Showing papers in "Siam Journal on Applied Mathematics in 1968"
TL;DR: A Steiner minimal tree for given points in the plane is a tree which interconnects these points using lines of shortest possible total length as mentioned in this paper, where the length of the shortest possible line is chosen.
Abstract: A Steiner minimal tree for given points $A_1 , \cdots ,A_n $ in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum lengt...
946 citations
297 citations
257 citations
TL;DR: An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way and under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's results.
Abstract: : Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.
206 citations
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TL;DR: Matrix theorem giving necessary and sufficient conditions for leaving cone invariant, considering finite dimensional spaces as mentioned in this paper, considering finite-dimensional spaces, is used to define the conditions for cone invariance.
Abstract: Matrix theorem giving necessary and sufficient conditions for leaving cone invariant, considering finite dimensional spaces
172 citations
TL;DR: In this article, two methods for computing optimal decision sequences and their cost functions are presented for solving a broad class of shortest-route problems and a third solution technique is shown to apply to certain, but not all, of these Markov renewal programs.
Abstract: : Two methods are presented for computing optimal decision sequences and their cost functions. The first method, called 'policy iteration,' is an adaption of an iterative scheme that is widely used for sequential decision problems. The second method is to specify a linear programming problem whose solution determines an optimal policy and its cost function. A third solution technique is shown to apply to certain, but not all, of these Markov renewal programs. As a byproduct of the development, new techniques are provided for solving a broad class of shortest-route problems. (Author)
153 citations
TL;DR: In this paper, the authors used ray methods to calculate high frequency scattering by objects in an unbounded medium, but they have not been used much for waveguide problems, and they treated by ray methods reflection from an open-ended parallel plane waveguide propagating several modes.
Abstract: Ray methods have been widely used to calculate high frequency scattering by objects in an unbounded medium, but they have not been used much for waveguide problems. To show that they can be used for waveguides, we treat by ray methods reflection from an open-ended parallel plane waveguide propagating several modes. The incident mode is decomposed into two plane waves whose scattering by the edges at the termination produces the reflected field. The singly diffracted cylindrical wave originating at each. edge, which is known from the asymptotic theory of diffraction by a single wedge, is represented by means of diffracted rays. Then the stem of the fields on the multiply reflected rays is converted into modal form. This yields formulas for the reflection coefficients in the various modes due to single diffraction. In addition double and multiple diffraction are also taken into account, yielding improved formulas for the reflection coefficients. The results of extensive numerical calculations for $TE$ and $...
132 citations
TL;DR: Gaussian channels capacity, studying supremum of information transmission rates with small error probability, is studied in the context of knowledge of discrete-time information flows.
Abstract: Gaussian channels capacity, studying supremum of information transmission rates with small error probability
112 citations
TL;DR: In this article, it was shown that the sampling theorem is still true if f (t) is the Fourier transform of a distribution with support in the interval (-Q, Q2).
Abstract: This result is due originally to Whittaker [1]. Either the expansion (2) or the fact that a bandlimited function is detered by its sample values f (nwr/Q) is frequently called the sampling theorem. Kotelnikov, Shannon and later workers have exploited the sampling theorem and its extensions in developing communication theory. It is also possible to reverse the implication in proceeding from (1) to (2) and to use (2) as an interpolation series. That is, given a suitably restricted sequence of numbers {f(nar/2)}, one can construct a function f(t) by the series (2) which interpolates values between these numbers. The function so constructed is bandlimited; that is, it satisfies (1). It has been suggested by Hamming 12, p. 276] that this kind of bandlimited interpolation function is more useful than a polynomial for some purposes. It is natural to inquire whether the sampling theorem is still true if f (t) is the Fourier transform of a distribution with support in the interval (-Q, Q2). If we put g (co) = a (co a), where a is the Dirac distribution and where < a < Q, the Fourier transform of g (w) is f (t) = e'it. Substitution in (2) produces the equation
TL;DR: In this article, it was shown that the nucleolus of a constant-sum weighted majority game G is a system of weights for G; moreover, if G is homogeneous then the nuclelus of G supplies homogeneous weights.
Abstract: : It is shown that the nucleolus of a constant-sum weighted majority game G is a system of weights for G; moreover, if G is homogeneous then the nucleolus of G supplies homogeneous weights. Minimum and minimal integer weights are discussed with respect to their relations to the kernel and the nucleolus.
TL;DR: The most interesting results of this paper are those concerning Gaussian random fields which are both homogeneous and Markov, and the Markovian property.
