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Map Folding

2013
*
Canadian Conference on Computational Geometry
*

Otherwise, the

dblp:conf/cccg/NishatW13
fatcat:d3cf5qfqmbcrzmni3z6rgihu74
*map*is unfoldable. ... If a*map*has a configuration such that all the faces of the*map*are stacked on a unit square and the paper does not self-intersect, then it is flat foldable, and the linear ordering of the faces is called ... A final flatfolded state of P is also called a final flat-*folded*state of C. Figure 2 shows an example of a flat*folded*state of a 6 × 8*map*. ...##
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Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings
[article]

2020
*
arXiv
*
pre-print

by them we construct new

arXiv:2003.04147v8
fatcat:a62mtgsakrgt7ovs2ysyr7f4bq
*fold**maps*. ... manifolds via Morse functions,*fold**maps*and more general generic*maps*. ... We can apply this to some explicit*fold**maps*obtained in Theorems 1 and 2 and so on. Acknowledgement. ...##
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On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections
[article]

2021
*
arXiv
*
pre-print

Classes of special generic

arXiv:2105.00575v2
fatcat:5sifdngkdbe63ofbyrekxulu5e
*maps*and round*fold**maps*are important classes of*fold**maps*. ... Round*fold**maps*are Morse functions obtained as doubles of Morse functions, or*fold**maps*such that the set of all the singular points are embeddings and that the images are concentric. ... In other words, we consider lifting given*fold**maps*to new*fold**maps*. ...##
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A Map-Folding Problem

1968
*
Mathematics of Computation
*

An algorithm is described to compute the number of ways of

doi:10.2307/2004779
fatcat:dkxatvi2urcx5jte5b6asa2aya
*folding*a one-dimensional*map*, and a table of values given. ... In how many ways can a*map*be*folded*up? ... In defiance of the well-known law that*maps*never*fold*up so that the front cover is on top, we shall call such a*folding*'normal'-figure (7) is normal. ...##
###
A map-folding problem

1968
*
Mathematics of Computation
*

An algorithm is described to compute the number of ways of

doi:10.1090/s0025-5718-1968-0221957-8
fatcat:vwsxajm6k5egrmws3p44t3hbjy
*folding*a one-dimensional*map*, and a table of values given. ... In how many ways can a*map*be*folded*up? ... In defiance of the well-known law that*maps*never*fold*up so that the front cover is on top, we shall call such a*folding*'normal'-figure (7) is normal. ...##
###
Explicit round fold maps on some fundamental manifolds
[article]

2014
*
arXiv
*
pre-print

generic

arXiv:1310.8407v7
fatcat:zas5gvia7ba4rabx5cxijptscq
*maps*on spheres are regarded as round*fold**maps*whose singular value sets are connected. ... Round*fold**maps*were introduced as stable*fold**maps*with singular value sets, defined as the set consisting of all the singular values, of concentric spheres by the author in 2013; for example, some special ...*fold**maps*into R 4 . ...##
###
Constructing fold maps by surgery operations and their Reeb spaces
[article]

2019
*
arXiv
*
pre-print

More precisely, we investigate

arXiv:1508.05630v14
fatcat:iaqhmtmj7rfxpecnenra53iqeu
*fold**maps*obtained by surgery operations to fundamental*fold**maps*and especially, homology groups of Reeb spaces, which are defined as the space of all connected components ... Studies of this paper are especially motivated by the stream of studies of*fold**maps*satisfying good (differential) topological properties such as special generic*maps*, which were defined in 1970s and ... more complex*fold**maps*. ...##
###
Constructions of round fold maps on circle bundles
[article]

2014
*
arXiv
*
pre-print

One of the fundamental and important problems in the theory of

arXiv:1307.7270v5
fatcat:aiq7jwm4vrbnziiuflbgx252iy
*fold**maps*is to construct explicit*fold**maps*, which are often difficult. ... In this paper, we construct new examples of round*fold**maps*, which are defined as stable*fold**maps*with singular value sets of concentric spheres introduced by the author on manifolds having the structures ... round*fold**maps*. ...##
###
Existence theorems of fold-maps
[article]

