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We generalise the surface-complexity of closed 3-manifolds to the compact case. This generalisation preserves the (natural generalisation of the) properties holding in the closed case: the surface-complexity on compact 3-manifolds is a natural number measuring how much the manifolds are complicated, it is subadditive under connected sum and it is finite-to-one on $$\mathbb {P}^2$$ -irreducible and boundary-irreducible manifolds without essential annuli and Mobius strips. Moreover, for these manifolds, it equals the minimal number of cubes in an ideal cubulation of the manifold, except for a finite number of cases. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity.

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model for further investigations is the M\"obius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" M\"obius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns., Comment: Revised version prior to publication. Accepted for publication in Portugaliae Mathematica

From the work of M.C. Escher to the universal recycling logo, no mathematical object is more ubiquitous than the Mobius strip. Introduced independently by Listing and Mobius in 1858, the Mobius str...

We show that computation of the SPERNER problem is PPA-complete on the Mobius band under proper boundary conditions, settling a long term open problem. Further, the same computational complexity results extend to other discrete fixed points on the Mobius band, such as the Brouwer fixed point problem, the DPZP fixed point problem and a simple version of the Tucker problem, as well as the projective plane and the Klein bottle. We expect it opens up a new route for further studies on the related combinatorial structures.

The structure of the 9 vertex obstructive graphs for the nonorientable surface of the genus 2 is established by the method of j-transformations of the graphs. The problem of establishing the structural properties of 9 vertex obstruction graphs for the surface of the undirected genus 2 by the method of j-transformation of graphs is considered. The article has an introduction and 5 sections. The introduction contains the main definitions, which are illustrated, to some extent, in Section 1, which provides several statements about their properties. Sections 2 – 4 investigate the structural properties of 9 vertex obstruction graphs for an undirected surface by presenting as a j-image of several graphs homeomorphic to one of the Kuratovsky graphs and at least one planar or projective-planar graph. Section 5 contains a new version of the proof of the statement about the peculiarities of the minimal embeddings of finite graphs in nonorientable surfaces, namely, that, in contrast to oriented surfaces, cell boundaries do not contain repeated edges. Also in section 5 the other properties peculiar to embeddings of graphs to non-oriented surfaces and the main result are given. The main result is Theorem 1. Each obstruction graph H for a non-oriented surface N2 of genus 2 satisfies the following. 1. An arbitrary edge u,u = (a,b) is placed on the Mebius strip by some minimal embedding of the graph H in N3 and there exists a locally projective-planar subgraph K of the graph H \ u which satisfies the condition: (tK({a,b},N3)=1)˄(tK\u({a,b},N2)=2), where tK({a,b},N) is the number of reachability of the set {a,b} on the nonorientable surface N; 2. There exists the smallest inclusion of many different subgraphs Ki of a 2-connected graph H homeomorphic to the graph K+e, where K is a locally planar subgraph of the graph H (at least K+e is homemorphic to K5 or K3,3), which covers the set of edges of the graph H. Keywords: graph, Klein surface, graph structure, graph obstruction, non-oriented surface, Möbius strip.

Using a technique which is inspired by topology, we construct original examples of 3 - and 4 -edge critical graphs. The 3 -critical graphs cover all even orders starting from 26 ; the 4 -critical graphs cover all even orders starting from 20 and all the odd orders. In particular, the 3 -critical graphs are not isomorphic to the graphs provided by Goldberg for disproving the Critical Graph Conjecture. Using the same approach we also revisit the construction of some fundamental critical graphs, such as Goldberg’s infinite family of 3 -critical graphs, Chetwynd’s 4 -critical graph of order 16 and Fiol’s 4 -critical graph of order 18 .

We explicitly classify all $$\mathbb {S}^1$$ -invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$ . This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $$\mathbb {S}^1$$ -invariant metrics on the annulus and Mobius band. First, we determine the supremum of the kth normalized Steklov eigenvalue among all $$\mathbb {S}^1$$ -invariant metrics on the Mobius band for each $$k \ge 1$$ , and show that it is achieved by the induced metric from a free boundary minimal embedding of the Mobius band into $${\mathbb {B}}^4$$ by kth Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of $$\mathbb {S}^1$$ -invariant metrics on the annulus and Mobius band are the induced metrics on explicit free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^3$$ and $${\mathbb {B}}^4$$ , including some new families of free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^4$$ . Finally, we prove that these are the only $$\mathbb {S}^1$$ -invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$ .

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\tan(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in $\mathbb{R}\times \mathbb{R}P^3$. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in $S^1$ to prove that $1/3$ is a sharp lower bound on the probability that a M\"obius strip filling the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.

