scholarly journals Global defect topology in nematic liquid crystals

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160265 ◽  
Author(s):  
Thomas Machon ◽  
Gareth P. Alexander
Keyword(s):  
Homology Group ◽  
Double Cover ◽  
Explicit Computation ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
General Domain ◽  
Global Classification ◽  
Local Characterization ◽  
Global Defect

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in R 3 , showing that the distinct homotopy classes have a 1–1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.

scholarly journals Homotopy Classification of Mappings of a 4-Dimensional Complex into a 2-Dimensional Sphere

1953 ◽  
Vol 5 ◽  
pp. 127-144 ◽  
Author(s):  
Nobuo Shimada
Keyword(s):  
Dimensional Sphere ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Cup Product

Steenrod [1] solved the problem of enumerating the homotopy classes of maps of an (n + 1)-complex K into an n-sphere Sn utilizing the cup-i-product, the far-reaching generalization of the Alexander-Čech-Whitney cup product [7] and the Pontrjagin *-product [5].

scholarly journals On the homotopy classification of proper Fredholm maps into a Hilbert space

2020 ◽  
Vol 2020 (759) ◽  
pp. 161-200 ◽  
Author(s):  
Alberto Abbondandolo ◽  
Thomas O. Rot
Keyword(s):  
Model Space ◽  
Complete Classification ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Infinite Dimensional ◽  
Ambient Manifold ◽  
Alternative Approach ◽  
Framed Cobordism ◽  
Fredholm Maps

AbstractWe classify the homotopy classes of proper Fredholm maps from an infinite-dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli–Smale mod 2 degree and the absolute value of the oriented degree.

scholarly journals Homotopy classification of filtered complexes

1973 ◽  
Vol 15 (3) ◽  
pp. 298-318 ◽  
Author(s):  
Ross Street
Keyword(s):  
Abelian Groups ◽  
Classification Theorem ◽  
Homotopy Equivalence ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Homology Groups ◽  
Chain Complexes ◽  
Kunneth Formula ◽  
Homology Functor

The homology functor from the category of free abelian chain complexes and homotopy classes of maps to that of graded abelian groups is full and replete (surjective on objects up to isomorphism) and reflects isomorphisms. Thus such a complex is determined to within homotopy equivalence (although not a unique homotopy equivalence) by its homology. The homotopy classes of maps between two such complexes should therefore be expressible in terms of the homology groups, and such an expression is in fact provided by the Künneth formula for Hom, sometimes called ‘the homotopy classification theorem’.

scholarly journals On a Homotopy Classification of Mappings of an (n+1) Dimensional Complex Into an Arcwise Connected Topological Space Which is Aspherical in Dimensions Less Thann (n >2)

1951 ◽  
Vol 3 ◽  
pp. 67-72 ◽  
Author(s):  
Nobuo Shimada ◽  
Hiroshi Uehara
Keyword(s):  
Topological Space ◽  
Nauk Sssr ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Continuous Mappings ◽  
Arcwise Connected ◽  
New Type

Pontrjagin classified mappings of a three dimensional sphere into anndimensional complex, where he made use of a new type of product of cocycles. By the aid of the generalized Pontrjagin’s product of cocycles Steenrod enumerated effectively all the homotopy classes of mappings of an (n+1) dimensional complex into annsphere. According to the recent issue of the Mathematical Reviews it is reported that M. M. Postnikov extended Steenrod’s case to the case where an arcwise connected topological space which is aspherical in dimensions less thann, takes place of annsphere. (Postnikov M. M., Classification of continuous mappings of an(n+1)dimensional complex into a connected topological space which is aspherical in dimensions less thann. Doklady Akad. Nauk SSSR (N.S.) 71., 1027-1028, 1950 (Russian. No. proof is given.)) But here in Japan no details are yet to hand. We intend to give a solution to this problem in case wheren>2, and also to give an application concerning the(n+ 3)-extension cocycle.

Homotopy classification and the generalized Swan homomorphism

2009 ◽  
Vol 4 (3) ◽  
pp. 491-536 ◽  
Author(s):  
F.E.A. Johnson
Keyword(s):  
Finite Group ◽  
Homotopy Theory ◽  
Group Cohomology ◽  
Projective Modules ◽  
Finitely Generated ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Algebraic Homotopy Theory ◽  
Fundamental Paper

AbstractIn his fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [18], [21]). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form here is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → J → Fn → … → F0 → F0 → M → 0 where each Fr is finitely generated free.

scholarly journals Global classification of NPGCs

2014 ◽  
Vol 13 (12) ◽  
pp. 888-888
Author(s):  
Sarah Crunkhorn
Keyword(s):  
Global Classification

scholarly journals An inshore–offshore sorting system revealed from global classification of ocean litter

2021 ◽  
Vol 4 (6) ◽  
pp. 484-493
Author(s):  
Carmen Morales-Caselles ◽  
Josué Viejo ◽  
Elisa Martí ◽  
Daniel González-Fernández ◽  
Hannah Pragnell-Raasch ◽  
...  
Keyword(s):  
Global Classification ◽  
Sorting System

Toward a global classification of mast cell activation diseases

2011 ◽  
Vol 127 (5) ◽  
pp. 1311 ◽  
Author(s):  
Gerhard J. Molderings ◽  
Jürgen Homann ◽  
Martin Raithel ◽  
Thomas Frieling
Keyword(s):  
Mast Cell ◽  
Cell Activation ◽  
Mast Cell Activation ◽  
Global Classification

scholarly journals Cultural Issues Related to ICD-11 Mental, Behavioural and Neurodevelopmental Disorders

10.17816/cp67 ◽  
2021 ◽  
Vol 2 (2) ◽  
pp. 7-15
Author(s):  
Pratap Sharan ◽  
Gagan Hans
Keyword(s):  
Neurodevelopmental Disorders ◽  
Cultural Issues ◽  
Cultural Variations ◽  
Cultural Challenges ◽  
Global Classification ◽  
Specific Disorders ◽  
Successive Generations ◽  
Classificatory System

The challenge of producing a classificatory system that is truly representative of different regions and cultural variations is difficult. This can be conceptualized as an ongoing process, achievable by constant commitment in this regard from various stakeholders over successive generations of the classificatory systems. The objective of this article is to conduct a qualitative review of the process and outcome of the efforts that resulted in the ICD-11 classification of mental, behavioural and neurodevelopmental disorders becoming a global classification. The ICD-11 represents an important, albeit iterative, advance in the classification of mental, behavioural and neurodevelopmental disorders. Significant changes have been incorporated in this regard, such as the introduction of new, culturally-relevant categories, modifications of the diagnostic guidelines, based on culturally informed data and the incorporation of culture-related features for specific disorders. Notwithstanding, there are still certain significant shortcomings and areas for further improvement and research. Some of the key limitations of ICD-11 relate to the paucity of research on the role of culture in the pathogenesis of illnesses. To ensure a classificatory system that is fair, reliable and culturally useful, there is a need to generate empirical evidence on diversity in the form of illnesses, as well as mechanisms that explain these in all the regions of the world. In this review, we try to delineate the various cultural challenges and their influences in the formulation of ICD-11, along with potential shortcomings and areas in need of more improvement and research in this regard.

scholarly journals Homotopy classification of elliptic operators on stratified manifolds

2006 ◽  
Vol 73 (3) ◽  
pp. 407-411 ◽  
Author(s):  
V. E. Nazaikinskii ◽  
A. Yu. Savin ◽  
B. Yu. Sternin
Keyword(s):  
Elliptic Operators ◽  
Homotopy Classification
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