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Concept in mathematics
Affine arithmetic (AA) is a model for self-validated numerical analysis. In AA, the quantities of interest are represented as affine combinations (affine
Affine_arithmetic
Branch of elementary mathematics
§ Arithmetic.) More sophisticated methods of dealing with uncertain values include interval arithmetic and affine arithmetic. Interval arithmetic describes
Arithmetic
Generalization of algebraic variety
geometric tools. Arakelov theory overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding to Archimedean valuations
Scheme_(mathematics)
Mathematical object studied in the field of algebraic geometry
An irreducible affine algebraic set is also called an affine variety. (Some authors use the phrase affine variety to refer to any affine algebraic set
Algebraic_variety
Method for bounding the errors of numerical computations
REC (International Workshop on Reliable Engineering Computing). Affine arithmetic INTLAB (Interval Laboratory) Automatic differentiation Multigrid method
Interval_arithmetic
Brazilian software programmer
and Mozilla as hunspell). After moving to UNICAMP, Jorge developed affine arithmetic, a model for self-validated computation (which he had conceived in
Jorge_Stolfi
positive definiteness of a given matrix) Root-finding algorithms Affine arithmetic Solving ODEs rigorously (This feature includes external tools such
INTLAB
Type of substitution cipher
The affine cipher is a type of monoalphabetic substitution cipher, where each letter in an alphabet is mapped to its numeric equivalent, encrypted using
Affine_cipher
Real numbers with + and - infinity added
see Floating-point arithmetic § Infinities and IEEE floating point Some authors use Affinely extended real number system and Affinely extended real number
Extended_real_number_line
Branch of mathematics
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:
Algebraic_geometry
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
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Euclidean geometry without distance and angles
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance
Affine_geometry
types. Safe numerics on GitHub Computer-assisted proof Interval arithmetic Affine arithmetic INTLAB (Interval Laboratory) Automatic differentiation wikibooks:Numerical
Validated_numerics
Branch of algebraic geometry
mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is
Arithmetic_geometry
Method of defining surface detail on a computer-generated graphic or 3D model
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
Texture_mapping
Mathematical space with two coordinates
mathematical spaces have additional arithmetical structure associated with their points. A vector plane is an affine plane whose points, called vectors
Two-dimensional_space
German mathematician
the arithmetic geometry and representation theory research group at the University of Münster. Before that she was a professor working on arithmetic geometry
Eva_Viehmann
Field of mathematics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex
Arithmetic_dynamics
Mathematics of varieties with integer coordinates
these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems of fundamental importance in Diophantine geometry
Diophantine_geometry
Basic framework of mathematics
and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry, and his logic served as the foundation of mathematics
Foundations_of_mathematics
Area of mathematics
transformed to a linear system as long as a particular solution is known. Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas
Dynamical_systems_theory
Type of zeta function
mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes
Arithmetic_zeta_function
Map raising elements to the pth power, in characteristic p
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
Frobenius_endomorphism
Field of mathematics which studies incidence structures
Every affine plane can be uniquely extended to a projective plane. The order of a finite affine plane is k, the number of points on a line. An affine plane
Incidence_geometry
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Class of mathematical expression
dividend (numerator). The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplied by the divisor
Division_by_zero
committed by doing only a finite numbers of steps Well-posed problem Affine arithmetic Unrestricted algorithm Summation: Kahan summation algorithm Pairwise
List of numerical analysis topics
List_of_numerical_analysis_topics
Branch of mathematics
shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works
Geometry
Symmetry group of a configuration in space
faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations
Space_group
Integer side lengths of a right triangle
algebraic variety of rational points on the unit circle is birational to the affine line over the rational numbers. The unit circle is thus called a rational
Pythagorean_triple
Number with a real and an imaginary part
when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing
Complex_number
Well-quasi-ordering of finite trees
statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion). In 2004, the result was generalized
Kruskal's_tree_theorem
Mathematics independent of applications
the gap between "arithmetic", now called number theory, and "logistic", now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for
Pure_mathematics
Mathematical set with some added structure
of a line, thereby reducing geometry to arithmetic. Three-dimensional Euclidean space is defined to be an affine space whose associated vector space of
Space_(mathematics)
Type of mathematical object
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
Group_scheme
Algebraic variety with a group structure
ISBN 978-1107167483, MR 3729270 Milne, J. S., Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups Mumford, David (1970), Abelian
Algebraic_group
Mathematical structure
is an affine Weyl group, the Coxeter complex is a subdivision of the affine plane and one speaks of affine, or Euclidean, buildings. An affine building
