Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B n , C n and D n

The Kostka–Foulkes polynomials related to a root system can be defined as alternating sums running over the Weyl group associated to . By restricting these sums over the elements of the symmetric group when is of type or , we obtain again a class of Kostka–Foulkes polynomials. When is of type or there exists a duality between these polynomials and some natural -multiplicities and in tensor products [11]. In this paper we first establish identities for the which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials with nonnegative integer coefficients. Moreover these coefficients are branching coefficients This allows us to clarify the connection between the -multiplicities and the polynomials defined by Shimozono and Zabrocki. Finally we show that and coincide up to a power of with the one dimension sum introduced by Hatayama and co-workers when all the parts of are equal to , which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.Presented by P. Littelmann.     

Maxwell’s equations in inhomogeneous bi-anisotropic materials: Existence,uniqueness and stability for the initial value problem

In the present paper inhomogeneous bi-anisotropic materials characterized by matrices of electric permittivity, magnetic permeability and magnetoelectric characteristics are considered. All elements of these matrices are functions of the position in three dimensional space. The time-dependent Maxwell’s equations describe the electromagnetic wave propagation in these materials. Maxwell’s equations together with zero initial data are analyzed and a statement of the initial value problem (IVP) is formulated. This IVP is reduced to the IVP for a symmetric hyperbolic system of partial differential equations of the first order. Applying the theory of a symmetric hyperbolic system, new existence, uniqueness and stability estimate theorems have been obtained for the IVP of Maxwell’s equations in inhomogeneous bi-anisotropic materials.     

Second-order consensus for heterogeneous multi-agent systems in the cooperation–competition network: A hybrid adaptive and pinning control approach

In this paper, a second-order consensus problem is investigated for the heterogeneous agents in the cooperation–competition network, with the parameters of the agents’ dynamics being uncertain. By introducing an exosystem, a hybrid adaptive and pinning control strategy is firstly proposed to ensure that all the agents follow a desired trajectory which is a function of the exosystem state. Several effective sufficient conditions are obtained by applying algebraic graph theory and the Barbalat lemma. Besides, the switching topologies between the competition agents are also considered. With the help of the Barbalat-like lemma, it is found that second-order consensus problem of the heterogeneous agents can be solved even when the competition sub-network is dynamic. Finally, simulation results are presented to validate the effectiveness of the theoretical analysis.     

Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

We investigate boundedness of the evolutione ^itH in the sense ofL ²(ℝ³→L ²(ℝ³) as well asL ¹(ℝ³→L ^∞(ℝ³) for the non-selfadjoint operator where μ>0 andV ₁, V₂ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)–A4) below, but without imposing any restrictions on the edges±μ of the essential spectrum. Our goal is to develop an “axiomatic approach,” which frees the linear theory from any nonlinear context in which it may have arisen. This work was initiated in June of 2004, while the first author visited Caltech, and he wishes to thank that institution for its hospitality and support. The first author was partially supported by the NSF grant DMS-0303413. The second author was partially supported by a Sloan fellowship and the NSF grant DMS-0300081. The authors thank Avy Soffer for his interest in this work.     

Holistic oil field value management: using system dynamics for ‘intermediate level’ and ‘value-based’ modelling in the oil industry

System dynamics has been seen primarily as a strategic tool, most effectively used at the highest level of strategy to identify robust policy interventions under a wide range of scenarios. However, an alternative, complementary and powerful role is emerging. This is at an ‘intermediate level’ in organisations to coordinate and integrate policies across the value chain. It is at this level where business value, as defined by the discounted value of future free cash flow, is both created and destroyed. This paper introduces the need for ‘intermediate-level’ and ‘value-based’ modelling and emphasises the natural role of system dynamics in supporting a methodology to fulfil the need. It describes the development of an approach and its application in the oil industry to coordinate the response of people and tools within operational, financial and commercial functions across the value chain to address a variety of problems and issues.     

Two problems in the 筭數書 Suanshu shu (Book of Mathematics): Geometric relations between circles and squares and methods for determining their mutual relations

Much research has been devoted to two problems, Yi yuancai fang (From a circular timber [find] a square) and Yi fangcai yuan (From a square timber [find] a circle), both of which appear in the Suanshu shu, an early Han dynasty mathematical work written on bamboo slips, excavated from tomb 247 at Zhangjiashan in Hubei Province, China. In this article, the geometric relations between circles and squares and the methods for determining their mutual relations in these two problems are interpreted in a different way, and an alternative approach is offered for reconciling these two problems.     

Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to $$ $$

We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, we present the general regular solution to Cauchy’s additive functional equation on restricted lower-dimensional convex domains. This provides a k-dimensional generalization of the so-called Interval Lemma, allowing us to deduce affine properties of the function from certain additivity relations. Next, we study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. We then give a theory of non-extremality certificates in the form of perturbation functions. We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function.     

Generalizing the Keller�CSegel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in?the?Presence of Attraction and Repulsion Between Competitive Interacting Species

In this paper we extend the famous Keller–Segel model for the chemotactic movement of motile species to some multi-species chemotaxis equations. The presented multi-species chemotaxis models are more general than those introduced so far and also include some interaction effects that have not been studied before. For example, we consider multi-species chemotaxis models with attraction and repulsion between interacting motile species. For some of the presented new models we give sufficient conditions for the existence of Lyapunov functionals. These new results are related to those of Wolansky (Scent and sensitivity: equilibria and stability of chemotactic systems in the absence of conflicts, preprint, 1998; Eur. J. Appl. Math. 13:641–661, 2002). Furthermore, a linear stability analysis is performed for uniform steady states, and results for the corresponding steady state problems are established. These include existence and nonexistence results for non-constant steady state solutions in some special cases.     

Is There Still a Place for Independent Research into Issues of Public Policy in England and Wales in the 1980's? A Case Study from the Field of Health Care: Modelling Hospital Costs

This paper is concerned with the viability of independent research into issues of public policy in a climate which is becoming increasingly hostile. The general argument is illustrated by one specific example relating to the development of models of hospital inpatient costs in the National Health Service. In a recent publication by staff of the Operational Research Service of the Department of Health and Social Security a model (denoted the C-G-N model) of hospital costs is put forward. However, no mention is made of an alternative model (denoted the A-B model) published some two years previously and we have therefore decided to compare the two models in terms of general structure, fit to the available observational data and applicability. It is shown that the C-G-N model was applied by its authors in a form which involves an heteroscedastic residual error term. When this model is fitted to a representative data set, with an appropriate weighting factor which produces homoscedastic errors, substantial differences are found in the corresponding parameter estimates. The C-G-N model in its original and modified versions is then compared with the A-B model, and it is shown that the latter is more parsimonious in terms of parameters, is applicable to a wider range of types of hospital and fits the available empirical data somewhat better than either version of the former. Furthermore, there are substantial differences in the conclusions which might be drawn on the basis of the various models in several important classes of application. An outline is given of a more complex model, denoted the component model, which incorporates features of both the C-G-N and A-B models, provides a better fit to the empirical data than either and has a structure which is more satisfactory for a wide range of applications.