Mathematics. Dictionary
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A group in which the group operation is commutative.
The first of the two terms by which a point is referred to, in a system of fixed rectilinear coordinate (Cartesian coordinate) axes.
For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, \sqrt{a^2+b^2}. Denoted by vertical bars | \, |.
The branch of mathematics concerned with algebraic structures, such as groups, rings, and fields.
An angle greater than 0 degrees and less than 90 degrees.
A triangle all of the angles of which are acute.
The arithmetic operation of adding.
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are inverse problems: given the size of the sumset A + B is small, what can we say about the structures of A and B? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for |A + B| in terms of |A| and |B|.…
A unary function that preserves the addition operation; a unary function f such that for any x and y: f(x + y) = f(x) + f(y).
Either of a pair of angles that share a common side
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to cer…
a vector space having no origin
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring k [ x 1 , … , x n ] . {\displaystyle k[x_{1},\ldots ,x_{n}].} An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime.…
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984.…
In differential geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces. One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ k is either a non-negative integer or infinite.…
Elementary algebra: A system for representing and manipulating unknown quantities (variables) in equations.
A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).
An algebraic variety of one.
A mathematical equation having an algebraic expression on one or both sides, such as 2x + 7y = 3.
A branch of mathematics that studies algebraic varieties (solution sets of polynomial equations) and their generalisations, using techniques from both algebra (chiefly commutative algebra) and geometry.
A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.
Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra.
The branch of mathematics that uses tools from abstract algebra to study topological spaces.
A collection of ordered steps that solve a mathematical problem. A precise step-by-step plan for a computational procedure that possibly begins with an input value and yields an output value in a finite number of steps.
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt(n).
An infinite series in the form of \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n, where a_n > 0 for all n.
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have x ( x y ) = ( x x ) y {\displaystyle x(xy)=(xx)y} ( y x ) x = y ( x x ) {\displaystyle (yx)x=y(xx)} for all x and y in the algebra. Every associative algebra is alternative, given that alternativity is simply a weak form of associativity. However, so too are some strictly non-associative algebras such…
The line perpendicularly connecting a figure’s vertex, especially a triangle’s apex, to the side opposite to the vertex.
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English, although nowadays German mathematicians use the term "Mittelbare Gruppe") in response to the Banach–Tarski paradox.…
Anabelian 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 geometric object, can help to recover X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck.…
Any smooth (infinitely differentiable) function f, defined on an open set D\subseteq \mathbb{C}\ (\textit{ or }\subseteq\mathbb{R}), whose value in some neighbourhood of any given point x_0 \in D is given by the Taylor series \textstyle \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n}.
The subbranch of geometry that investigates geometrical figures via the mathematical properties of their representations in a coordinate system.
A figure formed by two rays which start from a common point (a plane angle) or by three planes that intersect (a solid angle).
A ray that divides an angle into two equal parts.
Rotation opposite to the direction of the hands of a clock; counterclockwise.
A function whose derivative is a given function; an indefinite integral
An imprecise solution or result that is adequate for a defined purpose.
The study of how functions can best be approximated with simpler functions, and how the errors introduced in this way can be quantitatively characterized.
A continuous part of the circumference of a circle (circular arc) or of another curve.
Any of several single-valued or multivalued functions that are inverses of the cosine function.
Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin−1
Any of several single-valued or multivalued functions that are inverses of the tangent function. Symbol: arctan, tan-1
A measure of the extent of a surface; it is measured in square units.
A series of propositions organized so that the final proposition is a conclusion which is intended to follow logically from the preceding propositions, which function as premises.
Of, relating to, or using arithmetic; arithmetical.
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} ).} They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Final…
The measure of central tendency of a set of values computed by dividing the sum of the values by their number.
A sequence in which each term except the first is obtained from the previous by adding a constant value, known as the common difference of the arithmetic progression.
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.…
A straight line which a curve approaches arbitrarily closely as it goes to infinity. The limit of the curve; its tangent "at infinity".
Any of several methods for describing and analyzing the behaviour of a system at its limits
In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of l 2 {\displaystyle l^{2}} -Betti numbers.
An isomorphism of a mathematical object or system of objects onto itself.
Any measure of central tendency, especially any mean, the median or the mode.
One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.
A set of axioms from which theorems can be derived.
A fixed one-dimensional figure, such as a line or arc, with an origin and orientation and such that its points are in one-to-one correspondence with a set of numbers; an axis forms part of the basis of a space or is used to position and locate data in a graph (a coordinate axis)
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