Evaluating platform architectures within ecosystems: modeling the relation to indirect value

This thesis establishes a framework for understanding the role of a supplier within the context of a business ecosystem. Suppliers typically define their business in terms of capturing value by meeting the demands of direct customers. However, the framework recognises the importance of understanding how a supplier captures indirect value by meeting the demands of indirect customers. These indirect customers increasingly use a supplier’s products and services over time in combination with those of other suppliers. This type of indirect demand is difficult for the supplier to anticipate because it is asymmetric to their own definition of demand. Customers pay the costs of aligning products and services to their particular needs by expending time and effort, for example, to link disparate social technologies or to coordinate healthcare services to address their particular condition. The accelerating tempo of variation in individual needs increases the costs of aligning products and services for customers. A supplier’s ability to reduce its indirect customers’ costs of alignment represents an opportunity to capture indirect value. The hypothesis is that modelling the supplier's relationship to indirect demands improves the supplier’s ability to identify opportunities for capturing indirect value. The framework supports the construction and analysis of such models. It enables the description of the distinct forms of competitive advantage that satisfy a given variety of indirect demands, and of the agility of business platforms supporting that variety of indirect demands. Models constructed using this framework are ‘triply-articulated’ in that they articulate the relationships among three sub-models: (i) the technical behaviours generating products and services, (ii) the social entities managing their supply, and (iii) the organisation of value defined by indirect customers’ demands. The framework enables the derivation from such a model of a layered analysis of the risks to which the capture of indirect value exposes the supplier, and provides the basis for an economic valuation of the agility of the supporting platform architectures. The interdisciplinary research underlying the thesis is based on the use of tools and methods developed by the author in support of his consulting practice within large and complex organisations. The hypothesis is tested by an implementation of the modeling approach applied to suppliers within their ecosystems in three cases: (a) UK Unmanned Airborne Systems, (b) NATO Airborne Warning and Control Systems, both within their respective theatres of operation, and (c) Orthotics Services within the UK's National Health Service. These cases use this implementation of the modeling approach to analyse the value of platforms, their architectural design choices, and the risks suppliers face in their use. The thesis has implications for the forms of leadership involved in managing such platform-based strategies, and for the economic impact such strategies can have on their larger ecosystem. It informs the design of suppliers’ platforms as system-of-system infrastructures supporting collaborations within larger ecosystems. And the ‘triple-articulation’ of the modelling approach makes new demands on the mathematics of systems modeling

Fixed poin sets in digital topology, 1

[EN] In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology. We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(Cn) where Cn is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including Cn, in which F(X) does not equal {0, 1, . . . , #X}. We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.Boxer, L.; Staecker, PC. (2020). Fixed poin sets in digital topology, 1. Applied General Topology. 21(1):87-110. https://doi.org/10.4995/agt.2020.12091OJS87110211C. 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, Continuous maps on digital simple closed curves, Applied Mathematics 1 (2010), 377-386. https://doi.org/10.4236/am.2010.15050L. 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, 2019 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, Remarks on fixed point assertions in digital topology, Applied General Topology 20, no. 1 (2019), 135-153. https://doi.org/10.4995/agt.2019.10474J. 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/s10851-015-0578-8B. Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 18 (1983). https://doi.org/10.1090/conm/014E. Khalimsky, Motion, deformation, and homotopy in finite spaces, in Proceedings IEEE Intl. Conf. on Systems, Man, and Cybernetics (1987), 227-234.A. Rosenfeld, "Continuous" functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6P. C. Staecker, Some enumerations of binary digital images, arXiv:1502.06236, 2015

