Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

The original algorithm was described only for natural numbers and geometric lengths ( real numbers ), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials in one variable.

The Euclidean algorithm has been generalized further to other mathematical structures, such as knots and multivariate polynomials.

generalized the algorithm to digital waveguide synthesis, which could also be used to model acoustic waves in tubes and on drum membranes.

This generalized version of the game is NP-complete ; it is unlikely that any algorithm more efficient than a brute-force search exists that can find solutions for arbitrary generalized FreeCell configurations.

replaced this with a more efficient idea that later generalized as the " subgroup algorithm " ( in which form it works just as well for permutations and rotations ).

The Rete algorithm provides a generalized logical description of an implementation of functionality responsible for matching data tuples (" facts ") against productions (" rules ") in a pattern-matching production system ( a category of rule engine ).

Its original version, due to Gary L. Miller, is deterministic, but the determinism relies on the unproven generalized Riemann hypothesis ; Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm.

One drawback is that, like Freenet, DHTs only directly support exact-match search, rather than keyword search, although Freenet's routing algorithm can be generalized to any key type where a closeness operation can be defined.

A generalized algorithm to correct for the pressure level difference between the finger and brachial sites within an individual patient was developed and this correction worked under all circumstances that it was tested, even when it was not designed for it, since it applied general physiological principles.

Bruun's algorithm is a fast Fourier transform ( FFT ) algorithm based on an unusual recursive polynomial-factorization approach, proposed for powers of two by G. Bruun in 1978 and generalized to arbitrary even composite sizes by H. Murakami in 1996.

The Bruun factorization, and thus the Bruun FFT algorithm, was generalized to handle arbitrary even composite lengths, i. e. dividing the polynomial degree by an arbitrary radix ( factor ), as follows.

Lloyd's Method I algorithm, originally described in 1957, can be generalized in a straighforward way for application to vector data.

The generalized Euclidean algorithm identifies the greatest common right or left divisor of two Hurwitz quaternions, where the " size " of the remainder is measured by the norm.

While Peterson's original formulation worked with only two processes, the algorithm can be generalized for more than two.

This is further generalized by DNA sequence alignment algorithms such as the Smith – Waterman algorithm, which make an operation's cost depend on where it is applied.

* 1996-Bruun's algorithm generalized to arbitrary even composite sizes by H. Murakami

In 1992, Deutsch and Jozsa produced a deterministic algorithm which was generalized to a function which takes bits for its input.

Miriam had not yet goaded him into mentioning her directly, but one can feel the generalized anger in Wright's remarks to reporters when he was asked, one morning on arrival in Chicago, what he thought of the city as a whole.

Although the Af calculation is obvious by analogy with that for gravitational field and osmotic pressure, it is interesting to confirm it by a method which can be generalized to include related effects.

Note that the acid can be the charged species and the conjugate base can be neutral in which case the generalized reaction scheme could be written as HA < sup >+</ sup > H < sup >+</ sup > + A.

Anxiety is a generalized mood that can occur without an identifiable triggering stimulus.

Associativity can be generalized to n-ary operations.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

The Bessel functions can be expressed in terms of the generalized hypergeometric series as

( This can be generalized to non-integer order by contour integration or other methods.

The exponent r can be generalized to an arbitrary real number as follows: if x > − 1, then

The notion can be generalized to other spaces as described below.

The effects of this study can be observed in humans with amnesia, indicating the role of the hippocampus in developing episodic memories that can be generalized to similar situations.

It can also be generalized to explain variables across the economy, for example, total output ( estimated as real GDP ) and the general price level, as studied in macroeconomics.

The same construction can be generalized to the field of fractions of any integral domain.

The result can be considered as a type of generalized geometry, projective geometry, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.

η is the symbol for the linear predictor of a generalized linear model, and can also be used to denote the median of a population.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

In the presence of non-spherical errors, the generalized least squares estimator can be shown to be BLUE.

This result can sometimes substitute for the still-unproved generalized Riemann hypothesis.

What can be even more confusing is that certain syndromes, such as West syndrome, featuring seizures such as infantile spasms, can be classified as idiopathic, syndromic, or cryptogenic depending on cause and can arise from both focal or generalized epileptic lesions.

In addition, it would be difficult to imagine chronic generalized emphysema occurring in a cow, considering its marked lobular development but, conversely, not difficult to imagine this occurring in the horse or the dog.

If this be true, the possibility exists that an occlusive lesion of the bronchial arteries might cause widespread degeneration of supportive tissue similar to that seen in generalized emphysema.

In addition, a detailed interspecies survey of the incidence of generalized pulmonary emphysema in mammals would be interesting and pertinent.

It has been shown to be less effective than benzodiazepines in the treatment of generalized anxiety disorder, while producing less side-effects.

Cognitive behavioral therapy ( CBT ) has been found to be effective treatment for panic disorder, social anxiety disorder, generalized anxiety disorder, and obsessive-compulsive disorder.

CBT has been shown to be effective in the treatment of generalized anxiety disorder, and possibly more effective than pharmacological treatments in the long term.

Benzodiazepines have robust efficacy in the short-term management of generalized anxiety disorder ( GAD ), but were not shown to be effective in producing long-term improvement overall.

All the above approaches tend to be generalized to the form of integrated computational models of a synthetic / abstract intelligence, in order to be applied to the explanation and improvement of individual and social / organizational decision-making and reasoning.

These fundamental similarities and differences provide a unifying theme, sometimes allowing the principles learned from studying one cell type to be extrapolated and generalized to other cell types.

Not surprisingly, it is impractical for generalized correctness, which probably cannot even be defined, much less proven.

Most libraries create a separate fiction section to allow shelving in a more generalized fashion than Dewey provides for, or to avoid the space that would be taken up in the 800s, or simply to allow readers to find preferred authors by alphabetical order of surname.