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 basic Bruun algorithm for powers of two factorizes z < sup > N </ sup >- 1 recursively via the rules:

* 1978-Bruun's algorithm proposed for powers of two by Georg Bruun

* Danish composer Jesper Kyd was commissioned by Danish Film Festival founders Christian Ditlev Bruun and Lene Pels Jorgensen to provide a new score for the Danish Film Festival: Los Angeles.

Conrad Malte-Brun ( 12 August 1755 – 14 December 1826 ), born Malthe Conrad Bruun, was a Danish-French geographer and journalist.

From August 1892 he was married to actress Constance Bruun, but she died in October 1894.

This was countered by Nils Henrik Bruun, a constructor from Bergen, who was willing to construct all tunnels on the railway for less than the budgeted sum.

When Jebsen in addition was willing to act as personal guarantee for Bruun in case of his death, the majority in the parliament shifted.

Bruun also concludes that Cicero, who remained the legal defender of Caelius, ultimately used the conceptual phrase aqua inceste uterere in “ referring to the commonly known possession of a water supply by some brothels in Rome, while at the same time implying that Clodia was a prostitute .” The body of Bruun ’ s Water for Roman Brothels is subdivided into multiple different subtopics ; the first one devoted to Cicero ’ s personification of Appius Claudius Caecus.

Brunn provides Frontius ’ De aquaeductu Urbis Romae from AD 100 as an example of the “ various illegal uses to which public water in Rome was being diverted .” Christer Bruun suggests that as a recent find by a contemporary author, Caelius actually gave a speech in 50 BC when he was a curule aedile, ultimately proclaiming “ the worst misappropriation of public water in Rome ” which was due “ all the brothels, were enjoying an illegal supply of running water .” Furthermore, Bruun concludes that although these possible arguments can explain why Cicero attempted to connect Clodia to immorality and water, he simply used this argumentation to suggest Clodia ’ s case against Caelius was unfounded.

Thomas Bruun Eriksen ( born 13 February 1979 ) was a Danish professional road bicycle racer who ended his career after the 2005 UCI ProTour season.

He was then given a scholarship, the " Gustav Bruun Endowment " of.

Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms.

This section covers the last design series of the Finnish markka, designed in the 1980s by Finnish designer Erik Bruun and issued in 1986.

Bruun in 1876, who suggested that Prester John might be found among the kings of Georgia, which, at the time of Crusades, experienced military resurgence challenging the Muslim power.

* Zim, Herbert Spencer ; Robbins, Chandler S .; Bruun, Bertel ( 2001 ) Birds of North America: A Guide to Field Identification Golden Publishing.

* Ole Bruun.

Ole Bruun and Arne Kalland ( Surrey: Curzon, 1995 ) 173 – 88.

* Ole Bruun.

* Ole Bruun.

Auctioneers Bruun Rasmussen held an auction of COBRA artists on April 3, 2006 in Copenhagen.

* Flipper Scow ( in Swedish ) The Danish constructed ( Peer Bruun ) dinghy class.

* Zim, Herbert Spencer ; Robbins, Chandler S .; Bruun, Bertel ( 2001 ): Birds of North America: A Guide to Field Identification.

Bruun and Hans J. Holm.

* Bruun, Geoffrey.

Bruun argues that within § 34 of Pro Caelio Cicero powerfully employs “ the oratorical technique of “ personification ” or “ speech in character ” ( prosopopeia ) and for a while pretended, apparently both by gestures and by voice, to be one of Clodia ’ s most famous ancestors, the censor Appius Claudius Caecus .” According to Bruun, Appius proclaims to have spurred three major civic accomplishments, while for each Cicero attempts to point out a reason why Clodia should be ashamed of herself for immorality connected with the Appian works.

Bruun finds this passage not sufficient, suggesting instead “ it seems baffling that the use of water, should have been connected to Clodia ’ s allegedly loose morals .” Again dispelling Cicero ’ s connection of Clodia to water and sexual immorality, Bruun proclaims this to be antithetical to Clodia ’ s case by stating, “ the evidence from the Roman world for ritual cleansing with water after sexual “ pollution ” is very meager and different in character .” Bruun argues within his next sub-point that more compelling evidence exists on Clodia ’ s immorality in connection with water in the late Roman Republican period, ultimately by providing an analysis on Marcus Caleius Rufus ’ speech on illegal water conduits.

Sophie Amalienborg, gouache by Johan Jacob Bruun ( 1740 )

by Buchan Telfer with notes by P. Bruun ( London, 1879 ); Joseph von Hammer-Purgstall, " Berechtigung d. orientalischen Namen Schiltbergers ," in Denkschriften d. Konigl.

The best-known FFT algorithms depend upon the factorization of N, but there are FFTs with O ( N log N ) complexity for all N, even for prime N. Many FFT algorithms only depend on the fact that is an th primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms.

Specifically it takes time, demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is thus in the complexity class BQP.

However, a solution has been long in coming, and the factorization problem has been, thus, practically insoluble.

) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial, here into real-coefficient polynomials of the form and

* Integer factorization ( see Shor's algorithm )

The AKS primality test, published in 2002, proves that primality testing also lies in P, while factorization may or may not have a polynomial-time algorithm.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The most well-known use of the Cooley – Tukey algorithm is to divide the transform into two pieces of size at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general ( as was known to both Gauss and Cooley / Tukey ).

The Rader-Brenner algorithm ( 1976 ) is a Cooley – Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability ; it was later superseded by the split-radix variant of Cooley – Tukey ( which achieves the same multiplication count but with fewer additions and without sacrificing accuracy ).

In this sense the GCD problem is analogous to e. g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete.

When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known ; an effort concluded in 2009 by several researchers factored a 232-digit number ( RSA-768 ), utilizing hundreds of machines over a span of 2 years.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

However, this is not the case with a special-purpose factorization algorithm, since it may not apply to the smaller factors that occur during decomposition, or may execute very slowly on these values.

This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization, but is useful for illustrating concepts.

Large-scale quantum computers could be able to solve certain problems much faster than any classical computer by using the best currently known algorithms, like integer factorization using Shor's algorithm or the simulation of quantum many-body systems.

Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory.

Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems, including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation.

For example, to compute one prime factor of the natural number N in polynomial time ( no polynomial time factorization algorithm is known in traditional complexity theory ; see integer factorization ):

Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm ( an algorithm which runs on a quantum computer ) for integer factorization formulated in 1994.

The aim of the algorithm is to find a square root of one, other than and ; such a will lead to a factorization of, as in other factoring algorithms like the quadratic sieve.

Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm.

The Lenstra elliptic curve factorization or the elliptic curve factorization method ( ECM ) is a fast, sub-exponential running time algorithm for integer factorization which employs elliptic curves.