The method became known as the royal road to the calculus of variations.More recently the same idea has been taken into the theory of optimal control.The method can also be extended to multiple integrals.…

Either two or three players could participate in a game.In two-person play, the first player will win if there are two optimal players.…

Moreover, in operator theory there are specific instances of Krein spaces called Pontryagin spaces.Later in his career he worked in optimal control theory.His maximum principle is fundamental to the modern theory of optimization.…

Additional hours of sleep do not add to the initial level of performance.Thus this study suggests that full sleep may not be important for optimal performance of memory.Another study shows that people who experience SWS during the first half of their sleep cycle compared to subjects who did not, showed better recall of information.…

A typical example for an approximation algorithm is the one for vertex cover in graphs: find an uncovered edge and add both endpoints to the vertex cover, until none remain.It is clear that the resulting cover is at most twice as large as the optimal one.This is a constant factor approximation algorithm with a factor of 2.…

Some slime molds and small nematodes can migrate along amazingly small temperature gradients of less than 0.1C/cm.They apparently use this behavior to move to an optimal level in soil.Thigmotaxis is the response of an organism to physical contact or to the proximity of a physical discontinuity in the environment (e.g. rats preferring to swim near the edge of a water maze).…

This was due to the pattern of interaction: Typically an individual user entered bursts of information followed by long pauses but a group of users working at the same time would mean that the pauses of one user would be filled by the activity of the others.Given an optimal group size, the overall process could be very efficient.Similarly, small slices of time spent waiting for disk, tape, or network input could be granted to other users.…

Cousteau changed course a few degrees off the optimal course to the center of the strait, and the porpoises followed for a few minutes, then diverged toward mid-channel again.It was evident that they knew where the optimal course lay, even if the humans did not.Cousteau concluded that the cetaceans had something like sonar, which was a relatively new feature on submarines.…

Further, the solutions of the LP relaxations can be used to provide a worst-case estimate of how far from optimality the returned solution is.Finally, branch and bound methods can be used to return multiple optimal solutions.Lenstra in 1983 showed, that when number of variables is fixed, integer programming problem can be solved in a polynomial time.…

Investment in health is costly as consumers must trade off time and resources devoted to health, such as exercising at a local gym, against other goals.These factors are used to determine the optimal level of health that an individual will demand.The model makes predictions over the effects of changes in prices of health care and other goods, labour market outcomes such as employment and wages, and technological changes.…

The optimal size (and composition) of teams is debated and will vary depending on the task at hand.At least one study of problem-solving in groups showed an optimal size of groups at four members.Other works estimate the optimal size between 5-12 members.…

Microalgae such as Spirulina, a health food supplement, also can be cultivated.Deep-ocean water can be combined with surface water to deliver water at an optimal temperature.Non-native species such as salmon, lobster, abalone, trout, oysters, and clams can be raised in pools supplied by OTEC-pumped water.…

At each level of income level, say m, there would an optimal choice for each of the goods.

The resulting proof gives an optimal strategy for every possible position on the board.However, these proofs aren't as helpful in understanding deeper reasons why some games are solvable as a draw, and other, seemingly very similar games are solvable as a win.…

In voting systems, the Smith set, named after John H. Smith, is the smallest non-empty set of candidates in a particular election such that each member defeats every other candidate outside the set in a pairwise election.The Smith set provides one standard of optimal choice for an election outcome.Voting systems that always elect a candidate from the Smith set pass the Smith criterion and are said to be "Smith-efficient".…

In an equilibrium state we see that markets creating negative externalities of production will overproduce that good.As a result, the socially optimal production level would be lower than that observed.When marginal social costs of production are less than that of the private cost function, we see the occurrence of a positive externality of production.…

In an equilibrium state we see that markets creating positive externalities of production will under produce that good.As a result, the socially optimal production level would be greater than that observed.Of great importance in the theory of marginal cost is the distinction between the marginal private and social costs.…

A norm gives a person a rule of thumb for how they should behave.However, a rational person only acts according to the rule if it is optimal for them.…

In each of these areas, there is considerable debate as to the optimal treatment.One of the major reasons for this is the lack of large-scale clinical trials comparing different treatments.…

While the packaging of melatonin often warns against use in children, available studies suggest that melatonin is an efficacious and safe treatment for ADHD and sleep-onset insomnia.However larger and longer studies are needed to establish long-term safety and optimal dosing.Melatonin appears to cause very few side-effects in the short term, up to three months, at low doses.…