Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Climate and infectious disease: Use of remote sensing for detection of Vibrio cholerae by indirect measurement

It has long been known that cholera outbreaks can be initiated when Vibrio cholerae, the bacterium that causes cholera, is present in drinking water in sufficient numbers to constitute an infective dose, if ingested by humans. Outbreaks associated with drinking or bathing in unpurified river or brackish water may directly or indirectly depend on such conditions as water temperature, nutrient concentration, and plankton production that may be favorable for growth and reproduction of the bacterium. Although these environmental parameters have routinely been measured by using water samples collected aboard research ships, the available data sets are sparse and infrequent. Furthermore, shipboard data acquisition is both expensive and time-consuming. Interpolation to regional scales can also be problematic. Although the bacterium, V. cholerae, cannot be sensed directly, remotely sensed data can be used to infer its presence. In the study reported here, satellite data were used to monitor the timing and spread of cholera. Public domain remote sensing data for the Bay of Bengal were compared directly with cholera case data collected in Bangladesh from 1992–1995. The remote sensing data included sea surface temperature and sea surface height. It was discovered that sea surface temperature shows an annual cycle similar to the cholera case data. Sea surface height may be an indicator of incursion of plankton-laden water inland, e.g., tidal rivers, because it was also found to be correlated with cholera outbreaks. The extensive studies accomplished during the past 25 years, confirming the hypothesis that V. cholerae is autochthonous to the aquatic environment and is a commensal of zooplankton, i.e., copepods, when combined with the findings of the satellite data analyses, provide strong evidence that cholera epidemics are climate-linked.

Biochemical evolution II: Origin of life in tubular microstructures on weathered feldspar surfaces

Mineral surfaces were important during the emergence of life on Earth because the assembly of the necessary complex biomolecules by random collisions in dilute aqueous solutions is implausible. Most silicate mineral surfaces are hydrophilic and organophobic and unsuitable for catalytic reactions, but some silica-rich surfaces of partly dealuminated feldspars and zeolites are organophilic and potentially catalytic. Weathered alkali feldspar crystals from granitic rocks at Shap, north west England, contain abundant tubular etch pits, typically 0.4–0.6 μm wide, forming an orthogonal honeycomb network in a surface zone 50 μm thick, with 2–3 × 106 intersections per mm2 of crystal surface. Surviving metamorphic rocks demonstrate that granites and acidic surface water were present on the Earth’s surface by ∼3.8 Ga. By analogy with Shap granite, honeycombed feldspar has considerable potential as a natural catalytic surface for the start of biochemical evolution. Biomolecules should have become available by catalysis of amino acids, etc. The honeycomb would have provided access to various mineral inclusions in the feldspar, particularly apatite and oxides, which contain phosphorus and transition metals necessary for energetic life. The organized environment would have protected complex molecules from dispersion into dilute solutions, from hydrolysis, and from UV radiation. Sub-micrometer tubes in the honeycomb might have acted as rudimentary cell walls for proto-organisms, which ultimately evolved a lipid lid giving further shelter from the hostile outside environment. A lid would finally have become a complete cell wall permitting detachment and flotation in primordial “soup.” Etch features on weathered alkali feldspar from Shap match the shape of overlying soil bacteria.

Correlates of the desired family size among Indian communities.

The People of India database of the Anthropological Survey of India documents 631 cultural, ecological, and economic traits of the 4635 communities to which the entire Indian population is assigned. Focusing on 1342 communities of South India, we looked for correlates of low (1 or 2 children) and high (4 or more children) desired family size (DFS) reported as the norm for any given community by key informants. We found 10 cultural and 18 economic traits to be significantly correlated to high DFS and 21 cultural and 9 economic traits to low DFS. The economic traits so identified are compatible with high family size being desired by parents who have little capability of investing in quality of offspring, but whose children contribute economically from an early age. In contrast, communities desiring low family size are part of the modern intensive agriculture/organized industry/services sector and invest heavily in educating their children. A composite index based on 27 economic traits (CEI) has a high predictive value with respect to the DFS for the entire set of 4635 Indian communities. The 31 cultural traits highly correlated to high or low DFS constitute 5 clusters that can be identified as characterizing scheduled tribes, scheduled castes, rural and landless lower castes, urban upper castes, and Moslems. Whereas economic traits have similar influence on DFS within each of these ethnic categories, Moslems demonstrate a significantly higher DFS for lower values of CEI.