Behold! The Midpoint Rule Is Better Than the Trapezoidal Rule for Concave Functions

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic functions in another. Similar results are also given for the midpoint rule.

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Time and Latitude in South Africa

International Astronomical Union Colloquium ◽

10.1017/s0252921100026816 ◽

1972 ◽

Vol 1 ◽

pp. 27-38

Author(s):

J. Hers

Keyword(s):

South Africa ◽

Cape Town ◽

Astronomical Observations ◽

Royal Observatory ◽

The Republic ◽

Astronomical Determination ◽

Limited Accuracy ◽

Better Than

In South Africa the modern outlook towards time may be said to have started in 1948. Both the two major observatories, The Royal Observatory in Cape Town and the Union Observatory (now known as the Republic Observatory) in Johannesburg had, of course, been involved in the astronomical determination of time almost from their inception, and the Johannesburg Observatory has been responsible for the official time of South Africa since 1908. However the pendulum clocks then in use could not be relied on to provide an accuracy better than about 1/10 second, which was of the same order as that of the astronomical observations. It is doubtful if much use was made of even this limited accuracy outside the two observatories, and although there may – occasionally have been a demand for more accurate time, it was certainly not voiced.

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Localization of immunolabels by electron microscopy and image averaging

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100075208 ◽

1983 ◽

Vol 41 ◽

pp. 282-283

Author(s):

J. Frank ◽

P.-Y. Sizaret ◽

A. Verschoor ◽

J. Lamy

Keyword(s):

Electron Microscopy ◽

Single Particle ◽

Attachment Site ◽

Uranyl Acetate ◽

Limulus Polyphemus ◽

Resolution Limit ◽

Electron Microscopic ◽

Image Averaging ◽

Top View ◽

Better Than

The accuracy with which the attachment site of immunolabels bound to macromolecules may be localized in electron microscopic images can be considerably improved by using single particle averaging. The example studied in this work showed that the accuracy may be better than the resolution limit imposed by negative staining (∽2nm).The structure used for this demonstration was a halfmolecule of Limulus polyphemus (LP) hemocyanin, consisting of 24 subunits grouped into four hexamers. The top view of this structure was previously studied by image averaging and correspondence analysis. It was found to vary according to the flip or flop position of the molecule, and to the stain imbalance between diagonally opposed hexamers (“rocking effect”). These findings have recently been incorporated into a model of the full 8 × 6 molecule.LP hemocyanin contains eight different polypeptides, and antibodies specific for one, LP II, were used. Uranyl acetate was used as stain. A total of 58 molecule images (29 unlabelled, 29 labelled with antl-LPII Fab) showing the top view were digitized in the microdensitometer with a sampling distance of 50μ corresponding to 6.25nm.

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Closing the GAP—A 5Å Scanning Microscope

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100061938 ◽

1969 ◽

Vol 27 ◽

pp. 6-7

Author(s):

A. V. Crewe

Keyword(s):

Normal Form ◽

The Other ◽

Resolving Power ◽

Scanning Microscope ◽

Conventional Microscope ◽

Other Hand ◽

Closing The Gap ◽

Thermal Sources ◽

Better Than ◽

Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

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Electron Microscopic Observation of the Longitudinal Organization of Poly (p-Phenylene Terephthalamide) Fibers

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100055734 ◽

1982 ◽

Vol 40 ◽

pp. 680-681

Author(s):

Li Li-Sheng ◽

L.F. Allard ◽

W.C. Bigelow

Keyword(s):

Electron Microscopy ◽

Mechanical Properties ◽

Scanning Electron Microscopy ◽

Microscopic Observation ◽

Electron Microscopic Observation ◽

Electron Microscopic ◽

Aromatic Polyamides ◽

Radial Arrangement ◽

Scanning Electron ◽

Better Than

The aromatic polyamides form a class of fibers having mechanical properties which are much better than those of aliphatic polyamides. Currently, the accepted morphology of these fibers as proposed by M.G. Dobb, et al. is a radial arrangement of pleated sheets, with the plane of the pleats parallel to the axis of the fiber. We have recently obtained evidence which supports a different morphology of this type of fiber, using ultramicrotomy and ion-thinning techniques to prepare specimens for transmission and scanning electron microscopy.

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A High Angle, Double Tilting Cold Stage for the AEI EM7

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100052262 ◽

1974 ◽

Vol 32 ◽

pp. 450-451

Author(s):

P.R. Swann ◽

A.E. Lloyd

Keyword(s):

Habit Plane ◽

Temperature Control ◽

Tilt Angle ◽

Cooling System ◽

Thermal Drift ◽

High Angle ◽

Cold Stage ◽

High Degree ◽

Better Than

Figure 1 shows the design of a specimen stage used for the in situ observation of phase transformations in the temperature range between ambient and −160°C. The design has the following features a high degree of specimen stability during tilting linear tilt actuation about two orthogonal axes for accurate control of tilt angle read-out high angle tilt range for stereo work and habit plane determination simple, robust construction temperature control of better than ±0.5°C minimum thermal drift and transmission of vibration from the cooling system.

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Thermal Considerations in Lens Regulator Systems

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100068783 ◽

1970 ◽

Vol 28 ◽

pp. 358-359

Author(s):

K.C. Newton

Keyword(s):

Electron Microscope ◽

Thermal Effects ◽

Environmental Control ◽

Reference Networks ◽

Better Than

Thermal effects in lens regulator systems have become a major problem with the extension of electron microscope resolution capabilities below 5 Angstrom units. Larger columns with immersion lenses and increased accelerating potentials have made solutions more difficult by increasing the power being handled. Environmental control, component choice, and wiring design provide answers, however. Figure 1 indicates with broken lines where thermal problems develop in regulator systemsExtensive environmental control is required in the sampling and reference networks. In each case, stability better than I ppm/min. is required. Components with thermal coefficients satisfactory for these applications without environmental control are either not available or priced prohibitively.

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