The geometry of whips

In this paper we study geometric aspects of the space of arcs parametrized by unit speed in the $L^2$ metric. Physically this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic equation is the nonlinear, nonlocal wave equation $\eta_{tt} = \partial_s(\sigma \eta_s)$, with $\lvert \eta_s\rvert\equiv 1$ and $\sigma$ given by $\sigma_{ss}- \lvert \eta_{ss}\rvert^2 \sigma = -\lvert \eta_{st}\rvert^2$, with boundary conditions $\sigma(t,1)=\sigma(t,-1)=0$ and $\eta(t,0)=0$. We prove that the space of arcs is a submanifold of the space of all curves, that the orthogonal projection exists but is not smooth, and as a consequence we get a Riemannian exponential map that it continuous and even differentiable but not $C^1$. This is related to the fact that the curvature is positive but unbounded above, so that there are conjugate points at arbitrarily short times along any geodesic. We also compare this metric to an $L^2$ metric introduced by Michor and Mumford for shape recognition on the homogeneous space $\text{Imm}(I, \mathbb{R}^2)/\mathcal{D}(I)$ of immersed curves modulo reparametrizations; we show it has some similar properties (such as nonnegative but unbounded curvature and a nonsmooth exponential map), but that the $L^2$ metric on the arc space yields a genuine Riemannian distance.Comment: 24 page

Eliminating intermediate lists in pH using local transformations

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (p. 68-69).by Jan-Willem Maessen.M.Eng

Cicer L., a monograph of the genus, with special reference to the chickpea (Cicer arietinum L.), its ecology and cultivation

1. The history of the chickpea or gram, Cicer arietinum L., has been described from Homer's time and the earliest finds, 5450 B.C. in Hacilar, Turkey, up to the present day. The crop was first domesticated in Asia Minor and was introduced in India either from Central Asia or Asia Minor, the two main centres of origin. Some forms were even introduced rather recently. Ethiopia is a secondary centre of domestication; connections with Egypt or Asia remain speculative. Several pieces of evidence oppose the opinion of DE CANDOLLE (1882) that the ancient Egyptians and Jews had only known the chickpea for two millenia.Medical uses, no longer widely practised, are discussed. The spread to the present areas of cultivation is described and mapped.2. The genus Cicer L. has been revised. Popov (1929) accepted 22 species, now 39 species (8 annual, 31 perennial) are known. One species is described for the first time: C. multijugum from Afghanistan. A key to the species is prepared. The species, arranged alphabetically, are described and accompanied by detailed illustrations. The synonymy and typifications are given, as well as notes on geography, ecology and morphology. The geographical distribution of each species of the genus, occurring in Central Asia, Asia Minor and the Medi terranean, is presented in maps. It is stressed that the variability and geography of many species is not known sufficiently. The poor availability of fresh material of the wild species is a handicap to research.The relation to the other genera in Vicieae is discussed. Cicer occupies a somewhat peculiar place with its glandular hairs, inflated fruits and seed shape. The infraspecific classification in the cultivated species is reviewed; an informal classification is presented on base of the work of POPOVA (1937) without rejecting the older varieties distinguished by JAUBERT and SPACH, and ALEFELD.3. The importance of the chickpea as the third pulse crop in the world after beans and peas is presented in a map, graphs and tables. The crop ranks l5th among all crops in area occupied yearly. Yields, at present an average of about 700 kg per ha, are highest in Egypt (1670 kg) and Turkey (1220 kg). About 83 of the world production is in the Indian subcontinent.The weather is the main reason for fluctuation in area. The partial recession in area, due to the expanding new cereal cultivars, will be met by higher yields per unit area and aided by higher prices.4. Some anatomical particulars, e.g. the glandular hairs, are shortly reviewed.5. The chickpea is generally cultivated in a traditional way. The resistance to drought (deep roots) and ability to grow in poor soils has not increased the care of the crop. However, with good soil preparation, proper sowing on rows, cultivation and fertilization the crop can yield reasonably. The sowing date is very important. Sowing early in the growth season is to be preferred, except in case of wilt disease. Plant density, sowing depth and sowing seed are discussed. Irrigations, needed in some countries, should be practised with care so as not to induce soil anaeroby.Often the chickpea is grown mixed with wheat or mustard, against crop failures and for utilization of different soil layers. In rotation the chickpea is a well esteemed crop. It has maintained soil fertility at a certain level for centuries in the densely populated areas of India. The plants are harvested mainly by hand. Threshing machines need good adjustment to prevent breakage of seeds. Storage is an important problem, since much loss may occur.6. Ecological trials were carried out on light, daylength, temperature and relative humidity. The photosynthesis rate varied from 250-500 μg CO 2 -uptake per cm 2and per h at about 26°C, but at 18°C, the rate was not much less. Leaves of two-weeks old are the most effective in photosynthesis and may use twice as much CO 2 as the four-week old leaves. Estimated calculated production appeared to be 12-14 tons of total dry matter, or about 5-7 tons of grains, similar to the highest yield ever obtained on a small plot.The chickpea is a quantitative LD plant. Under 16-h days the flowering was advanced by e.g. 20-35 days, if compared with 9-h days. Short days did not prevent flowering. Dry matter yield was improved in LD. The influence of the photoperiodic effect alone of the daylengths following different sowing dates on flowering and yield is small. Increasing photoperiods appeared to be more favourable than decreasing ones.The optimum temperature for early vegetative growth ranges from 21-29°C (night and day) to 24-32°C for different cultivars. Over the entire growth period the optimum temperature is somewhat lower, 18-26°C and 21-29°C, which is also optimum for flowering.The relative humidity was found to have little influence on fruit-setting. A decrease in light intensity of 25 % of the available amount during May and June, however, was found to decrease the number of pods by 25-50%.Data on soils and nutrients are summarized. As yet the chickpea does not respond to dressings of more than 10 kg N and 30 kg P 2 O 5 per ha. Moderately heavy soils are preferred, but both heavy and light soils are used in some areas.Growth substances usually have a negative influence on the growth of chickpeas. Scarcity of practical trials prohibits any recommendation.Topping appears to be an old practice to stimulate branching. Regeneration, however, takes a long time and is only sufficient under optimum conditions and if applied at an early stage.7. Breeding has not yet improved yields over large areas. A review on cytogenetics is given. Some new reports on the somatic number of chromosomes of some wild species are added. As crossing technique is a delicate operation, hybridization on a large scale is at present not possible, but pollination at an early stage without emasculation may be a solution. The introduction of new cultivars has not been very successful because they have not shown large differ ences with local cultivars.8. The most important insect pests of the chickpea are the podborer and the pulse beetles, which are described in some detail. Geographical distribution and way of control is given. All reported pests are mentioned. Nematode attacks seem to be underestimated at present. Rats may cause important damage in stores.9. The diseases of the chickpea, their occurrence, possible way of control are described. Most damage is done by wilt, caused by both a soil fungus and by physiological drought, and blight. Several other diseases such as rust and foot rots are not yet serious over large areas. As for pests, chemical control is often uneconomic.10. The chickpea is mainly used as human food, whether fresh, boiled, or roasted in many preparations. As a part of balanced foods it can form an important supplement to the protein nutrition of children. The proteins of chickpea constitute an important part of the protein intake in India. The chemical composition of the seeds (e.g. up to nearly 30% of protein) is given, as well as the amounts of essential amino acids.Except sometimes for methionine and for tryptophan the chickpea appears to be an excellent source of amino acids