Building_(mathematics)
Theory in number theory
geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety X, or some related
Anabelian_geometry
Branch of mathematics
distinguished a new symbolical algebra, distinct from the old arithmetical algebra. Whereas in arithmetical algebra a − b {\displaystyle a-b} is restricted to a
Abstract_algebra
alignment and parallelism. Affine geometry of curves The study of curve properties that are invariant under affine transformations. Affine differential geometry
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Algebraic curve in mathematics
y 2 = x 3 − x {\displaystyle y^{2}=x^{3}-x} over F71 has 72 points (71 affine points including (0,0) and one point at infinity) over this field, whose
Elliptic_curve
Chinese mathematician (born 1982)
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
Xinwen_Zhu
Natural number
Three non-collinear points determine a unique plane in a three dimensional affine space, and a unique circle in a Euclidean plane. An object has rotational
3
Geometric system with a finite number of points
finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types
Finite_geometry
Open problem on 3x+1 and x/2 functions
Collatz conjecture. 3x + 1 semigroup Arithmetic dynamics Juggler sequence Modular arithmetic Residue-class-wise affine group It is also known as the 3n +
Collatz_conjecture
morphisms of affine schemes. Fontaine, Jean-Marc; Messing, William (1987), "p-adic periods and p-adic étale cohomology", Current trends in arithmetical algebraic
Syntomic_topology
\phi ^{*}:\operatorname {Spec} (R^{p})\to \operatorname {Spec} (R)} of affine schemes. Even in cases where R p = R {\displaystyle R^{p}=R} this is not
Arithmetic and geometric Frobenius
Arithmetic_and_geometric_Frobenius
Points with no three in a line
In affine geometry, a cap set is a subset of the affine space Z 3 n {\displaystyle \mathbb {Z} _{3}^{n}} (the n {\displaystyle n} -dimensional affine space
Cap_set
Academic subfield of computer science
Combinatorics Discrete geometry Graph theory Geometry Algebraic Affine Analytic Arithmetic Complex Computational Convex Differential Discrete Euclidean Finite
Theory_of_computation
Field of knowledge
Euclid's Elements. Mathematics was primarily divided into geometry and arithmetic until the 16th and 17th centuries, when algebra and infinitesimal calculus
Mathematics
of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
intersection, union, set difference emptiness check convex hull (integer) affine hull integer projection computing the lexicographic minimum using parametric
Integer_set_library
Chinese mathematician
of affine Deligne–Lusztig varieties", Annals of Mathematics (2) 179 (2014), 367–404. X. He and S. Nie, "Minimal length elements of extended affine Weyl
He_Xuhua
Concept in algebraic geometry
meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring
Normal_scheme
Surface described by a 4th-degree polynomial
specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form
Quartic_surface
Straight figure with zero width and depth
since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. In the Greek deductive geometry of Euclid's Elements, a general
Line_(geometry)
Real numbers with an added point at infinity
and ∞. The projectively extended real number line is distinct from the affinely extended real number line, in which +∞ and −∞ are distinct. Unlike most
Projectively extended real line
Projectively_extended_real_line
Mathematical concept
In the study of the arithmetic of elliptic curves, the j-line over a ring R is the coarse moduli scheme attached to the moduli problem sending a ring R
J-line
Linear map or polynomial function of degree one
distinguishing such a linear function from the other concept, the term affine function is often used. In linear algebra, mathematical analysis, and functional
Linear_function
American mathematician
for his work in spectral problems in number theory, probability, and Arithmetic combinatorics. He is a Presidential Professor of Mathematics at the CUNY
Alexander_Gamburd
Indian-American mathematician (born 1983)
the Institute for Advanced Study and Princeton University and works in arithmetic geometry and commutative algebra. Bhatt graduated with a B.S. in Applied
Bhargav_Bhatt_(mathematician)
Theories in mathematical logic
fragments of Peano arithmetic. The case n = 1 has about the same strength as primitive recursive arithmetic (PRA). Exponential function arithmetic (EFA) is IΣ0
List_of_first-order_theories
Geometric model of the physical space
space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It is sometimes
Three-dimensional_space
Style of teaching mathematics in 1960s Britain
theory and logic, non-cartesian co-ordinate systems, matrix mathematics, affine transforms, Euclidean vectors, and non-decimal number systems. The SMP published
School_Mathematics_Project
Arithmetical concept
interpretation of intuitionistic logic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed
Dialectica_interpretation
Branch of mathematics
by the commutative Gelfand representation. Similarly, the category of affine schemes in algebraic geometry is dual to the category of commutative rings
Noncommutative_geometry
Algorithm for generating pseudo-randomized numbers
implemented and fast, especially on computer hardware which can provide modular arithmetic by storage-bit truncation. The generator is defined by the recurrence
Linear_congruential_generator
Basic concepts of algebra
arithmetic: arithmetic deals with specified numbers, whilst algebra introduces numerical variables (quantities without fixed values). In arithmetic,
Elementary_algebra
First edition of the IEEE 754 floating-point standard
all 1 bits. fraction = all 0 bits. Some operations of floating-point arithmetic are invalid, such as taking the square root of a negative number. The
IEEE_754-1985
Algebraic variety in a projective space
by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X (e.g., singularity) reduces to that of an affine variety.
