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Journal articles on the topic 'Homotopy'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Pan, Binfeng, Xun Pan, and Yangyang Ma. "A quadratic homotopy method for fuel-optimal low-thrust trajectory design." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, no. 5 (2018): 1741–57. http://dx.doi.org/10.1177/0954410018761965.

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Solving fuel-optimal low-thrust trajectory problems is a long-standing challenging topic, mainly due to the existence of discontinuous bang–bang controls and small convergence domain. Homotopy methods, the principle of which is to embed a given problem into a family of problems parameterized by a homotopic parameter, have been widely applied to address this difficulty. Linear homotopy methods, the homotopy functions of which are linear functions of the homotopic parameter, serve as useful tools to provide continuous optimal controls during the homotopic procedure with an energy-optimal low-thr

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2

Kareem, Azhar. "Fuzzy Relative Homotopy and Fuzzy Weak Equivalence with Some Results." Wasit Journal for Pure sciences 3, no. 3 (2024): 9–15. http://dx.doi.org/10.31185/wjps.431.

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In this paper, first introduce my concept fuzzy homotopic and fuzzy homotopic relative and weproved that the relation fuzzy homotopic relative is a fuzzy equivalence relation. Secondly , weintroduce the concepts fuzzy homotopy equivalence, fuzzy fundamental group and fuzzy weak homotopy equivalence. We have proven some important theorems.

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3

Koceić-Bilan, Nikola, and Ivančica Mirošević. "On classification of morphisms by box-homotopy." Acta mathematica Spalatensia 1, no. 1 (2021): 97–103. http://dx.doi.org/10.32817/ams.1.1.8.

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In [1] the authors proposed a generalization of the notion of homotopy, a relation called to be box-homotopic, proven to be an equivalence relation on Top(X,Y) and well-adjusted with the composition. In this article we prove that all the mappings of Top(X,Y) are box-homotopic, that is, the classification of morphisms by the box-homotopy relation is the coarsest.

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4

Staecker, P. Christopher. "Digital homotopy relations and digital homology theories." Applied General Topology 22, no. 2 (2021): 223. http://dx.doi.org/10.4995/agt.2021.13154.

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In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.<br /><br />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.<br /><br />We also consider four different digital homology theories: a simplicial homology

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5

Side, Syafruddin, Maya Sari Wahyuni, and Muh Rifki. "Solusi Numerik Model SIR pada Penyebaran Penyakit Hepatitis B dengan Metode Perturbasi Homotopi di Provinsi Sulawesi Selatan." Journal of Mathematics, Computations, and Statistics 3, no. 2 (2020): 79. http://dx.doi.org/10.35580/jmathcos.v3i2.20122.

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Penelitian ini membahas mengenai solusi secara numerik dari model SIR pada penyebaran penyakit Hepatitis B dengan Metode Perturbasi Homotopi. Data yang digunakan adalah data sekunder dari penelitian Rosdiana (2015) yang berupa model SIR dan jumlah penderita Hepatitis B di Provinsi Sulawesi Selatan tahun 2015 dari Dinas Kesehatan Provinsi Sulawesi Selatan. Pembahasan dimulai dari penentuan solusi umum dengan Metode Perturbasi Homotopi, penentuan parameter, simulasi dan analisis hasil. Setelah dilakukan analisis dari simulasi numerik terlihat bahwa Metode Perturbasi Homotopi dapat digunakan untu

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6

Kazemi-Baneh, M. Z. "Homotopic Chain Maps Have Equals-Homology andd-Homology." International Journal of Mathematics and Mathematical Sciences 2016 (2016): 1–5. http://dx.doi.org/10.1155/2016/5647548.

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The homotopy of chain maps on preabelian categories is investigated and the equality of standard homologies andd-homologies of homotopic chain maps is established. As a special case, ifXandYare the same homotopy type, then theirnthd-homologyR-modules are isomorphic, and ifXis a contractible space, then itsnthd-homologyR-modules forn≠0are trivial.

