scholarly journals On the mode of approach to zero of the coefficients of a Fourier series

1917 ◽  
Vol 93 (654) ◽  
pp. 455-467
Keyword(s):  
Fourier Series ◽  
General Theory ◽  
Practical Interest ◽  
Present Communication ◽  
Single Instance ◽  
Order Of Magnitude ◽  
Real Importance ◽  
Derived Series ◽  
Insight Into ◽  
De La Vallée Poussin

1. In the present communication I give a number of results on the mode of approach to zero of the coefficients of a Fourier series, to which I have already made allusion in my paper on “The Order of Magnitude of the Coefficients of a Fourier Series.” These results are not merely curious, they have a real importance, and give one an insight into the nature of these series, which cannot easily be gained without them. Indeed, while the earlier paper leads, as I showed in a subsequent communication, to the discovery of classes of derived series of Fourier series, which, although not themselves Fourier series, none the less converge, and are utilisable in a similar manner, and is therefore in a certain sense of practical interest, the present paper does something towards the elucidation of the general theory of the convergence of Fourier series themselves, as well as of their derived series. It will be sufficient to give a single instance. I have recently shown that the well-known test of Dirichlet for the convergence of a Fourier series admits of a remarkable generalisation. It follows from Theorem. 1, given below (7), that the convergence secured by that test and by its generalisation alike possess what may be called greater strength than the rival tests of Dini and de la Vallée Poussin.

scholarly journals On Fourier series and functions of bounded variation

1913 ◽  
Vol 88 (606) ◽  
pp. 561-568 ◽  
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
General Theorem ◽  
General Term ◽  
The Other ◽  
Functions Of Bounded Variation ◽  
Trivial Case ◽  
Present Communication ◽  
Other Hand ◽  
Derived Series

1. In a previous communication to the Society I have pointed out that the succession of constants obtained by multiplying together two successions of Fourier constants in the manner which naturally suggests itself is a succession of Fourier constants, and I have discussed the summability of the function with new constants are associated. We may express the matter in another way by saying that I have shown that the use of the Fourier constants of an even function g(x) as convergence factors in the Fourier series of a function f(x) changes the latter series into a series which is associated with the new series is increased. The use of the Fourier constants of an odd function as convergence factors, on the other hand, has the effect of changing the allied series of the Fourier series of f (x) into a Fourier series, even when the allied series is not itself a Fourier series. It at once suggests itself that the former of the two statements in this form of the result is not the most that can be said. Indeed, the series, whose general term is cos nx , and whose coefficients are accordingly unity, may clearly take the place of the Fourier series of g(x) , although it is not a Fourier series. On the other hand, it is the derived series of the Fourier serious of a function of bounded variation, which is, moreover, odd. We are thus led to ask ourselves whether this is not the trivial case of a general theorem. In the present communication I propose to show, among other things, that the answer to this questions is in affirmative. The following theorems are, in fact, true:—

scholarly journals On the order of magnitude of the coefficients of a fourier series

1917 ◽  
Vol 93 (647) ◽  
pp. 42-55 ◽  
Keyword(s):  
Fourier Series ◽  
Continuous Function ◽  
Present Communication ◽  
General Sense ◽  
Cosine Series ◽  
Order Of Magnitude ◽  
Bounded Functions ◽  
Lebesgue Integration ◽  
Made In

§ 1. Riemann’s theorem that the coefficients of a Fourier series converge to zero was shown by Lebesgue to still hold when integration is understood to be in the general sense now employed, absolutely convergent, or Lebesgue integration. Little progress has, however, been made in the determination of the order of magnitude of the coefficients. It has, indeed, been proved that, when the function has bounded variation, na n and nb n are bounded functions of n , and that, when the function is a continuous function of such a type as satisfies a condition of Lipschitz, n q a n and n q b n converge to zero, where q is a positive number not greater than unity, depending on the particular Lipschitz condition satisfied by the function. As regards the second of these results, involving the satisfying of a condition of Lipschitz, it is to be remarked that, in well-known series of the type Σ n -q cos nx and Σ n -q sin nx , the functions of which they are the Fourier series do not, in any interval containing the origin, satisfy any condition of Lipschitz, being, indeed, unbounded. In the present communication I obtain a number of theorems corresponding to each of these two results, including them as particular cases, and, at the same time, leading to the known properties of the simple sine and cosine series above referred to.

