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Journal articles on the topic 'Geometry Constructions'

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Published: 4 June 2021
Last updated: 2 February 2022

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1

Gulwani, Sumit, Vijay Anand Korthikanti, and Ashish Tiwari. "Synthesizing geometry constructions." ACM SIGPLAN Notices 47, no. 6 (August 6, 2012): 50. http://dx.doi.org/10.1145/2345156.1993505.

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2

Gulwani, Sumit, Vijay Anand Korthikanti, and Ashish Tiwari. "Synthesizing geometry constructions." ACM SIGPLAN Notices 46, no. 6 (June 4, 2011): 50–61. http://dx.doi.org/10.1145/1993316.1993505.

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3

Janičić, Predrag. "Geometry Constructions Language." Journal of Automated Reasoning 44, no. 1-2 (June 24, 2009): 3–24. http://dx.doi.org/10.1007/s10817-009-9135-8.

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4

Sal'kov, N., and Nina Kadykova. "Representation of Engineering Geometry Development in “Geometry and Graphics” Journal." Geometry & Graphics 8, no. 2 (August 17, 2020): 82–100. http://dx.doi.org/10.12737/2308-4898-2020-82-100.

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In the paper "On the Increasing Role of Geometry", published in the electronic "Journal of Natural Science Research" in 2017, it was outspoken a hypothesis that now, at the time of innovative technologies, the importance of geometry is constantly increasing. The significance of geometry is also demonstrated by numerous Ph.D. and doctoral dissertations in the specialty No 05.01.01 - “Engineering Geometry and Computer Graphics”. It can be affirmed that all and everyone dissertations of technical and technological profile contain a geometric component to one degree or another. The "Geometry and Graphics" journal turned 8 (it was founded in June 2012). During this time, on its pages have been published numerous scientific papers, developing namely geometry and its branches: from simplest geometric constructions based on new properties of both lines and surfaces, to imaginary elements. Investigations were conducted in the following areas: “New Directions in Geometry”, “Fractal Geometry”, “Multidimensional Geometry”, “Geometric Constructions”, “Construction and Research of Surfaces”, “Imaginary Geometry”, “Practical Application of Geometry”, “Computer Graphics”, “Descriptive Geometry as Basis of other Branches of Geometry” ,”Geometry of Phase Spaces”. The journal publishes both recognized scientists and candidate for Ph.D. and doctor degrees. The considered array of papers clearly confirms the statement of the majority of authors, published in the journal, about geometry continuous development, which knocks out the ground for skeptics who decided that geometry is the science of the past centuries. As long as objects with shapes and surfaces surround us, geometry will be in demand. This, as they say, is unequivocal.
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Boyer, Charles P., Krzysztof Galicki, and Liviu Ornea. "Constructions in Sasakian geometry." Mathematische Zeitschrift 257, no. 4 (April 20, 2007): 907–24. http://dx.doi.org/10.1007/s00209-007-0151-2.

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6

Roberti, Joseph V. "Some Challenging Constructions." Mathematics Teacher 79, no. 4 (April 1986): 283–87. http://dx.doi.org/10.5951/mt.79.4.0283.

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In his book A Survey of Geometry, Howard Eves (1966, 183) states, “The Greek geometers of antiquity devised a game— which judged on … [challenge, variety and simplicity] must surely stand at the very top of any list of games to be played alone.”
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7

Giamati, Claudia. "Conjectures in Geometry and The Geometer's Sketchpad." Mathematics Teacher 88, no. 6 (September 1995): 456–58. http://dx.doi.org/10.5951/mt.88.6.0456.

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The Geometer's Sketchpad lets the user explore simple, as well as highly complex, theorems and relations in geometry. Although the user interface takes time to learn, students can use it to test a wide variety of conjectures once they have mastered this construction tool. The Geometer's Sketchpad has the ability to record students' constructions as A scripts. The most useful aspect of scripting one's constructions is that students can test whether their constructions work in general or whether they have discovered a special case.
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8

Chaoping Xing, H. Niederreiter, and Kwok Yan Lam. "Constructions of algebraic-geometry codes." IEEE Transactions on Information Theory 45, no. 4 (May 1999): 1186–93. http://dx.doi.org/10.1109/18.761259.