We consider the optimal flows on noncompact surfaces with boundary, which have a minimum number of fixed points and all these points lie on the boundary of the surface. It is proved that the flow is optimal if it has a single sink and a single source. We describe the structures of the optimal flows on a simply connected region, on a Mobius strip, on a torus with hole, and on a Klein bottle with hole.

Recently, the problem about Pfaffian property of graphs has attracted much attention in the matching theory. Its significance stems from the fact that the number of perfect matchings in a graph G can be computed in polynomial time if G is Pfaffian. The Mobius strip with unique surface structure strip has potentially significant chirality properties in molecular chemistry. In this paper, we consider the Pfaffian property of graphs on the Mobius strip based on topological resolution. A sufficient condition for Pfaffian graphs on the Mobius strip is obtained. As its application, we characterize Pfaffian quadrilateral lattices on the Mobius strip.

In this paper we prove existence and uniqueness of the Bjorling problem for the class of immersed surfaces in $$\mathbb {R}^3$$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Mobius strip for an arbitrary large class of prescribed functions. In particular, we use the Bjorling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Mobius strip in $$\mathbb {R}^3$$ .

The second part of the paper is devoted to enumeration of r -regular maps on the torus up to all its homeomorphisms (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to these homeomorphisms leads to maps on the Klein bottle, the annulus and the Mobius band. Using 3- and 4-regular maps as an example we describe the technique of enumerating quotient maps on surfaces with a boundary. Obtained recurrence relations are used to enumerate unsensed r -regular maps on the torus for various r .

Connection between external skin and inner core of the building is paramount for the case study of Eisenman’s Max Reinhart house. External envelope is generated by a series of surfaces perpendicular to a Moebius strip running through the whole building. Each square is constrained to a rotation in the space, so that a certain amount of overlapping, and subsequent cutting is generated. External surface is created by the extrusion of squares into boxes, which partially overlap themselves. The internal structure of the building is characterized by horizontal slices of the resulting volume. The work analyses the geometrical features of the Mobius strip by writing the discrete parametric equations that define this shape driver. A tentative equation system was worked out with a circular generator element. Through the variation of the parameter connected with the ordinate, it was possible to create a Moebius strip with an elliptic generator. This ellipsis lays on a vertical plane perpendicular to the ground. The resulting surface generated with the elliptical Moebius strip resulted to be ruled. There is a direct connection between urban landscape and building shape, both in the definition of geometric parameters and the constraint of morphogenetic characterization. The parametric geometry has been worked out with Grasshopper to generate an accurate solution within the given constraints.

The Klein bottle plays a crucial role in the main modern sciences. This surface was first described in 1882 by the German mathematician Felix Klein. In this paper we describe a technique to obtain the parameterization of the Klein bottle. This technique uses isometric transformations (translations and rotations) and the moving frame associated with the unit circumference lying on the xy-plane. The process we follow is to start with the parametrization of the Euclidean cylinder, then continue with the parameterization of the Mobius strip, after that with the parameterization of the torus of revolution and finally, in a natural way, we describe the aforementioned technique. With the parameterization of the Klien bottle obtained, it is easy to show that it can be obtained by gluing two Mobius strips. Additionally, the parameterizations of the n-twisted and n-turns Klein bottles are obtained. All geometric calculations and geometric interpretations are performed with the Mathematica symbolic calculus system.

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, M\"obius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,\pi) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues., Comment: Minor changes. Final version. Accepted for publication in the Proceedings of the American Mathematical Society

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

Mobius regular maps are surface embeddings of graphs with doubled edges such that (i) the automorphism group of the embedding acts regularly on flags and (ii) each doubled edge is a center of a Mobius band on the surface. In this paper, we classify Mobius regular maps of order pq for any two primes p and q, where p ≠ q.

The paper is devoted to multiple solutions of the classical problem on stationary configurations of an elastic rod on a plane; we describe boundary values for which there are more than two optimal configurations of a rod (optimal elasticae). We define sets of points where three or four optimal elasticae come together with the same value of elastic energy. We study all configurations that can be translated into each other by symmetries, i.e., reflections at the center of the elastica chord and reflections at the middle perpendicular to the elastica chord. For the first symmetry, the ends of the rod are directed in opposite directions, and the corresponding boundary values lie on a disk. For the second symmetry, the boundary values lie on a Mobius strip. As a result, we study both sets numerically and in some cases analytically; in each case, we find sets of points with several optimal configurations of the rod. These points form the currently known part of the reachability set where elasticae lose global optimality.

