If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.

If a polynomial time algorithm calls as a subroutine polynomially many polynomial time algorithms, the resulting algorithm is still polynomial time.

If a " weak " character is followed by another " weak " character, the algorithm will look at the first neighbouring " strong " character.

If the algorithm is an asymmetric one, the enciphering key is different from, but closely related to, the deciphering key.

If one key cannot be deduced from the other, the asymmetric key algorithm has the public / private key property and one of the keys may be made public without loss of confidentiality.

If a suitably sized quantum computer capable of running Grover's algorithm reliably becomes available, it would reduce a 128-bit key down to 64-bit security, roughly a DES equivalent.

If a naive rendering algorithm is used without any filtering, high frequencies in the image function will cause ugly aliasing to be present in the final image.

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

If a is smaller than b, the first step of the algorithm swaps the numbers.

If the arguments are both greater than zero then the algorithm can be written in more elementary terms as follows:

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

If the algorithm is unable to reduce A to triangular form, then A is not invertible.

If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i. e., that can factor it in time O ( b < sup > k </ sup >) for some constant k. There are published algorithms that are faster than O (( 1 + ε )< sup > b </ sup >) for all positive ε, i. e., sub-exponential.

If the true function is simple, then an " inflexible " learning algorithm with high bias and low variance will be able to learn it from a small amount of data.

If the desired output values are often incorrect ( because of human error or sensor errors ), then the learning algorithm should not attempt to find a function that exactly matches the training examples.

* Robustness: If the model, cost function and learning algorithm are selected appropriately the resulting ANN can be extremely robust.

If using efficient sorting such as mergesort or heapsort this is an O ( n log n ) average and worst-case algorithm.

The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.

If we apply the run-length encoding ( RLE ) data compression algorithm to the above hypothetical scan line, we get the following:

If not, is there an algorithm to determine how many are required?

If the user wanted to see a font at another size, the Font Manager would find the closest match and apply a basic scaling algorithm.

If a particle visits every point on its trajectory then the motion is an algorithm for each point.

If two processes attempt to enter a critical section at the same time, the algorithm will allow only one process in, based on whose turn it is.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If and we have for all v, w in V, then we say that B is symmetric.

If a relation is Euclidean and reflexive, it is also symmetric and transitive.

If a game has a board, the board is usually irregular rather than uniform or symmetric ( like Risk rather than chess or Scrabble ); the board is often random ( like Settlers of Catan ) or has random elements ( like Tikal ).

* If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S < sub > p </ sub >.

If the waves are propagating through an anisotropic ( i. e., not rotationally symmetric ) medium, for example a crystal, then the phase velocity vector and group velocity vector may point in different directions.

If G is a group and X is a set then a group action may be defined as a group homomorphism h from G to the symmetric group of X.

If confirmed, this would indicate that such symmetric fusion reactions should be modelled as ' hot fusion ' reactions rather than ' cold fusion ' ones, as first suggested.

If the quantum state is initially symmetric ( antisymmetric ), it will remain symmetric ( antisymmetric ) as the system evolves.

If the particles are bosons, they occupy a totally symmetric state, which is symmetric under the exchange of any two particle labels:

If the particles are bosons ( fermions ), the state after the measurement must remain symmetric ( antisymmetric ), i. e.

If they did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields ; see magnetic monopoles for details.

If the distribution is symmetric then the mean is equal to the median and the distribution will have close to zero skewness.

< li > If is a symmetric matrix of the form, and the strict inequality holds

If the problem is cylindrically symmetric, the natural solutions of the paraxial wave equation are Laguerre-Gaussian modes.

If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the shell theorem.

( If there is torsion, then the tensor is no longer symmetric.

If A is a real matrix, this is equivalent to A < sup > T </ sup > = A ( that is, A is a symmetric matrix ).

If A and B are n × n real symmetric matrices that commute, then there exists a basis of such that every element of the basis is an eigenvector for both A and B.

If A < sup >− 1 </ sup > exists, it is symmetric if and only if A is symmetric.

If Sym < sub > n </ sub > denotes the space of symmetric matrices and Skew < sub > n </ sub > the space of skew-symmetric matrices then since and

If A has real entries and is symmetric ( or more generally, has complex-valued entries and is Hermitian ) and positive definite, then A can be decomposed as