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Page "Weight (representation theory)" ¶ 8
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weight and on
He grabbed her by the shoulders and went down on one knee, taking her weight so that some of the wind was driven out of him.
The street that is full now of traffic and parked cars then and for many years drowsed on an August afternoon in the shade of the curbside trees, and silence was a weight, almost palpable, in the air.
There are millions who accept this doctrine, but few indeed are those who accept it so truly that the fate of humanity lies as a weight on their souls night and day.
He feels, suddenly, the weight of the fat that is on him.
It went right on creaking under his own considerable weight, and all it needed, Harold thought, was for somebody to fling himself back in a fit of laughter and that would be the end of it.
The ledger was full of most precise information: date of laying, length of incubation period, number of chick reaching the first week, second week, fifth week, weight of hen, size of rooster's wattles and so on, all scrawled out in a hand that looked more Chinese than English, the most jagged and sprawling Alex had ever seen.
Now when Henri was just 12 he was only 4' 10'' '' tall and weighed an astounding 72 pounds, and his greatest desire was to pack on some weight.
Start with the class standing in a circle, with weight on the right foot and the left extended a little way into the circle.
beef calves -- to 1.5 grams daily depending on weight of animal.
These widely advertised products, which are used primarily for washing clothes, are based on high-sudsing, synthetic organic actives ( sodium alkylbenzenesulfonates ) and contain up to 50% by weight of sodium tripolyphosphate or a mixture of sodium tripolyphosphate and tetrasodium pyrophosphate.
At the present time we do not know by what biochemical mechanism TSH acts on the thyroid, but for bio-assay of the hormone there are a number of properties by which its activity may be estimated, including release of iodine from the thyroid, increase in thyroid weight, increase in mean height of the follicular cells and increase in the thyroidal uptake of Af.
Besides, auditors do not count on faculty load with the same weight as regularly enrolled students.
Final ratings were made on the basis of a point system which was developed after studying the distributions of actual behaviors recorded and assigning weight values to each type of behavior that was deviant from the discovered norms.
Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.
Is it not the obligation of us older citizens to lend our weight to being creative on the character side and to hasten our own maturing process??
All these evaluations are then totted up and tabulated, by adding up the Hits and Significants, with the weight placed on those in the sitter's own reading.
Leg cramps, one person tells me, were relieved by standing barefoot with the weight of the body on the heel and pressing down hard.
He was fuzzy in his mind and, for a moment, helpless on the lobby floor, but he was conscious, and free of the weight of Roberts' body.
In the quantity of 900 calories a day, anyone will lose weight on it -- 20, 30 or 40 lbs. ''.
And the bed that sagged in a certain place where all the weight had been put too many times before and the walls fine and thin for overhearing talk in the next room when Gratt went out for ice, the sound coming through the walls like something on the other side of the curtain, so you knew they heard you when they were quiet and while you lay wondering what they had heard you listened.
The unbearable weight on his chest??
This is an anchor which relies solely on being a heavy weight.
Lowering a concentrated, heavy weight down the anchor line – rope or chain – directly in front of the bow to the seabed, behaves like a heavy chain rode and lowers the angle of pull on the anchor.
Common forms of antibacterial misuse include excessive use of prophylactic antibiotics in travelers and failure of medical professionals to prescribe the correct dosage of antibacterials on the basis of the patient's weight and history of prior use.
In any case the vests were abandoned by many soldiers due to their weight on long marches as well as the stigma they got for being cowards from their fellow troops.

weight and Lie
If G is a Lie group or an algebraic group, then a multiplicative character θ: G F < sup >×</ sup > induces a weight χ
Let V be a representation of a Lie algebra g over a field F and let λ be a weight of g. Then the weight space of V with weight λ: g F is the subspace
Similarly, we can define a weight space V < sub > λ </ sub > for any representation of a Lie group or an associative algebra.
Similar definitions apply to a Lie group G, a maximal commutative Lie subgroup H and any representation V of G. Clearly, if λ is a weight of the representation V of G, it is also a weight of V as a representation of the Lie algebra g of G.
A weight λ of the Lie group G is called integral, if for each t in h such that.
The definition can be extended further to tensor densities of weight w for any real w. If T is such a tensor density, then its Lie derivative is a tensor density of the same type and weight.
Semisimple Lie algebras are classified in terms of the weights of the adjoint representation, the so-called root system ; in a similar manner all finite-dimensional irreducible representations can be understood in terms of weights ; see weight ( representation theory ) for details.
Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U < sub > q </ sub >( G ), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators ( where λ is an element of the weight lattice, i. e. for all i ), and and ( for simple roots, ), subject to the following relations:

weight and algebra
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to Fa linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.
If G is a Kac-Moody algebra, then in any irreducible highest weight representation of U < sub > q </ sub >( G ), with highest weight, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of U ( G ) with equal highest weight.

