. "A Complete Graphical Calculus for Spekkens’ Toy Bit Theory." Foundations of Physics 46, no. 1 (October 7, 2015): 70–103. http://dx.doi.org/10.1007/s10701-015-9957-7.

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27

Peschanski, Frédéric, Hanna Klaudel, and Raymond Devillers. "A Decidable Characterization of a Graphical Pi-calculus with Iterators." Electronic Proceedings in Theoretical Computer Science 39 (October 28, 2010): 47–61. http://dx.doi.org/10.4204/eptcs.39.4.

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28

Nechita, Ion, and Satvik Singh. "A graphical calculus for integration over random diagonal unitary matrices." Linear Algebra and its Applications 613 (March 2021): 46–86. http://dx.doi.org/10.1016/j.laa.2020.12.014.

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29

Fuchs, Jürgen. "The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms." Journal of Nonlinear Mathematical Physics 13, sup1 (January 2006): 44–54. http://dx.doi.org/10.2991/jnmp.2006.13.s.6.

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30

FUCHS, Jurgen. "The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms." Journal of Non-linear Mathematical Physics 13, Supplement (2006): 44. http://dx.doi.org/10.2991/jnmp.2006.13.supplement.6.

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31

Burella, Glen, Paul Watts, Vincent Pasquier, and Jiří Vala. "Graphical Calculus for the Double Affine Q-Dependent Braid Group." Annales Henri Poincaré 15, no. 11 (December 8, 2013): 2177–201. http://dx.doi.org/10.1007/s00023-013-0289-x.

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32

Gadducci, Fabio, and Alberto Lluch Lafuente. "Graphical Verification of a Spatial Logic for the π-calculus." Electronic Notes in Theoretical Computer Science 154, no. 2 (May 2006): 31–46. http://dx.doi.org/10.1016/j.entcs.2005.03.031.

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33

JAFARI, S. AKBAR. "GRAPHICAL SOLUTION OF THE ISING MODEL ON HONEYCOMB LATTICE." International Journal of Modern Physics B 23, no. 03 (January 30, 2009): 395–401. http://dx.doi.org/10.1142/s0217979209049620.

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In this work, we present a detailed graphical solution for the Ising model on the honeycomb lattice. In view of the mapping between the calculation of the partition function for generalizations of the Ising model and the calculus of resonating valence bond (RVB) states at zero temperature, our calculation may be of relevance to the RVB physics on the honeycomb lattice.
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34

Dagan, Miriam, Pavel Satianov, and Mina Teicher. "Improving Calculus Learning Using a Scientific Calculator." Open Education Studies 2, no. 1 (November 18, 2020): 220–27. http://dx.doi.org/10.1515/edu-2020-0125.

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AbstractThis article discusses the use of a scientific calculator in teaching calculus by using representations of mathematics notions in different sub-languages (analytical, graphical, symbolical, verbal, numerical and computer language). Our long-term experience shows that this may have a positive and significant effect on the enhancement of conceptual understanding of mathematical concepts and approaches. This transcends the basic computational uses, and implies a potential for real improvement in the learning success, cognitive motivation and problem solving skills of the student. We illustrate the steps we have taken towards doing this through some examples.
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35

Demana, Franklin, and Bert K. Waits. "Around the Sun in a Graphing Calculator." Mathematics Teacher 82, no. 7 (October 1989): 546–50. http://dx.doi.org/10.5951/mt.82.7.0546.

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The topics of polar equations and parametric equations are found in almost all precalculus and calculus textbooks. They furnish important a lgebraic representations of real-world problems. For example, polar equations are used to describe the e lliptical orbit of planets around the sun (Kepler's first law), and parametric equations are frequently used to describe the trajectory of moving objects. These topics are seldom given more than superficial treatment in high school and college precalculus and calculus courses because of the difficulty of obtaining appropriate graphical representations.
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36

WESTBURY, BRUCE W. "Invariant tensors for the spin representation of (7)." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 1 (January 2008): 217–40. http://dx.doi.org/10.1017/s0305004107000722.

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AbstractWe give a graphical calculus for the invariant tensors of the eight dimensional spin representation of the quantum groupUq(B3). This leads to a finite confluent presentation of the centraliser algebras of the tensor powers of this representation and a construction of a cellular basis.
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37

Carette, Titouan, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. "Completeness of Graphical Languages for Mixed State Quantum Mechanics." ACM Transactions on Quantum Computing 2, no. 4 (December 31, 2021): 1–28. http://dx.doi.org/10.1145/3464693.

