The Gibbs Phenomenon for Taylor Means and for [F, Dn] Means

1960 ◽  
Vol 12 ◽  
pp. 660-673 ◽  
Author(s):  
Chester L. Miracle
Keyword(s):  
Gibbs Phenomenon ◽  
Left Hand ◽  
Right Hand ◽  
Image Position

The Gibbs phenomenon may be described, quite generally, as follows. Let a sequence {fn(x)} (n = 0, 1, 2, … ,) converge to a function f(x) for x in the interval x0 < x < x0+ h. We say that {fn(x)} displays the Gibbs phenomenon in a right-hand neighbourhood of the point X0, ifA similar definition holds for a left-hand neighbourhood. If {fn(x)} displays the Gibbs phenomenon at both sides of x0, we say simply that {fn(x)} displays Gibbs phenomenon at the point X0.

scholarly journals A Fourth Century Head in Central Museum, Athens

1895 ◽  
Vol 15 ◽  
pp. 194-201 ◽  
Author(s):  
E. F. Benson
Keyword(s):  
Fourth Century ◽  
Present Condition ◽  
Left Hand ◽  
Right Hand ◽  
Image Position ◽  
The Right

There is among the fourth century works in the Central Museum at Athens a head found at Laurium. It is made of Parian marble but it has been completely discoloured by slag or refuse from the lead mines, and is now quite black. In its present condition it is quite impossible to obtain a satisfactory photograph of it, and the reproduction given of it in the figure is from a cast.It has been published, as far as I am aware, only in M. Kavvadias' catalogue. There it is described as a head of the Lykeian Apollo. This identification rests solely on a passage of Lucian, who mentions a statue of the Lykeian Apollo in the gymnasium at Athens.He says of it ( 7)—It will be seen from a glance at the photograph that the grounds for this identification are very slender. The left hand with the bow does not exist, and the only reason for supposing therefore that this is a head of the Lykeian Apollo consists in the fact that the right hand of the statue rests on the head. This in itself seems insufficient and, among other reasons, it is I think rendered impossible by the phrase For the hand is not idly resting, it is not a tired hand; the posture of the fingers is firm and energetic.

An inequality relating means

1944 ◽  
Vol 40 (3) ◽  
pp. 253-255
Author(s):  
J. Bronowski
Keyword(s):  
Arithmetic Mean ◽  
Arithmetic Means ◽  
Geometric Means ◽  
Left Hand ◽  
Right Hand ◽  
Image Position ◽  
The Right

1. Let a, b be positive constants; and let y1, y2, …, yn be real exponents, not all equal, having arithmetic mean y defined by(here, and in what follows, the summation ∑ extends over the values i = 1, 2, …, n). Then it is clear thatsince the right-hand sides are the geometric means of the positive numbers whose arithmetic means stand on the left-hand sides. I know of no results, however, which relate the ratios and and I have had occasion recently to require such results. This note gives an inequality between these ratios, subject to certain restrictions on a and b.

A New Fragment of Greek Music in Cairo

1931 ◽  
Vol 51 (1) ◽  
pp. 91-100 ◽  
Author(s):  
J. F. Mountford
Keyword(s):  
Extreme Right ◽  
Bottom Edge ◽  
Left Hand ◽  
Right Hand ◽  
Formed Part ◽  
Image Position ◽  
The Right

The papyrus numbered 59533 in the Catalogue Général of the Cairo Museum is a mere scrap (13 cm. × 12 cm.), on one side of which is written a fragmentary text with suprascript musical signs:The writing is along the fibres, that is to say, on the recto if the scrap originally formed part of a roll; the verso is blank; and the papyrus has been folded horizontally. The right-hand and the top seem to be the original edges of a sheet or roll; the left-hand is clearly defective, and though the bottom edge may be the original edge of a sheet, it is not the bottom of a roll. The writing is carried to the extreme right-hand edge, without a margin. Below the text there is some scribbling which seems to have no particular significance and is probably to be taken as a probatio pennae rather than as a signature.

scholarly journals CONSISTENCY PROOF OF A FRAGMENT OF PV WITH SUBSTITUTION IN BOUNDED ARITHMETIC

2018 ◽  
Vol 83 (3) ◽  
pp. 1063-1090
Author(s):  
YORIYUKI YAMAGATA
Keyword(s):  
Propositional Logic ◽  
The Other ◽  
Bounded Arithmetic ◽  
Consistency Proof ◽  
Left Hand ◽  
Right Hand ◽  
Other Hand ◽  
Image Position

AbstractThis article presents a proof that Buss’s $S_2^2$ can prove the consistency of a fragment of Cook and Urquhart’s PV from which induction has been removed but substitution has been retained. This result improves Beckmann’s result, which proves the consistency of such a system without substitution in bounded arithmetic $S_2^1$.Our proof relies on the notion of “computation” of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and either its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven.This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss’s $S_2^1$. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies.

