Fixed point sets in digital topology, 2

[EN] We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.Boxer, L. (2020). Fixed point sets in digital topology, 2. Applied General Topology. 21(1):111-133. https://doi.org/10.4995/agt.2020.12101OJS111133211C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976.L. Boxer, Digitally Continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146L. Boxer, Fixed points and freezing sets in digital topology, Proceedings, Interdisciplinary Colloquium in Topology and its Applications in Vigo, Spain; 55-61.L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17, no. 2 (2016), 159-172. https://doi.org/10.4995/agt.2016.4704L. Boxer and I. Karaca, Fundamental groups for digital products, Advances and Applications in Mathematical Sciences 11, no. 4 (2012), 161-180.L. Boxer and P. C. Staecker, Fundamental groups and Euler characteristics of sphere-like digital images, Applied General Topology 17, no. 2 (2016), 139-158. https://doi.org/10.4995/agt.2016.4624L. Boxer and P. C. Staecker, Fixed point sets in digital topology, 1, Applied General Topology, to appear.G. Chartrand and L. Lesniak, Graphs & Digraphs, 2nd ed., Wadsworth, Inc., Belmont, CA, 1986.J. Haarmann, M. P. Murphy, C. S. Peters and P. C. Staecker, Homotopy equivalence in finite digital images, Journal of Mathematical Imaging and Vision 53 (2015), 288-302. https://doi.org/10.1007/s10851-015-0578-8S.-E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018E. Khalimsky, Motion, deformation, and homotopy in finite spaces, in Proceedings IEEE Intl. Conf. on Systems, Man, and Cybernetics, 1987, 227-234.A. Rosenfeld, Digital topology, The American Mathematical Monthly 86, no. 8 (1979), 621-630. https://doi.org/10.1080/00029890.1979.11994873A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-

Digital homotopy relations and digital homology theories

[EN] In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories.We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.Staecker, PC. (2021). Digital homotopy relations and digital homology theories. Applied General Topology. 22(2):223-250. https://doi.org/10.4995/agt.2021.13154OJS223250222H. Arslan, I. Karaca and A. Öztel, Homology groups of n-dimensional digital images, in: Turkish National Mathematics Symposium XXI (2008), 1-13.L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10, no. 1 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Generalized normal product adjacency in digital topology, Appl. Gen. Topol. 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, I. Karaca and A. Öztel, Topological invariants in digital images, J. Math. Sci. Adv. Appl. 11, no. 2 (2011), 109-140.L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Appl. Gen. Topol. 20, no. 1 (2019), 135-153. https://doi.org/10.4995/agt.2019.10474O. Ege and I. Karaca, Fundamental properties of digital simplicial homology groups, American Journal of Computer Technology and Application 1 (2013), 25-41.S.-E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171, no. 1-3 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.S. S. Jamil and D. Ali, Digital Hurewicz theorem and digital homology theory, arxiv eprint 1902.02274v3.T. Kaczynski, K. Mischaikow and M. Mrozek, Computing homology. Algebraic topological methods in computer science (Stanford, CA, 2001), Homology Homotopy Appl. 5, no. 2 (2003), 233-256. https://doi.org/10.4310/HHA.2003.v5.n2.a8I. Karaca and O. Ege, Cubical homology in digital images, International Journal of Information and Computer Science, 1 (2012), 178-187.D. W. Lee, Digital singular homology groups of digital images, Far East Journal of Mathematics 88 (2014), 39-63.G. Lupton, J. Oprea and N. Scoville, A fundamental group for digital images, preprint.W. S. Massey, A Basic Course in Algebraic Topology,Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4939-9063-4A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-