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There exist several graphical languages for quantum information processing, like quantum circuits, ZX-calculus, ZW-calculus, and so on. Each of these languages forms a †-symmetric monoidal category (†-SMC) and comes with an interpretation functor to the †-SMC of finite-dimensional Hilbert spaces. In recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of how to extend these languages beyond pure quantum mechanics to reason about mixed states and general quantum operations, i.e., completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map that allows one to get rid of a quantum system, an operation that is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction , which transforms any †-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction, we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However, this construction fails for some fringe cases like Clifford+T quantum mechanics, as the category does not have enough isometries.
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38

Collins, Benoît, and Ion Nechita. "Random Quantum Channels I: Graphical Calculus and the Bell State Phenomenon." Communications in Mathematical Physics 297, no. 2 (February 21, 2010): 345–70. http://dx.doi.org/10.1007/s00220-010-1012-0.

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39

Licata, Anthony, Daniele Rosso, and Alistair Savage. "A graphical calculus for the Jack inner product on symmetric functions." Journal of Combinatorial Theory, Series A 155 (April 2018): 503–43. http://dx.doi.org/10.1016/j.jcta.2017.11.020.

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40

Danielli, D., N. Garofalo, and D. M. Nhieu. "Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips." Mathematische Zeitschrift 265, no. 3 (April 21, 2009): 617–37. http://dx.doi.org/10.1007/s00209-009-0533-8.

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41

Heid, M. Kathleen. "Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool." Journal for Research in Mathematics Education 19, no. 1 (January 1988): 3–25. http://dx.doi.org/10.5951/jresematheduc.19.1.0003.

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During the first 12 weeks of an applied calculus course, two classes of college students (n=39) studied calculus concepts using graphical and symbol-manipulation computer programs to perform routine manipulations. Only the last 3 weeks were spent on skill development. Class transcripts, student interviews, field notes, and test results were analyzed for patterns of understanding. Students showed better understanding of course concepts and performed almost as well on a final exam of routine skills as a class of 100 students who had practiced the skills for the entire 15 weeks.
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42

Blasiak, Pawel, Gérard H. E. Duchamp, and Karol A. Penson. "Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity." Advances in Mathematical Physics 2018 (October 22, 2018): 1–9. http://dx.doi.org/10.1155/2018/9575626.

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We give a purely combinatorial proof of the Glaisher-Crofton identity which is derived from the analysis of discrete structures generated by the iterated action of the second derivative. The argument illustrates the utility of symbolic and generating function methodology of modern enumerative combinatorics. The paper is meant for nonspecialists as a gentle introduction to the field of graphical calculus and its applications in computational problems.
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43

Stefanini, Luciano, Maria Letizia Guerra, and Benedetta Amicizia. "Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity." Axioms 8, no. 4 (October 14, 2019): 113. http://dx.doi.org/10.3390/axioms8040113.

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We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear) space of compact real intervals and we apply corresponding concepts for the analysis and calculus of interval-valued functions. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in calculus. Concepts related to convergence and limits, continuity, gH-differentiability and monotonicity of interval-valued functions are introduced and analyzed in detail. Graphical examples and pictures accompany the presentation. A companion Part II of the paper will present additional properties (max and min points, convexity and periodicity).
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44

Hena, Hasna, Jenita Jahangir, and Md Showkat Ali. "Electromagnetics in Terms of Differential Forms." Dhaka University Journal of Science 67, no. 1 (January 30, 2019): 1–4. http://dx.doi.org/10.3329/dujs.v67i1.54564.

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The calculus of differential forms has been applied to electromagnetic field theory in several papers and texts, some of which are cited in the references. Differential forms are underused in applied electromagnetic research. Differential forms represent unique visual appliance with graphical apprehension of electromagnetic fields. We study the calculus of differential forms and other fundamental principle of electromagnetic field theory. We hope to show in this paper that differential forms make Maxwell’s laws and some of their basic applications more intuitive and are a natural and powerful research tool in applied electromagnetics. Dhaka Univ. J. Sci. 67(1): 1-4, 2019 (January)
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45

Borlaug, Victoria A. "From Algebra to Calculus—a Tonka® Toy Does the Trick." Mathematics Teacher 86, no. 4 (April 1993): 282–87. http://dx.doi.org/10.5951/mt.86.4.0282.

