On Tchebycheff Quadrature

1965 ◽  
Vol 17 ◽  
pp. 652-658 ◽  
Author(s):  
Paul Erdös ◽  
A. Sharma
Keyword(s):  
Algebraic Polynomials ◽  
History Of ◽  
Image Position

Tchebycheff proposed the problem of finding n + 1 constants A, x1, x2, . . , xn ( — 1 ≤ x1 < x2 < . . . < xn ≤ +1) such that the formula(1)is exact for all algebraic polynomials of degree ≤n. In this case it is clear that A = 2/n. Later S. Bernstein (1) proved that for n ≥ 10 not all the xi's can be real. For a history of the problem and for more references see Natanson (4).

scholarly journals Errata

1962 ◽  
Vol 3 (1) ◽  
pp. 139-139
Keyword(s):  
East Asia ◽  
South East Asia ◽  
History Of ◽  
New Type ◽  
Image Position ◽  
Printing Machine

Readers of this Journal will recall the provocative article in Vol. 2, No. 2 by John Smail entitled An Autonomous History of South-East Asia. This article has aroused considerable comment. It is all-the-more unfortunate then that it was marred by fifty or more misprints and omissions. With this issue of the Journal we have changed to a new type and printing machine, and we hope such errors as committed before will remain merely the follies of our youth. We attach a list of the more important of the misprints in Mr. Smail's article.

1. On the Anatomy of a new Species of Polyodon, the Polyodon Gladius of Martens, taken from the river Yang-tsze-Kiang, 450 miles above Woosung. Part I., being its External Characters and Structure

1875 ◽  
Vol 8 ◽  
pp. 50-51 ◽  
Author(s):  
P. D. Handyside
Keyword(s):  
New Species ◽  
History Of ◽  
Image Position ◽  
A New Species

The position of this new species of Ganoid, under our commonly accepted classification, the author gave as follows:—After referring to the Polyodon folium of Laépède (the P. reticulata of Shaw, the Planirostra spatula of Owen), the paddle-fish or spoon-bill sturgeon of the Ohio and Mississippi and their tributaries, as a well-known species of the genus in question, Dr Handyside went on to state that the new species now to be described was first observed on a Chinese fishmonger's stall at Woosung, 12 miles from Shanghai, and had since been found in the Yang-tsze-Kiang, and, as was alleged, in the northern Japanese sea. He then sketched the history of the Polyodontidoæ family, and narrated the researches of Lacépède, Von Martens, Blakiston, Kaup, and Duméril.

The early history of the Greek alphabet: new evidence from Eretria and Methone

Antiquity ◽  
2016 ◽  
Vol 90 (353) ◽  
pp. 1238-1254 ◽  
Author(s):  
John K. Papadopoulos
Keyword(s):  
Early History ◽  
History Of ◽  
Greek Alphabet ◽  
Image Position ◽  
New Evidence

Abstract

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Famous Conjecture ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
History Of ◽  
Indefinite Quadratic ◽  
Image Position

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

X.—The Theory of Compound Determinants in the Historical Order of its Development up to 1860.

1908 ◽  
Vol 28 ◽  
pp. 197-209
Author(s):  
Thomas Muir
Keyword(s):  
Special Form ◽  
Early Stage ◽  
Lagrange’S Equation ◽  
Cauchy's Theorem ◽  
History Of ◽  
The Subject ◽  
Image Position

Determinants whose elements are themselves determinants made their appearance at a very early stage in the history of the subject, the first foreshadowing of them being contained in Lagrange's “équation identique et très remarquable” of 1773, namely,whereThis, viewed as a result in determinants, is a case of Cauchy's theorem of 1812 regardingthe adjugate, and the adjugate of course is an instance of the special form to which we have now come. Jacobi's theorem regarding any minor of the adjugate has a like history and may be similarly classified. Passing from the case of the adjugate, where each element is a primary minor of the original determinant, Cauchy also considered the determinants of other “systèmes dérivés,” that is to say, the determinants whose elements are the secondary, ternary, … minors of the original, and gave the theorem that the product of the determinants of two “complementary derived systems” is a power of the original determinant, the index of the power being where n is the order of the original determinant and p the order of each element of one of the “derived systems.”

Uniform approximate solutions of certain nth-order differential equations. I

1963 ◽  
Vol 59 (1) ◽  
pp. 95-110 ◽  
Author(s):  
J. Heading
Keyword(s):  
Differential Equations ◽  
Approximate Solutions ◽  
Second Order ◽  
Large Parameter ◽  
Integral Methods ◽  
History Of ◽  
Transcendental Functions ◽  
Linear Differential ◽  
Image Position ◽  
Complete History

Exact analytical solutions of certain second-order linear differential equations are often employed as approximate solutions of other second-order differential equations when the solutions of this latter equation cannot be expressed in terms of the standard transcendental functions. The classical exposition of this method has been given by Jeffreys (6); approximate solutions of the equation (using Jeffreys's notation)are given in terms of solutions either of the equationor of the equationwhere h is a large parameter. A complete history of this technique is given in the author's recent text An introduction to phase-integral methods (Heading (5)).

