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# Direct proof

About: Direct proof is a research topic. Over the lifetime, 1520 publications have been published within this topic receiving 25515 citations.

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TL;DR: A short and direct proof of this recursion relation for tree-level scattering amplitudes based on properties of tree- level amplitudes only is given.

Abstract: Recently, by using the known structure of one-loop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering amplitudes has been deduced. Here, we give a short and direct proof of this recursion relation based on properties of tree-level amplitudes only.

1,605 citations

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TL;DR: A certain analogy is found to exist between a special case of Fisher's quantity of information I and the inverse of the “entropy power” of Shannon and this constitutes a sharpening of the uncertainty relation of quantum mechanics for canonically conjugated variables.

Abstract: A certain analogy is found to exist between a special case of Fisher's quantity of information I and the inverse of the “entropy power” of Shannon (1949, p. 60). This can be inferred from two facts: (1) Both quantities satisfy inequalities that bear a certain resemblance to each other. (2) There is an inequality connecting the two quantities. This last result constitutes a sharpening of the uncertainty relation of quantum mechanics for canonically conjugated variables. Two of these relations are used to give a direct proof of an inequality of Shannon (1949, p. 63, Theorem 15). Proofs are not elaborated fully. Details will be given in a doctoral thesis that is in preparation.

792 citations

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TL;DR: From a direct proof of the universal approximation capabilities of perceptron type networks with two hidden layers, estimates of numbers of hidden units are derived based on properties of the function being approximation and the accuracy of its approximation.

685 citations

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TL;DR: Rabin has proved that two-way finite automata, which are allowed to move in both directions along their input tape, are equivalent to one-way automata as far as the classification of input tapes is concerned.

Abstract: Rabin has proved 1,2 that two-way finite automata, which are allowed to move in both directions along their input tape, are equivalent to one-way automata as far as the classification of input tapes is concerned. Rabin's proof is rather complicated and consists in giving a method for the successive elimination of loops in the motion of the machine. The purposeo f this note is to give a short, direct proof of the result.

308 citations

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TL;DR: In this paper, the authors present formulas for crossed and nested classifications, based on a model of sufficient generality and flexibility that the necessary assumptions concern only the selection of the levels of the factors and not the behavior of what is being experimented upon.

Abstract: 1. Summary. The assumptions appropriate to the application of analysis of variance to specific examples, and the effects of these assumptions on the resulting interpretations, are today a matter of very active discussion. Formulas for average values of mean squares play a central role in this problem, as do assumptionis about interactions. This paper presents formulas for crossed (and, incidentally, for nested and for non-interacting completely randomized) classifications, based on a model of sufficient generality and flexibility that the necessary assumptions concern only the selection of the levels of the factors and not the behavior of what is being experimented upon. (This means, in particular, that the average response is an arbitrary function of the factors.) These formulas are not very complex, and specialize to the classical results for crossed and nested classifications, when appropriate restrictions are made. Complete randomization is only discussed for the elementary case of "no interactions with experimental units" and randomized blocks are not discussed. In discussion and proof, we give most space to the two-way classification with replication, basing our direct proof more closely on the proof independently obtained by Cornfield [17], than on the earlier proof by Tukey [201. We also treat the three-way classification in detail. Results for the general factorial are also stated and proved. The relation of this paper to other recent work, published and unpublished, is discussed in Section 4 (average values of mean squares) and in Section 11 (various types of linear models). INITIAL DISCUSSION 2. Introduction. During the last years of the last decade it was relatively easy to believe that the analysis of variance was well understood. Eisenhart's summary article of 1947 [5], when combined with the work of Pitman [13] and Welch [15] on the randomization approach (work published in 1937-1938, which ever since has been far too much neglected), seemed to provide a simple, easily understandable account of the foundations. But as the years have passed, both statisticians and users of analysis of variance have gradually become aware of a number of areas in which we needed to deepen our understanding. One of these is the relation of formulas for average values of mean squares to assumptions. These are of central importance, since the choice of an "error term" as a basis for either

283 citations