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.

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.

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 ).

In 1828, Carl Friedrich Gauss proved his Theorema Egregium ( remarkable theorem in Latin ), establishing an important property of surfaces.

Gauss proved that if m > 2

The origins of class field theory lie in the quadratic reciprocity law proved by Gauss.

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.

Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings.

Gauss proved that the sample-mean minimizes the expected squared-error loss-function ( while Laplace proved that a median-unbiased estimator minimizes the absolute-error loss function ).

Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796.

* Although Gauss proved that the regular 17-gon is constructible, he did not actually show how to do it.

* Why Gauss could not have proved necessity of constructible regular polygons

In 1934, Hans Heilbronn proved the Gauss Conjecture.

In section VII, article 358, Gauss proved what can be interpreted as the first non-trivial case of the Riemann Hypothesis for curves over finite fields ( the Hasse – Weil theorem ).

Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, .. Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line " ΕΥΡΗΚΑ!

Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a regular lattice arrangement is

He specialized in number theory and analysis, and proved several results that eluded even Gauss.

Gauss proved this using his theory of equivalence relations by showing that the quadratic w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 represents all natural numbers.

Gauss proved that for every value D, there are only finitely many classes of binary quadratic forms with discriminant D. Their number is the class number of discriminant D. He described an algorithm, called reduction, for constructing a canonical representative in each class, the reduced form, whose coefficients are the smallest in a suitable sense.

While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method.

The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1795.

This method ( and the general idea of an FFT ) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered ( Heideman & Burrus, 1984 ) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 ( and subsequently rediscovered several times in limited forms ).

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.

The method is named after Carl Friedrich Gauss, but it was not invented by him.

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.

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

The method of least squares was first described by Carl Friedrich Gauss around 1794.

Unable to compute its orbit with existing methods, the renowned mathematician Carl Friedrich Gauss developed a new method of orbit calculation that allowed astronomers to locate it again.

The least-squares method was first described by Carl Friedrich Gauss around 1794.

Gauss did not publish the method until 1809, when it appeared in volume two of his work on celestial mechanics, Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium.

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.

The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination before Carl Friedrich Gauss ( 1777 – 1855 ).

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.

Developed by Carl Gauss, who began research on the treatment in 1903, it was also sometimes known as the " Freiburg method ".

* Carl Friedrich Gauss publishes Theoria motus corporum coelestium in sectionibus conicis solem ambientum in Hamburg, introducing the Gaussian gravitational constant and containing an influential treatment of the least squares method.

* The 18-year-old Carl Friedrich Gauss develops the basis for the method of least squares analysis.

Claus Tondering's algorithm uses a variant of the method of congruence used by Gauss, thereby shifting month-numbers by the same amount, and arriving at

Adrain, Gauss, and Legendre all motivated the method of least squares by the problem of reconciling disparate physical measurements ; in the case of Gauss and Legendre, the measurements in question were astronomical, and in Adrain's case they were survey measurements.

* Gauss – Seidel method: This is the earliest devised method.

In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation ( Gaussian optics ).

Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula.

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

In 1838 Johann Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li ( x ) ( under the slightly different form of a series, which he communicated to Gauss ).

In a handwritten note on a reprint of his 1838 paper " Sur l ' usage des séries infinies dans la théorie des nombres ", which he mailed to Carl Friedrich Gauss, Johann Peter Gustav Lejeune Dirichlet conjectured ( under a slightly different form appealing to a series rather than an integral ) that an even better approximation to π ( x ) is given by the offset logarithmic integral function Li ( x ), defined by

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

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.

Gauss constructs the order-3 periods, corresponding to the cyclic group ( Z / pZ )< sup >×</ sup > of non-zero residues modulo p under multiplication and its unique subgroup of index three.

From there he went to the University of Heidelberg in 1848, and soon after to the University of Göttingen, where he studied under Gauss and Weber, and where Stern awakened in him a strong interest in historical research.

He received his PhD in mathematics ( number theory ) in 1969 from the University of California, Berkeley under Derrick Henry Lehmer for a thesis entitled " Proof of a Conjecture of Gauss on Class Number Two ".

It is the Gauss curvature of the-section at p ; here-section is a locally-defined piece of surface which has the plane as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of under the exponential map at p.

Having studied under Carl Friedrich Gauss, he became a teacher at the gymnasium in Hof, tutoring, among others, Carl Culmann and Philipp Ludwig von Seidel.

After going on to Harvard College and graduating in 1844, he studied mathematics and astronomy under C. F. Gauss at Göttingen, Germany, during which time he published approximately 20 papers on the observation and motion of comets and asteroids.

" Gauss Research Laboratory Inc " ( GRL-INC ) is the managing organization of the Puerto Rico's top level domain under the website.

Discovered in February 1902 by the German Antarctic Expedition under Erich von Drygalski, who named it as his expedition ship in honour of Carl Friedrich Gauss.

He was a precursor of the German school of mathematical thinking, which under Carl Friedrich Gauss and his followers largely determined the lines on which mathematics developed during the nineteenth century.

One of the deepest discoveries of Gauss was the existence of a natural composition law on the set of classes of binary quadratic forms of given discriminant, which makes this set into a finite abelian group called the class group of discriminant D. Gauss also considered a coarser notion of equivalence, under which the set of binary quadratic forms of a fixed discriminant splits into several genera of forms and each genus consists of finitely many classes of forms.

Dissecting the Holocaust was edited and coauthored by Rudolf under the nom de plume Ernst Gauss.