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7

Lyubashenko, V., and A. Matsui. "Homotopy equivalence of normalized and unnormalized complexes, revisited." Algebra and Discrete Mathematics 32, no. 2 (2021): 253–66. http://dx.doi.org/10.12958/adm1879.

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We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the Dold-Kan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the unnormalized complex. We prove that this idempotent is homotopic to identity via homotopy which is expressed via faces and degeneracies. Hence, the normalized and unnormalized complex are homotopy isomorphic to each other. We provide explicit formulae for the homotopy.

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8

FIEDLER, THOMAS, and ARNAUD MORTIER. "ON HOMOTOPIES WITH TRIPLE POINTS OF CLASSICAL KNOTS." Journal of Knot Theory and Its Ramifications 21, no. 04 (2012): 1250038. http://dx.doi.org/10.1142/s0218216511009911.

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We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant v2(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that

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9

Dyer, Eldon, and Joseph Roitberg. "Homotopy-epimorphisms, homotopy-monomorphisms and homotopy-equivalences." Topology and its Applications 46, no. 2 (1992): 119–24. http://dx.doi.org/10.1016/0166-8641(92)90127-l.

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10

Kang, Min, and Sang-Eon Han. "Compression of Khalimsky topological spaces." Filomat 26, no. 6 (2012): 1101–14. http://dx.doi.org/10.2298/fil1206101k.

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Aiming at the study of the compression of Khalimsky topological spaces which is an interesting field in digital geometry and computer science, the present paper develops a new homotopy thinning suitable for the work. Since Khalimsky continuity of maps between Khalimsky topological spaces has some limitations of performing a discrete geometric transformation, the paper uses another continuity (see Definition 3.4) that can support the discrete geometric transformation and a homotopic thinning suitable for studying Khalimsky topological spaces. By using this homotopy, we can develop a new homotop

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11

KOTORII, YUKA. "THE MILNOR $\bar{\mu}$ INVARIANTS AND NANOPHRASES." Journal of Knot Theory and Its Ramifications 22, no. 02 (2013): 1250142. http://dx.doi.org/10.1142/s0218216512501428.

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Two link diagrams are link homotopic if one can be transformed into the other by a sequence of Reidemeister moves and self-crossing changes. Milnor introduced invariants under link homotopy called [Formula: see text]. Nanophrases, introduced by Turaev, generalize links. In this paper, we extend the notion of link homotopy to nanophrases. We also generalize [Formula: see text] to the set of those nanophrases that correspond to virtual links.

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12

Ashraf, Samia, Madiha Akram, and Amna Amanat Ali. "Subdivision-based homotopy equivalence of digital circles." Applied General Topology 26, no. 1 (2025): 431–45. https://doi.org/10.4995/agt.2025.22288.

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A direct translation of the notion of homotopy equivalence from algebraic topology to digital images leads to a much more rigid definition in the context of digital topology. This results in two digital circles of different radii being not homotopic. A more suitable idea of equivalences, called "subdivision-based homotopy equivalence" which is introduced by Lupton et al., involves the concept of subdivision of digital images. We prove that, in this sense, any digital circle is subdivision-based homotopic to the Diamond, the prototypical digital circle of radius 1, which implies, as a consequen

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13

Mukherjee, Goutam. "Equivariant homotopy epimorphisms, homotopy monomorphisms and homotopy equivalences." Bulletin of the Belgian Mathematical Society - Simon Stevin 2, no. 4 (1995): 447–61. http://dx.doi.org/10.36045/bbms/1103408700.

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14

Kawauchi, Akio. "Smooth Homotopy 4-Sphere." WSEAS TRANSACTIONS ON MATHEMATICS 22 (September 26, 2023): 690–701. http://dx.doi.org/10.37394/23206.2023.22.76.

