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 ).

Despite initial opposition from her parents and difficulties presented by a sexist society, she gained education from books in her father's library and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss.

The first letter, dated 21 November 1804, discussed Gauss ' Disquisitiones and presented some of Germain's work on Fermat's Last Theorem.

The quantity P − P * presented above is a quadratic Gauss sum mod p, the simplest non-trivial example of a Gauss sum.

In recognition of his achievements, the Berlin Geographical Society presented him with the Ferdinand von Richthofen Medal in 1933 ; the same honor was also awarded to Erich von Drygalski for his Gauss Expedition to the Antarctic ; and to Alfred Philippson for his research on the Aegean Region.

In fact out of the three people that can be credited with discovery of hyperbolic geometry-Gauss, Lobachevsky and Bolyai, Lobachevsky rightfully deserves having his name attached to it, since Gauss never published his ideas and out of the latter two Lobachevsky was the first who duly presented his views to the world mathematical community.

He accepted and presented his credentials in Canberra on August 23, 2001 as the 22nd representative of the U. S. president, 63 years after the first, Clarence E. Gauss, a professional diplomat from Connecticut, presented his credentials to Australia's governor-general of the time, Lord Gowrie, on January 12, 1940.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

The theorem was conjectured by Euler and Legendre and first proven by Gauss.

* Gauss – Lucas theorem

She asked Gauss if her approach to the theorem was worth pursuing.

The Gauss – Markov theorem states that the estimate of the mean having minimum variance is given by:

If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that

* 1942 – Joseph Leo Doob states his theorem on Gauss – Markov processes

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic ; see the Gauss – Bonnet theorem.

The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the ( angular ) defect ; the analog for the Gauss – Bonnet theorem is Descartes ' theorem on total angular defect.

This result is known as the Gauss – Markov theorem.

* The Gauss – Markov theorem.

The Gauss – Markov theorem shows that, when this is so, is a best linear unbiased estimator ( BLUE ).

* Theorema Egregium − The " remarkable theorem " discovered by Gauss which showed there is an intrinsic notion of curvature for surfaces.

The Gauss – Bonnet theorem or Gauss – Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry ( in the sense of curvature ) to their topology ( in the sense of the Euler characteristic ).

It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.

Thinking of curvature as a measure, rather than as a function, Descartes ' theorem is Gauss – Bonnet where the curvature is a discrete measure, and Gauss – Bonnet for measures generalizes both Gauss – Bonnet for smooth manifolds and Descartes ' theorem.

On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.

The easiest way to compute the rank of a matrix A is given by the Gauss elimination method.

Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors.

In this way the result of Gauss can be understood in current terms ; for actual calculation of the equations to be solved, the periods can be squared and compared with the ' lower ' periods, in a quite feasible algorithm.

Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and treat it in a systematic way.

Though Gauss had up to that point been financially supported by his stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support.

His friend Farkas Wolfgang Bolyai with whom Gauss had sworn " brotherhood and the banner of truth " as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry.

Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines.

Gauss plunged into a depression from which he never fully recovered.

See also the letter from Robert Gauss to Felix Klein on 3 September 1912.

His inventions were based on the printing mechanism from Hughes ' instrument, a distributor invented by Bernard Meyer during 1871, and the five-unit code devised by Carl Friedrich Gauss and Wilhelm Weber.

Several other experiments followed, with André-Marie Ampère, who in 1820 discovered that the magnetic field circulating in a closed-path was related to the current flowing through the perimeter of the path ; Carl Friedrich Gauss ; Jean-Baptiste Biot and Félix Savart, both of which in 1820 came up with the Biot-Savart Law giving an equation for the magnetic field from a current-carrying wire ; Michael Faraday, who in 1831 found that a time-varying magnetic flux through a loop of wire induced a voltage, and others finding further links between magnetism and electricity.

The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century.

* Excerpt from Siege of the South Pole ; includes picture of Gauss under sail

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

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

In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite ; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.

Property ( 3 ) means that every Gauss – Markov process can be synthesized from the standard Wiener process ( SWP ).

While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita who first codified the notion of a tensor.

He studied mathematics and astronomy from 1811 at the University of Göttingen under Carl Friedrich Gauss ; but he enlisted in the Hanseatic Legion for the campaign of 1813 – 1814, and became lieutenant of artillery in the Prussian army in 1815.

Meanwhile, the famous mathematician Carl Friedrich Gauss was entrusted from 1821 to 1825 with the triangulation of the kingdom of Hanover, for which he developed the method of least squares to find the best fit solution for problems of large systems of simultaneous equations given more real-world measurements than unknowns.

The other difference formulas, such as those of Stirling, Bessel and Gauss, can be derived from Newton's, using Newton's terms, with data points and x values renamed in keeping with the choice of x zero, and based on the fact that they must add up to the same sum value as Newton's ( With Stirling that is so when polynomial degree is even.

There he attended classes at the Collège de France and at the Faculté des sciences de Paris, learning mathematics from Hachette among others, while undertaking private study of Gauss ' Disquisitiones Arithmeticae, a book he kept close for his entire life.

Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an unusual amount of praise that " Dirichlet showed excellent talent ".

And only recently has it been possible to complete the definition of ν ( ℓ ′) by developing and proving the validity of an exact series expansion for this function ( by starting from known special-case solutions of the Gauss hypergeometric differential equation ).