Why are there no infinite left-sided decimal expansions?

2021 ◽  
Vol 105 (562) ◽  
pp. 78-86
Author(s):  
A. C. Paseau
Keyword(s):  
Reverse Process

It was my nine-year-old daughter who got me interested in the title question. As she appreciates, multiplying an integer by a power of 10 is a cinch. To multiply 34 by 100, simply add two zeros at the end: 34 × 100 = 3400. Dividing 3400 by 100 is the reverse process: remove two zeros to obtain 34. More generally, to multiply an integer by 10N, for non-negative N, add N zeros to the end of its decimal notation, and to divide an integer by 10N remove N zeros from its end — so long as it has them. Easy-peasy; my daughter knows all that.

Die kinetische Energie ionisierter MolekülfragmenteIII. Quasithermische Ionen von einfachen Paraffinen

1964 ◽  
Vol 19 (7-8) ◽  
pp. 911-925
Author(s):  
Rolf Taubert
Keyword(s):  
Activation Energy ◽  
Statistical Theory ◽  
Mass Spectra ◽  
Transition Energy ◽  
Bond Cleavage ◽  
Theoretical Data ◽  
Initial Energy ◽  
Transition Energies ◽  
Hydrogen Shift ◽  
Reverse Process

From the monotonic increase of the average initial energy of paraffin fragment ions formed by electron impact the existence of unimolecular dissociation sequences is concluded as it is assumed by the statistical theory of mass spectra. For primary decomposition steps the total kinetic energies set free during the respective dissociation processes (transition energies) have been deduced from measured initial energies. The transition energies obtained in this way may be compared with theoretical data for the translational energies in the transition states (statistical energies) as calculated by means of the statistical theory of mass spectra. In case of—C bond cleavage theoretical data are close to the experimental values (≈ 0.1 ev). In case of a C—H bond cleavage, however, theoretical values are always lower than the experimental ones.In rearrangement reactions an activation energy for the reverse process may exist, which should show up, at least partly, in the transition energy. For a primary H2-abstraction process the activation energy for the reverse process aE can be calculated from thermochemical data. The observed transition energies are always lower than the calculated values. A C—C skeleton rearrangement also shows some influence of aΕ on the transition energy. The absolute effect, however, is small—less than 0.1 ev. For hydrogen shift reactions no indication of an aE-contribution was found.

scholarly journals Electromagnetic Detectors of Gravitational Waves

1974 ◽  
Vol 64 ◽  
pp. 54-58
Author(s):  
V. B. Braginsky ◽  
L. P. Grishchuk ◽  
A. G. Doroshkievich ◽  
Ya. B. Zel'Dovich ◽  
I. D. Novikov ◽  
...  
Keyword(s):  
Gravitational Waves ◽  
Electromagnetic Waves ◽  
Reverse Process

Our group is investigating highfrequency gravitational waves (GW). The most promising approach to detection and laboratory generation of such GW seems to be through the transformation of GW into electromagnetic waves (EMW), and the reverse process: EMW→GW. The effects are small of course.

scholarly journals A detailed study of the "radioactive decay" of, and the penetration of α-particles into, a simplified one-dimensional nucleus

1929 ◽  
Vol 124 (795) ◽  
pp. 493-501 ◽  
Keyword(s):  
Simple Model ◽  
Common Knowledge ◽  
Radioactive Decay ◽  
Particle Emission ◽  
Decay Process ◽  
Characteristic Energy ◽  
One Dimensional ◽  
Reverse Process ◽  
The Law ◽  
Α Particles

1. Introduction .—Gamow's elegant deduction by general arguments of the law of radioactive decay by α-particle emission and his subsequent investigations on artificial disintegration suggested to us the desirability of investigating as closely as possible any simple model of a decaying nucleus as a verification of his general approximations. For the model chosen the exact investigation of the decay process is almost trivial. Since we obtained this, now some time ago, Dr. Gamow informed us that he had also obtained equivalent detailed results. Still more recently such results have been published by Kudar. We shall not therefore dwell upon them here. The application of the same ideals, however, to the reverse process of penetration presents points of very definite interest, which we think are well worth discussion. The main point that arises is that the chance of penetration α-particle is or is not equal to a characteristic energy of the nucleus itself. This is a point which is not dealt with by Gamow in his paper. We have discussed it with him, and now put forward the results we have obtained. Since the solution of the decay problem is required in the main discussion of the penetration of α-particles into the nucleus it is included here in 2 for reference. We must emphasise that we claim no novelty, except of detail, for the work of 2; the general lines by now are a matter of fairly common knowledge.

Hellenistic Eponymous Cities and Ethnics

2009 ◽  
Author(s):  
P. M. Fraser
Keyword(s):  
New World ◽  
Documentary Evidence ◽  
Classical World ◽  
Reverse Process ◽  
Urban Settlements ◽  
History Of ◽  
Peripheral Regions

Chapter 6 showed the long history of metonomasy, which is preserved in a number of entries in documentary evidence and particularly in Stephanus, relating to cities and communities of the Classical world. It also investigated the reverse process, by which ethnics of cities that had for one reason or another ceased to exist as independent bodies continued to be used, particularly (but not exclusively) in peripheral regions such as Egypt. This chapter looks forward to the new world, particularly the early Hellenistic age, which brought into being new urban settlements, with politically eponymous titles.

Dance Musicalization and the Choreomusicking Body

Dancing Women ◽  
2020 ◽  
pp. 27-58
Author(s):  
Usha Iyer
Keyword(s):  
Musical Composition ◽  
Reverse Process ◽  
New Models ◽  
Hindi Cinema ◽  
The Many ◽  
Historical Accounts ◽  
Hindi Film

Chapter 1 presents a dance-centered taxonomy of musical numbers, which clarifies how dance promotes agency and authorship. Reconsidering the term “song picturization,” which suggests the primacy of the song as setting the agenda for the visuals, this chapter proposes that in the case of certain dance numbers or famed dancer-actors, a reverse process of “dance musicalization” is at work, in which a desired dance vocabulary precedes and influences the conceptualization of the song. This disruption of given logics of production and authorship spurs the conceptualization of a multi-bodied “choreomusicking body,” which directs our attention to the many on- and off-screen bodies laboring to produce the song-and-dance number, and fundamentally shifts ideological readings of narrative and spectacle in popular Hindi cinema. Employing choreomusicological theory, historical accounts of dancer-actors’ influence on musical composition, and spectatorial responses to the music-dance composite, this chapter proposes new models for theorizing the Hindi film song-and-dance sequence.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (03) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn (w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn , which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN –1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ 2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (3) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

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