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9

Milman, V., and L. Rotem. "“Irrational” constructions in Convex Geometry." St. Petersburg Mathematical Journal 29, no. 1 (December 27, 2017): 165–75. http://dx.doi.org/10.1090/spmj/1487.

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10

Evered, Lisa. "Tape Constructions." Mathematics Teacher 80, no. 5 (May 1987): 353–56. http://dx.doi.org/10.5951/mt.80.5.0353.

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Construction problems long have been a favorite subject in geometry. Indeed, students find constructions a welcome diversion from the formal deductive approach to geometry. In classical construction problems only the use of ruler and compass is allowed, and the ruler is used merely as a straightedge, not for measuring or marking off distances. This restriction to ruler and compass goes back to antiquity. The Greeks, however, did not hesitate to use other instruments when the need arose. For example, a ruler in the form of a right angle was used to solve certain problems such as “doubling the cube” (Courant and Robbins 1941).
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11

Sanders, Cathleen V. "Sharing Teaching Ideas: Geometric Constructions: Visualizing and Understanding Geometry." Mathematics Teacher 91, no. 7 (October 1998): 554–56. http://dx.doi.org/10.5951/mt.91.7.0554.

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Geometric constructions can enrich students’ visualization and comprehension of geometry, lay a foundation for analysis and deductive proof, provide opportunities for teachers to address multiple intelligences, and allow students to apply their creativity to mathematics.
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12

Hasson, Assaf. "Some questions concerning Hrushovski's amalgamation constructions." Journal of the Institute of Mathematics of Jussieu 7, no. 4 (October 2008): 793–823. http://dx.doi.org/10.1017/s1474748008000224.

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AbstractIn order to construct a counterexample to Zilber's conjecture—that a strongly minimal set has a degenerate, affine or field-like geometry—Ehud Hrushovski invented an amalgamation technique which has yielded all the exotic geometries so far. We shall present a framework for this construction in the language of standard geometric stability and show how some of the recent constructions fit into this setting. We also ask some fundamental questions concerning this method.
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13

Škoda, Zoran. "Some Equivariant Constructions in Noncommutative Algebraic Geometry." gmj 16, no. 1 (March 2009): 183–202. http://dx.doi.org/10.1515/gmj.2009.183.

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Abstract We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.
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14

Perham, Arnold E., C. S. V., Bernadette H. Perham, and Faustine L. Perham. "Creating a Learning Environment for Geometric Reasoning." Mathematics Teacher 90, no. 7 (October 1997): 521–42. http://dx.doi.org/10.5951/mt.90.7.0521.

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Geometry teachers can introduce the theorems of Euclidean geometry in various ways. Complementing the standbys of the past, namely, compass-and-straightedge constructions and manipulatives, are geometry-construction software, spreadsheets, and programmable graphing calculators.
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15

VIEIRA, C. M. O. B., and J. D. SILVA. "CONSTRUÇÕES GEOMÉTRICAS PARA O ENSINO DE GEOMETRIA NA 1ª SÉRIE DO ENSINO MÉDIO." Revista SODEBRAS 15, no. 180 (December 2020): 53–57. http://dx.doi.org/10.29367/issn.1809-3957.15.2020.180.53.

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16

Peterson, Blake E., and James H. Jordan. "Integer Geometry: Some Examples and Constructions." Mathematical Gazette 81, no. 490 (March 1997): 18. http://dx.doi.org/10.2307/3618764.

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17

Pandiscio, Eric A. "Alternative Geometric Constructions: Promoting Mathematical Reasoning." Mathematics Teacher 95, no. 1 (January 2002): 32–36. http://dx.doi.org/10.5951/mt.95.1.0032.

