Biorthogonal polynomials: Recent developments

An important application of biorthogonal polynomials is in the generation of polynomial transformations that map zeros in a predictable way. This requires the knowledge of the explicit form of the underlying biorthogonal polynomials.The most substantive set of parametrized Borel measures whose biorthogonal polynomials are known explicitly are theMöbius quotient functions (MQFs), whose moments are Möbius functions in the parameter. In this paper we describe recent work on the characterization of MQFs, following two distinct approaches. Firstly, by restricting the attention to specific families of Borel measures, of the kind that featured in [4], it is possible sometimes to identify all possible MQFs by identifying a functional relationship between weight functions for different values of the parameter. Secondly, provided that the coefficients in Möbius functions are smooth (in a well defined sense), it is possible to prove that the weight function obeys a differential relationship that, in specific cases, allows an explicit characterization of MQFs. In particular, if all such coefficients are polynomial, the MQFs form a subset of generalized hypergeometric functions.Dedicated to Syvert P. Nørsett on the occasion of his 50th birthdayThis paper has been written during the author's visit to California Institute of Technology, Pasadena.     

The r-cubical Lattice and a Generalization of the cd-index

In this paper we generalize thecd-index of the cubical lattice to anr-cd-index, which we denote byΨ(r). The coefficients ofΨ(r) enumerate augmented Andrér-signed permutations, a generalization of Purtill's work relating thecd-index of the cubical lattice and signed André permutations. As an application we use ther-cd-index to determine that the extremal configuration which maximizes the Möbius function of arbitrary rank selections, where all ther_i's are greater than one, is the odd alternating ranks, {1, 3, 5, ...}.     

Möbius Categories as Reduced Standard Division Categories of Combinatorial Inverse Monoids

The aim of this paper is to characterize inverse monoids such that their reduced standard division categories C_F(S) are Möbius categories and special Möbius categories. We give also a general technique for the evaluation of the Möbius function of C_F(S).     

Möbius hypersurfaces in S S+1with three distinct principal curvatures

On a hypersurface of a unit sphere without umbilical points, we know that three Möbius invariants can be defined and analogous to Euclidean case, we have the concepts of Möbius isoparametric and isotropic hypersurfaces. In this paper, we study the relationship between Euclidean geometry and Möbius geometry, and prove that a hypersurface in a sphere with constant length of the second fundamental form is Euclidean isoparametric if and only if it is Möbius isoparametric. When restricting to the case of three distinct principal curvatures, we show that such a hypersurface is either Möbius isoparametric or isotropic if the Blaschke tensor has constant eigenvalues.     

Cohen-Macaulay types of subgroup lattices of finite abelianp-groups

For a minimal free resolution of a Stanley-Reisner ring constructed from the order complex of a modular lattice. T. Hibi showed that its last Betti number (called the Cohen-Macaulay type) is computed by means of the Möbius function of the given modular lattice. Using this result, we consider the Stanley-Reisner ring of the subgroup lattice of a finite abelianp-group associated with a given partition, and show that its Cohen-Macaulay type is a polynomial inp with integer coefficients.     

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

Characteristic for reflexive relations

Poincaré characteristic for reflexive relations (oriented graphs) is defined in terms of homology and is not invariant under transitive closure. Formulas for the Poincaré characteristic of products, joins, and bounded products are given. Euler's definition of characteristic extends to certain filtrations of reflexive relations which exist iff there are no oriented loops. Euler characteristic is independent of filtration, agrees with Poincaré characteristic, and is unique. One-sided Möbius characteristic is defined as the sum of all values of a one-sided inverse of the zeta function. Such one-sided inverses exist iff there are no local oriented loops (although there may be global oriented loops). One-sided Möbius characteristic need not be Poincaré characteristic, but it is when a one-sided local transitivity condition is satisfied. A two-sided local transitivity condition insures the existence of the Möbius function but Möbius inversion fails for non-posets.     

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L²-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FPR) is a class in the projective class group K₀(RΓ); the functorial Euler characteristic and functorial L²-Euler characteristic are respectively its RΓ-rank and L²-rank. We also extend the second author's K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez and Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L²-Euler characteristic. Some of Leinster's results on Möbius–Rota inversion are special cases of the K-theoretic Möbius inversion.     

A Rational-Function Identity Related to the Murnaghan–Nakayama Formula for the Characters of S n

The Murnaghan–Nakayama formula for the characters of S _n is derived from Young's seminormal representation, by a direct combinatorial argument. The main idea is a rational function identity which when stated in a more general form involves Möbius functions of posets whose Hasse diagrams have a planar embedding. These ideas are also used to give an elementary exposition of the main properties of Young's seminormal representations.     

