Affine arithmetic (AA) is a model for self-validated numerical analysis. In AA, the quantities of interest are represented as affine combinations (affine forms) of certain primitive variables, which stand for sources of uncertainty in the data or approximations made during the computation.
Affine arithmetic is meant to be an improvement on interval arithmetic (IA), and is similar to generalized interval arithmetic, first-order Taylor arithmetic, the center-slope model, and ellipsoid calculus — in the sense that it is an automatic method to derive first-order guaranteed approximations to general formulas.
Affine arithmetic is potentially useful in every numeric problem where one needs guaranteed enclosures to smooth functions, such as solving systems of non-linear equations, analyzing dynamical systems, integrating functions, differential equations, etc. Applications include ray tracing, plotting curves, intersecting implicit and parametric surfaces, error analysis (mathematics), process control, worst-case analysis of electric circuits, and more.
Affinearithmetic (AA) is a model for self-validated numerical analysis. In AA, the quantities of interest are represented as affine combinations (affine...
§ Arithmetic.) More sophisticated methods of dealing with uncertain values include interval arithmetic and affinearithmetic. Interval arithmetic describes...
mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is...
The affine cipher is a type of monoalphabetic substitution cipher, where each letter in an alphabet is mapped to its numeric equivalent, encrypted using...
irreducible affine algebraic set is also called an affine variety.: 3 (Some authors use the phrase affine variety to refer to any affine algebraic set...
the positive definiteness of a given matrix) root-finding algorithm Affinearithmetic Solving ODEs rigorously (This feature includes external tools such...
triangles for rendering and affine mapping is used on them. The reason this technique works is that the distortion of affine mapping becomes much less noticeable...
and Mozilla as hunspell). After moving to UNICAMP, Jorge developed affinearithmetic, a model for self-validated computation (which he had conceived in...
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance...
such as when H is closed in G and both are affine. The multiplicative group Gm has the punctured affine line as its underlying scheme, and as a functor...
see Floating-point arithmetic § Infinities and IEEE floating point Some authors use Affinely extended real number system and Affinely extended real number...
an affine variety to A1, we can define regular maps from one affine variety to another. First we will define a regular map from a variety into affine space:...
committed by doing only a finite numbers of steps Well-posed problem Affinearithmetic Unrestricted algorithm Summation: Kahan summation algorithm Pairwise...
of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme...
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes...
representation theory, and arithmetic geometry. He is particularly interested in questions related to (finite and affine) Weyl groups and flag varieties...
Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average...
faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations...
interpretation of intuitionistic logic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed...
Choose an open affine subset U = Spec A of X. The ring A is an Fp-algebra, so it admits a Frobenius endomorphism. If V is an open affine subset of U, then...
based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory. Fast Fourier transforms are widely...
and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry, and his logic served as foundations of mathematics...
289–337. "Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian", Advances in Mathematics 221 (2009), No. 2, 570–600. "Affine Grassmannians...
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex...
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