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It is shown that every homotopy 4-disk with boundary 3-sphere is diffeomorphic to the 4-disk, so that every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere. As a consequence, it is also shown that any (smoothly) embedded 3-sphere in the 4-sphere splits the 4-sphere into two components of 4-manifolds which are both diffeomorphic to the 4-ball. The argument used for the proof also shows that any two homotopic diffeomorphisms of the stable 4-sphere are smoothly isotopic if one diffeomorphism allows a local diffeomorphism change, so that they are smoothly concordant and piecewise-linearl

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15

Lightfoot, Ash. "Detecting Whitney disks for link maps in the four-sphere." Journal of Knot Theory and Its Ramifications 26, no. 12 (2017): 1750077. http://dx.doi.org/10.1142/s0218216517500778.

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It is an open problem whether Kirk’s [Formula: see text]-invariant is the complete obstruction to a link map [Formula: see text] being link homotopic to the trivial link. The link homotopy invariant associates to such a link map [Formula: see text] a pair [Formula: see text], and we write [Formula: see text]. With the objective of constructing counterexamples, Li proposed a link homotopy invariant [Formula: see text] such that [Formula: see text] is defined on the kernel of [Formula: see text] and which also obstructs link null-homotopy. We show that, when defined, the invariant [Formula: see

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16

Ihsan, Hisyam, Ahmad Zaki, and Nur Syuaiba. "Solusi Numerik Model Matematika SIRI Metode Perturbasi Homotopi dalam Penggunaan E-money Sistem E-parking." Journal of Mathematics Computations and Statistics 5, no. 1 (2022): 20. http://dx.doi.org/10.35580/jmathcos.v5i1.32246.

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Penelitian ini merupakan penelitian terapan mengenai penerapan metode Perturbasi Homotopi untuk mencari solusi numerik model matematika SIRI dalam penggunaan E-money sistem E-parking dengan metode Perturbasi Homotopi. Data yang digunakan adalah data yang diperoleh dengan membagikn angket kepada 236 responden secara acak di lokasi penelitian yaitu Mall Panakkukang, Mall Nipah dan Mall Ratu Indah. Pembahasan dimulai dari penentuan solusi umum dengan metode Perturbasi Homotopi, penentuan parameter, simulasi dan analisis hasil. Dalam penelitian ini diperoleh grafik pergerakan dari model SIRI denga

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17

Møller, Jesper Michael. "Samelson Products In Spaces of Self-Homotopy Equivalences." Canadian Journal of Mathematics 42, no. 1 (1990): 95–108. http://dx.doi.org/10.4153/cjm-1990-006-7.

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The homotopy groups of any group-like space are equipped with a Samelson product satisfying, up to sign, the identities of a graded Lie bracket. We shall compute the Samelson product in two kinds of spaces of selfhomotopy equivalences arising when adding a homotopy or a homology group to a space.First, let A→ X be a cofibration with a Moore space M(G,n) as cofibre. For the monoid autA (X) of maps under A homotopic (rel. A) to the identity, the Samelson product is a pairingπn+i(G;X)⨂πn+j(G;X) → πn+i+j(G;X)of homotopy groups with coefficients [1] in G. Theorem 2.1 computes this pairing in terms

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18

Li, Bang-He, and Gui-Song Li. "Immersions with non-zero normal vector fields." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 2 (1992): 281–85. http://dx.doi.org/10.1017/s0305004100070961.

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Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

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19

Borat, Ayse. "Digital homotopic distance between digital functions." Applied General Topology 22, no. 1 (2021): 183. http://dx.doi.org/10.4995/agt.2021.14542.

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<p>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.</p>

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20

Chen, Beifang, Shing-Tung Yau, and Yeong-Nan Yeh. "Graph homotopy and Graham homotopy." Discrete Mathematics 241, no. 1-3 (2001): 153–70. http://dx.doi.org/10.1016/s0012-365x(01)00115-7.

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21

Hong, Lin, and Shen Wenhuai. "Homotopy epimorphisms in homotopy pullbacks." Topology and its Applications 59, no. 1 (1994): 73–77. http://dx.doi.org/10.1016/0166-8641(94)90100-7.