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Construction tools in most high school Euclidean geometry classes have typically been limited to a compass for drawing circular arcs and a straightedge for drawing line segments. The strengths of these tools include both mathematical precision and a long history of use. However, alternatives can provide fresh possibilities for engaging students in the mathematical reasoning that lies at the heart of traditional geometry (Gibb 1982; Robertson 1986). This article proposes that a single task completed with a variety of construction tools fosters a greater sense of mathematical contemplation than multiple tasks done with the same tool. The premises are simple: each tool fosters different mathematical ideas, and using multiple tools not only requires understanding of a greater breadth and depth of geometric concepts but also highlights the connections that exist among different ideas.
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18

Burger, William F. "Geometry." Arithmetic Teacher 32, no. 6 (February 1985): 52–56. http://dx.doi.org/10.5951/at.32.6.0052.

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The elementary school mathemalics curriculum contains no substitute for the study of informal concepts in geometry. in geometry, children organize and structure their spatial experiences. Also, geometry provides a vehicle for developing mathematical reasoning abilities about visual concepts, for example, through the study of planar shapes. In this article, I shall focus on how reasoning can be developed through the study of two-dimensional shapes, their properties, and the relationships among them. Additional topics, such as tessellations with shapes, motions and symmetry, congruence, similarity, geometric constructions, and measurement, are also highly useful in developing reasoning in geometry. The Bibliography includes many materials that teachers have recommended on these topics.
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19

Bannister, Nicole, and Benjamin J. Sinwell. "Connecting Hexagon Constructions and Similarity." Mathematics Teacher 112, no. 4 (January 2019): 320. http://dx.doi.org/10.5951/mathteacher.112.4.0320.

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Our favorite lesson helps geometry students learn to use diagrams for thinking and communicating (Sinclair, Pimm and Skelin 2012) and motivates a discussion about similarity. The lesson connects sidewalk chalk-andstring (C&S) constructions of regular hexagons with a dynamic geometry software (DGS) approach.
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20

Thiessen, Diane, and Margaret Matthias. "Selected Children's Books For Geometry." Arithmetic Teacher 37, no. 4 (December 1989): 47–51. http://dx.doi.org/10.5951/at.37.4.0047.

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What experiences have your students had with geometry both outside and within the classroom? Do your students view the study of geometry as interesting and functional? Do they appreciate the beauty of geometric design? How is geometry presented in your textbook? Are the lessons on exploring shapes limited to having the students name such figures as triangles and squares or spheres and cubes? Are the lessons on area limited to substituting measures in formulas? Are the lessons on constructions limited to step-by-step instructions? Has geometry been reduced to the study of definitions? Or are the lessons designed with activities in which students develop geometric concepts and relationships through explorations?
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21

Tan, S. T., M. M. F. Yuen, and K. M. Yu. "Parameterized Object Design Using a Geometrical Construction Representation." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 207, no. 1 (February 1993): 21–30. http://dx.doi.org/10.1243/pime_proc_1993_207_058_02.

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This paper proposes a geometrical construction interpretation of variational geometry for achieving parameterized object definition. Vector geometry is used to represent the various kinds of geometrical entities while dimensions are defined to have directional sense. Geometrical constructions are then used to link up all geometrical entities and dimensions of a solid in a meaningful way, to represent the variational geometry properly. The geometrical construction scheme is implemented in a prototype modeller using polyhedral solids as examples.
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22

Geretschläger, Robert. "Euclidean Constructions and the Geometry of Origami." Mathematics Magazine 68, no. 5 (December 1, 1995): 357. http://dx.doi.org/10.2307/2690924.

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23

Azad, H., and A. Laradji. "88.70 Some impossible constructions in elementary geometry." Mathematical Gazette 88, no. 513 (November 2004): 548–51. http://dx.doi.org/10.1017/s0025557200176302.

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24

Geretschläger, Robert. "Euclidean Constructions and the Geometry of Origami." Mathematics Magazine 68, no. 5 (December 1995): 357–71. http://dx.doi.org/10.1080/0025570x.1995.11996356.

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25

Heath, Daniel J. "Straightedge and Compass Constructions in Spherical Geometry." Mathematics Magazine 87, no. 5 (December 2014): 350–59. http://dx.doi.org/10.4169/math.mag.87.5.350.

