WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

In earlier studies of passive remote sensing of shallow-water
bathymetry, bottom depths were usually derived by empirical
regression. This approach provides rapid data processing, but it
requires knowledge of a few true depths for the regression parameters
to be determined, and it cannot reveal in-water constituents. In
this study a newly developed hyperspectral, remote-sensing reflectance
model for shallow water is applied to data from computer simulations
and field measurements. In the process, a remote-sensing
reflectance spectrum is modeled by a set of values of absorption,
backscattering, bottom albedo, and bottom depth; then it is compared
with the spectrum from measurements. The difference between the two
spectral curves is minimized by adjusting the model values in a
predictor–corrector scheme. No information in addition to the
measured reflectance is required. When the difference reaches a
minimum, or the set of variables is optimized, absorption coefficients
and bottom depths along with other properties are derived
simultaneously. For computer-simulated data at a wind speed of 5
m/s the retrieval error was 5.3% for depths ranging from 2.0 to
20.0 m and 7.0% for total absorption coefficients at 440 nm
ranging from 0.04 to 0.24 m-1. At a wind speed of 10
m/s the errors were 5.1% for depth and 6.3% for total absorption at
440 nm. For field data with depths ranging from 0.8 to 25.0 m
the difference was 10.9% (R2 = 0.96, N =
37) between inversion-derived and field-measured depth values
and just 8.1% (N = 33) for depths greater than
2.0 m. These results suggest that the model and the method
used in this study, which do not require in situ calibration
measurements, perform very well in retrieving in-water optical
properties and bottom depths from above-surface hyperspectral
measurements.

/pdf/hyperspectral-remote-sensing-for-shallow-waters-2-deriving-3zvpy4j9ar.pdf

Hyperspectral remote sensing for shallow waters: 2. Deriving bottom depths and water properties by optimization

We used simplifying assumptions to derive analytical formulae expressing the reflectance of shallow waters as a function of observation depth and of bottom depth and albedo. These formulae also involve two apparent optical properties of the water body: a mean diffise attenuation coefficient and a hypothetical reflectance which would be observed if the bottom was infinitely deep. The validity of these approximate formulae was tested by comparing their outputs with accurate solutions of the radiative transfer obtained under the same boundary conditions by Monte Carlo simulations. These approximations were also checked by comparing the reflectance spectra for varying bottom depths and compositions determined in coastal lagoons with those predicted by the formulae. These predictions were based on separate determinations of the spectral albedos of typical materials covering the floor, such as coral sand and various green or brown algae. The simple analytical expressions are accurate enough for most practical applications and also allow quantitative discussion of the limitations of remote-sensing techniques for bottom recognition and bathymetry. As early as 1944, Duntley used a spectrograph mounted in a glass-bottomed boat or flown in an airplane to analyze radiances emerging from the ocean and shallow waters. He evidenced the influence of the water depth on the spectral composition of the upward flux (Duntley 1963). Using a Monte Carlo technique, Plass and Kattawar (1972) calculated the radiative field in the atmosphere+cean system and in particular examined the dependence of the upward flux on the albedo of the ocean floor. Gordon and Brown (1974) studied the diffuse reflectance of a shallow ocean using Monte Carlo simulations and a probabilistic approach. Gordon and Brown provided an analysis based on photon history of the light field as modified by the presence of a reflecting bottom. In, addition, Ackleson and Klemas (1986) developed a two-flow model that simulated the light field within a canopy of bottom-adhering plants. A single scattering approximation for irradiance reflectance was also

/pdf/diffuse-reflectance-of-oceanic-shallow-waters-influence-of-123v20h9sg.pdf

Diffuse reflectance of oceanic shallow waters: influence of water depth and bottom albedo

For existing triboelectric nanogenerators (TENGs), it is important to explore unique methods to further enhance the output power under realistic environments to speed up their commercialization. We report here a practical TENG composed of three layers, in which the key layer, an electric double layer, is inserted between a top layer, made of Al/polydimethylsiloxane, and a bottom layer, made of Al. The efficient charge separation in the middle layer, based on Volta's electrophorus, results from sequential contact configuration of the TENG and direct electrical connection of the middle layer to the earth. A sustainable and enhanced output performance of 1.22 mA and 46.8 mW cm-2 under low frequency of 3 Hz is produced, giving over 16-fold enhancement in output power and corresponding to energy conversion efficiency of 22.4%. Finally, a portable power-supplying system, which provides enough d.c. power for charging a smart watch or phone battery, is also successfully developed.

http://scholarworks.unist.ac.kr/bitstream/201301/20625/1/ncomms12985.pdf

Boosted output performance of triboelectric nanogenerator via electric double layer effect

Bathymetric mapping will be most straightforward where water quality and atmospheric conditions are invariant over the scene. Under these conditions, both depth and an effective attenuation coefficient of the water over several different bottom types may be retrieved from passive, multispectral imagery. As scenes become more complex-with changing water type and variable atmospheric conditions-it is probable that a strictly spectral analysis will no longer be sufficient to extract depth from multispectral imagery. In these cases an independent source of information will be required. The most likely sources for such information are spatial and temporal variations in image data.

Bathymetric mapping with passive multispectral imagery.

: In this paper we use the linear multiband algorithm on two channels of a Landsat 4 TM scene and demonstrate its improved performance over the two-band ratio method. The Landsat scene (scene 5032614162) taken 21 Jan. 1985) contains Isla de Vieques, an island off the southeast coast of Puerto Rico in the Caribbean Sea. The water clarity in this area is extremely good, and bottom reflectances in the blue and green channels (TM bands one and two) are detectable for depths up to 25 m. Bottom reflectances in the red channel (TM band three) are detectable for depths up to approx 6 m; sensor noise (striping) over the water renders this band useless at deeper depths. Reprints.

Bathymetry calculations with landsat 4 TM imagery under a generalized ratio assumption

http://people.kzoo.edu/barth/math280/articles/lion_wildebeest_zebra.pdf

Lion, wildebeest and zebra: A predator–prey model

Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described. For the linear model using Hooke's Law, the motion of each weight is described by a fourth- order linear differential equation. A nonlinear model is also described and damping and external forcing are considered. The model has many features that permit the meaningful introduction of many concepts including: accuracy of numerical algorithms, dependence on parameters and initial conditions, phase and synchronization, periodicity, beats, linear and nonlinear resonance, limit cycles, harmonic and subharmonic solutions. These solutions produce a wide variety of interesting motions and the model is suitable for study as a computer laboratory project in a beginning course on differential equations or as an individual or a small-groupundergraduate research p roject.

http://lab.fs.uni-lj.si/lat/uploads/oip/coupled_spring.pdf

Coupled spring equations

(2001). The Gibbs' phenomenon. International Journal of Mathematical Education in Science and Technology: Vol. 32, No. 1, pp. 73-89.

The Gibbs' phenomenon

(1989). The Butterfly Curve. The American Mathematical Monthly: Vol. 96, No. 5, pp. 442-443.

Temple H. Fay

Papers

Bathymetry calculations with landsat 4 TM imagery under a generalized ratio assumption

Lion, wildebeest and zebra: A predator–prey model

Coupled spring equations

The Gibbs' phenomenon

The Butterfly Curve