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22

Vallette, Bruno. "Homotopy theory of homotopy algebras." Annales de l'Institut Fourier 70, no. 2 (2020): 683–738. http://dx.doi.org/10.5802/aif.3322.

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23

Ghorbal, Sonia. "Homotopy monomorphisms and homotopy pushouts." Topology and its Applications 79, no. 2 (1997): 173–76. http://dx.doi.org/10.1016/s0166-8641(96)00170-8.

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24

Mather, Michael. "Homotopy monomorphisms and homotopy pushouts." Topology and its Applications 81, no. 2 (1997): 159–62. http://dx.doi.org/10.1016/s0166-8641(97)00023-0.

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25

Bingham, N. H., and A. J. Ostaszewski. "Homotopy and the Kestelman–Borwein–Ditor Theorem." Canadian Mathematical Bulletin 54, no. 1 (2011): 12–20. http://dx.doi.org/10.4153/cmb-2010-093-4.

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AbstractThe Kestelman–Borwein–Ditor Theorem, on embedding a null sequence by translation in (measure/category) “large” sets has two generalizations. Miller replaces the translated sequence by a “sequence homotopic to the identity”. The authors, in a previous paper, replace points by functions: a uniform functional null sequence replaces the null sequence, and translation receives a functional form. We give a unified approach to results of this kind. In particular, we show that (i) Miller's homotopy version follows fromthe functional version, and (ii) the pointwise instance of the functional ve

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26

Manuilov, V., and K. Thomsen. "Extensions of $C^*$-algebras and translation invariant asymptotic homomorphisms." MATHEMATICA SCANDINAVICA 100, no. 1 (2007): 131. http://dx.doi.org/10.7146/math.scand.a-15018.

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Let $A$, $B$ be $C^*$-algebras; $A$ separable, $B$ $\sigma$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathsf{R})\otimes A$ to $B$ and show that the Connes-Higson construction applied to any extension of $A$ by $B$ is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of $A$ by $B$ out of such a translation invariant asymptotic homomorphism. This leads to our main result; that the homotopy classes of extensions coincide with the homotopy classes

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27

Iglesias-Zemmour, Patrick. "Variations of Integrals in Diffeology." Canadian Journal of Mathematics 65, no. 6 (2013): 1255–86. http://dx.doi.org/10.4153/cjm-2012-044-5.

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AbstractWe establish a formula for the variation of integrals of differential forms on cubic chains in the context of diffeological spaces. Then we establish the diffeological version of Stokes’ theorem, and we apply that to get the diffeological variant of the Cartan–Lie formula. Still in the context of Cartan–De Rham calculus in diffeology, we construct a chain-homotopy operator K, and we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the chain-homotopy operator that is used in symplectic diffeology to construct the moment map.

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28

Saeed, Rostam K., and Rebwar S. Muhammad. "Solving Coupled Hirota System by Using Homotopy Perturbation and Homotopy Analysis Methods." Journal of Zankoy Sulaimani - Part A 17, no. 2 (2015): 201–18. http://dx.doi.org/10.17656/jzs.10394.

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29

Arkhipov, Sergey, and Daria Poliakova. "Homotopy characters as a homotopy limit." Homology, Homotopy and Applications 26, no. 2 (2024): 1–20. http://dx.doi.org/10.4310/hha.2024.v26.n2.a1.

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30

Khinich, V. A., and V. V. Shekhtman. "The homotopy limit of homotopy algebras." Russian Mathematical Surveys 41, no. 3 (1986): 213–14. http://dx.doi.org/10.1070/rm1986v041n03abeh003351.

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31

Cole, Michael. "Many homotopy categories are homotopy categories." Topology and its Applications 153, no. 7 (2006): 1084–99. http://dx.doi.org/10.1016/j.topol.2005.02.006.

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32

Bergner, Julia E. "Homotopy fiber products of homotopy theories." Israel Journal of Mathematics 185, no. 1 (2011): 389–411. http://dx.doi.org/10.1007/s11856-011-0116-3.

