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 ).

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 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.

In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism ( including finding a representation for the unit of magnetism in terms of mass, length and time ) and the discovery of Kirchhoff's circuit laws in electricity.

When his second wife died in 1831 after a long illness, one of his daughters, Therese, took over the household and cared for Gauss until the end of his life.

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.

Gauss a letter to Schumacher, 12 July 1831 )

To this discrimination Vincenzo Brunacci ( 1810 ), Carl Friedrich Gauss ( 1829 ), Siméon Poisson ( 1831 ), Mikhail Ostrogradsky ( 1834 ), and Carl Jacobi ( 1837 ) have been among the contributors.

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

Gauss knew of both of these results.

In 1830, Carl Friedrich Gauss, the German mathematician, unified the work of these two scientists to derive the Young – Laplace equation, the formula that describes the capillary pressure difference sustained across the interface between two static fluids.

The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.

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.

Extending this to circles immersed in the plane – the immersion condition is precisely the condition that the derivative does not vanish – the Whitney – Graustein theorem classified these by turning number by considering the homotopy class of the Gauss map and showing that this satisfies an h-principle ; here again order 0 is more complicated.

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.

Midway through these sessions, Gauss left the band, so Edwards played some drums himself, until a replacement was found in Kellii Scott.

In these times Bolyai became a close friend of Carl Friedrich Gauss.

By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Gauss.

The biographical sections give relevant information about the lives of mathematicians who worked in these areas, including Euler, Gauss, Dirichlet, Lobachevsky, Chebyshev, Vallée-Poussin, Hadamard, as well as Riemann himself.

Most prominent among these titles were Datu, meaning a chieftain or leader, awarded in this order to Shishir Inocalla, Kelly Worden and Ric " Bong " Jornales ( of Arnis Sikaran ) ( all in the 1980s ), Dieter Knuettel ( 1996 ), Tim Hartman and David Hoffman ( both in 2000 ); and Master of Tapi-Tapi, awarded to Jeff Delaney, Chuck Gauss, Jim Ladis, Gaby Roloff, Randi Schea, Ken Smith, and Brian Zawilinski.

Their nonuniversality can be observed by noticing that, if any of these are used to code the Gauss – Kuzmin distribution or the Zeta distribution with parameter s = 2, expected codeword length is infinite.

Zach noted that " without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again ".

Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it.

Mathematical historian Eric Temple Bell estimated that, had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.

The problem of their multiplicity was solved by McClellan and Parks ( 1972 ), although it was later shown to have been equivalent to a problem solved by Gauss ( Dickinson and Steiglitz, 1982 ).

The puzzle was originally proposed in 1848 by the chess player Max Bezzel, and over the years, many mathematicians, including Gauss, have worked on this puzzle and its generalized n-queens problem.

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

Gauss gave the first proof that seems to have been known in Europe ( the third after Adrain's ) in 1809.

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 >).

Many famous mathematicians have studied such problems, including Euler, Legendre, and Gauss.

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.

* The universe of BattleTech and MechWarrior have an Assault class battlemech named Fafnir, recognizable for its two shoulder mounted Heavy Gauss Rifles.

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.

Of all of Gauss ' children, Wilhelmina was said to have come closest to her father's talent, but regrettably, she died young.

Starting with Gauss ' law for electricity ( also one of Maxwell's equations ) in differential form, we have:

Other extremal principles of classical mechanics have been formulated, such as Gauss ' principle of least constraint and its corollary, Hertz's principle of least curvature.

Gauss, Bolyai and Lobachevski independently constructed hyperbolic geometry, where parallel lines have an angle of parallelism that depends on their separation.

* Examples related to dynamical systems arising from number theory, such as the Gauss shift on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.

The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, ( that is along a parabolic line ) the Gauss map will have a fold catastrophe.