This is the basis for the probabilistic Solovay – Strassen primality test and its refinement the Miller – Rabin primality test.

Euler probable primes ( P = 1 / 2, Solovay – Strassen algorithm ).

The Miller – Rabin primality test and Solovay – Strassen primality test are more sophisticated variants which detect all composites ( once again, this means: for every composite number n, at least 3 / 4 ( Miller – Rabin ) or 1 / 2 ( Solovay – Strassen ) of numbers a are witnesses of compositeness of n ).

The Solovay – Strassen primality test uses another equality: Given an odd number n, choose some integer a < n, if

The subsequent discovery of the Solovay – Strassen and Miller – Rabin algorithms put PRIMES in coRP.

similar to the Fermat primality test and the Solovay – Strassen primality test.

Just like the Fermat and Solovay – Strassen tests, the Miller – Rabin test relies on an equality or set of equalities that hold true for prime values, then checks whether or not they hold for a number that we want to test for primality.

It can be shown that for any odd composite n, at least ¾ of the bases a are witnesses for the compositeness of n. The Miller – Rabin test is strictly stronger than the Solovay – Strassen primality test in the sense that for every composite n, the set of strong liars for n is a subset of the set of Euler liars for n, and for many n, the subset is proper.

On the other hand, the Solovay – Strassen primality test declares n probably prime with a probability at most 2 < sup >− k </ sup >.

Fast primality testing is key in the successful implementation of most public-key cryptography, and in 2003 Miller, Rabin, Robert M. Solovay, and Volker Strassen were given the Paris Kanellakis Award for their work on primality testing.

For instance, the Solovay – Strassen primality test is used to determine whether a given number is a prime number.

Consider again the Solovay – Strassen algorithm which is ( 1 / 2 )- correct false-biased.

Well-known Monte Carlo algorithms include the Solovay – Strassen primality test, the Miller – Rabin primality test, and certain fast variants of the Schreier – Sims algorithm in computational group theory.

Strassen is also known for his 1977 work with Robert M. Solovay on the Solovay – Strassen primality test, the first method to show that testing whether a number is prime can be performed in randomized polynomial time and one of the first results to show the power of randomized algorithms more generally.

In 1999 Strassen was awarded the Cantor medal, and in 2003 he was co-recipient of the Paris Kanellakis Award with Robert Solovay, Gary Miller, and Michael Rabin for their work on randomized primality testing.

The Solovay – Strassen primality test, developed by Robert M. Solovay and Volker Strassen, is a probabilistic test to determine if a number is composite or probably prime.

To solve this, Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible to define the truth of statements about the constructible universe.

The theoretical interest in NP-completeness was also enhanced by the work of Theodore P. Baker, John Gill, and Robert Solovay who showed that solving NP-problems in Oracle machine models requires exponential time.

In 1970, Solovay demonstrated that the existence of a non-measurable set for Lebesgue measure is not provable within the framework of Zermelo – Fraenkel set theory in the absence of the Axiom of Choice, by showing that ( assuming the consistency of an inaccessible cardinal ) there is a model of ZF, called Solovay's model, in which countable choice holds, every set is Lebesgue measurable and in which the full axiom of choice fails.

It is also consistent with ZF + DC that every set of reals is Lebesgue measurable ; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption ( the existence of an inaccessible cardinal ).

It is well associated with a jingle featured in almost every advertisement since 1984, with lyrics by Susan Spiegel Solovay and Bill Vernick, and music by Leslie Pearl.

Following an initial observation of Robert Solovay, Scott formulated the concept of Boolean-valued model, as Solovay and Petr Vopěnka did likewise at around the same time.

In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all subsets of the reals are measurable.

A finer equivalence relation, Solovay equivalence, can be used to characterize the halting probabilities among the left-c. e.

Zero sharp was defined by Silver and Solovay as follows.

According to Reuters, McMahon failed to pay divorce attorney Norman Solovay $ 275, 168, according to a lawsuit filed in the Manhattan federal court.

McMahon and his wife, Pamela, hired Solovay to represent Linda Schmerge, his daughter from another relationship, in a " matrimonial matter ," said Solovay's lawyer, Michael Shanker.

Mac Lane supervised the Ph. Ds of, among many others, David Eisenbud, William Howard, Irving Kaplansky, Michael Morley, Anil Nerode, Robert Solovay, and John G. Thompson.

1981a, Review of Robert M. Solovay, Provability Interpretations of Modal Logic ," Journal of Symbolic Logic 46: 661-662.

* Robert M. Solovay

It was pioneered by Robert M. Solovay in 1976.

As describes, the time for finding the fundamental solution using the continued fraction method, with the aid of the Schönhage – Strassen algorithm for fast integer multiplication, is within a logarithmic factor of the solution size, the number of digits in the pair ( x < sub > 1 </ sub >, y < sub > 1 </ sub >).

: Magma contains asymptotically-fast algorithms for all fundamental integer and polynomial operations, such as the Schönhage – Strassen algorithm for fast multiplication of integers and polynomials.

: Magma contains asymptotically-fast algorithms for all fundamental dense matrix operations, such as Strassen multiplication.

The running time for multiplying rectangular matrices ( one m × p-matrix with one p × n-matrix ) is O ( mnp ), however, more efficient algorithms exist, such as Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as " fast matrix multiplication ".

It was the key, for example, to Karatsuba's fast multiplication method, the quicksort and mergesort algorithms, the Strassen algorithm for matrix multiplication, and fast Fourier transforms.

Due to its overhead, Toom – Cook is slower than long multiplication with small numbers, and it is therefore typically used for intermediate-size multiplications, before the asymptotically faster Schönhage – Strassen algorithm ( with complexity Θ ( n log n log log n )) becomes practical.

An example is the Strassen algorithm for fast matrix multiplication, as well as the Hamming ( 7, 4 ) encoding for error detection and recovery in data transmissions.

In 1969, Strassen shifted his research efforts towards the analysis of algorithms with a paper on Gaussian elimination, introducing Strassen's algorithm, the first algorithm for performing matrix multiplication faster than the O ( n < sup > 3 </ sup >) time bound that would result from a naive algorithm.

Volker Strassen giving the Knuth Prize lecture at SODA 2009

Volker Strassen ( born April 29, 1936 ) is a German mathematician, a professor emeritus in the department of mathematics and statistics at the University of Konstanz.

In 1968, Strassen moved to the Institute of Applied Mathematics at the University of Zurich, where he remained for twenty years before moving to the University of Konstanz in 1988.

The seal of Johann Strassen ( 1411 ) and piety dating from 1500 ( currently on display at the National Museum of the State ), provided the basis for the municipal coat of arms created in 1976 and whose description is hieratic " Cloche d ' or ".