2003
*
arXiv
*
pre-print

A smooth

arXiv:math/0309141v2
fatcat:jtfkefocmjfmtfeqnlsfsq7q2e
*map*having only*fold*singularities is called a*fold*-*map*. We will give effective conditions for a continuous*map*to be homotopic to a*fold*-*map*from the viewpoint of the homotopy principle. ... A smooth*map*f : N → P is called a*fold*-*map*if it has only*fold*singularities. ... for*fold*-*maps*in the existence level holds. ...##
###
Return maps of folded nodes and folded saddle-nodes

2008
*
Chaos
*

These return

doi:10.1063/1.2790372
pmid:18377089
fatcat:32kq3tv75zfjphd2tkmh2vzrcq
*maps*are approximated by rank-1*maps*, and the local twisting of trajectories near a*folded*node gives rise to multiple turning points in the approximating one dimensional*maps*. ... This paper presents numerical investigations of the flow*map*between two cross sections that lie on opposite sides of a*folded*node. It finds that this*map*is a perturbation of a one-dimensional*map*. ... The apparent discontinuity in values of the half-return*map*occurs on the stable manifold of the*folded*saddle. ...##
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FOLDING TRANSFORMATIONS AND HKY MAPPINGS

2011
*
Journal of Nonlinear Mathematical Physics
*

By considering the autonomous limit of these relations we derive

doi:10.1142/s1402925111001179
fatcat:h3gg5oqzyrc4dfchljdzorllhe
*folding*-like transformations which relate QRT*mappings*to HKY ones. ... We construct the invariants of the latter*mappings*and show how they can be extended beyond the ones given by the strict application of the*folding*transformation. ... From*Folding*Transformations to HKY*Mappings*Several*folding*transformations of discrete Painlevé equations have been derived in [11] . ...##
###
Existence theorems of fold-maps

2004
*
Japanese journal of mathematics
*

The following lemma is a consequence
of the fundamental
properties
of

doi:10.4099/math1924.30.29
fatcat:6uh2paeur5ezlakouzuk36v3bm
*fold**maps*([E1, 3.8 and 3.9] and [Sal, Lemma 3.1]). LEMMA 3.1. Let n_??_p_??_1. Let f:N•¨P be a*fold*-*map*. ... Any continuous*map*f:Sn•¨Sp is homotopic to a*fold*-*map*which*folds*exactly on the boundaries of a finite number of disjointly embedded disks of dimension p within an embedded dish ...##
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Cobordism invariants of fold maps
[article]

2007
*
arXiv
*
pre-print

This is a survey paper of author's results on cobordism groups and semigroups of

arXiv:0709.0350v1
fatcat:aggdmqueuze3jcv4pg3vzsj6ce
*fold**maps*and simple*fold**maps*. ... The results include: establishing a relation between*fold**maps*and immersions through geometrical invariants of cobordism classes of*fold**maps*and simple*fold**maps*in terms of immersions with prescribed ... A smooth*map*f : Q n+q → N n is called a*fold**map*if f has only*fold*singularities. ...##
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Fold maps on 4-manifolds

2003
*
Commentarii Mathematici Helvetici
*

We give a complete characterization of those closed orientable 4-manifolds which admit smooth

doi:10.1007/s00014-003-0758-9
fatcat:vrfmp3uhmnb7tlalksno2is7z4
*maps*into R 3 with only*fold*singularities. ... We also clarify the relationship between the existence problem of*fold**maps*and that of linearly independent vector fields on manifolds. Mathematics Subject Classification (2000) . ... A smooth*map*f is a*fold**map*1 if all its singularities are of*fold*type. ...##
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Birational maps of 3-folds
[article]

2013
*
arXiv
*
pre-print

We show that 3-

arXiv:1304.5949v1
fatcat:hw4gmnuxarbjviy4ujhroftrzm
*fold*terminal flips and divisorial contractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-*fold*or divisorial contractions to points with minimal ... Together with some recent advances on geometry of 3-*folds*, for example, m-th canonical*maps*is birational for m ≥ 73 and the canonical volume ≥ 1 2660 (cf. ... To give a quick tour of known results: Mori and then Cutkosky classified birational*maps*from a nonsingular and Gorenstein 3-*fold*respectively [18, 6] , and Tziolas has a series of works on divisorial ...
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