Refined common knowledge logics or logics of common information

 In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on [6] we introduce formal axiomatic systems for common information logics and prove that these systems are sound and complete. Analyzing the common information operation we show that it can be understood as greatest open fixed points for knowledge formulas. Using obtained results we explore monotonicity, omniscience problem, and inward monotonocity, describe their connections and give dividing examples. Also we find algorithms recognizing these properties for some particular cases. Received: 21 October 2000 / Published online: 2 September 2002 Key words or phrases: Multi-agent systems – Non-standard logic – Knowledge representation – Common knowledge – Common information – Fixed points, Kripke models – Modal logic     

Stability and General Logics

In this paper we make an attempt to study classes of models by using general logics. We do not believe that L_ww is always the best logic for analyzing a class of models. Let K be a class of models and L a logic. The main assumptions we make about K and C are that K has the L-amalgamation property and, later in the paper, that K does not omit L-types. We show that, if modified suitably, most of the results of stability theory hold in this context. The main difference is that existentially closed models of K play the role that arbitrary models play in traditional stability theory. We prove e. g. a structure theorem for the class of existentially closed models of K assuming that K is a trivial superstable class with ndop.     

Knowledge management: a review of the field and of OR's contribution

This paper examines the field of knowledge management (KM) and identifies the role of operational research (OR) in key milestones and in KM's future. With the presence of the OR Society journal Knowledge Management Research and Practice and with the INFORMS journal Organization Science, OR may be assumed to have an explicit and a leading role in KM. Unfortunately, the origins and the evidence of recent research efforts do not fully support this assumption. We argue that while OR has been inside many of the milestones there is no explicit recognition of its role and while OR research on KM has considerably increased in the last 5 years, it still forms a rather modest explicit contribution to KM research. Nevertheless, the depth of OR's experience in decision-making models and decision support systems, soft systems with hard systems and in risk management suggests that OR is uniquely placed to lead future KM developments. We suggest that a limiting aspect of whether OR will be seen to have a significant profile will be the extent to which developments are recognized as being informed by OR.     

Diameter in ultra‐small scale‐free random graphs

It is well known that many random graphs with infinite variance degrees are ultra‐small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k^{?(τ ? 1)} with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to and , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order precisely when the minimal forward degree d_fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.     

Phase coexistence in Ising, Potts and percolation models

We study phase coexistence (separation) phenomena in Ising, Potts and random cluster models in dimensions d3 below the critical temperature. The simultaneous occurrence of several phases is typical for systems with appropriately arranged (mixed) boundary conditions or for systems satisfying certain physically natural constraints (canonical ensembles). The various phases emerging in these models define a partition, called the empirical phase partition, of the space. Our main results are large deviations principles for (the shape of) the empirical phase partition. More specifically, we establish a general large deviation principle for the partition induced by large (macroscopic) clusters in the Fortuin–Kasteleyn model and transfer it to the Ising–Potts model where we obtain a large deviation principle for the empirical phase partition induced by the various phases. The rate function turns out to be the total surface free energy (associated with the surface tension of the model and with boundary conditions) which can be naturally assigned to each reasonable partition. These LDP-s imply a weak law of large numbers: asymptotically, the law of the phase partition is determined by an appropriate variational problem. More precisely, the empirical phase partition will be close to some partition which is compatible with the constraints imposed on the system and which minimizes the total surface free energy. A general compactness argument guarantees the existence of at least one such minimizing partition. Our results are valid for temperatures T below a limit of slab-thresholds conjectured to agree with the critical point T_c. Moreover, T should be such that there exists only one translation invariant infinite volume state in the corresponding Fortuin–Kasteleyn model; a property which can fail for at most countably many values and which is conjectured to be true for every T≠T_c.     

Method for Solving the Multidimensional n‐Wave Resonant Equations and Geometry of Generalized Darboux‐Manakov‐Zakharov Systems

The intrinsic geometric properties of generalized Darboux‐Manakov‐Zakharov systems of semilinear partial differential equations (1) for a real‐valued function u(x₁, …, x_n) are studied with particular reference to the linear systems in this equation class. System (1) is overdetermined and will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive tools for explicitly constructing involutive systems of the form (1) , essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multidimensional n‐wave resonant interaction system and its modified version. For each integer n≥ 3 and nonnegative integer k, our procedure constructs solutions of the n‐wave resonant interaction system depending on at least k arbitrary functions each of one variable. The construction of these solutions relies only on differentiation, linear algebra, and the solution of ordinary differential equations.     