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26

Robertson, Jack M. "Geometric Constructions Using Hinged Mirrors." Mathematics Teacher 79, no. 5 (May 1986): 380–86. http://dx.doi.org/10.5951/mt.79.5.0380.

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The debate over what geometry to teach has raged without resolution or respite for years, with nearly any position having its proponents and detractors. However, in all these exchanges I have yet to hear the suggestion that the teaching of geometric constructions be replaced by something better, for a number of possible reasons. Most of the constructions we teach are actually quite useful. They give the secondary school student, starved for a Piagetian concrete-operational experience, something tangible. The constructions offer an opportunity for problem solving if properly presented. Probably most significant, however, is the simple fact that they are inherently interesting, as an extensive literature on the problems attests.
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27

Ligeti, Peter, Peter Sziklai, and Marcella Takáts. "Generalized threshold secret sharing and finite geometry." Designs, Codes and Cryptography 89, no. 9 (June 23, 2021): 2067–78. http://dx.doi.org/10.1007/s10623-021-00900-9.

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AbstractIn the history of secret sharing schemes many constructions are based on geometric objects. In this paper we investigate generalizations of threshold schemes and related finite geometric structures. In particular, we analyse compartmented and hierarchical schemes, and deduce some more general results, especially bounds for special arcs and novel constructions for conjunctive 2-level and 3-level hierarchical schemes.
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28

Grannell, M. J., T. S. Griggs, and J. ?ir�? "Recursive constructions for triangulations." Journal of Graph Theory 39, no. 2 (January 11, 2002): 87–107. http://dx.doi.org/10.1002/jgt.10014.

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29

Anderson, Lara B., Xin Gao, and Mohsen Karkheiran. "Extending the geometry of heterotic spectral cover constructions." Nuclear Physics B 956 (July 2020): 115003. http://dx.doi.org/10.1016/j.nuclphysb.2020.115003.

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30

Alderson, T. L., and Keith E. Mellinger. "Constructions of Optical Orthogonal Codes from Finite Geometry." SIAM Journal on Discrete Mathematics 21, no. 3 (January 2007): 785–93. http://dx.doi.org/10.1137/050632257.

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31

Ebert, G. L. "Constructions in Finite Geometry Using Computer Algebra Systems." Journal of Symbolic Computation 31, no. 1-2 (January 2001): 55–70. http://dx.doi.org/10.1006/jsco.1998.1006.

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32

Dreiling, Keith M. "Delving Deeper: Triangle Construction in Taxicab Geometry." Mathematics Teacher 105, no. 6 (February 2012): 474–78. http://dx.doi.org/10.5951/mathteacher.105.6.0474.

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33

Noakes, Lyle, and Tomasz Popiel. "Geometry for robot path planning." Robotica 25, no. 6 (November 2007): 691–701. http://dx.doi.org/10.1017/s0263574707003669.

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SUMMARYThere have been many interesting recent results in the area of geometrical methods for path planning in robotics. So it seems very timely to attempt a description of mathematical developments surrounding very elementary engineering tasks. Even with such limited scope, there is too much to cover in detail. Inevitably, our knowledge and personal preferences have a lot to do with what is emphasised, included, or left out.Part I is introductory, elementary in tone, and important for understanding the need for geometrical methods in path planning. Part II describes the results on geometrical constructions that imitate well-known constructions from classical approximation theory. Part III reviews a class of methods where classicalcriteriaare extended to curves in Riemannian manifolds, including several recent mathematical results that have not yet found their way into the literature.
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34

BEESON, MICHAEL. "CONSTRUCTIVE GEOMETRY AND THE PARALLEL POSTULATE." Bulletin of Symbolic Logic 22, no. 1 (March 2016): 1–104. http://dx.doi.org/10.1017/bsl.2015.41.

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AbstractEuclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom (dating from 1795), which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions.
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35

Boykov, A. A. "Development and application of the geometry constructions language to building computer geometric models." Journal of Physics: Conference Series 1901, no. 1 (May 1, 2021): 012058. http://dx.doi.org/10.1088/1742-6596/1901/1/012058.

