Showing papers on "Metric (mathematics) published in 1974"
TL;DR: The algorithm, introduced here, lends itself to computer programming and provides a method to compute evolutionary distance which is shorter than the other methods currently in use.
Abstract: This paper gives a formal definition of the biological concept of evolutionary distance and an algorithm to compute it. For any set S of finite sequences of varying lengths this distance is a real-valued function on $S \times S$, and it is shown to be a metric under conditions which are wide enough to include the biological application. The algorithm, introduced here, lends itself to computer programming and provides a method to compute evolutionary distance which is shorter than the other methods currently in use.
523 citations
TL;DR: In this paper sufficient conditions for local and superlinear convergence to a Kuhn—Tucker point are established for a class of algorithms which may be broadly defined and comprise a quadratic programming algorithm for repeated solution of a subproblem and a variable metric update to develop the Hessian in the subproblem.
Abstract: In this paper sufficient conditions for local and superlinear convergence to a Kuhn—Tucker point are established for a class of algorithms which may be broadly defined and comprise a quadratic programming algorithm for repeated solution of a subproblem and a variable metric update to develop the Hessian in the subproblem. In particular the DFP update and an update attributed to Powell are shown to provide a superlinear convergent subclass of algorithms provided a start is made sufficiently close to the solution and the initial Hessian in the subproblem is sufficiently close to the Hessian of the Lagrangian at this point.
448 citations
TL;DR: In this article, it was shown that if a graph T contains no triangles, then there can be no four-point condition and it follows that there can also be no circuit in T. Necessity follows immediately, and to prove sufficiency, assume that the graph, T, contains a circuit, and choose points x, y, z, t in the circuit such that distances d(x, y), d(y, z), d (z, t), and d(t, x) are all either q or q + 1.
352 citations
TL;DR: A new criterion is introduced for comparing the convergence properties of variable metric algorithms, focusing on stepwise descent properties, and conditions are derived for these parameters that guarantee monotonic improvement in the single-step convergence rate.
Abstract: This part of the paper introduces some possible implementations of Self-Scaling Variable Metric algorithms based on the theory presented in Part I. These implementations are analyzed theoretically ...
189 citations
TL;DR: In this paper, a new criterion is introduced for comparing the convergence properties of variable metric algorithms, focusing on stepwise descent properties, which is a bound on the rate of decrease in the function value at each iterative step (single-step convergence rate).
Abstract: A new criterion is introduced for comparing the convergence properties of variable metric algorithms, focusing on stepwise descent properties. This criterion is a bound on the rate of decrease in the function value at each iterative step (single-step convergence rate). Using this criterion as a basis for algorithm development leads to the
158 citations
112 citations
TL;DR: In this article, it was shown that for every separable metric space X, there is a mapT:X →co satisfyingd(x, y)≦Tx−Ty≦Kd(k, k) for every x, y ∈ X.
Abstract: It is shown that there is a constantK so that, for every separable metric spaceX, there is a mapT:X →co satisfyingd(x, y)≦‖Tx−Ty‖≦Kd(x, y) for everyx, y ∈ X.
106 citations
TL;DR: This application demonstrates the importance of the metric triangle inequality property, results in an ancestral discrimination criterion and suggests that the chemical distance between proteins is linear with genetic coding distance.
Abstract: A precisely stated algorithm is given for reconstructing phylogenetic relationships from protein amino acid sequence data under the restriction that all distance measures be proper metrics. In conjunction with a general sequence metric, the algorithm is applied to the cytochrome c data used in earlier studies. This application demonstrates the importance of the metric triangle inequality property, results in an ancestral discrimination criterion and suggests that the chemical distance between proteins is linear with genetic coding distance.
89 citations
TL;DR: Numerical results obtained for the SSVM algorithm indicate that with proper parameter selection it is superior to the DFP algorithm, particularly for high-dimensional problems.