Fundamental groups and Euler characteristics of sphere-like digital images

[EN] The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but dierent notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in the paper [15] by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results.Boxer, L.; Staecker, PC. (2016). Fundamental groups and Euler characteristics of sphere-like digital images. Applied General Topology. 17(2):139-158. doi:10.4995/agt.2016.4624.SWORD139158172Boxer, L. (1994). Digitally continuous functions. Pattern Recognition Letters, 15(8), 833-839. doi:10.1016/0167-8655(94)90012-4Boxer, L. (2005). Properties of Digital Homotopy. Journal of Mathematical Imaging and Vision, 22(1), 19-26. doi:10.1007/s10851-005-4780-yBoxer, L. (2006). Homotopy Properties of Sphere-Like Digital Images. Journal of Mathematical Imaging and Vision, 24(2), 167-175. doi:10.1007/s10851-005-3619-xBoxer, L. (2006). Digital Products, Wedges, and Covering Spaces. Journal of Mathematical Imaging and Vision, 25(2), 159-171. doi:10.1007/s10851-006-9698-5Boxer, L. (2010). Continuous Maps on Digital Simple Closed Curves. Applied Mathematics, 01(05), 377-386. doi:10.4236/am.2010.15050Chen, L., & Zeng, T. (2014). A Convex Variational Model for Restoring Blurred Images with Large Rician Noise. Journal of Mathematical Imaging and Vision, 53(1), 92-111. doi:10.1007/s10851-014-0551-yHan, S.-E. (2007). Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces. Information Sciences, 177(16), 3314-3326. doi:10.1016/j.ins.2006.12.013Han, S.-E. (2008). Equivalent (k0,k1)-covering and generalized digital lifting. Information Sciences, 178(2), 550-561. doi:10.1016/j.ins.2007.02.004Kong, T. Y. (1989). A digital fundamental group. Computers & Graphics, 13(2), 159-166. doi:10.1016/0097-8493(89)90058-7Rosenfeld, A. (1979). Digital Topology. The American Mathematical Monthly, 86(8), 621. doi:10.2307/2321290Rosenfeld, A. (1986). ‘Continuous’ functions on digital pictures. Pattern Recognition Letters, 4(3), 177-184. doi:10.1016/0167-8655(86)90017-

