scholarly journals New Telyakovskii-type estimates via the Boolean sum approach

1996 ◽  
Vol 54 (1) ◽  
pp. 131-146
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Present Note ◽  
Linear Operators ◽  
Convolution Type ◽  
The One ◽  
Boolean Sums ◽  
Boolean Sum

In the present note the magnitude of constants in Telyakovskii-type theorems is investigated. Our general approach to construct the linear operators yielding good constants is the one via Boolean sums. Explicit values for the constants in question are given for general convolution-type operators; the classical Fejér-Korovkin kernel is then used as an example for which one obtains rather small values. Furthermore, also an asymptotic assertion is derived which indicates the room left for improvement of the main results. This leads to a natural conjecture concluding this article.

scholarly journals Approximation by Boolean sums of linear operators: Telyakovskiǐ-type estimates

1990 ◽  
Vol 42 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Upper Bound ◽  
Algebraic Polynomial ◽  
Present Note ◽  
Type Inequality ◽  
Linear Operators ◽  
Polynomial Operators ◽  
Boolean Sums ◽  
General Conditions

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

scholarly journals Pointwise estimates for higher order convexity preserving polynomial approximation

1994 ◽  
Vol 36 (2) ◽  
pp. 213-233 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Polynomial Approximation ◽  
Convex Functions ◽  
Higher Order ◽  
Second Order ◽  
Convolution Type ◽  
Shape Preservation ◽  
Moduli Of Continuity ◽  
Pointwise Estimates ◽  
Higher Order Convexity ◽  
Boolean Sum

AbstractDeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

Notes on Ptolemy

1952 ◽  
Vol 14 (1) ◽  
pp. 78-86
Author(s):  
J. Ph. Vogel
Keyword(s):  
Present Note ◽  
The Other ◽  
Trade Routes ◽  
Guiding Principle ◽  
Saint Martin ◽  
The One

It has been recognized long ago that Ptolemy's topography of Cis-Gangetic India was based on trade-routes. Nearly a century ago Vivien de Saint Martin spoke of “the almost exclusive employment of itineraries of merchants and caravans indicating on each route the series of daily stations”. We may compare the lists of stations inserted in books on India of the seventeenth and eighteenth centuries. One of the earliest examples we find in Joannes de Laet's little volume De Imperio Magni Mogolis (Leiden, 1631), p. 57. The author evidently derived his lists from the itineraries of the English merchants Richard Steel and John Crowther. The French jeweller and traveller Jean-Baptiste Tavernier has included full lists of stations along the principal trade-routes in his Indian Travels (ch. iv–xii). In the first place he deals with the two routes from Surat to Agra, all-important to European traders, the one by the Tāptī valley and Mālwā, the other by way of Aḥmadābād and Rājpūtānā. If we keep in mind that Ptolemy must have used similar lists, it will go far to explain the disconcerting fact that so many among the localities in his tables are not known from indigenous sources, either literary or epigraphical, whereas famous towns have been omitted. His tables are fundamentally lists of stages, and this must be our guiding principle in unravelling the riddle of Ptolemy's topography. The present note is an attempt to demonstrate this in some detail.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

Edward Jenner, F. R. S. and the cuckoo

1974 ◽  
Vol 28 (2) ◽  
pp. 235-240 ◽  
Keyword(s):  
Small Town ◽  
Present Note ◽  
The Other ◽  
150Th Anniversary ◽  
The Subject ◽  
The One ◽  
Medical Historian ◽  
National Biography

In January 1973 the 150th anniversary of the death of Edward Jenner, the originator of vaccination, was modestly commemorated in the small town of Berkeley, Gloucestershire, where he was born and lived for most of his life (1). Jenner was himself the subject of controversy during and after his lifetime, on the one hand lauded as a selfless saviour of mankind, and on the other denounced as a self-seeking charlatan; his discovery likewise has virtually never ceased to be a subject of debate (2). There is no dearth of books and articles about Jenner and vaccination (3) and the subject of the present note is an unconnected and very minor discovery of Jenner’s—nevertheless one which is not only intrinsically interesting to naturalists, but also pertinent to any study of the character and integrity of its author, for the discovery was still being disputed more than a century after is was made. Edward Jenner was the first to publish, in 1788, an accurate account of what happens to the unfortunate young of the parent bird in whose nest a cuckoo deposits one of her eggs. Yet to consult the Dictionary of National Biography for details of Jenner’s life is to read, in an article published in 1892, that the ‘absurdity’ of this account had been demonstrated by the naturalist Charles Waterton. A few years earlier Charles Creighton (1847-1927), the epidemiologist and medical historian, whose intense opposition to vaccination led him to denigrate Jenner in every conceivable way, had described his paper on the cuckoo as mainly ‘a tissue of inconsistencies and absurdities’ (4).

