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### Map Folding

Rahnuma Islam Nishat, Sue Whitesides
2013 Canadian Conference on Computational Geometry
Otherwise, the 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.  ...

### Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings [article]

Naoki Kitazawa
2020 arXiv   pre-print
by them we construct new 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.  ...

### On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections [article]

Naoki Kitazawa
2021 arXiv   pre-print
Classes of special generic 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.  ...

### A Map-Folding Problem

W. F. Lunnon
1968 Mathematics of Computation
An algorithm is described to compute the number of ways of 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

W. F. Lunnon
1968 Mathematics of Computation
An algorithm is described to compute the number of ways of 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]

Naoki Kitazawa
2014 arXiv   pre-print
generic 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]

Naoki Kitazawa
2019 arXiv   pre-print
More precisely, we investigate 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]

Naoki Kitazawa
2014 arXiv   pre-print
One of the fundamental and important problems in the theory of 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]

Yoshifumi Ando
2003 arXiv   pre-print
A smooth 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

John Guckenheimer
2008 Chaos
These return 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.  ...

### FOLDING TRANSFORMATIONS AND HKY MAPPINGS

B. GRAMMATICOS, A. RAMANI, R. WILLOX
2011 Journal of Nonlinear Mathematical Physics
By considering the autonomous limit of these relations we derive 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  .  ...

### Existence theorems of fold-maps

Yoshifumi ANDO
2004 Japanese journal of mathematics
The following lemma is a consequence of the fundamental properties of 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  ...

### Cobordism invariants of fold maps [article]

Boldizsar Kalmar
2007 arXiv   pre-print
This is a survey paper of author's results on cobordism groups and semigroups of 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.  ...

### Fold maps on 4-manifolds

Osamu Saeki
2003 Commentarii Mathematici Helvetici
We give a complete characterization of those closed orientable 4-manifolds which admit smooth 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.  ...

### Birational maps of 3-folds [article]

Jungkai Alfred Chen
2013 arXiv   pre-print
We show that 3-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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