Abstract: This paper considers some aspects of two-dimensional random fields with a view toward application in information processing problems involving two-dimensional data. In particular, we call attention to two possible properties which have important implications in terms of representations. They are (a) second order homogeneity with respect to some groups of transformation; (b) the Markovian property. The most interesting results of this paper are those concerning Gaussian random fields which are both homogeneous and Markov.
TL;DR: In this paper, the class of adjoint boundary value problems associated with ODEs is extended to include problems which may have discontinuities in the solution or some of its derivatives at a finite number of interior points.
Abstract: In this paper the class of adjoint and in particular self-adjoint boundary value problems associated with ordinary linear differential equations is extended to include problems which may have discontinuities in the solution or some of its derivatives at a finite number of interior points.
TL;DR: A theory of combinatorial information retrieval systems for file organization is developed that allows for efficient retrieval of records relevant to a query involving t or fewer attributes of binary-valued attributes.
Abstract: This paper develops a theory of combinatorial information retrieval systems for file organization. Geometric and algebraic methods are employed to construct some combinatorial configurations. These configurations are used for constructing combinatorial filing systems—for files with n binary-valued attributes. These systems use some redundancy in storage and allow for efficient retrieval of records relevant to a query involving t or fewer attributes $(t < n)$.
TL;DR: A new method for computing the equilibrium composition of a multicomponent single phase chemical system is described, based on the duality between geometric programming and the minimization of the Gibbs free energy function.
Abstract: Geometric programming algorithm method for computing chemical systems equilibrium composition reduces minimization variables
TL;DR: In this article, the concept of the pseudoinverse is widened so as to admit, for any given matrix, of a class of pseudoinverses from which that member may be uniquely selected which has the most convenient properties for a particular application.
Abstract: A singular or rectangular matrix does not have an inverse in the usual sense. Nevertheless a matrix having properties which are closely akin to those of an inverse may be defined for such matrices. This matrix, the pseudoinverse or generalized inverse, has hitherto been uniquely defined for any given matrix. In this paper the concept of the pseudoinverse is widened so as to admit, for any given matrix, of a class of pseudoinverses from which that member may be uniquely selected which has the most convenient properties for a particular application. The conventional pseudoinverse is included in the widened definition. Much of the analysis can be applied to bounded linear operators on Hilbert space.8
TL;DR: The statistics of placing vertices of a graph on a square array of points is developed for several classes of graphs and a lower bound is devised for the average value within each class of the lowest placement distance of a graphs.
Abstract: The statistics of placing vertices of a graph on a square array of points is developed for several classes of graphs Distribution functions for the lengths to be associated with edges are given A lower bound is devised for the average value within each class of the lowest placement distance of a graph For one class of graphs, this is compared with values devised from actual placements of a random set of representatives, and it can be seen that the lower bound devised here comes close to the actual value for the average minimum distanceA highly approximate formula is derived for one class of graphs (the simplest of the three classes considered) and is given by (19): \[ {\tilde{\mathcal{L}}}_{\min } = N\frac{{C^{1/2 - C /2N}}}{{e^{1 - C/2N}}} \] where C is the number of vertices in the graph and N is the number of edges, while ${\tilde{\mathcal{L}}}_{\min} $ is our lower bound on the average of the minimum distance of the placements of the graphsThe relationship to the quadratic assignment problem is s
TL;DR: Primitive root codes /error correcting codes/ consisting of cyclic codes and forming linear mapping of Reed-Solomon codes is proposed.
Abstract: Primitive root codes /error correcting codes/ consisting of cyclic codes and forming linear mapping of Reed-Solomon codes
TL;DR: In this paper, the authors considered the problem of radiative transfer in a semi-infinite atmosphere, bounded by a totally absorbing wall and illuminated by a beam of radiation of intensity $J_0 (α rt)$, and derived an analytic solution by expanding the kernel of the integral equation and the source function in Fourier cosine series.
Abstract: The transport equation of radiative transfer for the isotropic scattering of radiation in a homogeneous axially symmetric medium is discussed. The integral equation for the source function, which replaces the Milne equation in a one-dimensional medium, is derived from the governing integro-partial differential equation for the intensity of radiation.The problem of the transfer of radiation in a semi-infinite atmosphere, bounded by a totally absorbing wall and illuminated by a beam of radiation of intensity $J_0 ( \alpha rt)$, is discussed. The analytic solution is obtained by expanding the kernel of the integral equation and the source function in Fourier cosine series. The effect of the wall on the solution of the problem is determined by inverting the series solution by the application of contour integration.