Geometric combinatorial algebras: cyclohedron and simplex

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto–Reutenauer algebra of permutations and the Loday–Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.     

The structure of a class of Z-local rings

A local ring R is called Z-local if J(R) = Z(R) and J(R)2 = 0. In this paper the structure of a class of Z-local rings is determined.     

The structure of the Mitchell order—I

We isolate here a wide class of well-founded orders called tame orders, and show that each such order of cardinality at most κ can be realized as the Mitchell order on a measurable cardinal κ, from a consistency assumption weaker than o(κ) = κ⁺.     

Stratified structure of the Universe and Burgers' equation—a probabilistic approach

Summary The model of the potential turbulence described by the 3-dimensional Burgers' equation with random initial data was developped by Zeldovich and Shandarin, in order to explain the existing Large Scale Structure of the Universe. Most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results (the Gaussian scenario), under suitable moment assumptions on the initial velocity field. These results and some open questions are discussed in Sect. 2, where we concentrate on the Gaussian model and the shot-noise model. In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise field with high amplitudes) which reveals an intermittent large time behaviour, with the velocity determined by the position of the largest initial fluctuation (discounted by the heat kernelg(t,x·)) in a neighborhood ofx. The asymptoties of such local maximum ast can be analyzed with the help of the theory of records (Sect. 4). Finally, in Sect. 5 we introduce a global definition of a point process oft-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit ast.     

Localization with respect to a class of maps I — Equivariant localization of diagrams of spaces

Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [4, 13, 24, 35]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces [12]. This model category is not cofibrantly generated [8]. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the classes of maps which induce ordinary localizations on the generalized fixed-points sets. During the preparation of this paper the author was a fellow of Marie Curie Training Site hosted by Centre de Recerca Matemàtica (Barcelona), grant no. HPMT-CT-2000-00075 of the European Commission.     

Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support

We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.     

Performance evaluation of demodulation with diversity—a combinatorial approach II: bijective methods

This paper is devoted to the presentation of a combinatorial approach for analyzing the performance of a generic family of demodulation methods used in mobile telecommunications. We show that a fundamental formula in this context is in fact highly connected with a slight modification of a very classical bijection of Knuth between pairs of Young tableaux of conjugate shapes and $\{0,1\}$-matrices. These considerations allowed us to obtain the first explicit expressions for several important specializations of the performance evaluation formula that we studied.     

Global stability of a class of Nicholson’s blowflies model with patch structure and multiple time-varying delays

This paper is concerned with a class of the generalized Nicholson’s blowflies model with patch structure and multiple time-varying delays, which is defined on the nonnegative function space. Under proper conditions, we employ a novel proof to establish some criteria of the global stability of positive equilibrium for this model. Moreover, we give an example to illustrate our main result.     

Projection pressure and Bowen’s equation for a class of self-similar fractals with overlap structure

Let {Si}il=1 be an iterated function system (IFS) on Rd with attractor K. Let π be the canonical projection. In this paper, we define a new concept called projection pressure Pπ(φ) for ∈ C(Rd) under certain affine IFS, and show the variational principle about the projection pressure. Furthermore, we check that the unique zero root of projection pressure still satisfies Bowen’s equation when each Si is the similar map with the same compression ratio. Using the root of Bowen’s equation, we can get the Hausdorff dimension of the attractor K.     

Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras

We develop a theory of multigraded (i.e., ℕ ^l -graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar et al. (Compos. Math. 142:1–30, 2006). In particular we introduce the notion of canonical k-odd and k-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-peaks map.     

Agency and structure: a social simulation of knowledge-intensive industries

Modern knowledge-intensive economies are complex social systems where intertwining factors are responsible for the shaping of emerging industries: the self-organising interaction patterns and strategies of the individual actors (an agency-oriented pattern) and the institutional frameworks of different innovation systems (a structure-oriented pattern). In this paper, we examine the relative primacy of the two patterns in the development of innovation networks, and find that both are important. In order to investigate the relative significance of strategic decision making by innovation network actors and the roles played by national institutional settings, we use an agent-based model of knowledge-intensive innovation networks, SKIN. We experiment with the simulation of different actor strategies and different access conditions to capital in order to study the resulting effects on innovation performance and size of the industry. Our analysis suggests that actors are able to compensate for structural limitations through strategic collaborations. The implications for public policy are outlined.     

Directed graphs and combinatorial properties of completely regular semigroups

In this paper, we investigate the divisibility graphs and power graphs of completely regular semigroups. We give the structures of these two kinds of graphs and describe a combinatorial property of completely regular semigroups defined in terms of divisibility graphs and power graphs, respectively.     

Uniformly Distributed Distances – a Geometric Application of Janson's Inequality

denote the distances determined by n points in the plane. It is shown that , where the minimum is taken over all point sets with minimal distance . This bound is asymptotically tight. Received: September 4, 1997     

When is a spherical body of constant diameter of constant width?

Aequationes mathematicae - We prove that a smooth convex body of diameter $$\delta < \frac{\pi }{2}$$ on the d-dimensional unit sphere $$S^d$$ is of constant diameter $$\delta $$ if and only...