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Thomas' algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant (either by rows or columns) or symmetric positive definite; [1] [2] for a more precise characterization of stability of Thomas' algorithm, see Higham Theorem 9.12. [3]
A real hyperelliptic curve of genus g over K is defined by an equation of the form : + = where () has degree not larger than g+1 while () must have degree 2g+1 or 2g+2. This curve is a non singular curve where no point (,) in the algebraic closure of satisfies the curve equation + = and both partial derivative equations: + = and ′ = ′ ().
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution [1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category.
An example where it does not is given by the isolated singularity of x 2 + y 3 z + z 3 = 0 at the origin. Blowing it up gives the singularity x 2 + y 2 z + yz 3 = 0. It is not immediately obvious that this new singularity is better, as both singularities have multiplicity 2 and are given by the sum of monomials of degrees 2, 3, and 4.
It provides a raster of 2.5′×2.5′ and an accuracy approaching 10 cm. 1'×1' is also available [7] in non-float but lossless PGM, [5] [8] but original .gsb files are better. [9] Indeed, some libraries like GeographicLib use uncompressed PGM, but it is not original float data as was present in .gsb format.
The resolution rule can be traced back to Davis and Putnam (1960); [1] however, their algorithm required trying all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson 's syntactical unification algorithm , which allowed one to instantiate the formula during the proof "on ...
Bernstein polynomials can be generalized to k dimensions – the resulting polynomials have the form B i 1 (x 1) B i 2 (x 2) ... B i k (x k). [1] In the simplest case only products of the unit interval [0,1] are considered; but, using affine transformations of the line, Bernstein polynomials can also be defined for products [a 1, b 1] × [a 2 ...
Here, m=2 and there are 10 subsets of 2 indices, however, not all of them are bases: the set {3,5} is not a basis since columns 3 and 5 are linearly dependent. The set B ={2,4} is a basis, since the matrix A B = ( 5 4 1 5 ) {\displaystyle A_{B}={\begin{pmatrix}5&4\\1&5\end{pmatrix}}} is non-singular.