However, beyond this theoretical significance, efficient and scalable algorithms for SAT that were developed over the last decade have contributed to dramatic advances in our ability to automatically solve problem instances involving tens of thousands of variables and millions of constraints.

" Difficult ", in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem.

An application of Euclidean solid geometry is the determination of packing arrangements, such as the problem of finding the most efficient packing of spheres in n dimensions.

This new proof is similar to Appel and Haken's but more efficient because it reduced the complexity of the problem and required checking only 633 reducible configurations.

This argument, first proposed by Max Weber, says that the efficient exchange and use of resources can be maintained only through the price mechanism in free markets ( see economic calculation problem ).

The problem is simply whether a complex system, in which many components function unitedly together, and in which each component is uniquely necessary to the efficient functioning of the whole, could ever arise by random processes.

The Diffie – Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Its security is connected to the ( presumed ) extreme difficulty of factoring large integers, a problem for which there is no known efficient ( i. e. practicably fast ) general technique.

In various branches of mathematics, a useful construction is often viewed as the “ most efficient solution ” to a certain problem.

A laser could be used as a photon rocket engine, and would solve the reflection / collimation problem, but lasers are absolutely less efficient at converting energy into light than blackbody radiation is — though one should also note the benefits of lasers vs blackbody source, including unidirectional controllable beam and the mass and durability of the radiation source.

Informally, the " P vs. NP " question asks whether every optimization problem whose answers can be efficiently verified for correctness / optimality can be solved optimally with an efficient algorithm.

In some problem areas they are shown to be more efficient than traditional evolutionary algorithms.

Drag may also present a problem in flight, and the aerodynamically efficient body shapes of birds highlight this point.

It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic.

The idea of using an initial property is to set up the problem in terms of some auxiliary category E, and then identify that what we want is to find an initial object of E. This has an advantage that the optimization — the sense that we are finding the most efficient solution — means something rigorous and is recognisable, rather like the attainment of a supremum.

The existence of a morphism between R → S < sub > 1 </ sub > and R → S < sub > 2 </ sub > implies that S < sub > 1 </ sub > is at least as efficient a solution as S < sub > 2 </ sub > to our problem: S < sub > 2 </ sub > can have more adjoined elements and / or more relations not imposed by axioms than S < sub > 1 </ sub >.

Therefore, the assertion that an object R → R * is initial in E, that is, that there is a morphism from it to any other element of E, means that the ring R * is a most efficient solution to our problem.

Finding an efficient way to parallelize the solution to some P-complete problem would show that NC = P. It can also be thought of as the " problems requiring superlogarithmic space "; a log-space solution to a P-complete problem ( using the definition based on log-space reductions ) would imply L = P.

However, like the drinking-age problem above, the specific length of time would need to be different for every product in order to be efficient ; a 20-year term is used because it is difficult to tell what the number should be for any individual patent.

It can be argued that it is difficult to apply the standard efficient market hypothesis ( efficient market theory ) to understand the stock market bubble that ended in 2000 and collapsed thereafter ; however, advocates of rational expectations say that the problem of ascertaining all the pertinent effects of the stock-market crash is a great challenge.

The problem control process aims to handle problems in an efficient way.

" Informally, hypothesis boosting problem asks whether an efficient learning algorithm that outputs an hypothesis whose performance is only slightly better than random guessing a weak learner implies the existence of an efficient algorithm that outputs an hypothesis of arbitrary accuracy a strong learner.

In 1861, Mr. Brown's attention was called to yet another basic production problem -- the manufacture of twist drills.

Tank and vehicle production was a constant problem for Germany ; indeed, late in the war many panzer " divisions " had no more than a few dozen tanks.

This controller chip, however, was the seat of some early problems with 1581 drives when the first production runs were recalled due to a high failure rate ; the problem was quickly corrected.

The problem of planning production is the knowledge problem explained by Hayek ( 1937, 1945 ) The planning could either be done in a decentralised fashion, requiring some mechanism to make the individual plans coherent, or centrally, requiring a lot of information.

But central planners face the knowledge problem in forming a comprehensive plan for production.

The Swedish industry did not have this problem, as its production was more in balance with the market, and more importantly, the quality of its films was now superior to those from Denmark.

With Allen Newell, Simon developed a theory for the simulation of human problem solving behavior using production rules.

Another problem was that although Speer and Goebbels were allies, their agendas conflicted: Speer wanted absolute priority in the allocation of labour to be given to arms production, while Goebbels sought to press every able-bodied male into the army.

The demonstration of the relation between commodities ' unit values and their respective prices is known in Marxian terminology as the transformation problem or the transformation of values into prices of production.

Illicit production of cannabis for domestic and international drug markets continues to ask as an international problem, as do rumours that the country operates as a transit point for opiates from Southeast Asia to the West

In language similar to Traynor's, the Directive stated that " liability without fault on the part of the producer is the sole means of adequately solving the problem, peculiar to our age of increasing technicality, of a fair apportionment of the risks inherent in modern technological production.

" The structure of a dissertation consists in an introduction that elucidates the basic definitions involved in the question as set, followed by an argument or thesis, a counter-argument or antithesis, and a resolving argument or synthesis that is not a compromise between the former but the production of a new argument, ending with a conclusion that does not sum up the points but opens onto a new problem.

Stuttering is generally not a problem with the physical production of speech sounds or putting thoughts into words.

Whereas reusable SSTOs would reduce per launch costs by making a reusable high-tech vehicle that launches frequently with low maintenance, the " mass production " approach views the technical advances as a source of the cost problem in the first place.

This tendency within the 1918 left communists emphasized that the problem with capitalist production was that it treated workers as objects.

However, while the critique from the right emphasized moral degeneracy ascribed to sexual and racial influences within popular culture, Adorno located the problem not with the content, but with the objective realities of the production of mass culture and its effects, e. g. as a form of reverse psychology.

Language posed a problem ; four-letter curse words were uncommon in the theatre at the time, and slang expressions were avoided for fear they would be dated by the time the production opened.

Dwarfing genes enable the carbon that is fixed in the plant during photosynthesis to be diverted towards seed production, and they also help prevent the problem of lodging.

( SMT can pose quite a problem, at least an inconvenience, for prototyping in general ; most of the characteristics of SMT that are advantages for mass production are difficulties for prototyping.

Henry Bessemer worked on the problem of manufacturing cheap steel for the purposes of ordnance production from 1850 to 1855 when he patented his method.

However, diabetes insipidus is either a problem with the production of antidiuretic hormone ( central diabetes insipidus ) or kidney's response to antidiuretic hormone ( nephrogenic diabetes insipidus ), whereas diabetes mellitus causes polyuria via a process called osmotic diuresis, due to the high blood sugar leaking into the urine and taking excess water along with it.

Furthermore, local authorities frequently exaggerated production numbers, hiding and intensifying the problem for several years.

But the rest problem and social welfare function selection, as well as the subjective factors in behavioral finance, has led to a closer analysis of factors of production.