The X-ray transform on the periodic slab [ 0 , 1 ] × T n , n ≥ 0 , has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless n = 0 . We characterize the kernel of the geodesic X-ray transform for L 2 -regular m -tensors for any m ≥ 0 . The characterization extends to more general manifolds, twisted slabs, including the Mobius strip as the simplest example.

If the closed wire frame of a soap film having the shape of a Möbius strip is pulled apart and gradually deformed into a planar circle, the soap film transforms into a two-sided orientable surface. In the presence of a finite-time twist singularity, which changes the linking number of the film's Plateau border and the centreline of the wire, the topological transformation involves the collapse of the film toward the wire. In contrast to experimental studies of this process reported elsewhere, we use a numerical approach based on the immersed boundary method, which treats the soap film as a massless membrane in a Navier-Stokes fluid. In addition to known effects, we discover vibrating motions of the film arising after the topological change is completed, similar to the vibration of a circular membrane.

The full scan condition on a language L introduced in [1] ensures that a word w must be completely inspected in order to decide whether or not w lies in L. We strengthen the condition by replacing the factor words in that definition by single letters. After examining the general case for both arbitrary and regular languages, we investigate the two-letter alphabet to find a complete description of the corresponding languages, which may be coded as infinite binary strings. Regularity of these languages corresponds to the associated numbers being rational and we find the minimal automata in all cases, which may be pictured as a cylinder of tape with a protruding end, although this cylinder is replaced by a Mobius strip for a special class of rational languages.

The main theme of this chapter is the study of extremal eigenvalue problems and its relations to minimal surface theory. We describe joint work with R. Schoen on progress that has been made on the Steklov eigenvalue problem for surfaces with boundary, and in higher dimensions. For surfaces, the Steklov eigenvalue problem has a close connection to free boundary minimal surfaces in Euclidean balls. Specifically, metrics that maximize Steklov eigenvalues are characterized as induced metrics on free boundary minimal surfaces in \(\mathbb {B}^n\). We discuss the existence of maximizing metrics for surfaces of genus zero, and explicit characterizations of maximizing metrics for the annulus and Mobius band. We also give an overview of results on existence, uniqueness, and Morse index of free boundary minimal surfaces in \(\mathbb {B}^n\).

We will present 2 different analytical representations of only one general idea—this is the representation of complex movements using the superposition of certain elementary displacements! Despite of the analytical and structural similarity of these representations, they describe fundamentally different geometric figures (in statics) and trajectories of motion (in dynamics). In previous articles [1, 2, 3, 4, 5, 6, 7, 8, 9] a wide class of geometric figures—“Generalized Twisting and Rotated” bodies \(GRT^n_m\) in short—was defined through their analytic representation. In particular cases, this analytic representation gives back many classical objects (torus, helicoid, helix, Mobius strip ... etc.). The aim of this article is to consider some geometric properties of a wide subclass of the generally defined surfaces. We show some geometric properties of GRT and GML—surfaces.

The original motivation to study Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena. The cutting of GML bodies and surfaces, with the Möbius strip as one special case, is related to the field of knots and links, and classifications were obtained for GML with cross sectional symmetry of 2, 3, 4, 5 and 6. The general case of cutting GML bodies and surfaces, in particular the number of ways of cutting, could be solved by reducing the 3D problem to planar geometry. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.

We obtain Hurwitz numbers as the number of Feynman diagrams of a certain type divided by the order of the automorphism group of the diagram., Comment: a number of corrections

In this pedagogical paper, John S. Bell's amusing example of Dr. Bertlmann's socks is reconsidered, first within a toy model of a two-dimensional one-sided world of a non-orientable M\"obius strip, and then within a real world of three-dimensional quaternionic sphere, S^3, which results from an addition of a single point to R^3 at infinity. In the quaternionic world, which happens to be the spatial part of a solution of Einstein's field equations of general relativity, the singlet correlations between a pair of entangled fermions can be understood as classically as those between Dr. Bertlmann's colorful socks., Comment: 18 pages, 7 figures; An appendix is added proving the equivalence of the conservation of spin angular momentum and the twists in the Hopf bundle of S^3; minor improvements

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $\pi_0\mathcal{S}(f,\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\to\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\mathbb{R}^2\to\mathbb{R}$ without multiple factors. Together with previous results of the second author this allows to compute similar groups for certain classes of smooth functions $f:N\to\mathbb{R}$ on non-orientable compact surfaces $N$.