Estimation of the Time Delay Associated With Damping Controlled Fluidelastic Instability in a Normal Triangular Tube Array

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
John Mahon ◽  
Craig Meskell
Keyword(s):  
Time Delay ◽  
Cross Flow ◽  
Fluid Forces ◽  
Fluidelastic Instability ◽  
Order Of Magnitude ◽  
Sample Frequency ◽  
Pitch Ratio ◽  
Parameterized Model ◽  
Semi Empirical ◽  
Insight Into

Fluidelastic instability (FEI) produces large amplitude self-excited vibrations close to the natural frequency of the structure. For fluidelastic instability caused by the damping controlled mechanism, there is a time delay between tube motion and the resulting fluid forces but magnitude and physical cause of this is unclear. This study measures the time delay between tube motion and the resulting fluid forces in a normal triangular tube array with a pitch ratio of 1.32 subject to air cross-flow. The instrumented cylinder was forced to oscillate in the lift direction at three excitation frequencies for a range of flow velocities. Unsteady surface pressures were monitored with a sample frequency of 2 kHz at the mid plane of the instrumented cylinder. The instantaneous fluid forces were obtained by integrating the surface pressure data. A time delay between the tube motion and resulting fluid forces was obtained. The nondimensionalized time delay was of the same order of magnitude assumed in the semi-empirical quasi-steady model (i.e., τ2 = 0.29 d/U). Although, further work is required to provide a parameterized model of the time delay which can be embedded in a model of damping controlled fluidelastic forces, the data already provides some insight into the physical mechanism responsible.

scholarly journals Origin of the Low Magnetic Moment in Fe2AlTi: An Ab Initio Study

Materials ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 1732 ◽  
Author(s):  
Martin Friák ◽  
Anton Slávik ◽  
Ivana Miháliková ◽  
David Holec ◽  
Monika Všianská ◽  
...  
Keyword(s):  
Ab Initio ◽  
Magnetic Moment ◽  
Energy Surface ◽  
Magnetic Moments ◽  
Local Magnetic Moment ◽  
Spin Configuration ◽  
Spin Effects ◽  
Order Of Magnitude ◽  
Huge Impact ◽  
Insight Into

The intermetallic compound Fe 2 AlTi (alternatively Fe 2 TiAl) is an important phase in the ternary Fe-Al-Ti phase diagram. Previous theoretical studies showed a large discrepancy of approximately an order of magnitude between the ab initio computed magnetic moments and the experimentally measured ones. To unravel the source of this discrepancy, we analyze how various mechanisms present in realistic materials such as residual strain effects or deviations from stoichiometry affect magnetism. Since in spin-unconstrained calculations the system always evolves to the spin configuration which represents a local or global minimum in the total energy surface, finite temperature spin effects are not well described. We therefore turn the investigation around and use constrained spin calculations, fixing the global magnetic moment. This approach provides direct insight into local and global energy minima (reflecting metastable and stable spin phases) as well as the curvature of the energy surface, which correlates with the magnetic entropy and thus the magnetic configuration space accessible at finite temperatures. Based on this approach, we show that deviations from stoichiometry have a huge impact on the local magnetic moment and can explain the experimentally observed low magnetic moments.

scholarly journals On the series of Legendre

1918 ◽  
Vol 94 (660) ◽  
pp. 292-295
Keyword(s):  
Fourier Series ◽  
General Type ◽  
General Term ◽  
Present Communication ◽  
Legendre Series ◽  
Normal Functions ◽  
Sturm Liouville ◽  
To Come ◽  
In Virtue Of

In recent communications to the Society, I have confined myself largely to the Theory of Fourier Series, partly because much seemed to me still to require doing in this subject, partly because I believed its thorough investigation to be the natural preparation for the study of other series of normal functions. It has, indeed, been known for some time that the behaviour of, for instance, series of Sturm-Liouville functions exactly corresponds to that of Fourier series. The introduction that I have recently made into Analysis of what I have called restricted Fourier series enables us to notably extend the range of such analogies. I propose in the present communication to illustrate this remark with reference to series of Legendre coefficients. Whereas Fourier series may be said to be “naturally unrestricted,” in virtue of the fact that the convergence of the integrated series to an integral necessarily involves the tendency towards zero of its own general term, so that the consideration of the more general type of series does not at once suggest itself, Legendre series may be said to come into being “restricted,” even when the coefficients are expressible in what may be called the Fourier form by means of integrals involving Legendre’s coefficients. In other words, such series correspond precisely to restricted Fourier series, instead of to ordinary Fourier series like the analogous series of Sturm-Liouville functions.

A MODEL HIERARCHY FOR IONOSPHERIC PLASMA MODELING

2004 ◽  
Vol 14 (03) ◽  
pp. 393-415 ◽  
Author(s):  
CHRISTOPHE BESSE ◽  
PIERRE DEGOND ◽  
FABRICE DELUZET ◽  
JEAN CLAUDEL ◽  
GÉRARD GALLICE ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Plasma Density ◽  
Maxwell Equations ◽  
Ionospheric Plasma ◽  
Practical Interest ◽  
Plasma Modeling ◽  
Model Hierarchy ◽  
Great Practical Interest ◽  
Two Fluid ◽  
Insight Into

This paper deals with the modeling of the ionospheric plasma. Starting from the two-fluid Euler–Maxwell equations, we present two hierarchies of models. The MHD hierarchy deals with large plasma density situations while the dynamo hierarchy is adapted to lower density situations. Most of the models encompassed by the dynamo hierarchy are classical ones, but we shall give a unified presentation of them which brings a new insight into their interrelations. By contrast, the MHD hierarchy involves a new (at least to the authors) model, the massless-MHD model. This is a diffusion system for the density and magnetic field which could be of great practical interest. Both hierarchies terminate with the "classical" Striation model, which we shall investigate in detail.

