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 r-color 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....
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. ...
CRUDE OIL DISTILLATION (FRACTIONATION)
1. Description The first step in the refining process is the separation of crude oil into various fractions or straight-run 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. ...
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 larger-diameter columns are used to maintain comparable vapor velocities at the reduced pressures, the equipment is also similar.
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 …
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 protein-bound
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
b-D-glucans, which are of particular interest because of
their pharmacological properties.
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).
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 real-time systems range from the anti-lock braking controller in automobiles to the vital-sign monitor in hospital intensive-care units.
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 Rogers-Ramanujan continued fraction 1+ qz| q 2 z| q 3 z| + + + ··· |1 |1 |1
provides a counterexample to the conjecture. We also highlight some...
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.
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).
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.
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 H-NMR, IR spectrum, the structure of the products was identified and keeps the same as of chitosan. By LC-MS analysis, it proved that chitosan was decomposed to five fractions, the smallest one about D- glucosamine 3 units (at least).
Poor Families in America’sHealth Care Crisis examines the implications
of the fragmented and two-tiered health insurance system in the United
States for the health care access of low-income 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.
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.
Antifreeze glycoproteins (AFGPs) constitute the major
fraction of protein in the blood serum of Antarctic noto-thenioids and Arctic cod. Each AFGP consists of a varying
number of repeating units of (Ala-Ala-Thr)n, withminor
sequence variations, and the disaccharide b-D-galactosyl-(1fi3)-a-N-acetyl-D-galactosamine joined as a glycoside to
the hydroxyl oxygen of the Thr residues. These compounds
allow the fish to survive in subzero ice-laden polar oceans by
kineticallydepressing the temperatureatwhich icegrows ina