Remarks on fixed point assertions in digital topology

[EN] Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that some of the published assertions based on these tools are incorrect or trivial; we offer improvements on others.Boxer, L.; Staecker, PC. (2019). Remarks on fixed point assertions in digital topology. Applied General Topology. 20(1):135-153. https://doi.org/10.4995/agt.2019.10474SWORD135153201S. Banach, Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fundamenta Mathematicae 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181C. 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://link.springer.com/article/10.1023/AL. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18 (2) (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19 (1) (2018), 21-53. https://doi.org/10.4995/agt.2018.7146L. Boxer, O. Ege, I. Karaca, J. Lopez, and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied GeneralTopology 17(2) (2016), 159-172. https://doi.org/10.4995/agt.2016.4704S.K. Chatterjea, Fixed point theorems, Comptes rendus de l'Acadmie bulgare des Sciences 25 (1972), 727-730.U.P. Dolhare and V.V. Nalawade, Fixed point theorems in digital images and applications to fractal image compression, Asian Journal of Mathematics and Computer Research 25 (1) (2018), 18-37. http://www.ikpress.org/abstract/6915M. Edelstein, An extension of Banach's contraction principle, Proceedings of the American Mathematical Society 12 (1) (1961), 7-10. https://doi.org/10.1090/s0002-9939-1961-0120625-6O. Ege and I. Karaca, The Lefschetz Fixed Point Theorem for Digital Images, Fixed Point Theory and Applications 2013:253 2013. https://doi.org/10.1186/1687-1812-2013-253O. Ege and I. Karaca, Banach fixed point theorem for digital images, Journal of Nonlinear Sciences and Applications, 8 (2015), 237-245. https://doi.org/10.22436/jnsa.008.03.08O. Ege and I. Karaca, Digital homotopy fixed point theory, Comptes Rendus Mathematique 353 (11) (2015), 1029-1033. https://doi.org/10.1016/j.crma.2015.07.006O. Ege and I. Karaca, Nielsen fixed point theory for digital images, Journal of Computational Analysis and Applications 22 (5) (2017), 874-880.J. Haarmann, M.P. Murphy, C.S. Peters, and P.C. Staecker, Homotopy equivalence of finite digital images, Journal of Mathematical Imaging andVision 53, (3), (2015), 288-302. https://doi.org/10.1007/s10851-015-0578-8G. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993), 381-396. https://doi.org/10.1006/cgip.1993.1029S-E Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 7391. https://doi.org/10.1016/j.ins.2004.03.018S-E Han, Banach fixed point theorem from the viewpoint of digital topology, Journal of Nonlinear Science and Applications 9 (2016), 895-905. https://doi.org/10.22436/jnsa.009.03.19S-E Han, The fixed point property of an M-retract and its applications, Topology and its Applications 230, 139-153. https://doi.org/10.1016/j.topol.2017.08.026A. Hossain, R. Ferdausi, S. Mondal, and H. Rashid, Banach and Edelstein fixed point theorems for digital images, Journal of Mathematical Sciences and Applications 5 (2) (2017), 36-39. https://doi.org/10.12691/jmsa-5-2-2D. Jain, Common fixed point theorem for intimate mappings in digital metric spaces, International Journal of Mathematics Trends and Technology 56 (2) (2018), 91-94. https://doi.org/10.14445/22315373/ijmtt-v56p511B. Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 18 (1983). https://bookstore.ams.org/conm-14K. Jyoti and A. Rani, Digital expansions endowed with fixed point theory, Turkish Journal of Analysis and Number Theory 5 (5) (2017), 146-152. https://doi.org/10.12691/tjant-5-5-1K. Jyoti and A. Rani, Fixed point theorems for β−ψ−φ-expansive type mappings in digital metric spaces, Asian Journal of Mathematics and Computer Research 24 (2) (2018), 56-66. http://www.ikpress.org/abstract/6855R. Kannan, Some results on fixed points, Bulletin of the Calcutta Mathematical Society 60 (1968), 71-76.L.N. Mishra, K. Jyoti, A. Rani, and Vandana, Fixed point theorems with digital contractions image processing, Nonlinear Science Letters A 9 (2) (2018), 104-115. http://www.nonlinearscience.com/paper.php?pid=0000000271C. Park, O. Ege, S. Kumar, D. Jain, and J. R. Lee, Fixed point theorems for various contraction conditions in digital metric spaces, Journal of Computational Analysis and Applications 26 (8) (2019), 1451-1458.S. Reich, Some remarks concerning contraction mappings, Canadian Mathematical Bulletin, 14 (1971), 121-124. https://doi.org/10.4153/cmb-1971-024-9B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, II, Indian Journal of Pure and Applied Mathematics 24 (11) (1993), 691-703.A. Rosenfeld, 'Continuous' functions on digital images, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for α − ψ-contractive mappings, Nonlinear Analysis: Theory, Methods & Applications 75 (4) (2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014T. Zamfirescu, Fixed point theorems in metric spaces, Archiv der Mathematik 23 (1972), 292-298. https://doi.org/10.1007/bf0130488

Digitally Continuous Multivalued Functions, Morphological Operations and Thinning Algorithms

In a recent paper (Escribano et al. in Discrete Geometry for Computer Imagery 2008. Lecture Notes in Computer Science, vol. 4992, pp. 81–92, 2008) we have introduced a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach, which uses multivalued functions, provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In this work we develop properties of this family of continuous functions, now concentrating on morphological operations and thinning algorithms. We show that our notion of continuity provides a suitable framework for the basic operations in mathematical morphology: erosion, dilation, closing, and opening. On the other hand, concerning thinning algorithms, we give conditions under which the existence of a retraction F:X⟶X∖D guarantees that D is deletable. The converse is not true, in general, although it is in certain particular important cases which are at the basis of many thinning algorithms