The Rates of Change of the Eigenvalues of a Lambda Matrix and their Use in Flutter Investigations

1966 ◽  
Vol 70 (662) ◽  
pp. 364-365 ◽  
Author(s):  
D. L. Woodcock
Keyword(s):  
Dynamical System ◽  
Variable Parameter ◽  
Equation Of Motion ◽  
Left Hand ◽  
Rates Of Change ◽  
Right Hand ◽  
Image Position ◽  
Derivatives Of

Consider a lambda matrix F(ƛ, δ). This may, for example, result from the equation of motion of a dynamical system, δ being a measure of the magnitude of some possible modification to the system or some other variable parameter. Let λ be an eigenvalue and p’ and q the corresponding left hand and right hand eigenvectors. ThenOne wishes to determine the derivatives of ƛ with respect to δ. Expressions for the first two derivatives will be obtained on the assumptions that F is simply degenerate and that ƛ is not a repeated root.

scholarly journals Some Inequalities for Stop-Loss Premiums

1977 ◽  
Vol 9 (1-2) ◽  
pp. 75-83 ◽  
Author(s):  
H. Bühlmann ◽  
B. Gagliardi ◽  
H. U. Gerber ◽  
E. Straub
Keyword(s):  
Continuous Function ◽  
Random Variable ◽  
Left Hand ◽  
Right Hand ◽  
Image Position ◽  
The Right ◽  
Stop Loss

In this paper any given risk S (a random variable) is assumed to have a (finite or infinite) mean. We enforce this by imposing E[S−] < ∞.Let then v(t) be a twice differentiate function withand let z be a constant with o ≤ z ≤ 1.We define the premium P as followsor equivalentlyNotation: v−(∞) = ∞.The definitions (1) and (equivalently) (2) are meaningful because of theLemma: a) E[v(S − zQ)] exists for all Q∈(− ∞, + ∞).b) The set{Q∣ − ∞ < Q < + ∞, E[v(S−zQ)]>v((1−z)Q)} is not empty.Proof: a) b) Because of a) E[v(S−zQ)] is always finite or equal to + ∞ If v(− ∞) = − ∞ then E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q. The left hand side of the inequality is a nonincreasing continuous function in P (strictly decreasing if z > 0), while the right hand side is a nondecreasing continuous function in Q (strictly increasing if z > 1).If v(− ∞) = c finite then E[v(S − zQ)] > c(otherwise S would need to be equal to − ∞ with probability 1) and again E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q.

From Two-Star Instruments to Modern Star Trackers

1977 ◽  
Vol 30 (1) ◽  
pp. 27-34
Author(s):  
H. C. Freiesleben
Keyword(s):  
Local Time ◽  
The Sun ◽  
Left Hand ◽  
The Earth ◽  
Right Hand ◽  
Celestial Navigation ◽  
History Of ◽  
East And West ◽  
Image Position

For a long while now positions on the surface of the Earth have been fixed by observing the altitudes of two heavenly bodies. The equations:include on their left-hand side two observed zenith distances and on their right-hand sides the required values of latitude and longitude. The history of celestial navigation records many different solutions to the problem, one of them being the observation of equal altitudes of the same body east and west of the meridian, usually the Sun, or of two different stars. If only the local time is required it is not even necessary to measure the exact altitude.

On an inequality for the normal distribution arising in bioequivalence studies

1999 ◽  
Vol 36 (1) ◽  
pp. 279-286
Author(s):  
Yi-Ching Yao ◽  
Hari Iyer
Keyword(s):  
Normal Distribution ◽  
Random Variable ◽  
Normal Random Variable ◽  
Individual Bioequivalence ◽  
Left Hand ◽  
Right Hand ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Image Position ◽  
The Right

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

On an inequality for the normal distribution arising in bioequivalence studies

1999 ◽  
Vol 36 (01) ◽  
pp. 279-286 ◽  
Author(s):  
Yi-Ching Yao ◽  
Hari Iyer
Keyword(s):  
Normal Distribution ◽  
Random Variable ◽  
Normal Random Variable ◽  
Individual Bioequivalence ◽  
Left Hand ◽  
Right Hand ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Image Position ◽  
The Right

For (μ,σ2) ≠ (0,1), and 0 &lt; z &lt; ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 &lt; z &lt; ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

Editorial Note

1946 ◽  
Vol 11 (1) ◽  
pp. 2-2
Keyword(s):  
Editorial Note ◽  
Speech Sounds ◽  
Left Hand ◽  
Hand Column ◽  
Right Hand ◽  
Infant Speech ◽  
The Right

In the article “Infant Speech Sounds and Intelligence” by Orvis C. Irwin and Han Piao Chen, in the December 1945 issue of the Journal, the paragraph which begins at the bottom of the left hand column on page 295 should have been placed immediately below the first paragraph at the top of the right hand column on page 296. To the authors we express our sincere apologies.

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