Unit fractions

We prove an old conjecture of Erd˝s and Graham on sums of unit fractions: o There exists a constant b 0 such that if we rcolor the integers in [2, br ], then there exists a monochromatic set S such that n∈S 1/n = 1. 1. Introduction We will prove a result on unit fractions which has the following corollary. Corollary. There exists a constant b so that for every partition of the integers in [2, br ] into r classes, there is always one class containing a subset S with the property n∈S 1/n = 1....
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REFINING OPERATIONS Petroleum refining operations can be separated into five basic areas: 1. Fractionation (distillation) is the separation of crude oil in atmospheric and vacuum distillation towers into different "fractions" or "cuts." 2. Conversion processes change the size and/or structure of hydrocarbon molecules. These processes include: − Decomposition by thermal and catalytic cracking; − Unification through alkylation and polymerization; − Alteration with isomerization and catalytic reforming. ...
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CRUDE OIL DISTILLATION (FRACTIONATION) 1. Description The first step in the refining process is the separation of crude oil into various fractions or straightrun cuts by distillation in atmospheric and vacuum towers. The main fractions or "cuts" obtained can be classified in order of decreasing volatility into gases, light distillates, middle distillates, gas oils, and residuum. 2. Atmospheric Distillation Tower At the refinery, the desalted crude feedstock is preheated using recovered process heat. ...
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VACUUM DISTILLATION TOWER In order to further distill the residuum from the atmospheric tower at higher temperatures, reduced pressure is required to prevent thermal cracking. The process takes place in one vacuum distillation tower. The principles of vacuum distillation resemble those of fractional distillation and, except that largerdiameter columns are used to maintain comparable vapor velocities at the reduced pressures, the equipment is also similar.
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To help Ss review whole knowledge which they leant from unit 1 to unit 7  Ss can have the skill of doing examination tests II  language contents A. Grammar : fractions, Relative clause B. Vocabulary :  haunted III Techniques : Communicative approach. IV Teaching aids : Textbooks, color chalk, photos, ruler …
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Basidiomycetes constitute a natural source of biologically active metabolites. Many basidiomycetes have been classified by the National Cancer Institute of the United States as antitumor agents exhibiting an immunomodulatory activity.1 The therapeutic activity is mainly related to polysaccharides or proteinbound polysaccharides, such as glucans, heterogalactans, and glucanproteins, which are present either in the mycelium or in the fruit body.2–6 Among these polysaccharides are bDglucans, which are of particular interest because of their pharmacological properties.
6p the_eye_1991 18092012 28 5 Download

About 95% of the renewable energy sold to consumers through green pricing programs was supplied from projects meeting the generally accepted industry definition of “new.” Renewable energy sold through green pricing programs in 2008 represents an equivalent renewable energy capacity of more than 1,500 MW, with more than 1,400 MW of this represented by “new” renewable energy resources (Table 11).
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Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : On generalized SrivastavaOwa fractional operators in the unit disk
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The correctness of many systems and devices in our modern society depends not only on the effects or results they produce but also on the time at which these results are produced. These realtime systems range from the antilock braking controller in automobiles to the vitalsign monitor in hospital intensivecare units.
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In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Pad´ approximants e converges uniformly in compact subsets of the ball omitting poles of f . There is also apparently a cruder version of the conjecture due to Pad´ himself, going e back to the early twentieth century. We show here that for carefully chosen q on the unit circle, the RogersRamanujan continued fraction 1+ qz q 2 z q 3 z + + + ··· 1 1 1 provides a counterexample to the conjecture. We also highlight some...
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It is known that nonergodic directions in a rational billiard form a subset of the unit circle with Hausdorﬀ dimension at most 1/2. Explicit examples realizing the dimension 1/2 are constructed using Diophantine numbers and continued fractions. A lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction.
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Chapter 14 Biomass Chronosequences of United States Forests: Implications for Carbon Storage and Forest Management Forest Management and Carbon Sequestration Forests account for a large fraction of the carbon stored in global soils and vegetation (Dixon et al. 1994).
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When the majority is very large compared to the minority  in the United States whites are nine times as numerous and have much more human and physical capital per capita than blacks  market discrimination by the majority hardly lowers their incomes, but may greatly reduce the incomes of the minority. However, when minority members are a sizable fraction of the total, discrimination by the majority injures them as well.
60p trinhcaidat 19042013 23 3 Download

Chitosan oligomers were prepared by oxidative degradation of chitosan (with hydrogen peroxide) under microwave irradiation. This is convenient, friendly and effective method to prepare chitosan oligomers. By 1 HNMR, IR spectrum, the structure of the products was identified and keeps the same as of chitosan. By LCMS analysis, it proved that chitosan was decomposed to five fractions, the smallest one about D glucosamine 3 units (at least).
5p uocvong02 24092015 11 3 Download

Poor Families in America’sHealth Care Crisis examines the implications of the fragmented and twotiered health insurance system in the United States for the health care access of lowincome families. For a large fraction of Americans, their jobs do not provide health insurance or other beneﬁts, and although government programs are available for children, adults without private health care coverage have few options.
270p lyly_5 22032013 25 2 Download

While the link between demographic change and stock prices is a positive question, the second issue that we investigate is normative: whether the United States is “saving enough for its old age.” The coming increase in the ratio of dependent elderly to working age adults will, ceteris paribus, lower the average level of consumption. In the face of this coming demographic shock, it may be optimal for the country to save a higher fraction of its income, in order to smooth consumption.
18p quaivattim 04122012 25 1 Download

Antifreeze glycoproteins (AFGPs) constitute the major fraction of protein in the blood serum of Antarctic notothenioids and Arctic cod. Each AFGP consists of a varying number of repeating units of (AlaAlaThr)n, withminor sequence variations, and the disaccharide bDgalactosyl(1fi3)aNacetylDgalactosamine joined as a glycoside to the hydroxyl oxygen of the Thr residues. These compounds allow the fish to survive in subzero iceladen polar oceans by kineticallydepressing the temperatureatwhich icegrows ina noncolligative manner....
12p tumor12 20042013 19 1 Download