Abstract: This paper addresses the problem of selecting the parameter in a family of algorithms for unconstrained minimization known as Self Scaling Variable Metric (SSVM) Algorithms. This family, that has some very attractive properties, is based on a two parameter formula for updating the inverse Hessian approximation, in which the parameters can take any values between zero and one. Earlier results obtained for SSVM algorithms apply to the entire family and give no indication of how the choice of parameter may affect the algorithm's performance. In this paper, we examine empirically the effect of varying the parameters and relaxing the line-search. Theoretical consideration also leads to a switching tule for these parameters. Numerical results obtained for the SSVM algorithm indicate that with proper parameter selection it is superior to the DFP algorithm, particularly for high-dimensional problems.
52 citations
TL;DR: A method is described and tested that transforms similarity measures into distances that meet just three conditions that can determine the true underlying distances and the form of the unknown monotone function relating the similarity measures to those distances without assuming that the underlying space has any particular Euclidean, Minkowskian, or even dimensional strucutre.
TL;DR: In this paper, continuity concepts for metric projections are introduced which are simpler and more general than the usual upper and lower semicontinuity, and are strong enough to generalize a number of known results yet weak enough so that now the converses of many of these generalizations are also valid.
TL;DR: Methods which can utilize information on metric traits associated with a familial condition, for example, blood pressure in hypertension or blood glucose levels in diabetes are informative in estimating risks for these diseases.
Abstract: The threshold model of disease liability (Falconer, 1965) has proved a useful tool for summarizing and comparing familial frequencies for diseases and conditions not inherited in a simple Mendelian manner. The model also provides a basis for estimating recurrence risks for specific family histories, given the population frequency of the condition and the correlation in liability between relatives (Smith, 1971; Curnow, 1972; Mendell & Elston, 1974). These methods use information only on the disease status of family members. Often there is additional information on items or traits associated with the condition and this could be included so as to improve the estimate of the recurrence risk in a specific family. The object of this paper is to derive methods which can utilize information on metric traits associated with a familial condition. For example, blood pressure in hypertension or blood glucose levels in diabetes are informative in estimating risks for these diseases. The usefulness of such information in practice will be examined, studying the size of possible changes in the estimates of risk and the increases in accuracy of the risk estimates obtained. PRINCIPLES
TL;DR: In this paper, optimal solutions to a special type of constrained location problem are characterized, where the solution is constrained to be within a maximum distance of each demand point, and an algorithm for its solution is developed and discussed.
Abstract: In many location problems, the solution is constrained to lie within a closed set. In this paper, optimal solutions to a special type of constrained location problem are characterized. In particular, the location problem with the solution constrained to be within a maximum distance of each demand point is considered, and an algorithm for its solution is developed and discussed.
TL;DR: For arbitrary summation methods, the authors obtained inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions.
Abstract: For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities.
TL;DR: In this article, a new definition of diffusion is proposed that would provide a continuous metric for finite systems, acoustical or otherwise, and approaches the current definition for very large irregular systems.
Abstract: The conventional definition of diffusion of sound in resonating systems has important major deficiencies. It is oriented toward acoustical spaces and is difficult to apply to the vibrations of structural systems. Second, it is impossible to reach the state of perfect diffusion in any finite space, and any reasonable metric of diffusion predicts finite system diffusion as very far from the ideal. Third, the definition applies only to systems of irregular geometry. A new definition of diffusion is proposed that would provide a continuous metric for finite systems, acoustical or otherwise, and approaches the current definition for very large irregular systems.
TL;DR: In this paper, the Kruskal metric is obtained in a systematic way, with possibilities of generalisation, and spherically symmetric universes are defined and solutions of Einstein's field equations in vacuo are explored in terms of suitable coordinates.
Abstract: Spherically symmetric universes are defined, and spherically symmetric solutions of Einstein's field equations in vacuo are explored in terms of suitable coordinates. The Kruskal metric is thus obtained in a systematic way, with possibilities of generalisation.
TL;DR: In this article, it was shown that the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, and it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces.
Abstract: In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl . 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X .
TL;DR: A computer‐implemented technique is described which edits the velocity analysis data, thereby reducing the interpreter’s task, and makes efficient use of computer time.
