3,332 research outputs found
Fixed poin sets in digital topology, 1
No full text[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
Digital homotopy relations and digital homology theories
No full text[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 -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-
A Mixed-Method Approach for Quantifying Illegal Fishing and Its Impact on an Endangered Fish Species
No full textIllegal harvest is recognized as a widespread problem in natural resource management. The use of multiple methods for quantifying illegal harvest has been widely recommended yet infrequently applied. We used a mixed-method approach to evaluate the extent, charac- ter, and motivations of illegal gillnet fishing in Lake Hovsgol National Park, Mongolia and its impact on the lake’s fish populations, especially that of the endangered endemic Hovsgol grayling (Thymallus nigrescens). Surveys for derelict fishing gear indicate that gillnet fishing is widespread and increasing and that fishers generally use 3–4 cm mesh gillnet. Interviews with resident herders and park rangers suggest that many residents fish for subsistence during the spring grayling spawning migration and that some residents fish commercially year-round. Interviewed herders and rangers generally agree that fish population sizes are decreasing but are divided on the causes and solutions. Biological monitoring indicates that the gillnet mesh sizes used by fishers efficiently target Hovsgol grayling. Of the five species sampled in the monitoring program, only burbot (Lota lota) showed a significant decrease in population abundance from 2009–2013. However, grayling, burbot, and roach (Rutilus ruti- lus) all showed significant declines in average body size, suggesting a negative fishing impact. Data-poor stock assessment methods suggest that the fishing effort equivalent to each resident family fishing 50-m of gillnet 11–15 nights per year would be sufficient to over- exploit the grayling population. Results from the derelict fishing gear survey and interviews suggest that this level of effort is not implausible. Overall, we demonstrate the ability for a mixed-method approach to effectively describe an illegal fishery and suggest that these methods be used to assess illegal fishing and its impacts in other protected areas.Peer reviewe
A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles
No full textWe give a formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles in terms of the Fox calculus. Our formula reduces the problem of computing the coincidence Reidemeister trace to the problem of distinguishing doubly twisted conjugacy classes in free groups
Typical elements in free groups are in different doubly-twisted conjugacy classes
No full textAbstractWe give an easily checkable algebraic condition which implies that two elements of a finitely generated free group are members of distinct doubly-twisted conjugacy classes with respect to a pair of homomorphisms. We further show that this criterion is satisfied with probability 1 when the homomorphisms and elements are chosen at random
Nielsen equalizer theory
No full textAbstractWe extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori
Christopher Dawson
No full textInspired with Jude P. Dougherty’s works in which he stresses the overruling importance of the classical, humanistic education and the central place and role of religion in the Western culture, the author presents Christopher Dawson’s analysis of the Western civilization and his demonstration of the central role of Christianity in it. The author traces the premises on which was based Dawson’s opinion that modern Western man might be absorbed by his technical inventions, to the point of losing his soul
On the uniqueness of the coincidence index on orientable differentiable manifolds
No full textAbstractThe fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are sufficient to characterize the coincidence index in the setting of continuous mappings on oriented differentiable manifolds, the most common setting for Nielsen coincidence theory
Remnant inequalities and doubly-twisted conjugacy in free groups
No full textAbstractWe give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms φ and ψ such that certain remnant words from φ are longer than the images of generators under ψ.Our first result is a remnant inequality condition which implies that two words u and v are not doubly-twisted conjugate. Further we show that if ψ is given and φ, u, and v are chosen at random, then the asymptotic probability that u and v are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if φ, u, and v are chosen at random, then u and v are not in the same singly-twisted conjugacy class with asymptotic probability 1.Our second result generalizes Kim’s “bounded solution length”. We give an algorithm for deciding doubly-twisted conjugacy relations in the case where φ and ψ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if φ has remnant words of length at least 2.As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of the sets of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism
sj-png-2-njn-10.1177_20571585241227583 - Supplemental material for Swedish nursing research: A bibliometric and content analysis revealing author and institute collaborations, impact, and topics
No full textSupplemental material, sj-png-2-njn-10.1177_20571585241227583 for Swedish nursing research: A bibliometric and content analysis revealing author and institute collaborations, impact, and topics by Christopher Holmberg in Nordic Journal of Nursing Research</p
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