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33

Morozov, A. I. "Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection." Nelineinaya Dinamika 17, no. 4 (2021): 465–73. http://dx.doi.org/10.20537/nd210408.

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According to the Nielsen – Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_{1}$) periodic homeomorphism; $T_{2}$) reducible non-periodic homeomorphism of algebraically finite order; $T_{3}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_{4}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_{1}$, $T_{2}$, $

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34

HIROSE, SUSUMU, and AKIRA YASUHARA. "REGULAR HOMOTOPIC DEFORMATION OF COMPACT SURFACE WITH BOUNDARY AND MAPPING CLASS GROUP." Journal of Knot Theory and Its Ramifications 20, no. 10 (2011): 1391–96. http://dx.doi.org/10.1142/s021821651100925x.

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A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in S3 to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given.

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35

Fisette, Robert, and Alexander Polishchuk. "-algebras associated with curves and rational functions on . I." Compositio Mathematica 150, no. 4 (2014): 621–67. http://dx.doi.org/10.1112/s0010437x13007574.

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AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater th

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36

Blackadar, Bruce. "The Homotopy Lifting Theorem for Semiprojective $C^*$-Algebras." MATHEMATICA SCANDINAVICA 118, no. 2 (2016): 291. http://dx.doi.org/10.7146/math.scand.a-23691.

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We prove a complete analog of the Borsuk Homotopy Extension Theorem for arbitrary semiprojective $C^*$-algebras. We also obtain some other results about semiprojective $C^*$-algebras: a partial lifting theorem with specified quotient, a lifting result for homomorphisms close to a liftable homomorphism, and that sufficiently close homomorphisms from a semiprojective $C^*$-algebra are homotopic.

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37

Hernandez, Emili, Marc Carreras, and Pere Ridao. "A Path Planning Algorithm for an AUV Guided with Homotopy Classes." Proceedings of the International Conference on Automated Planning and Scheduling 21 (March 22, 2011): 82–89. http://dx.doi.org/10.1609/icaps.v21i1.13457.

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The paper proposes a method that uses topological information to guide path planning in any 2D workspace. Our method builds a topological environment based on the workspace to compute homotopy classes, which topologically describe how paths go through the obstacles in the workspace. Then, the homotopy classes are sorted according to an heuristic estimation of their lower bound. Only those with smaller lower bound are used to guide a planner based on the Rapidly-exploring Random Tree (RRT), called Homotopic RRT (HRRT), to compute the path in the workspace. Simulated and real results with an Aut

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38

NAKANISHI, YASUTAKA, and YOSHIYUKI OHYAMA. "DELTA LINK HOMOTOPY FOR TWO COMPONENT LINKS, II." Journal of Knot Theory and Its Ramifications 11, no. 03 (2002): 353–62. http://dx.doi.org/10.1142/s0218216502001664.

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In this note, we will study Δ link homotopy (or self Δ-equivalence), which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition is given in the terms of Conway polynomials for two link types to be Δ link homotopic. A pair of numerical invariants δ1 and δ2 classifies all (ordered and oriented) prime 2-component link types with seven crossings or less up to Δ link homotopy. We will show here that for any pair of integers n1 and n2 there exists a 2-component link κ such that δ1(κ) = n1 and δ2(κ) = n2 provided that at least one of n1 and n2 is even.

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39

Robert-Nicoud, Daniel, and Felix Wierstra. "Homotopy morphisms between convolution homotopy Lie algebras." Journal of Noncommutative Geometry 13, no. 4 (2020): 1463–520. http://dx.doi.org/10.4171/jncg/351.

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40

Dydak, Jerzy, and Francisco Romero Ruiz del Portal. "Bimorphisms in pro-homotopy and proper homotopy." Fundamenta Mathematicae 160, no. 3 (1999): 269–86. http://dx.doi.org/10.4064/fm-160-3-269-286.

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41

Hardie, K. A., K. H. Kamps, and H. Marcum. "COMPUTING HOMOTOPY GROUPS OF A HOMOTOPY PULLBACK." Quaestiones Mathematicae 14, no. 2 (1991): 179–99. http://dx.doi.org/10.1080/16073606.1991.9631635.