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36

Сальков, Николай, and Nikolay Sal'kov. "Application of Dupin Cyclide’s Properties to Inventions." Geometry & Graphics 5, no. 4 (December 13, 2017): 37–43. http://dx.doi.org/10.12737/article_5a17fd233418b2.84489740.

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Dupin cyclide belongs to channel surfaces. These surfaces are the single known ones whose focal surfaces, i.e. surfaces consisting of point sets of centers of curvatures, have been degenerated into two confocal second order curves. In the works devoted to Dupin cyclide and published in the "Geometry and Graphics" journal, are presented various cyclides’ properties and demonstrated application of these surfaces in various industries, mostly in construction. Based on Dupin cyclides’ properties have been developed several inventions relating to drawing devices and having the opportunity to apply in various geometric constructions with the use of computer technologies. It is possible because the Dupin cyclides’ geometric properties suppose not only to create devices recognized as inventions, but also provide an opportunity to apply these properties to write programs for drawing v arious kinds of curves on a display screen: the second order curves, their equidistant in the direction of normals or tangents, as well as to perform other constructions. It should be said that in inventions belonging to technical areas, which include the drawing devices, the geometric component is always decisive. This position with the express aim of technical inventions can justify any copyright certificate of the USSR, any patent of Russia and foreign countries. Unfortunately, currently in schools geometry is not studied as a component of pupil’s intellectual horizons, that broadens his area of interests and teaches to analytical understanding the world, but as an inevitable, almost unnecessary appendage in preparation for the Unified State Examination.
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37

SCHIMMRIGK, ROLF. "ARITHMETIC SPACE–TIME GEOMETRY FROM STRING THEORY." International Journal of Modern Physics A 21, no. 31 (December 20, 2006): 6323–50. http://dx.doi.org/10.1142/s0217751x06034343.

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An arithmetic framework for string compactification is described. The approach is exemplified by formulating a strategy that allows to construct geometric compactifications from exactly solvable theories at c = 3. It is shown that the conformal field theoretic characters can be derived from the geometry of space–time, and that the geometry is uniquely determined by the two-dimensional field theory on the worldsheet. The modular forms that appear in these constructions admit complex multiplication, and allow an interpretation as generalized McKay–Thompson series associated to the Mathieu and Conway groups. This leads to a string motivated notion of arithmetic moonshine.
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38

Xia, Shu-Tao, Xin-Ji Liu, Yong Jiang, and Hai-Tao Zheng. "Deterministic Constructions of Binary Measurement Matrices From Finite Geometry." IEEE Transactions on Signal Processing 63, no. 4 (February 2015): 1017–29. http://dx.doi.org/10.1109/tsp.2014.2386300.

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39

Evans, David M., and Marco S. Ferreira. "The geometry of Hrushovski constructions, I: The uncollapsed case." Annals of Pure and Applied Logic 162, no. 6 (April 2011): 474–88. http://dx.doi.org/10.1016/j.apal.2011.01.008.

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40

Tong, Fenghua, Lixiang Li, Haipeng Peng, and Yixian Yang. "Deterministic Constructions of Compressed Sensing Matrices From Unitary Geometry." IEEE Transactions on Information Theory 67, no. 8 (August 2021): 5548–61. http://dx.doi.org/10.1109/tit.2021.3088090.

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41

BOS, ROGIER. "GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS." International Journal of Geometric Methods in Modern Physics 04, no. 03 (May 2007): 389–436. http://dx.doi.org/10.1142/s0219887807002077.

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We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.
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42

Walpuski, Thomas. "G2–instantons on generalised Kummer constructions." Geometry & Topology 17, no. 4 (August 22, 2013): 2345–88. http://dx.doi.org/10.2140/gt.2013.17.2345.

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43

Wiegand, Kevin, and Kai Zehmisch. "Two constructions of virtually contact structures." Journal of Symplectic Geometry 16, no. 2 (2018): 563–83. http://dx.doi.org/10.4310/jsg.2018.v16.n2.a5.

