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.     

The interaction transform for functions on lattices

The paper proposes a general approach of interaction between players or attributes. It generalizes the notion of interaction defined for players modeled by games, by considering functions defined on distributive lattices. A general definition of the interaction transform is provided, as well as the construction of operators establishing transforms between games, their Möbius transforms and their interaction indices.     

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.     

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 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.     

Rudin orthogonality problem on the Bergman space

In this paper, we study the Rudin orthogonality problem on the Bergman space, which is to characterize those functions bounded analytic on the unit disk whose powers form an orthogonal set in the Bergman space of the unit disk. We completely solve the problem if those functions are univalent in the unit disk or analytic in a neighborhood of the closed unit disk. As a consequence, it is shown that an analytic multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity n if and only if its symbol is a constant multiple of the n-th power of a Möbius transform, which was obtained via the Hardy space theory of the bidisk in Sun et al. (2008) [10].     

Möbius functions and semigroup representation theory

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon's approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role.     

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).     

Space and time inversions of stochastic processes and Kelvin transform

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.     

Explicit Inversion Formulas for the X–Ray Transform in {\mathbb{Z}^n}

In the present paper, we state and prove explicit inversion formulas for the X–Ray transform in the lattice ${\mathbb{Z}^n}$ by using certain arithmetical geometrical techniques.     

The homogeneous shifts

A bounded linear operator T on a complex Hilbert space is called homogeneous if the spectrum of T is contained in the closed unit disc and all bi-holomorphic automorphisms of this disc lift to automorphisms of the operator modulo unitary equivalence. We prove that all the irreducible homogeneous operators are block shifts. Therefore, as a first step in classifying all of them, it is natural to begin with the homogeneous scalar shifts.In this paper we determine all the homogeneous (scalar) weighted shifts. They consist of the unweighted bilateral shift, two one-parameter families of unilateral shifts (adjoints of each other), a one-parameter family of bilateral shifts and a two-parameter family of bilateral shifts. This classification is obtained by a careful analysis of the possibilities for the projective representation of the Möbius group associated with an irreducible homogeneous shift.     

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.

     

Comparison theorems for compound quadrature formulas

Summary The aim of this note is to compare the rates of convergence of quadrature processes. To this end, representation formulas for the remainders of one process in terms of a second one are developed. The main tool is the Möbius inversion. It turns out that for a large class of compound quadrature processes the rates of convergence are essentially the same.     

The general method to solve the inverse lattice problems in physics

By a system of linear equations on a multiplicative semigroup, we present a general mathematical method to determine the interatomic potential of various lattice structures in order to compute the performances of materials, and show the relation between the method and the Möbius inversion.     

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.     

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.     

Linear bounds for Calderón–Zygmund operators with even kernel on UMD spaces

It is well-known that several classical results about Calderón–Zygmund singular integral operators can be extended to X-valued functions if and only if the Banach space X has the UMD property. The dependence of the norm of an X-valued Calderón–Zygmund operator on the UMD constant of the space X is conjectured to be linear. We prove that this is indeed the case for sufficiently smooth Calderón–Zygmund operators with cancellation, associated to an even kernel. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytönen to extend the result to general Calderón–Zygmund operators.     

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.     

Axiomatizing Provable <Emphasis Type="Italic">n</Emphasis>-Provability

The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an ε₀-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.