scholarly journals A preliminary note on branched α-ray tracks

1921 ◽  
Vol 99 (700) ◽  
pp. 432-435 ◽  
Keyword(s):  
Present Note ◽  
Nuclear Collision ◽  
Preliminary Note ◽  
Reasonable Time ◽  
Preceding Article ◽  
Order Of Magnitude ◽  
Pass Through ◽  
The One ◽  
Reciprocating Expansion ◽  
Α Particles

According to Prof. Sir E. Rutherford, if about one hundred thousand α-rays from radium C pass through the air, on the average there will be one close nuclear collision, which produces a swiftly moving atom. Thus, if we take a great number of photographs of α-ray tracks compatible with the above order of magnitude, we might expect to find some evidence to indicate the disruption of atoms by the α-particles. The present note is to describe some preliminary trials in this direction. The reciprocating expansion apparatus of the preceding article is very convenient for taking a large number of photographs within a reasonable time. Direct eye-observation confirmed the existence of some branched tracks which differed greatly in configuration from spurred tracks like the one photographed by Mr. C. T. R. Wilson. Arrangements were then made to devise a suitable method of photographing such tracks and to show their orientation in space.

The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

2015 ◽  
Vol 69 (1) ◽  
Author(s):  
Mariusz Plaszczyk
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Present Note ◽  
Cotangent Bundles ◽  
Natural Transformations ◽  
The One

AbstractIf (M,g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T* M given by v → g(v,−) between the tangent TM and the cotangent T* M bundles of M. In the present note first we generalize this isomorphism to the one J

scholarly journals On a New Species of Mite (Tarsonemus) Injurious to Sugar-Canes in Barbados

1912 ◽  
Vol 3 (3) ◽  
pp. 325-328 ◽  
Author(s):  
Stanley Hirst
Keyword(s):  
New Species ◽  
Sugar Cane ◽  
Present Note ◽  
The Body ◽  
Detailed Account ◽  
Nominal Species ◽  
The Third ◽  
Sufficient Detail ◽  
The One ◽  
A New Species

Mr. A. D. Michael has already recorded* the presence of mites of the genus Tarsonemus upon diseased sugar-cane from Barbados. He states that two species belonging to this genus were present in the material sent to him, and that the larger of the two species was certainly identical with one which Mr. Bancroft found doing serious damage to sugar-cane in Queensland. Mr. Michael proposed the name Tarsonemus bancrofti for this larger species. So far as I am aware, no description of this nominal species has been published and I am obliged to rely on Mr. Bancroft's published sketches† for information concerning it. Unfortunately his drawings are not executed in sufficient detail and I am not certain that his mite is the same species as the one which is dealt with in the present note; but as the figures of the Queensland mite differ appreciably from the Barbados specimens, it seems advisable to describe the latter under another name (T. spinipes). The species of the genus Tarsonemus often resemble one another very closely in structure, and they cannot be recognised with certainty unless a fully detailed account of their principal characters, accompanied by careful drawings, is given. Dr. Bancroft gives drawings of both sexes of his mite. He does not figure any spines on the third leg of the male, but he shows a lobeshaped expansion, similar to that of T. spinipes, on the inner side of the short fourth leg. The hairs of the body are not depicted. According to his drawings, the body of the female resembles that of T. spinipes in being very long and narrow, but is apparently much narrower at the anterior end. He represents the two terminal setae of the fourth leg of the female as being both very long and slender, the outer one being seemingly almost as long as the inner. The size of T. bancrofti is not stated, nor is the scale of enlargement of the figures given.

The Committee as an Instrument of Coördination in the New Deal

1937 ◽  
Vol 31 (2) ◽  
pp. 301-310
Author(s):  
Mary C. Trackett
Keyword(s):  
New Deal ◽  
Present Note ◽  
Horizontal Integration ◽  
Concerted Action ◽  
Policy Framing ◽  
Administrative Management ◽  
The Past ◽  
The Government ◽  
The One ◽  
The New Deal

Of all criticisms of the New Deal, the one most frequently emphasized is the lack of coördination. Headlessness in policy-framing and sprawling aimlessness in policy execution are twin charges which the Administration has been forced to admit. The recent report of the President's Committee on Administrative Management is an indication that the Administration intends to leave to posterity a good record on this score; but both practitioners and students of government are well aware that no reorganization can be so complete, so perfect in its functional allotment of duties to departments, that the problem of horizontal integration will not still need to be faced and solved. This reminder is less an apologia than an indication of the frame of reference of the present note; those who have been administering the government for the past four years have never been unaware of the need for concerted action among the executive departments, and many attempts have been made to achieve it. A device often employed for the purpose has been the interdepartmental committee.

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