Projective_variety
Type of geometry
than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry
Projective_geometry
Injective polynomial functions are bijective
therefore bijective on affine space of the algebraic closure). A generically surjective rational map of n {\displaystyle n} -dimensional affine space over a Hilbertian
Ax–Grothendieck_theorem
In abstract algebra, Abhyankar's conjecture for affine curves is a conjecture of Shreeram Abhyankar posed in 1957, on the Galois groups of algebraic function
Abhyankar's_conjecture
fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation
Flat_topology
Type of functional equation (mathematics)
Combinatorics Discrete geometry Graph theory Geometry Algebraic Affine Analytic Arithmetic Complex Computational Convex Differential Discrete Euclidean Finite
Differential_equation
Array of numbers
matrices to represent objects; to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional
Matrix_(mathematics)
Russian-American mathematician
Jointly with Boris Feigin, Frenkel constructed the free field realizations of affine Kac–Moody algebras (these are also known as Wakimoto modules), defined the
Edward_Frenkel
used for the same purpose. 5. In Euclidean geometry and more generally in affine geometry, P Q → {\displaystyle {\overrightarrow {PQ}}} denotes the vector
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
between schemes is said to be quasi-compact if Y can be covered by open affine subschemes V i {\displaystyle V_{i}} such that the pre-images f − 1 ( V
Quasi-compact_morphism
Pictorial representation of symmetry
hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact
Coxeter–Dynkin_diagram
Type of polynomial
multilinear polynomial is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously
Multilinear_polynomial
Discrete subgroup in a locally compact topological group
Grigory Margulis states that in most cases all lattices are obtained as arithmetic groups. Lattices are also well-studied in some other classes of groups
Lattice_(discrete_subgroup)
Type of object in algebraic geometry
curves, where they showed that the moduli stack of stable curves of fixed arithmetic genus is a proper smooth Deligne–Mumford stack over Spec Z {\displaystyle
Deligne–Mumford_stack
Field arising from a quotient ring by a maximal ideal
of X {\displaystyle X} . By the definition of a scheme, we may find an affine neighbourhood U = Spec ( A ) {\displaystyle {\mathcal {U}}={\text{Spec}}(A)}
Residue_field
Subgroup of the group of invertible n×n matrices
equivalent to affine group schemes. (Every affine group scheme over a field k is pro-algebraic in the sense that it is an inverse limit of affine group schemes
Linear_algebraic_group
Geometric model of the planar projection of the physical universe
numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has
Euclidean_plane
resulting Hecke ring is isomorphic to the Hecke algebra of the affine Weyl group of G, or the affine Hecke algebra, where the indeterminate q has been specialized
Hecke_algebra_of_a_pair
Graphics mode on the Super NES video game console
(background mode 7) has a single layer that can be scaled and rotated. 2D affine transformations can produce any combination of translation, scaling, reflection
Mode_7
Two geometries based on axioms closely related to those specifying Euclidean geometry
As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel
Non-Euclidean_geometry
Study of Lie groups, Lie algebras and differential equations
Combinatorics Discrete geometry Graph theory Geometry Algebraic Affine Analytic Arithmetic Complex Computational Convex Differential Discrete Euclidean Finite
Lie_theory
Israeli mathematician
William; Rudnick, Zeev; Sarnak, Peter (1993). "Density of integer points on affine homogeneous varieties". Duke Mathematical Journal. 71: 143–179. CiteSeerX 10
Zeev_Rudnick
structure used to describe algebraic varieties. The witness set for an affine variety that is equidimensional consists of three pieces of information
Numerical_algebraic_geometry
French mathematician
MR 0801923. Zbl 1182.14049. Raynaud, Michel (1994). "Revêtements de la droite affine en caractéristique p > 0". Inventiones Mathematicae. 116 (1): 425–462. Bibcode:1994InMat
Michel_Raynaud
Topological concept in algebraic geometry
For example, the tame fundamental group of the affine line is zero. It turns out that every affine scheme X ⊂ A k n {\displaystyle X\subset \mathbf
Étale_fundamental_group
On heights of points on algebraic varieties over number fields
other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. Let F {\displaystyle F} be a number
Vojta's_conjecture
Theoretical object in mathematics
geometry, and the category of affine monoid schemes is dual to the category of multiplicative monoids, mirroring the duality of affine schemes and commutative
Field_with_one_element
Concept in mathematics
also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
C++ framework for compiler development
above, the affine dialect enables polyhedral analysis and optimizations, while the memref and arith dialects express memory and arithmetic operations
MLIR_(software)
Loss function used in robust regression
neighborhood, the Huber loss function has a differentiable extension to an affine function at points a = − δ {\displaystyle a=-\delta } and a = δ {\displaystyle
Huber_loss
Point not touching any other point
This makes it possible to define the arithmetic zeta function of such a scheme. Let X {\displaystyle X} be affine scheme (or equivalently, a spectral space)
Closed_point
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