* О суммахъ Гаусса и о законе взаимности символа Лежандра ( About Gauss sums and the reciprocity law of the Legendre symbol ) ( 1877 ),

His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function.

As is discussed in more detail below, the Gaussian periods are closely related to another class of sums of roots of unity, now generally called Gauss sums ( sometimes Gaussian sums ).

The Gauss sums are ubiquitous in number theory ; for example they occur significantly in the functional equations of L-functions.

( Gauss sums are in a sense the finite field analogues of the gamma function.

The Gaussian periods are related to the Gauss sums for which the character χ is trivial on H. Such χ take the same value at all elements a in a fixed coset of H in G. For example, the quadratic character mod p described above takes the value 1 at each quadratic residue, and takes the value-1 at each quadratic non-residue.

In other words, the Gaussian periods and Gauss sums are each other's Fourier transforms.

On the other hand, Gauss sums have nicer algebraic properties.

Examples of complete exponential sums are Gauss sums and Kloosterman sums ; these are in some sense finite field or finite ring analogues of the gamma function and some sort of Bessel function, respectively, and have many ' structural ' properties.

* the tendency for an initial segment of data to show some bias that drops out later ( one example in number theory being Kummer's conjecture on cubic Gauss sums )

In number theory, quadratic Gauss sums are certain finite sums of roots of unity.

Thus in the evaluation of quadratic Gauss sums one may always assume gcd ( a, c )= 1.

One has the following analogue of the quadratic reciprocity law for ( even more general ) Gauss sums

The values of Gauss sums with b = 0 and gcd ( a, c )= 1 are explicitly given by

* For b > 0 the Gauss sums can easily be computed by completing the square in most cases.

This can, for example, be proven as follows: Because of the multiplicative property of Gauss sums we only have to show that if n > 1 and a, b are odd with gcd ( a, c )= 1.

Since the discrete logarithm problem reduces to Gauss sum estimation, an efficient classical algorithm for estimating Gauss sums would imply an efficient classical algorithm for computing discrete logarithms, which is considered unlikely.

However, quantum computers can estimate Gauss sums to polynomial precision in polynomial time.

Footnotes referencing these are of the form " Gauss, BQ, § n ".

Footnotes referencing the Disquisitiones Arithmeticae are of the form " Gauss, DA, Art.

The opening ceremony of the International Congress of Mathematicians ( ICM ) is where the awards are presented: Fields Medals ( two to four medals are given since 1936 ), the Rolf Nevanlinna Prize ( since 1986 ), the Carl Friedrich Gauss Prize ( since 2006 ), and the Chern Medal Award ( since 2010 ).

Other standard iterative methods for matrix equation solutions can also be used, for example the Gauss – Seidel method, where updated values for each patch are used in the calculation as soon as they are computed, rather than all being updated synchronously at the end of each sweep.

There are two versions of the first message sent by Gauss and Weber: the more official one is based on a note in Gauss's own handwriting stating that " Wissen vor meinen – Sein vor scheinen " (" knowing before opining, being before seeming ") was the first message sent over the electromagnetic telegraph.

Gauss has also found some of Vega's errors in the calculations in the range of numbers, of which there are more than a million.

Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about " whole complex numbers " ( i. e. the Gaussian integers ) than they are as statements about ordinary whole numbers ( i. e. the integers ).

These are two forms of the lower-case Greek letter phi φ ( n ) is from Gauss ' 1801 treatise Disquisitiones Arithmeticae.

Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.

Carl Friedrich Gauss in 1796 showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes.

In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator.

These are the defining equations of the Gauss – Newton algorithm.

LLSQ solutions can be computed using direct methods, although problems with large numbers of parameters are typically solved with iterative methods, such as the Gauss – Seidel method.

There are several combinatorial analogs of the Gauss – Bonnet theorem.

Especially, Gauss had looked at the case of imaginary quadratic fields: he found exactly nine values of < math > D < 0 </ math > for which the ring of integers is a PID and conjectured that there are no further values.

In fact, although Gauss also conjectured that there are infinitely many primes such that the ring of integers of is a PID, to this day we do not even know whether there are infinitely many number fields ( of arbitrary degree ) such that is a PID!

Gauss – Markov stochastic processes ( named after Carl Friedrich Gauss and Andrey Markov ) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes.

Its three book awards are the Ralph Waldo Emerson award, the Christian Gauss Award, and the Phi Beta Kappa Award in Science.

In statistics, the Gauss – Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator ( BLUE ) of the coefficients is given by the ordinary least squares estimator.

The Gauss – Markov assumptions are

The Artin reciprocity law, which is a high level generalization of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field.

One must loop over elements twice ( so we get n < sup > 2 </ sup > passes through ) and for each pair of elements we loop through Gauss points in the elements producing a multiplicative factor proportional to the number of Gauss-points squared.

This is justified, if unsatisfactorily, by Gauss in his " Disquisitiones Arithmeticae ", where he states that all analysis ( i. e., the paths one travelled to reach the solution of a problem ) must be suppressed for sake of brevity.

An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points x < sub > i </ sub > and weights w < sub > i </ sub > for i = 1 ,..., n.

The Gauss map can always be defined locally ( i. e. on a small piece of the surface ).

The Gauss – Newton algorithm can be derived by linearly approximating the vector of functions r < sub > i </ sub >.

In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + i in Z, the ring of Gaussian integers.

# constructing a regular polygon whose number of sides is not the product of a power of two and any number of distinct Fermat primes ( i. e. that does not fulfill the same conditions proven to be sufficient by Carl Friedrich Gauss )