On metrics matched to the discrete memoryless channel

A sequence of metrics {D_N} is said to be additive and matched to a discrete memoryless channel (DMC) if D_N is the sum on its coordinates of N single letter metrics and if the maximum likelihood decoder for sequences of length N is a minimum D_N-distance decoder. Necessary and sufficient conditions on the transition probabilities of a DMC for the existence of a sequence of additive metrics matched to it are given. In the case of the binary channel these are shown to be equivalent to the channel being symmetric. Explicit transition probabilities are given for a large class of ternary DMCs with an associated sequence of additive matched metrics. The problem solved here may be considered a generalization of the problem of finding the DMCs matched to the Lee metric solved by Chiang and Wolf in 1971 (2).     

A control problem with structural choices

The case is considered in which, during the operation of an optimal control system, the optimizer, in addition to applying his usual control, may switch structures. Necessary and sufficient conditions are derived and emphasis is placed on the special characteristics of this problem. Continuous and discrete time set-ups are considered and the separation principle is shown not to hold for the linear quadratic case in the presence of noise.     

Water-clocks and time measurement in classical antiquity

The water-clock was the first mechanical device for time measurement to be used by the Greeks. Previously they had relied on sundials, or observation of the phases of the moon or the position of the sun in the zodiac to locate points of time within the day, month or year; and by the mid-fifth century BC they had made some progress with the difficult calendar problems due to the fact that the solar day, the lunar month, and the solar year are not commensurable with each other in whole numbers.     

The graph-theoretic field model—I. Modelling and formulations

The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure.This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.