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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.
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Generalizations and –
Generalizations of Bass – Serre theory such as the theory of complexes of groups.
Generalizations of the Nyquist – Shannon sampling theorem allow sampling of other band-limited passband signals instead of baseband signals – see undersampling.
Real Numbers, Generalizations of the Reals, and Theories of Continua, vi – xxxii.
* --------, " Pannenberg on Marxism: Insights and Generalizations ," The Christian Century ( 30 September 1987 ): 824 – 26.
Generalizations and theorem
Generalizations of Noether's theorem to superspaces are also available.
Since this theorem applies in infinite-dimensional ( Banach space ) settings, it is the tool used in proving the infinite-dimensional version of the inverse function theorem ( see " Generalizations ", below ).
and Arora, K., " Generalizations of Dixon's theorem on the sum of a 3F2 ", Math.
* R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years ( 1862 — 2012 ).
Generalizations and manifolds
Generalizations of this definition are possible, for instance to complex manifolds and algebraic varieties.
Generalizations and by
Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements.
Note that by definition, a linear combination involves only finitely many vectors ( except as described in Generalizations below ).
Generalizations of Bertrand's Postulate have also been obtained by elementary methods.
Tulsi was admitted by the Indian Institute of Science ( IISc )., where he wrote a 33-page long Ph. D. thesis on " Generalizations of the Quantum Search Algorithm ".
Generalizations have been studied by D. G. Higman ( coherent configurations ) and B. Weisfeiler ( distance regular graphs ).
Generalizations and ;
Generalizations to sinusoids of other phases, and to complex exponentials, are also common ; see plane wave.
Generalizations and .
Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings.
Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.
Generalizations of compactness include H-closed and the property of being an H-set in a parent space.
Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filter and Cauchy net.
Generalizations are possible to metrics other than Euclidean.
Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order.
In the theory of quadratic forms, the parabola is the graph of the quadratic form ( or other scalings ), while the elliptic paraboloid is the graph of the positive-definite quadratic form ( or scalings ) and the hyperbolic paraboloid is the graph of the indefinite quadratic form Generalizations to more variables yield further such objects.
Generalizations of these theories form the basis for understanding the closely related phenomenon of superfluidity, because they fall into the Lambda transition universality class, but the extent to which similar generalizations can be applied to unconventional superconductors as well is still controversial.
Generalizations.
See " Introduction to Methods and Generalizations " in T. Janssen and G. Redeker ( Eds ) ( 1999 ).
** 1. 31 Knowledge of Principles and Generalizations
Ma, Generalizations of tic-tac-toe,.
Generalizations to higher dimensions are also possible.
Generalizations of these approaches underlie the path integral formulation of quantum mechanics, and is used for calculating geodesic motion in general relativity.
Generalizations to N =( 2, 0 ) supersymmetry have also appeared.
Generalizations that are weak generally have more exceptions ( the number of exceptions to the generalization need not be a minority of cases ) and vice versa.
Gauss and –
* 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.
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