The last voyage of HMS Investigator, 1850–53, and the discovery of the North West Passage

Polar Record ◽  
1967 ◽  
Vol 13 (87) ◽  
pp. 753-768 ◽  
Author(s):  
J. H. Nelson
Keyword(s):  
Literary Style ◽  
North West ◽  
The North ◽  
Banks Island ◽  
History Of ◽  
Image Position

[The list of officers and crew of HMS Investigator, 1850–53, contains the name “James Nelson, AB”. Recently J. W. Nelson, a descendant of Nelson's, brought to the Institute two copy-books containing accounts by him of the last voyage of Investigator, the discovery of the North West Passage and the return of the surviving members of the ship's company to England after the abandonment of the ship in the Bay of Mercy, Banks Island, in June 1853. Both accounts were obviously written carefully after the author's return, presumably from diary notes. The handwriting is neat copperplate with alterations in another hand. Little is known of the origins or later history of James Nelson, whose command of contemporary literary style and facility of expression must have been unusual in an AB of the period. His naval Certificate of Service is as follows:

Hot plasma theory of whistler mode wave packet propagation along a non-uniform magnetic field

1969 ◽  
Vol 3 (4) ◽  
pp. 611-631 ◽  
Author(s):  
M. J. Houghton
Keyword(s):  
Magnetic Field ◽  
Wave Packet ◽  
Uniform Magnetic Field ◽  
Particle Distribution ◽  
Whistler Mode ◽  
Mode Wave ◽  
Whistler Mode Wave ◽  
History Of ◽  
Smallness Parameter ◽  
Image Position

We discuss the propagation of wave packets of the formin an infinite uniform plasma, where G(z, t) is a slowly varying function of space z and time t. One can very simply derive the equation of change of G(z, t) for the stable or unstable case. The terms in the equation are of physical interest and clearly define the limitations of linear theory. We then investigate the problem of whistler mode wave propagation in a collisionless Vlasov plasma in a given non-uniform magnetic field. We choose the electric field to be of a W.K.B. form and the particle distribution to be isotropic. We can express the perturbation in the particle distribution in terms of an integration along the zero-order par tide orbits (an integration overtime). These orbits can be found correct to a term linear in a smallness parameter ε (when ε equals zero we arrive back at a uniform magnetic field). The charge and current density due to the perturbation are related through Maxwell's equations to the electric and magnetic field of the wave in the usual self consistent Boltzmann—Vlasov description. We show that the contribution to the current arises from recent events in the history of a given particle because of the finite temperature of the plasma. This result leads to an expansion of slowly varying parameters which in turn gives rise to the equation governing the motion of the wave packet.

scholarly journals A Theory of the Consolidation of Snow

1966 ◽  
Vol 6 (43) ◽  
pp. 145-157 ◽  
Author(s):  
E. D. Feldt ◽  
G. E. H. Ballard
Keyword(s):  
Structural Parameters ◽  
Uniaxial Stress ◽  
Consolidation Rate ◽  
History Of ◽  
Predicted Values ◽  
Image Position ◽  
Coefficient Of Viscosity ◽  
Diagenetic History ◽  
Existing Data ◽  
Porosity Range

AbstractA consolidation theory is developed for an age-hardened snow under uniaxial stress in the porosity range of 35 to 55 per cent by considering one mechanism, viz. viscous flow of interparticle bonds. For a uniaxial stress σ the differential equation for porosity n in terms of time t is shown to be where a and ν are structural parameters and η is the coefficient of viscosity of ice. Comparison of this equation and the integrated form with existing data predicts consistent and reasonable values for a. The predicted values of η/ν range from 10−2 to 102 times the published values for, η which may indicate that ν, and hence the consolidation rate, is greatly affected by the diagenetic history of the snow and the conditions of experimentation.

Money, Credit and Bank Behaviour: Need for a New Approach

2010 ◽  
Vol 214 ◽  
pp. F73-F82 ◽  
Author(s):  
C.A.E. Goodhart
Keyword(s):  
Bank Lending ◽  
Developed Countries ◽  
Analytical Framework ◽  
The United States ◽  
Monetary Base ◽  
Money Multiplier ◽  
Core Content ◽  
History Of ◽  
Image Position

The standard approach, in teaching and textbooks, to explaining the determination of both the supply of money, and the provision of bank credit to the private sector, has been the money multiplier approach, whereby the Central Bank sets the high-powered monetary base, and then the stock of money is a multiple of that. The greatest book on Monetary History ever written, Friedman and Schwartz (1963), Monetary History of the United States, was constructed around this same analytical framework of the money multiplier, whereby M, the money supply, would increase by a large multiple of the change in the high-powered monetary base, H. M=H⋅(1+C/D)(R/D+C/D) Yet when the authorities in the major developed countries attempted to use this relationship to expand the money stock (and bank lending) by force-feeding the banks with base money (H), in the process of Quantitative Easing (QE) in 2009, the prior relationships collapsed.

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