Semiboolean SQS-skeins

We will present a counter example to the conjecture that the class of boolean SQS-skeins is defined by the equation q(x, u, q(y, u, z)) = q(q(x, u, y), u, z ). The SQS-skeins satisfying this equation will be seen to be exactly those SQS-skeins that correspond to Steiner quadruple systems whose derived Steiner triple systems are all projective geometries.     

Integrable (2+1)-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The Gibbons-Tsarev (GT) systems are most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g = 0 and g = 1 and also a new GT system corresponding to algebraic curves of genus g = 2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is “trivial.”     

Pitfalls of the Fourier Transform Method in Affine Models,and Remedies

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of A_n(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).     

Dissipativity and global attractors for a class of quasilinear parabolic systems

We show that a-priori weak L^p dissipativity implies strong L ^∞ dissipativity for a class of weakly coupled quasilinear parabolic systems satisfies general structure conditions. The existence of global attractors of general nonlinear reaction diffusion systems will be proven.     

An inversion theorem in Fermi surface theory

We prove a perturbative inversion theorem for the map between the interacting and the noninteracting Fermi surface for a class of many fermion systems with strictly convex Fermi surfaces and short‐range interactions between the fermions. This theorem gives a physical meaning to the counterterm function K that we use in the renormalization of these models: K can be identified as that part of the self‐energy that causes the deformation of the Fermi surface when the interaction is turned on. © 2000 Wiley & Sons, Inc.     

On the absence of eigenvalues of Maxwell and Lamé systems in exterior domains

The arguments showing non‐existence of eigensolutions to exterior‐boundary value problems associated with systems—such as the Maxwell and Lamé system—rely on showing that such solutions would have to have compact support and therefore—by a unique continuation property—cannot be non‐trivial. Here we will focus on the first part of the argument. For a class of second order elliptic systems it will be shown that L²‐solutions in exterior domains must have compact support. Both the asymptotically isotropic Maxwell system and the Lamé system with asymptotically decaying perturbations can be reduced to this class of elliptic systems. Copyright © 2004 John Wiley & Sons, Ltd.     

Necessary and Sufficient Conditions of Existence for Positive Solution to a Class of Quasilinear Elliptic Systems in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we establish the necessary and sufficient conditions of existence for a positive solution to a class of non-variational quasilinear elliptic systems in R ^N . The sufficient condition of existence result bases on the Mountain Pass Lemma and the sub-super solution methods, and the necessary condition is a consequence of a Picone’s identity. The system models some phenomena in different physical and other natural sciences: non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology and so on.     

Check Character Systems Using Quasigroups: II

In this part of the article we continue to research check character systems with one check character over quasigroups under check equations which have one permutation. These systems always detect all single errors (i.e. errors in only one component of a code word) and can detect some other errors arising during transmission of data. We study check character systems over T-quasigroups. These quasigroups are isotopic to abelian groups and generalize the well-known class of medial quasigroups. We establish some properties of a T-quasigroup so that the check character systems over it are able to detect transpositions, jump transpositions, twin errors and jump twin errors. We also give some models of T-quasigroups, which satisfy all of the required properties for detection of errors of each of the considered types. Communicated by: P. Wild     

Barwise's information frames and modal logics

 The paper studies Barwise's information frames and answers the John Barwise question: to find axiomatizations for the modal logics generated by information frames. We find axiomatic systems for (i) the modal logic of all complete information frames, (ii) the logic of all sound and complete information frames, (iii) the logic of all hereditary and complete information frames, (iv) the logic of all complete, sound and hereditary information frames, and (v) the logic of all consistent and complete information frames. The notion of weak modal logics is also proposed, and it is shown that the weak modal logics generated by all information frames and by all hereditary information frames are K and K4 respectively. To develop general theory, we prove that (i) any Kripke complete modal logic is the modal logic of a certain class of information frames and that (ii) the modal logic generated by any given class of complete, rarefied and fully classified information frames is Kripke complete. This paper is dedicated to the memory of talented mathematician John Barwise. Received: 7 May 2000 Published online: 10 October 2002 Key words or phrases: Knowledge presentation – Information – Information flow – Information frames – Modal logic-Kripke model     

Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

ANALYTIC COMPLETENESS THEOREM FOR ABSOLUTELY CONTINUOUS BIPROBABILITY MODELS

Hoover [2] proved a completeness theorem for the logic L(∫)??. The aim of this paper is to prove a similar completeness theorem with respect to product measurable biprobability models for a logic L(∫₁, ∫₂) with two integral operators. We prove: If T is a ∑₁ definable theory on ?? (a countable admissible set and ω ∈) and consistent with the axioms of L(∫₁, ∫₂), then there is an analytic absolutely continuous biprobability model in which every sentence in T is satified.