* 1777 – Carl Friedrich Gauss, German mathematician ( d. 1855 )

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

* Gauss – Kronrod quadrature formula

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 two monographs Gauss published on biquadratic reciprocity have consecutively-numbered sections: the first contains §§ 1 – 23 and the second §§ 24 – 76.

The Gauss – Legendre algorithm is an algorithm to compute the digits of π.

The method is based on the individual work of Carl Friedrich Gauss ( 1777 – 1855 ) and Adrien-Marie Legendre ( 1752 – 1833 ) combined with modern algorithms for multiplication and square roots.

Gauss – Jordan elimination, an extension of this algorithm, reduces the matrix further to diagonal form, which is also known as reduced row echelon form.

Gaussian elimination alone is sufficient for many applications, and requires fewer calculations than the Gauss – Jordan version.

At the end of the algorithm, if the Gauss – Jordan elimination ( zeros under and above the leading 1 ) is applied:

By the 1830s mathematics, physics, chemistry, and biology had emerged with world class science, led by Alexander von Humboldt ( 1769 – 1859 ) in natural science and Carl Friedrich Gauss ( 1777 – 1855 ) in mathematics.

File: Carl Friedrich Gauss. jpg | Carl Friedrich Gauss ( 1777 – 1855 )

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.

* Gauss – Lucas theorem

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.

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

However, in the pure Gauss – Bonnet gravity ( a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology ) exotic matter is not needed in order for wormholes to exist — they can exist even with no matter.

* May 6 – Carl Friedrich Gauss and Wilhelm Weber obtain permission to build an electromagnetic telegraph in Göttingen.

* March 30 – Carl Gauss obtains conditions for the constructibility by ruler and compass of regular polygons, and is able to announce that the regular 17-gon is constructible by ruler and compasses.

* July 10 – Carl Friedrich Gauss discovers that every positive integer is representable as a sum of at most 3 triangular numbers.

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.

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.

Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss – Bonnet formula does not work.

Then we can apply Gauss – Bonnet to the surface T formed by the inside of that triangle and the piecewise boundary given by the triangle itself.

A number of earlier results in spherical geometry and hyperbolic geometry over the preceding centuries were subsumed as special cases of Gauss – Bonnet.

This is the special case of Gauss – Bonnet, where the curvature is concentrated at discrete points ( the vertices ).

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.

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

Generalizations of the Gauss – Bonnet theorem to n-dimensional Riemannian manifolds were found in the 1940s, by Allendoerfer, Weil, and Chern ; see generalized Gauss – Bonnet theorem and Chern – Weil homomorphism.

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

Gauss's Theorema Egregium ( Latin: " Remarkable Theorem ") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.

As with the two-dimensional Gauss – Bonnet Theorem, there are generalizations when M is a manifold with boundary.

The Gauss – Bonnet Theorem can be seen as a special instance in the theory of characteristic classes.

An extremely far-reaching generalization of the Gauss – Bonnet Theorem is the Atiyah – Singer Index Theorem.

The 2-dimensional Gauss – Bonnet Theorem arises as the special case where the analytical index is defined in terms of Betti numbers and the topological index is defined in terms of the Gauss – Bonnet integrand.

He was awarded the Gauss medal of the Braunschweigische Wissenschaftliche Gesellschaft in 1996 for his work on Fermat's Last Theorem.

* A. Durner, On a Theorem of Gauss – Kuzmin – Lévy.

* E. Wirsing, On the Theorem of Gauss – Kuzmin – Lévy and a Frobenius-Type Theorem for Function Spaces.

Statue of Gauss at his birthplace, Braunschweig

While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the famous note: " ΕΥΡΗΚΑ!

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, who was 23 at the time, heard about the problem and tackled it.

" Four normal distribution | Gaussian distributions in statisticsThis unproved statement put a strain on his relationship with János Bolyai ( who thought that Gauss was " stealing " his idea ), but it is now generally taken at face value.

Grave of Gauss at Albanifriedhof in Göttingen, Germany.

Gauss also upheld religious tolerance, believing it wrong to disturb others who were at peace with their own beliefs.

* The Gauss Haus, an NMR center at the University of Utah,

However, the Theorema Egregium of Carl Friedrich Gauss showed that already for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same.

When the matter of honorary degrees came up at the University of Göttingen six years after Germain's death, Gauss lamented, “ proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree.

* Carl Friedrich Gauss pioneers the field of summation with the formula summing 1: n as ( n ( n + 1 ))/ 2, at the age of 7.

Carl Friedrich Gauss is credited with developing the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen.

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.

The average magnetic field strength in the Earth's outer core was measured to be 25 Gauss, 50 times stronger than the magnetic field at the surface.

The average magnetic field in the Earth's outer core was calculated to be 25 Gauss, 50 times stronger than the field at the surface.

The regular heptadecagon is a constructible polygon ( that is, one that can be constructed using a compass and unmarked straightedge ), as was shown by Carl Friedrich Gauss in 1796 at the age of 19.

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.

As Germany provided little opportunity to study higher mathematics at the time, with only Gauss at the University of Göttingen who was nominally a professor of astronomy and anyway disliked teaching, Dirichlet decided to go to Paris in May 1822.