In organic chemistry, there are strong connections between the number of the Kekule structures and chemical properties for many molecules. In this paper, we focus on some honeycomb lattices on Klein bottle and Mobius strip. We obtain explicit expressions of the number of Kekule structures of them by enumerating Pfaffians.

This paper aims to define and study a notion of orientability in the Heisenberg sense ( H -orientability) for the Heisenberg group H n . In particular, we define such notion for H -regular 1-codimensional surfaces. Analysing the behaviour of a Mobius Strip in H 1 , we find a 1-codimensional H -regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, H -orientability implies Euclidean-orientability. As a consequence, we conclude that non- H -orientable H -regular surfaces exist in H 1 .

The space of shapes of quadrilaterals can be identified with $\mathbb{CP}^{2}$. We deal with the subset of $\mathbb{CP}^{2}$ corresponding to convex quadrilaterals and the subset which corresponds to simple (that is, without self-intersections) quadrilaterals. We provide a complete description of the topological closures in $\mathbb{CP}^{2}$ of both spaces. Although the interior of each space is homeomorphic to a disjoint union $\mathbb{R}^{4}\sqcup \mathbb{R}^{4}$, their closures are topologically different. In particular, the boundary of the space corresponding to convex quadrilaterals is homeomorphic to a pair of three-dimensional spheres glued along a Möbius strip while the boundary of the space corresponding to simple quadrilaterals is more complicated.

In [4], the continuous cluster category was introduced. This is a topological category whose space of isomorphism classes of indecomposable objects forms a Moebius band. It was found in [4] that, in order to have a continuously triangulated structure on this category, one needs at least two copies of each indecomposable object forming a 2-fold covering space of the Moebius band. This paper classifies all continuous triangulations of finite coverings of the basic continuous cluster category. This includes the connected 2-fold covering of Igusa-Todorov [4], the disconnected 2-fold covering of Orlov [6] and a third unexpected continuously add-triangulated 2-fold covering of the Moebius strip category., 49 pages, 3 figures, earlier versions of this work were presented by the first author at the Auslander Conference at Woods Hole in May, 2015 and by the second author at a workshop at Tsinghua University in Beijing in July 2017. Version 1 was part of the first author's PhD thesis. Submitted to Topology and its Applications

We determine p-colorability of the paradromic rings. These rings arise by generalizing the well-known experiment of bisecting a Mobius strip. Instead of joining the ends with a single half twist, use $m$ twists, and, rather than bisecting ($n = 2$), cut the strip into $n$ sections. We call the resulting collection of thin strips $P(m,n)$. By replacing each thin strip with its midline, we think of $P(m,n)$ as a link, that is, a collection of circles in space. Using the notion of $p$-colorability from knot theory, we determine, for each $m$ and $n$, which primes $p$ can be used to color $P(m,n)$. Amazingly, almost all admit 0, 1, or an infinite number of prime colorings! This is reminiscent of solutions sets in linear algebra. Indeed, the problem quickly turns into a study of the eigenvalues of a large, nearly diagonal matrix. Our paper combines this explicit calculation in linear algebra with a survey of several ideas from knot theory including colorability and torus links., Comment: 24 pages, 18 figures

We prove the existence of graphs with empty intersection of their longest paths or cycles as subgraphs of lattices on the torus and the Mobius strip.

This article is about deformations minimising Kirchhoff's nonlinear bending energy of plates. An earlier (local) partial regularity result asserts that such deformations are smooth away from a singular set. Here we complement this by providing global results. They allow us to estimate the size of the singular set.More precisely, we study the geometry of planar regions; their boundaries constitute the central part of the singular set. Under natural assumptions on the given boundary data, we show that deformed configurations minimising the bending energy have only finitely many planar regions, and each of them is a polygon. We obtain explicit bounds on the number of such polygons and on the number of edges of each polygon in terms of the boundary data.

In this paper we present an explicit construction of the fundamental solution to the Dirac and Laplace operator on some non-orientable conformally flat manifolds and special orbifolds. We first treat a class of projective cylinders and tori where we can study monogenic sections with values in different pin bundles. Then we discuss the Mobius strip, the Klein bottle and higher dimensional generalizations of them. We present integral representation formulas together with some elementary tools of harmonic analysis on these manifolds resp. orbifolds.

