Beginning with the theory of recursive sets and functions described above, the field of recursion theory has grown to include the study of many closely related topics.

Field automorphisms are important to the theory of field extensions, in particular Galois extensions.

Acoustic theory is the field relating to mathematical description of sound waves.

A unified interpretation of antiparticles is now available in quantum field theory, which solves both these problems.

This is an example of renormalization in quantum field theory — the field theory being necessary because the number of particles changes from one to two and back again.

In quantum field theory, this process is allowed only as an intermediate quantum state for times short enough that the violation of energy conservation can be accommodated by the uncertainty principle.

These processes are important in the vacuum state and renormalization of a quantum field theory.

This section draws upon the ideas, language and notation of canonical quantization of a quantum field theory.

This technique is the most widespread method of computing amplitudes in quantum field theory today.

* Quantum field theory and the list of particles

His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations.

Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

His theory of schemes has become established as the best universal foundation for this major field, because of its great expressive power as well as technical depth.

The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis ( the symmetrical axis of the system ) are infinitely small, i. e. with infinitesimal objects, images and lenses ; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.

Until his return from Mexico in 1930 Whorf had been entirely an autodidact in linguistic theory and field methodology, yet he had already made a name for himself in Middle American linguistics.

Since a leg theory attack means the batsman is more likely to hit the ball on the leg side, additional fielders on that side of the field can be effective in preventing runs and taking catches.

* BCS theory correctly predicts the Meissner effect, i. e. the expulsion of a magnetic field from the superconductor and the variation of the penetration depth ( the extent of the screening currents flowing below the metal's surface ) with temperature.

BCS theory relates the value of the critical field at zero temperature to the value of the transition temperature and the density of states at the Fermi energy.

The analogy was completed when Hawking, in 1974, showed that quantum field theory predicts that black holes should radiate like a black body with a temperature proportional to the surface gravity of the black hole.

Unital Banach algebras over the complex field provide a general setting to study spectral theory.

Today, when mathematicians talk about " set theory " as a field, they usually mean axiomatic set theory.

Frundsberg had meanwhile mauled the heavily outnumbered Swiss infantry opposing him ; Tiercelin and Flourance were unable to hold their troops together, and the French foot began to flee the field.

British Colonel McDouall meanwhile placed a small force bearing muskets, rifles, and two field guns, behind low breastworks at the opposite end of the clearing.

Revson, meanwhile, was rocketing through the field from his last place start, and at the midway point, he had gone from twenty-third to seventh.

Those of our boys who were carrying slings, meanwhile, provoked the enemy into advancing onto the field with the result that, being the more provoked by the slinging of stones from a distance, the enemy ventured a major attack.

The field of computational complexity categorizes decidable decision problems by how difficult they are to solve.

The use of a sequence of experiments, where the design of each may depend on the results of previous experiments, including the possible decision to stop experimenting, is within the scope of Sequential analysis, a field that was pioneered by Abraham Wald in the context of sequential tests of statistical hypotheses.

Jardine wrote that Larwood had asked for the field, while Larwood said that it was Jardine's decision.

Runoff advocates counter that voters first preference is more important than lower preferences because that's where voters are putting the most effort of decision and that, unlike Condorcet methods, runoffs require a high showing among the full field of choices in addition to a strong showing in the final head-to-head competition.

As triage concepts become more sophisticated, triage guidance is also evolving into both software and hardware decision support products for use by caregivers in both hospitals and the field.

Anderson postponed his decision to run, lost his campaign manager, and struggled to raise money, but in late April 1979 he made the decision to enter the Republican primary anyway, joining a crowded field that included Robert Dole, John Connally, Howard Baker, George H. W. Bush and Ronald Reagan.

The main decision for a fielding captain is to strike a balance between setting an attacking field and a defensive field.

Ellsberg's dissertation in the field of decision theory was based on a set of experiments that showed that, in general, decisions under conditions of uncertainty or ambiguity may not be consistent with well defined subjective probabilities.

They said in a statement that the decision is part of a wider strategic plan being implemented to improve all aspects of the club's operations, both on and off the field.

DIA came of age in the 1980s by focusing heavily on the intelligence needs of both field commanders and national-level decision makers.

Maxtor had initially made efforts to get into the 2. 5-inch hard disk market ( notebook computer format ) but, in the beginning of 2005, new management made the surprising decision to discontinue development in this field.

The difficult position of the Allies was confirmed by the decision to send in the Russian Imperial Guard ; Grand Duke Constantine, Tsar Alexander's brother, commanded the Guard and counterattacked in Vandamme's section of the field, forcing a bloody effort and the only loss of a French standard in the battle ( the unfortunate victim was a battalion of the 4th Line Regiment ).

By 1953, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions, and the field was thereafter recognized by the IEEE as a systems analysis and engineering topic.

The news on the death of Georg Wilhelm Richmann, a professor in St. Petersburg, who was killed by a lightning in 1753 during his attempt at measuring the intensity of the electric field in the atmosphere, caused Diviš to become interested in atmospheric electricity, and to the decision to construct a " weather-machine " at Přímětice.

More generalized in the field of economics, cost is a metric that is totaling up as a result of a process or as a differential for the result of a decision.

The difficult position of the Allies was confirmed by the decision to send in the Russian Imperial Guard ; Grand Duke Constantine, Tsar Alexander's brother, commanded the Guard and counterattacked in Vandamme's section of the field, forcing a bloody effort and the loss of the only French standard in the battle ( the unfortunate victim was a battalion of the 4th Line Regiment ).

This decision resulted in limiting troops in the field to a single class of packaged ration that despite meal variances, was neither suited to varied field environments nor long-term use.

In 1949, at the fourth session of general UNESCO conference, a decision was accepted to begin the work for international law regulation in the field of cultural heritage protection in case of armed conflict.

1949 – 4th session of general UNESCO conference has taken a decision to begin a work for international-law regulation in the field of cultural values protection during the military actions.

Radio snapshot observations with the VLA ruled out one of these fields because it contained a bright radio source, and the final decision between the other two was made on the basis of the availability of guide stars near the field: Hubble observations normally require a pair of nearby stars on which the telescope's Fine Guidance Sensors can lock during an exposure, but given the importance of the HDF observations, the working group required a second set of back-up guide stars.

The contract between the two teams stated that if there was a dispute as to the condition of the field, the decision would belong to the home team.