Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.     

Tight MIP formulations for bounded up/down times and interval-dependent start-ups

Mathematical Programming - Switching machines on and off is an important aspect of unit commitment problems and production planning problems, among others. Here we study tight mixed integer...     

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules and then consider the classical Baker and τ-functions as operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map leads to an operator τ-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We link directly this cross-ratio with Fay’s trisecant identity for the τ-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall–Chaundy C*-algebra (Part I) provides a link to the study of the KP hierarchy.     

On the Prolate Spheroidal Wave Functions and Hardy’s Uncertainty Principle

We prove a weak version of Hardy’s uncertainty principle using properties of the prolate spheroidal wave functions. We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on \(L^2(\mathbb {R})\) . A weak version of Hardy’s uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.     

Rheology of lyotropic and thermotropic lamellar phases

We investigate the steady-state rheological behaviour of the lamellar phase of a lyotropic system (CpCl, hexanol, brine) and of a thermotropic system (8CB). Power laws characterize the behaviour of the imposed stress as a function of the measured shear rate and similarities are observed for both systems; the same regime γ˙∼σ ^m with m≈1.7 is obtained at low shear stresses corresponding to a texture of oily streaks oriented in the direction of the flow, as shown by microscopic observations. The “onion state” only exists in the case of dilute samples of the lyotropic lamellar phase; the stress then varies as γ˙∼σ ^m with m≈4.8, as already observed by other groups with different systems. Rheological measurements at different temperatures allow determination of different activation energies relating to the still badly understood processes involved in the different rheological regimes. We propose a model which reproduces the experimental power laws and which is based on an analogy with the theory of high-temperature creep in metals and alloys. Received: 19 October 1999/Accepted: 1 November 1999     

Weighted allocation rules for standard fixed tree games

In this paper we consider standard fixed tree games, for which each vertex unequal to the root is inhabited by exactly one player. We present two weighted allocation rules, the weighted down-home allocation and the weighted neighbour-home allocation, both inspired by the painting story in Maschler et al. (1995) . We show, in a constructive way, that the core equals both the set of weighted down-home allocations and the set of weighted neighbour allocations. Since every weighted down-home allocation specifies a weighted Shapley value (Kalai and Samet (1988)) in a natural way, and vice versa, our results provide an alternative proof of the fact that the core of a standard fixed tree game equals the set of weighted Shapley values. The class of weighted neighbour allocations is a generalization of the nucleolus, in the sense that the latter is in this class as the special member where players have all equal weights.     

Viscous linear instability of a sheared liquid-gas interface - Effect of a velocity deficit near the interface

The initial state of liquid atomization by a fast gas stream is considered by viscous linear spatial stability analysis for parallel two-fluid flow. The unbounded basic velocity profile is characterized by different asymptotic velocities and a velocity deficit near the interface. We examine the influence of the velocity deficit on the spatial growth rates of two competing modes originating from the Kelvin-Helmholtz and viscosity contrast mechanisms. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)