Digital homotopic distance between digital functions

[EN] In this paper, we define digital homotopic distance and give its relation with LS category of a digital function and of a digital image. Moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance.The author would like to thank Tane Vergili and the referees for their helpful suggestions. In particular, the author would like to thank the referee who contributed Proposition 3.2 and Example 4.3.Borat, A. (2021). Digital homotopic distance between digital functions. Applied General Topology. 22(1):183-192. https://doi.org/10.4995/agt.2021.14542OJS183192221C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976.A. Borat and T. Vergili, Digital Lusternik-Schnirelmann category, Turkish Journal of Mathematics 42, no 1 (2018), 1845-1852. https://doi.org/10.3906/mat-1801-94A. Borat and T. Vergili, Higher homotopic distance, Topological Methods in Nonlinear Analysis, to appear.L. Boxer, Digitally continuous functions, Pattern Recognit. Lett. 15 (1994), 883-839. https://doi.org/10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Homotopy properties of sphere-like digital images, J. Math. Imaging Vision, 24 (2006), 167-175. https://doi.org/10.1007/s10851-005-3619-xL. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146O. Cornea, G. Lupton, J. Oprea and D. Tanre, Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society 2003. https://doi.org/10.1090/surv/103M. Farber, Topological complexity of motion planning, Discrete and Computational Geometry 29 (2003), 211-221. https://doi.org/10.1007/s00454-002-0760-9S. E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018M. Is and I. Karaca, The higher topological complexity in digital images, Applied General Topology 21, no. 2 (2020), 305-325. https://doi.org/10.4995/agt.2020.13553I. Karaca and M. Is, Digital topological complexity numbers, Turkish Journal of Mathematics 42, no. 6 (2018), 3173-3181. https://doi.org/10.3906/mat-1807-101E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conference on Systems, Man, and Cybernetics (1987), 227-234.G. Lupton, J. Oprea and N. A. Scoville, Homotopy theory in digital topology, ArXiv: 1905.07783.G. Lupton, J. Oprea and N. A. Scoville, Subdivisions of maps of digital images, ArXiv: 1906.03170.E. Macias-Virgos and D. Mosquera-Lois, Homotopic distance between maps, Math. Proc. Cambridge Philos. Soc., to appear.G. Sabidussi, Graph multiplication, Math. Z. 72 (1960), 446-457. https://doi.org/10.1007/BF01162967T. Vergili and A. Borat, Digital Lusternik-Schnirelmann category of digital functions, Hacettepe Journal of Mathematics and Statistics 49, no. 4 (2020), 1414-1422. https://doi.org/10.15672/hujms.55979

The relationship between copyright and contract law

Contracts lie at the heart of the regulatory system governing the creation and dissemination of cultural products in two respects: (1) The exclusive rights provided by copyright law only turn into financial reward, and thus incentives to creators, through a contract with a third party to exploit protected material. (2) From a user perspective purchases of protected material may take the form of a licensing contract, governing behaviour after the initial transaction. Thus, a review of the relationship between copyright and contract law has to address both supply- and demand-side issues. On the supply side, policy concerns include whether copyright law delivers the often stated aim of securing the financial independence of creators. Particularly acute are the complaints by both creators and producers that they fail to benefit from the exponential increase in the availability of copyright materials on the Internet. On the demand side, the issue of copyright exceptions and their policy justification has become central to a number of reviews and consultations dealing with digital content. Are exceptions based on user needs or market failure? Do exceptions require financial compensation? Can exceptions be contracted out by licence agreements? This report (i) reviews economic theory of contracts, value chains and transaction costs, (ii) identifies a comprehensive range of regulatory options relating to creator and user contracts, using an international comparative approach, (iii) surveys the empirical evidence on the effects of regulatory intervention, and (iv) where no evidence is available, extrapolates predicted effects from theory

Algebraic Topology

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd

Some Reflections on Copyright Management Systems and Laws Designed to Protect Them

Copyright management systems (CMS)—technologies that enable copyright owners to regulate reliably and charge automatically for access to digital works—are the wave of the very near future. The advent of digital networks, which make copying and distribution of digital content quick, easy, and undetectable, has provided the impetus for CMS research and development. CMS are premised on the concept of trusted systems or secure digital envelopes that protect copyrighted content and allow access and subsequent copying only to the extent authorized by the copyright owner. Software developers are testing prototype systems designed to detect, prevent, count, and levy precise charges for uses that range from downloading to excerpting to simply viewing or listening to digital works. In a few years, for example, an individual seeking online access to a collection of short fiction might be greeted with a menu of options including: Open and view short story A — $0.50, or$0.40 for students doing assigned reading (verified based on roster submitted by instructor) Open and view short story B (by a more popular author) — $0.80, or$0.70 for students Download short story A (encrypted and copy-protected) — $1.35 Download short story B —$2.25 Download entire collection — $15.00 Extract excerpt from short story A —$0.03 per 50 words Extract excerpt from short story B — $0.06 per 50 words CMS also loom large on the legislative horizon. Copyright owners have argued that technological protection alone will not deter unauthorized copying unless the law provides penalties for circumventing the technology. Although a bill to protect CMS against tampering failed to reach a vote in Congress last year, the World Intellectual Property Organization\u27s recent adoption of treaty provisions requiring protection means that Congress must revisit the question soon. Part II describes these developments. The seemingly inexorable trend toward a digital CMS regime raises two questions, which the author addresses in parts III and IV, respectively. First, broadly drawn protection for CMS has the potential to proscribe technologies that have indisputably lawful uses and also to foreclose, as a practical matter, uses of copyrighted works that copyright law expressly permits. How may protection for CMS be drafted to avoid disrupting the current copyright balance? Second, and equally fundamental, CMS may enable both pervasive monitoring of individual reading activity and comprehensive private legislation designed to augment—and possibly alter beyond recognition—the default rules that define and delimit copyright owners\u27 rights. Given the unprecedented capabilities of these technologies, is it also desirable to set limits on their reach