While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stageira, who bore him a son whom he named after his father, Nicomachus.

While there is no generally accepted formal definition of " algorithm ," an informal definition could be " a set of rules that precisely defines a sequence of operations.

While this loss of precision may be tolerable and viewed as a minor flaw of Algorithm I, it is easy to find data that reveal a major flaw in the naive algorithm: Take the sample to be ( 10 < sup > 9 </ sup > + 4, 10 < sup > 9 </ sup > + 7, 10 < sup > 9 </ sup > + 13, 10 < sup > 9 </ sup > + 16 ).

* While the decision problem is NP-complete the optimization problem is NP-hard, its resolution is at least as difficult as the decision problem, and there is no known polynomial algorithm which can tell, given a solution, whether it is optimal ( which would mean that there is no solution with a larger, thus solving the decision problem NP-complete ).

While, in principle, any general-purpose lossless compression algorithm ( general-purpose meaning that they can accept any bitstring ) can be used on any type of data, many are unable to achieve significant compression on data that are not of the form for which they were designed to compress.

While the patent was initially rejected by the patent office as being a purely mathematical invention, following 12 years of appeals, Pardo and Landau won a landmark court case at the CCPA ( Predecessor Court of the Federal Circuit ) overturning the Patent Office in 1983 — establishing that " something does not cease to become patentable merely because the point of novelty is in an algorithm.

While algorithms such as Wu's algorithm are also frequently used in modern computer graphics because they can support antialiasing, the speed and simplicity of Bresenham's line algorithm means that it is still important.

While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the ( philosophical ) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.

While it is possible to solve any of these problems using the simplex algorithm, each specialization has more efficient algorithms designed to take advantage of its special structure.

While the rsync algorithm forms the heart of the rsync application that essentially optimizes transfers between two computers over TCP / IP, the rsync application supports other key features that aid significantly in data transfers or backup.

While z-buffering allows the geometry to be unsorted, sorting polygons by increasing depth ( thus using a reverse painter's algorithm ) allows each screen pixel to be rendered fewer times.

While creating this new path, the algorithm decides how many new ' nodes ' to add and whether or not these new nodes will actually create a new path.

While a clear description of the algorithm on computers appeared in 1946 in an article by John Mauchly, the idea of using a sorted list of items to facilitate searching dates back at least as far as Babylonia in 200 BC.

While the solution to a seven-node instance is trivial, this paper is the first known instance of the successful use of DNA to compute an algorithm.

While there, Rabin invented the Miller-Rabin primality test, a randomized algorithm that can determine very quickly ( but with a tiny probability of error ) whether a number is prime.

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

While the graph traversal in Fleury's algorithm is linear in the number of edges, i. e. O (| E |), we also need to factor in the complexity of detecting bridges.

( While one can " delete " an element from an array in constant time by somehow marking its slot as " vacant ", an algorithm that iterates over the elements may have to skip a large number of vacant slots ).

While it is OK for the case of the topmost vertex in the example or the vertex between crossing 4 and 5, the case of the rightmost vertex ( in the example ) requires that we count one intersection for the algorithm to work correctly.

While this algorithm allows point location in logarithmic time and is easy to implement, the space required to build the slabs and the regions contained within the slabs can be as high as O ( n² ), since each slab can cross a significant fraction of the segments.

While CSA algorithm uses 64-bit keys, most of the time only 48 bits of key are unknown, since bytes 3 and 7 are used as checksum bytes in CA systems, and may be easily recalculated.

While the BATON algorithm itself is secret, the public PKCS # 11 standard includes some general information about how it is used.

While it would appear that there is no key ( the key space being therefore empty ), in fact a constant key, implied in the algorithm, is used.

While not entirely rigorous, his proof that Morrison and Brillhart's continued fraction factoring algorithm ran in roughly steps was an important milestone in factoring and laid a foundation for much later work, including the current " champion " factoring algorithm, the Number Field Sieve.

While the pattern is uneven, some having gained more than others, nationalism has in fact served the Western peoples well.

While sovereignty has roots in antiquity, in its present usage it is essentially modern.

While it is hazardous to project the trend of history, it seems clear that a genuine community is painfully emerging in the Western world, particularly among the countries of Western Europe.

While he is worldly, the musician often cultivates public attitudes of childlike astonishment and naivete.

While some think we move too fast and others too slowly, Florida's record is a good one and stands out among the 50.

While white is the coolest summer shade, there are lots of pastel hues along with tintable fabrics that will blend with any wardrobe color.

While it is easy enough to ridicule Hawkins' pronouncement in Pleas Of The Crown from a metaphysical point of view, the concept of the `` oneness '' of a married couple may reflect an abiding belief that the communion between husband and wife is such that their actions are not always to be regarded by the criminal law as if there were no marriage.

While this may well be true in general, I believe it is also important to keep in mind that some recent developments suggest that over the next year or so military electronics may be one of the most strongly growing areas in an economy which is not expanding rapidly in other directions.

While clay is still pressed in mold, press three equally spaced holes 1/4'' '' deep, using pencil eraser, in bottom of clay to allow for proper drying and firing.

While assembled lid is still on design head, gently but firmly press it on plaster board.

While some regression is inevitable, it is discouraged rather than encouraged so that the transference does not follow the stages of planned regression associated with certain casework adaptations of the psychoanalytic model for insight therapy.

While other conditions might be even more effective in bringing about a change from immobility to mobility in Kohnstamm reactivity, it is our hypothesis that all such conditions would have as a common factor the capacity to induce an attitude in the subject which enabled him to divorce himself temporarily from feelings of responsibility for his behavior.

While we had expected that compulsive children in the unstructured school setting would have difficulty when compared to those in the structured, we were surprised to find that the achievement of the high compulsives within the schools where the whole-word method is used in beginning reading compares favorably with that of the low compulsives.

While roleplaying for testing is not too well understood at the present time, it represents one of the major uses of this procedure.

While the phonemes can be very easily stated, no one is likely to be satisfied with the statement until phonemic occurrences can be related in some way to morphemic units, i.e. until the morphophonemics is worked out, or at least far enough that it seems reasonable to expect success.

While the U. S. Department of Labor has a program of projecting industry and occupational employment trends and publishing current outlook statements, there is little tangible evidence that these projections have been used extensively in local curriculum planning.

While it had long been known in general, that `` the public is always wrong '', the use of odd-lot indices now puts the adage on a statistical basis.

While there may be several such industries to which the model of this paper is applicable, the authors make particular claim of relevance to the explanation of the course of wages and prices in the steel industry of the United States since World War 2.

While this influence is a complex matter, depending upon personality factors in the individual as well as upon his social-class experience, there probably are some general statements about social-class background and educational policy that can be made with a fair degree of truth.

While one element is announcing progress, another is delineating its problems.