Closed Unstretchable Knotless Ribbons and the Wunderlich Functional

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.

     

SUBMANIFOLDS IN S^n＋p WITH PARALLEL MOEBIUS FORM

In this paper,the rigidity theorems of the submanifolds in S^n p with parallel Moebius form and constant MObius scalar curvature are given.     

Möbius Polynomials of Face Posets of Convex Polytopes

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their f-vectors and some additional constraints.     

On Möbius form and Möbius isoparametric hypersurfaces

An umbilic-free hypersurface in the unit sphere is called MSbius isoparametric if it satisfies two conditions, namely, it has vanishing MSbius form and has constant MSbius principal curvatures. In this paper, under the condition of having constant MSbius principal curvatures, we show that the hypersurface is of vanishing MSbius form if and only if its MSbius form is parallel with respect to the Levi-Civita connection of its MSbius metric. Moreover, typical examples are constructed to show that the condition of having constant MSbius principal curvatures and that of having vanishing MSbius form are independent of each other.     

Möbius and triangle semigroups

The purpose of this paper is to describe the structures of the Möbius semigroup induced by the Möbius transformation group (?, SL(2,?)). In particular, we study stabilizer subsemigoups of Möbius semigroup via the triangle semigroup. In this work, we obtained a geometric interpretation of the least contraction coefficient function of the Möbius semigroup via the triangle semigroup and investigated an extension of stabilizer subsemigoups of the Möbius semigroup. Finally, we obtained a factorization of our stabilizer subsemigoups of the Möbius semigroup.     

On an invariant Möbius measure and the Gauss-Kuzmin face distribution

We study Möbius measures of the manifold of n-dimensional continued fractions in the sense of Klein. By definition any Möbius measure is invariant under the natural action of the group of projective transformations PGL(n + 1) and is an integral of some form of the maximal dimension. It turns out that all Möbius measures are proportional, and the corresponding forms are written explicitly in some special coordinates. The formulae obtained allow one to compare approximately the relative frequencies of the n-dimensional faces of given integer-affine types for n-dimensional continued fractions. In this paper we make numerical calculations of some relative frequencies in the case of n = 2.     

Spacelike Möbius Hypersurfaces in Four Dimensional Lorentzian Space Form

In this paper, we first set up an alternative fundamental theory of Möbius geometry for any umbilic-free spacelike hypersurfaces in four dimensional Lorentzian space form, and prove the hypersurfaces can be determined completely by a system consisting of a function W and a tangent frame {E_i}. Then we give a complete classification for spacelike Möbius homogeneous hypersurfaces in four dimensional Lorentzian space form. They are either Möbius equivalent to spacelike Dupin hypersurfaces or to some cylinders constructed from logarithmic curves and hyperbolic logarithmic spirals. Some of them have parallel para-Blaschke tensors with non-vanishing Möbius form.     

Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations

Starting from identities obtained by Möbius inversion, we prove some inequalities involving the ordinary and logarithmic summatory functions of the Möbius function.     

Möbius transformation and a version of Schwarz lemma in octonionic analysis

The octonion is a generalization of complex to noncommutative and nonassociative space which has closed relation with exception geometries, wave equation, Yang‐Mills equations, black hole, string theory, and special relativity. In this paper, the Möbius transformation in this manner is first introduced, and some properties are discussed about the transformation in octonionic analysis. Some technique lemmas will be given to solve the problems caused by the weak form of associativity. These versions of Schwarz lemma and Schwarz‐Pick lemma are first studied in octonionic setting which will invoke integral representation formula for harmonic function and Möbius transformations. This will generalize the corresponding results which appear in the classical function theory to nonassociative space and may give new energy for the development of physics.     

Mean value conjectures for rational maps

Let p be a polynomial in one complex variable. Smale's mean value conjecture estimates |p′(z)| in terms of the gradient of a chord from (z,?p(z)) to some stationary point on the graph of p. The conjecture does not immediately generalize to rational maps since its formulation is invariant under the group of affine maps, not the full Möbius group. Here we give two possible generalizations to rational maps, both of which are Möbius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture. Finally, we discuss some candidate extremal rational maps, namely rational maps all of whose critical points are fixed points.     

The Möbius function of permutations with an indecomposable lower bound

We show that the Möbius function of an interval in a permutation poset where the lower bound is sum (resp. skew) indecomposable depends solely on the sum (resp. skew) indecomposable permutations contained in the upper bound, and that this can simplify the calculation of the Möbius sum. For increasing oscillations, we give a recursion for the Möbius sum which only involves evaluating simple inequalities.     

On the Möbius Function of a Pointed Graded Lattice

In this paper, we compute the Möbius function of pointed integer partition and pointed ordered set partition using topological and analytic methods. We show that the associated order complex is a wedge of spheres and compute the associated reduced homology group for each subposet. In addition, we compute the Möbius function of pointed graded lattice and use our method to compute the Möbius function of pointed direct sum decomposition of vector spaces.     

The Möbius invariants for circle packings

Möbius invariants of general circle packings are defined in terms of cross ratios. The necessary and sufficient conditions of existence of circle packings are established by the techniques of Möbius invariants. It is shown that circle packings are uniquely determined, up to Möbius transformations, by their Möbius invariants. The rigidity of infinite circle packings with bounded degree is proved using the approach of Möbius invariants.     

Generalized Angles in Ptolemaic Möbius Structures. II

We continue studying the BAD class of multivalued mappings of Ptolemaic Möbius structures in the sense of Buyalo with controlled distortion of generalized angles. In Möbius structures we introduce a Möbius-invariant version of the HTB property (homogeneous total boundedness) of metric spaces which is qualitatively equivalent to the doubling property. We show that in the presence of this property and the uniform perfectness property, a single-valued mapping is of the BAD class iff it is quasimöbius.     

Catégories de Möbius et fonctorialités: un cadre général pour l'inversion de Möbius

This paper proposes a general setting for Möbius inversion which includes the cases of locally finite partially ordered sets and monoids with the finite decomposition property (f.d.p.). The unifying concept is that of categories with the f.d.p., called Möbius categories, which we characterize in four different ways. The established theory of incidence algebras, Möbius functions, product formulas, reduced algebras, etc., is carried over. Furthermore, the study of functors between Möbius categories yields results in two directions: firstly, inspired from the substitution of formal power series is the construction of a homomorphism between incidence algebras which allows the transfer of Möbius inversions; secondly, surjective functors often give rise to a reduced incidence algebra, thus shedding new light on many important reduced algebras in combinatorial theory.     

Möbius Regular Maps of Order pq

M?bius 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 M?bius band on the surface. In this paper, we classify M?bius regular maps of order pq for any two primes p and q, where p≠q.     

Möbius homogeneous hypersurfaces with two distinct principal curvatures in S n+1

The purpose of this paper is to classify the Möbius homogeneous hypersurfaces with two distinct principal curvatures in S ⁿ⁺¹ under the Möbius transformation group. Additionally, we give a classification of the Möbius homogeneous hypersurfaces in S ⁴.     

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials.     

The Möbius function of a composition poset

We determine the Möbius function of a poset of compositions of an integer. In fact, we give two proofs of this formula, one using an involution and one involving discrete Morse theory. This composition poset turns out to be intimately connected with subword order, whose Möbius function was determined by Björner. We show that, using a generalization of subword order, we can obtain both Björner’s results and our own as special cases.