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42

Lin, James P. "Homotopy commutativity, homotopy associativity and power spaces." Journal of Pure and Applied Algebra 134, no. 2 (1999): 133–62. http://dx.doi.org/10.1016/s0022-4049(97)00145-x.

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43

Franz, Matthias. "Homotopy Gerstenhaber algebras are strongly homotopy commutative." Journal of Homotopy and Related Structures 15, no. 3-4 (2020): 557–95. http://dx.doi.org/10.1007/s40062-020-00268-y.

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44

Golasiński, Marek, and Aniceto Murillo. "Homotopy cofibres, higher coassociativity and homotopy coalgebras." Journal of Mathematics of Kyoto University 48, no. 3 (2008): 631–38. http://dx.doi.org/10.1215/kjm/1250271387.

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45

Umble, Ronald N. "Homotopy conditions that determine rational homotopy type." Journal of Pure and Applied Algebra 60, no. 2 (1989): 205–17. http://dx.doi.org/10.1016/0022-4049(89)90128-x.

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46

GIBSON, ANDREW. "Factorization of homotopies of nanophrases." Mathematical Proceedings of the Cambridge Philosophical Society 152, no. 1 (2011): 55–90. http://dx.doi.org/10.1017/s0305004111000612.

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AbstractHomotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. For any homotopy given by a composite homotopy data triple we define a complete invariant of nanophrases. This invariant is used to show that equivalence of nanophrases under such a homotopy can be calculated just by using the homotopies given by its prime factors.

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47

Sorcar, Gangotryi. "Teichmüller space of negatively curved metrics on Gromov–Thurston manifolds is not contractible." Journal of Topology and Analysis 06, no. 04 (2014): 541–55. http://dx.doi.org/10.1142/s1793525314500204.

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In this paper we prove that for all n = 4k - 2, k ≥ 2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that [Formula: see text] is nontrivial. [Formula: see text] denotes the Teichmüller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity. Gromov–Thurston branched cover manifolds provide examples of negatively curved mani

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Ahmed Ghanawi Jasim. "Homotopy on Smooth Fuzzy Fréchet Manifold." Communications on Applied Nonlinear Analysis 31, no. 4s (2024): 362–70. http://dx.doi.org/10.52783/cana.v31.859.

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This paper presents definition to a fuzzy FF-smooth homotopy on FF-smooth fuzzy Fréchet manifold and proves that the fuzzy FF-smooth homotopy of a fuzzy path forms an equivalence relation. The researchers also expand the study to include three types of fuzzy FF-smooth homotopy of a fuzzy path, namely a maximal fuzzy FF-smooth homotopy, an internal fuzzy FF-smooth homotopy, and local fuzzy FF-smooth homotopy admitting equivalence relations and a structure of a group.

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49

Casals, Roger, Álvaro del Pino, and Francisco Presas. "Loose Engel structures." Compositio Mathematica 156, no. 2 (2020): 412–34. http://dx.doi.org/10.1112/s0010437x19007759.

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This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$-principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

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50

Lu, Ju-Hong, and Chun-Long Zheng. "Approximate Solution of Generalized Ginzburg-Landau-Higgs System via Homotopy Perturbation Method." Zeitschrift für Naturforschung A 65, no. 4 (2010): 301–4. http://dx.doi.org/10.1515/zna-2010-0406.

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Using the homotopy perturbation method, a class of nonlinear generalized Ginzburg-Landau-Higgs systems (GGLH) is considered. Firstly, by introducing a homotopic transformation, the nonlinear problem is changed into a system of linear equations. Secondly, by selecting a suitable initial approximation, the approximate solution with arbitrary degree accuracy to the generalized Ginzburg- Landau-Higgs system is derived. Finally, another type of homotopic transformation to the generalized Ginzburg-Landau-Higgs system reported in previous literature is briefly discussed.

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