A novel algorithm enumerating bent functions

Based on the relationship between the Walsh spectra of a Boolean function at partial points and the Walsh spectra of its subfunctions, and on the binary Möbius transform, a novel algorithm is developed, which can theoretically construct all bent functions. Practically we enumerate all bent functions in 6 variables. With the restriction on the algebraic normal form, the algorithm is also efficient in more variables case. For example, enumeration of all homogeneous bent functions of degree 3 in 8 variables can be done in one minute with a P4 1.7 GHz computer; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is computationally proved.     

On the quality of algorithms based on spline interpolation

The qualityq of a numerical algorithm using some specified information is the ratio of its error to the smallest possible error of an algorithm based on the same information. We use as information function values at equidistant points, periodicity and a bound for therth derivative. We show thatq is rather small, if the algorithm is based on spline interpolation.     

Irrationality of Power Series for Various Number Theoretic Functions

We study formal power series whose coefficients are taken to be a variety of number theoretic functions, such as the Euler, Möbius and divisor functions. We show that these power series are irrational over [X], and we obtain lower bounds on the precision of their rational approximations.     

A unique representation of polyhedral types. Centering via Möbius transformations

For n3 distinct points in the d-dimensional unit sphere there exists a Möbius transformation such that the barycenter of the transformed points is the origin. This Möbius transformation is unique up to post-composition by a rotation. We prove this lemma and apply it to prove the uniqueness part of a representation theorem for 3-dimensional polytopes as claimed by Ziegler (1995): For each polyhedral type there is a unique representative (up to isometry) with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere.Mathematics Subject Classification (2000): 52B10The author is supported by the DFG Research Center Mathematics for key technologies.     

On Critical Points of Proper Holomorphic Maps on The Unit Disk

We prove that a proper holomorphic map on the unit disk in thecomplex plane is uniquely determined up to post-compositionwith a Möbius transformation by its critical points. 1991Mathematics Subject Classification 30C99, 30F45.     

Topological structure of integral manifolds and period-doubling bifurcation

We investigate the topological structure of integral manifolds near a closed orbit of an autonomous differential system. We prove that under some circumstances these manifolds are homeomorphic to a Möbius strip. It is shown that the appearance of a period-doubling bifurcation in systems depending on a parameter is intimately connected with the occurence of a center manifold homeomorphic to a Möbius strip. Finally we demonstate that the period-doubling bifurcation can be treated as Hopf bifurcation on a Möbius strip.

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Zusammenfassung Wir untersuchen die topologische Struktur von Integralmannigfaltigkeiten in der Nähe einer geschlossenen Lösungskurve eines autonomen Differentialgleichungssystems. Wir beweisen, daß unter gewissen Umständen diese Mannigfaltigkeiten homöomorph zu einem Möbius-Band sind. Es wird gezeigt, daß das Auftreten einer Periodenverdopplungsbifurkation in parameterabhängigen Systemen eng mit der Existenz einer Zentrumsmannigfaltigkeit verknüpft ist, die homöomorph zu einem Möbius-Band ist. Abschließend demonstrieren wir, daß die Periodenverdopplungsbifurkation als Hopf-Bifurkation auf einem Möbius-Band behandelt werden kann.

     

A new characteristic of Möbius transformations by use of polygons having type A

In the hyperbolic plane Möbius transformations can be characterized by Lambert quadrilaterals, i.e., a continuous bijection which maps Lambert quadrilaterals to Lambert quadrilaterals must be Möbius. In this paper we generalize this result to the case of polygons with n sides having type A, that is, having exactly two non-right interior angle.     

Möbius inversion formulas for flows of arithmetic semigroups

We define a convolution-like operator which transforms functions on a space X via functions on an arithmetical semigroup S, when there is an action or flow of S on X. This operator includes the well-known classical Möbius transforms and associated inversion formulas as special cases. It is defined in a sufficiently general context so as to emphasize the universal and functorial aspects of arithmetical Möbius inversion. We give general analytic conditions guaranteeing the existence of the transform and the validity of the corresponding inversion formulas, in terms of operators on certain function spaces. A number of examples are studied that illustrate the advantages of the convolutional point of view for obtaining new inversion formulas.     

Discreteness and Convergence of Möbius Groups

In this paper we show that one can use a fixed nontrivial Möbius transformation as a test map to test the discreteness of a nonelementary Möbius group. We also establish two theorems in algebraic convergence.