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44

Jensen, Tommy R., and Gordon F. Royle. "Haj�s constructions of critical graphs." Journal of Graph Theory 30, no. 1 (January 1999): 37–50. http://dx.doi.org/10.1002/(sici)1097-0118(199901)30:1<37::aid-jgt5>3.0.co;2-v.

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45

Brand, Neal, and W. Cary Huffman. "Invariants and constructions of mendelsohn designs." Geometriae Dedicata 22, no. 2 (February 1987): 173–96. http://dx.doi.org/10.1007/bf00181265.

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46

Olejníková, Tatiana. "Geometry of Prismatic Tensegrity Constructions Composed of Three and Four-strut Cells." Selected Scientific Papers - Journal of Civil Engineering 9, no. 2 (November 1, 2014): 47–56. http://dx.doi.org/10.2478/sspjce-2014-0015.

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Abstract In the paper there is described geometry of double layer tensegrity constructions composed of prismatic cells with rhombic configuration of three or four strut bases so-called prismatic Tensegrity constructions. There are described bi-dimensional assemblies creating double layer grids of three or four-strut cells with a node-on-node junction. The grids can be planar, of one or two curvature constructions.
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47

SCHWERMER, JOACHIM, and CHRISTOPH WALDNER. "On the cohomology of uniform arithmetically defined subgroups in SU*(2n)." Mathematical Proceedings of the Cambridge Philosophical Society 151, no. 3 (July 18, 2011): 421–40. http://dx.doi.org/10.1017/s0305004111000430.

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AbstractWe study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU*(2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields non-vanishing (co)homology classes.In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms.
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48

Alexandrov, M., A. Schwarz, O. Zaboronsky, and M. Kontsevich. "The Geometry of the Master Equation and Topological Quantum Field Theory." International Journal of Modern Physics A 12, no. 07 (March 20, 1997): 1405–29. http://dx.doi.org/10.1142/s0217751x97001031.

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In Batalin–Vilkovisky formalism, a classical mechanical system is specified by means of a solution to the classical master equation. Geometrically, such a solution can be considered as a QP-manifold, i.e. a supermanifold equipped with an odd vector field Q obeying {Q, Q} = 0 and with Q-invariant odd symplectic structure. We study geometry of QP-manifolds. In particular, we describe some construction of QP-manifolds and prove a classification theorem (under certain conditions). We apply these geometric constructions to obtain in a natural way the action functionals of two-dimensional topological sigma-models and to show that the Chern–Simons theory in BV-formalism arises as a sigma-model with target space [Formula: see text]. (Here [Formula: see text] stands for a Lie algebra and Π denotes parity inversion.)
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49

Szarek, S. J., and N. Tomczak-Jaegermann. "Saturating constructions for normed spaces." Geometrical and Functional Analysis GAFA 14, no. 6 (December 2004): 1352–75. http://dx.doi.org/10.1007/s00039-004-0495-2.

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50

Kock, Anders. "Differential Calculus and Nilpotent Real Numbers." Bulletin of Symbolic Logic 9, no. 2 (June 2003): 225–30. http://dx.doi.org/10.2178/bsl/1052669291.

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Do there exist real numbers d with d2 = 0 (besides d = 0, of course)? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and followers did (completed in Hilbert's Grundlagen der Geometrie, [2, Chapter 3]): “geometrization of arithmetic”. (Picking two distinct points on the geometric line, geometric constructions in an ambient Euclidean plane provide structure of a commutative ring on the line, with the two chosen points as 0 and 1).Starting from the geometric side, nilpotent elements are somewhat reasonable, although Euclid excluded them. The sophist Protagoras presented a picture of a circle and a tangent line; the apparent little line segment D which tangent and circle have in common, are, by Pythagoras' Theorem, precisely the points, whose abscissae d (measured along the tangent) have d2 = 0. Protagoras wanted to use this argument for destructive reasons: to refute the science of geometry.A couple of millenia later, the Danish geometer Hjelmslev revived the Protagoras picture. His aim was more positive: he wanted to describe Nature as it was. According to him (or extrapolating his position), the Real Line, the Line of Sensual Reality, had many nilpotent infinitesimals, which we can see with our naked eyes.
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