For more than almost 200 years the Mobius strip and its “mysterious” property attracts the attention of mathematicians. After a “complete cut” of this surface, one object appears, but already with a fourfold twist. The generalization of this phenomenon to figures of a more complex configuration led to an “unexpected” result: after the cut of the generalized Mobius-Listing body, more than two geometric shapes may appear. In this paper, we consider all possible cases of a complete cut of the generalized Mobius-Listing body with a regular hexagon as radial section. In early works, together with different colleagues, on the basis of importance, they separately examined the case of Mobius-Listing’s bodies with a radial section of regular 3, 4 and 5 angular figures. Also, cases of similar bodies with a radial section of convex regular two and three angular figures were considered separately. One possible application of these results is assumed in the description of the properties of the middle surfaces in the theory of elastic shells [14] (Vekua, Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston, p. 287, 1985).

The slow divergence integral has proved to be an important tool in the study of slow-fast cycles defined on an orientable two-dimensional manifold (e.g. R 2 ). The goal of our paper is to study 1-canard cycle and 2-canard cycle bifurcations on a non-orientable two-dimensional manifold (e.g. the Mobius band) by using similar techniques. Our focus is on smooth slow-fast models with a Hopf breaking mechanism. The same results can be proved for a jump breaking mechanism and non-generic turning points. The slow-fast bifurcation problems on the Mobius band require the study of the 2-return map attached to such 1- and 2-canard cycles. We give a simple sufficient condition, expressed in terms of the slow divergence integral, for the existence of a period-doubling bifurcation near the 1-canard cycle. We also prove the finite cyclicity property of “singular” 1- and 2-homoclinic loops (“regular” 1-homoclinic loops of finite codimension have been studied by Guimond).

Given knots K and J, one can ask whether a single smoothing of a crossing in a diagram for K can convert it into a diagram for J. As an interesting example, Zekovic discovered that the torus knot T(2,5) can be converted into T(2,-5) with a single smoothing. On the other hand, Moore and Vasquez have shown that among torus knots of the form T(2,m), T(2,5) is the only one which can be converted into its mirror image with a single smoothing, assuming that m is square free. In this paper, Casson-Gordon theory is applied to extend the Moore-Vasquez result to all torus knots T(2,m), with the exception of T(2,9), which remains an open case. Applications of similar techniques to the general problem are also described, as are tools arising from Heegaard Floer theory., Comment: 8 pages, 1 figure, typographical corrections

In this paper, we numerically study quantum walks on two kinds of two-dimensional graphs: cylindrical strip and Mobius strip. The two kinds of graphs are typical two-dimensional topological graph. We study the crossing property of quantum walks on these two models. Also, we study its dependence on the initial state, size of the model. At the same time, we compare the quantum walk and classical walk on these two models to discuss the difference of quantum walk and classical walk.

We construct a closed orientable polyhedral surface of arbitrary genus that is embedded in three-dimensional Euclidean space and admits a one-parameter bending under which all its handles bend. This surface admits no other bendings. We also construct a flexible closed nonorientable polyhedral surface of arbitrary genus such that all its handles and Mobius strips bend during its bending.

The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the context of n-dimensional Riemannian manifolds, is re-examined from a kinematical viewpoint. A special type of frame, whose angular velocity is determined by the rate at which the tangent plane turns as one moves along a surface curve, is defined and is called a Levi-Civita frame. The surface may be orientable or not. Vectors and tensors fixed on Levi-Civita frames are parallel transported. Covariant differentiation of vectors and tensors along a surface curve can be expressed in terms of the corresponding corotational rates measured on Levi-Civita frames. Relevant results on ruled surfaces are also included.

We present some rigorous results about the bending behaviour of paper. By adapting these results to the Mobius strip, we obtain some qualitative properties of developable Mobius strips which minimize the bending energy. We also provide some numerical simulations which illustrate and strengthen the analytic results.

The following is a translation of Walter Wunderlich’s article “Uber ein abwickelbares Mobiusband”, which appeared in the Monatshefte fur Mathematik66 (1962), 276–289 and was dedicated to Prof. Dr. Paul Funk on the occasion of his 75th birthday. Wunderlich summarizes Sadowsky’s work (Sitzber. Preuss. Akad. Wiss. 22:412–415, 1930; Verhandlungen des 3. Internationalen Kongresses fur Technische Mechanik, II (Stockholm, 1930), pp. 444–451, Sveriges Litografiska Tryckerier, Stockholm, 1931) on developable Mobius bands and improves Sadowsky’s upper bound of the dimensionally-reduced variational description for determining the configuration of a Mobius band whose width is small in comparison to its length. Attempting to reproduce the equilibrium depiction of a band of finite width, using a rational-algebraic developable, Wunderlich then extends Sadowsky’s results by presenting perhaps the first successful model of a closed, analytic, developable Mobius band with associated thinness bounds. This translation makes Wunderlich’s work accessible to the broader research community at a time of growing interest in and relevance of thin-walled structural elements.