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This classroom presentation is designed to introduce and interpret the graphical representation of a Tonka® toy truck's forward and backward motion. It can be used to illustrate an application of slope in an algebra class or to introduce the derivative in a calculus class. In the presentation, the teacher moves the Tonka® truck up and down on the chalkboard using chalk to record the motion, asks students questions about the motion, and encourages discussion. The class is asked to pretend that the toy truck has a speedometer. Unlike speedometers in real trucks, this toy speedometer has negative values to represent backward motion, in addmon to its usual positive and zero values. (Speed is the absolute value of velocity. In this situation, the speedometer might better be renamed “velocity-ometer,” but that term would introduce vocabulary unfamiliar to the student.) The presentation leads students to develop a graphical representation of the truck's one-dimensional motion, creates graphs representing constant velocity, leads students to a definition of average velocity, and introduces the concept of instantaneous velocity.
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46

Kesner, Delia. "A fine-grained computational interpretation of Girard’s intuitionistic proof-nets." Proceedings of the ACM on Programming Languages 6, POPL (January 16, 2022): 1–28. http://dx.doi.org/10.1145/3498669.

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This paper introduces a functional term calculus, called pn, that captures the essence of the operational semantics of Intuitionistic Linear Logic Proof-Nets with a faithful degree of granularity, both statically and dynamically. On the static side, we identify an equivalence relation on pn-terms which is sound and complete with respect to the classical notion of structural equivalence for proof-nets. On the dynamic side, we show that every single (exponential) step in the term calculus translates to a different single (exponential) step in the graphical formalism, thus capturing the original Girard’s granularity of proof-nets but on the level of terms. We also show some fundamental properties of the calculus such as confluence, strong normalization, preservation of β-strong normalization and the existence of a strong bisimulation that captures pairs of pn-terms having the same graph reduction.
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47

Östlund, Olof-Petter. "A diagrammatic approach to link invariants of finite degree." MATHEMATICA SCANDINAVICA 94, no. 2 (June 1, 2004): 295. http://dx.doi.org/10.7146/math.scand.a-14444.

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In [5] M. Polyak and O. Viro developed a graphical calculus of diagrammatic formulas for Vassiliev link invariants, and presented several explicit formulas for low degree invariants. M. Goussarov [2] proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. The original note [5] contained no proofs, and it also contained some minor inaccuracies. This paper fills the gap in literature by presenting the material of [5] with all proofs and details, in a self-contained form. Furthermore, a compatible coalgebra structure, related to the connected sum of knots, is introduced on the algebra of based arrow diagrams with one circle.
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48

Backens, Miriam, and Aleks Kissinger. "ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity." Electronic Proceedings in Theoretical Computer Science 287 (January 31, 2019): 23–42. http://dx.doi.org/10.4204/eptcs.287.2.

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49

Ozsoyoglu, G., and H. Wang. "A relational calculus with set operators, its safety, and equivalent graphical languages." IEEE Transactions on Software Engineering 15, no. 9 (1989): 1038–52. http://dx.doi.org/10.1109/32.31363.

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50

Guo, Xiaoyi, Jianwei Zhou, Huantian Xie, and Ziwu Jiang. "MHD Peristaltic Flow of Fractional Jeffrey Model through Porous Medium." Mathematical Problems in Engineering 2018 (October 23, 2018): 1–10. http://dx.doi.org/10.1155/2018/6014082.

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The magnetohydrodynamic (MHD) peristaltic flow of the fractional Jeffrey fluid through porous medium in a nonuniform channel is presented. The fractional calculus is considered in Darcy’s law and the constitutive relationship which included the relaxation and retardation behavior. Under the assumptions of long wavelength and low Reynolds number, the analysis solutions of velocity distribution, pressure gradient, and pressure rise are investigated. The effects of fractional viscoelastic parameters of the generalized Jeffrey fluid on the peristaltic flow and the influence of magnetic field, porous medium, and geometric parameter of the nonuniform channel are presented through graphical illustration. The results of the analogous flow for the generalized second grade fluid, the fractional Maxwell fluid, are also deduced as special cases. The comparison among them is presented graphically.
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