History and International Relations

Mind-Society ◽  
2019 ◽  
pp. 228-256
Author(s):  
Paul Thagard
Keyword(s):  
World War I ◽  
General Theory ◽  
International Relations ◽  
Social Cognitive ◽  
Social Changes ◽  
Social Mechanisms ◽  
World War ◽  
Historical Explanation ◽  
Historical Developments ◽  
Insight Into

Historical explanation and the understanding of international relations can be enhanced by applying detailed psychological, neural, and social mechanisms to real-world events. By applying the method of social cognitive-emotional workups to the origins of World War I, this chapter shows the relevance of an integrated account of beliefs, concepts, values, rules, analogies, metaphors, emotions, inferences, and communication. The result transcends the limitations of purely narrative explanations in history and provides insight into why the field of international relations has lacked a satisfactory general theory. Explaining social changes in both groups and individuals requires understanding the communicative interactions of cognitive-emotional minds; the result is mechanistic-narrative explanation. Dealing with complex historical developments such as the outbreak of wars requires solution of the person–group problem.

The Fréchet Variation, Sector Limits, and Left Decompositions

1950 ◽  
Vol 2 ◽  
pp. 344-374 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Fourier Series ◽  
Variational Analysis ◽  
Cartesian Product ◽  
Multiple Fourier Series ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Variation Of A Function ◽  
De La Vallée Poussin

1. Introduction. The Fréchet variation of a function g defined over a 2-interval I2 was introduced by Fréchet to enable him to generalize Riesz's theorem on the representation of functionals linear over the space C [7]. Recently the authors have found this variation fundamental in the study of functionals bilinear over the Cartesian product A ⨯ B of two normed linear spaces with certain characteristic properties, and in the further use of this theory in spectral and variational analysis. The recent discovery by the authors of several new properties of the Fréchet variation has made it possible to to give new and natural tests for the convergence of multiple Fourier series generalizing the classical Jordan, de la Vallée Poussin, Dini, Young and Lebesgue tests under considerably less restrictive hypotheses than those now accepted.

The Place of Language in a Scientific Psychology

1990 ◽  
Vol 1 (1) ◽  
pp. 7-14 ◽  
Author(s):  
George A. Miller
Keyword(s):  
General Theory ◽  
Scientific Psychology ◽  
Crucial Problem ◽  
Insight Into ◽  
Methodological Difficulties

One of the psychologists’ great methodological difficulties is how they can make the events they wish to study publicly observable, countable, measurable. It is significant to note that the device most often used for conversion from private to public is language. Thus speech is a crucial problem for psychology. None of their other activities gives the same sort of insight into another person as does their language. Since people spend so many of their waking hours generating and responding to words, and since speech is such a typically human mode of adjustment, no general theory of psychology will be adequate if it does not take account of language.

scholarly journals Modeling the dynamics of recruitment and derecruitment in mice with acute lung injury

2008 ◽  
Vol 105 (6) ◽  
pp. 1813-1821 ◽  
Author(s):  
Christopher B. Massa ◽  
Gilman B. Allen ◽  
Jason H. T. Bates
Keyword(s):  
Acute Lung Injury ◽  
Lung Injury ◽  
Lung Recruitment ◽  
Mechanically Ventilated ◽  
Critical Opening ◽  
Injured Lung ◽  
Order Of Magnitude ◽  
Opening And Closing ◽  
Air Liquid Interface ◽  
Insight Into

Lung recruitment and derecruitment contribute significantly to variations in the elastance of the respiratory system during mechanical ventilation. However, the decreases in elastance that occur with deep inflation are transient, especially in acute lung injury. Bates and Irvin ( 8 ) proposed a model of the lung that recreates time-varying changes in elastance as a result of progressive recruitment and derecruitment of lung units. The model is characterized by distributions of critical opening and closing pressures throughout the lung and by distributions of speeds with which the processes of opening and closing take place once the critical pressures have been achieved. In the present study, we adapted this model to represent a mechanically ventilated mouse. We fit the model to data collected in a previous study from control mice and mice in various stages of acid-induced acute lung injury ( 3 ). Excellent fits to the data were obtained when the normally distributed critical opening pressures were about 5 cmH2O above the closing pressures and when the hyperbolically distributed opening velocities were about an order of magnitude greater than the closing velocities. We also found that, compared with controls, the injured mice had markedly increased opening and closing pressures but no change in the velocities, suggesting that the key biophysical change wrought by acid injury is dysfunction of surface tension at the air-liquid interface. Our computational model of lung recruitment and derecruitment dynamics is thus capable of accurately mimicking data from mice with acute lung injury and may provide insight into the altered biophysics of the injured lung.

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