Abstract: Extensive velocity analysis interpretation can impose a substantial man‐hour cost. A computer‐implemented technique is described which edits the velocity analysis data, thereby reducing the interpreter’s task. The technique uses graph theory to simulate the complex decision‐making inherent in the interpreter’s method of velocity analysis interpretation. The heart of the technique lies in defining a distance measure between candidate time‐velocity points from the velocity analysis. This metric is a function of the specific and potentially complex constraints imposed by the interpreter and of the weighted separation of the points. The graph theoretic techniques employed use the metric to join and, hence, select appropriate points from the candidate points while rejecting those which are invalid. Additional editing, based in part on implied interval velocities, further reduces the bulk of data presented to the interpreter. The method makes efficient use of computer time and has yielded encouraging results, a...
TL;DR: In this article, the structure of electromagnetic and gravitational fields near the singular ring of the Kerr metric is analyzed, and an approximate joint solution of the system of Maxwell-Einstein equations is given for a metric with the parameters of an electron.
Abstract: The structure of electromagnetic and gravitational fields near the singular ring of the Kerr metric is analyzed. An approximate joint solution of the system of Maxwell-Einstein equations is given for a metric with the parameters of an electron; this solution describes a microgeon, a model of an electron. A new interpretation of the two-sheet nature of the Kerr metric is given.
TL;DR: In this paper, the Minkowski-p metric was proposed to measure the similarity of the evaluation schemes of the other person's evaluation scheme to their own schemes, where the similarity is defined as the similarity between the schemes represented by the utility functions.
Abstract: It was proposed that people could evaluate the other person’s evaluation scheme in terms of the similarity (or dissimilarity) of the other person’s evaluation scheme to their own schemes. The metric for the dissimilarity of evaluation schemes was proposed to be the Minkowski-p metric for the utilities evaluated under different schemes. This model provides a basis for allocating importance to utility functions, where importance is to be interpreted as the similarity of the integrated scheme to the schemes represented by the utility functions. An experiment was done to investigate whether people’s intuitive judgments for the dissimilarity of evaluation schemes could be described by the Minkowski-p) metric; the results generally supported the model and also suggested that the metric would be the “city-block” metric, i.e., p-1 in the Minkowski-p metric.
01 Jan 1974
TL;DR: In this article, the results of three methods for multidimensional scaling (Torgerson's metric analysis, a non-metric method (TORSCA), and a method dealing with individual differences in multi-dimensional scaling (INDSCAL) were compared in experiments on rhythm experience and on perception of sound quality.
Abstract: .— The results of three methods for multidimensional scaling—Torgerson's metric analysis, a non-metric method (TORSCA), and a method dealing with individual differences in multidimensional scaling (INDSCAL)—were compared in experiments on rhythm experience and on perception of sound quality. The INDSCAL analysis seemed to be the most adequate method for treating the data in these experiments.
TL;DR: In this paper, a theorem of Lloyd is extended to the Lie metric case, which is a generalization of Lloyd's theorem for Lie matrices, and is used in this paper.
Abstract: A theorem of Lloyd is extended to the Lie metric case.
TL;DR: In this article, the authors used algebraic conditions on the continuity of the components of the metric tensor to get an approximate metric in four limiting forms relevant to a condensation in an expanding Einstein/de Sitter substratum.
Abstract: Algebraic conditions on the continuity of the components of the metric tensor are employed to get an approximate metric in four limiting forms relevant to a condensation in an expanding Einstein/de Sitter substratum. The metric of the condensation is in general spherically-symmetric, nonstatic and asymptotically flat, passing over into the usual Friedmann solution at large distances and late times. The line-element derived supersedes an earlier incorrect formulation of the problem by Einstein and Straus. The metric is applicable in particular to clusters of galaxies, wich cannot avoid being involved in the expansion of the Universe for the density-distributions relevant to average loose clusters as presently observed. It is likely that all clusters, including compact ones, are in a state of dynamical evolution, a conclusion which may remove the missing mass problem. The results found agree, in this respect, with recent work by Noerdlinger and Petrosian, and give effective Hubble parmeters for systems in an expanding substratum.