weight and g
At 2. 5 troy oz ( 78 g ) gold, this is the largest ( by weight ) coin ever produced by the U. S. Mint.
A slab of rock with an iron staple in it to attach a chain to would serve the purpose, as would any dense object of appropriate weight ( e. g. an engine block ).
The weight of a Roman pound is generally believed to have been 327. 45 g or possibly up to 5 g less.
Finally, weight of the chain is represented by ( 0, − λgs ) where λ is the mass per unit length, g is the acceleration of gravity and s is the length of chain between c and r.
If an arithmetic mean was used instead of a geometric mean, the financial viability is given more weight because its numeric range is larger-so a small percentage change in the financial rating ( e. g. going from 80 to 90 ) makes a much larger difference in the arithmetic mean than a large percentage change in environmental sustainability ( e. g. going from 2 to 5 ).
Each subunit has a molecular weight of about 17, 000 daltons, for a total molecular weight of the tetramer of about 64, 000 daltons ( 64, 458 g / mol ).
Approximate weight is 150 g and has good standing ability for up to 30 days after maturity.
The Krugerrand's actual weight is 1. 0909 troy ounces ( 33. 93 g ).
The remaining 8. 33 % of the coin's weight ( 2. 826 g ) is copper ( an alloy known historically as crown gold which has long been used for English gold sovereigns ), which gives the Krugerrand a more orange appearance than silver-alloyed gold coins.
Weight is the gravitational force acting on a given body — which differs depending on the gravitational pull of the opposing body ( e. g. a person's weight on Earth vs on the Moon ) — while mass is an intrinsic property of that body that never changes.
Currently the entire Earth contains over 75 billion tons ( 150 trillion pounds or about 6. 8 x 10 < sup > 13 </ sup > kilograms ) of biomass ( life ), which lives within various environments within the biosphere .< ref > The figure " about one-half of one percent " takes into account the following ( See, e. g.,, which takes global average weight as 60 kg.
), the total human biomass is the average weight multiplied by the current human population of approximately 6. 5 billion ( see, e. g., ): Assuming 60 – 70 kg to be the average human mass ( approximately 130 – 150 lb on the average ), an approximation of total global human mass of between 390 billion ( 390 × 10 < sup > 9 </ sup >) and 455 billion kg ( between 845 billion and 975 billion lb, or about 423 million – 488 million short tons ).
Normal penicillin has a molecular weight of 313 to 334 g / mol ( latter for penicillin G ).
In the liver hepatocytes, glycogen can compose up to eight percent of the fresh weight ( 100 – 120 g in an adult ) soon after a meal.
Silver's atomic weight is 107. 8682 ( 2 ) g / mol.
Methyl-2-cyanoacrylate ( CH < sub > 2 </ sub >= C ( CN ) COOCH < sub > 3 </ sub > or C < sub > 5 </ sub > H < sub > 5 </ sub > NO < sub > 2 </ sub >) has a molecular weight equal to 111. 1, a flashpoint of 79 ° C, and a density of 1. 1 g / ml.
FIVB regulations state that the ball must be spherical, made of leather or synthetic leather, have a circumference of 65 – 67 cm, a weight of 260 – 280 g and an inside pressure of 0. 30 – 0. 325 kg / cm < sup > 2 </ sup >.
In particular, Newton considered weight to be relative to another object causing the gravitational pull, e. g. the weight of the Earth towards the Sun.
Alkylresorcinols are phenolic lipids present in high amounts in the bran layer ( e. g. pericarp, testa and aleurone layers ) of wheat and rye ( 0. 1-0. 3 % of dry weight ).
For example, in the brewing industry, the Plato table, which lists sucrose concentration by weight against true SG, were originally ( 20 ° C / 4 ° C ) that is based on measurements of the density of sucrose solutions made at laboratory temperature ( 20 ° C ) but referenced to the density of water at 4 ° C which is very close to the temperature at which water has its maximum density of ρ () equal to 0. 999972 g / cm < sup > 3 </ sup > ( or 62. 43 lb < sub > m </ sub >· ft < sup >− 3 </ sup >).
This weight is equal to the mass of liquid displaced multiplied by g, which in the case of the reference liquid is ρ < sub > ref </ sub > Vg.

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