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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics and augmented-reality given the need to associate pictures with coordinates (e.g. CT scans).

Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

The study of manifolds requires working knowledge of calculus and topology.

Method (Ancient Greek: μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to:

INFO: Full form for LIS is List of Insignificant Sets in Maths category
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INFO: Full form for HCF is Highest Common Factor in Maths category
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INFO: Full form for FLM is For The Learning of Mathematics in Maths category
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INFO: Full form for CLS is Constrained Least Squares in Maths category
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INFO: Full form for CLS is Closed Linear Span in Maths category
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INFO: Full form for CLS is Conditional Least Squares in Maths category
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INFO: Full form for UC is Uncoupled Chains in Maths category
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INFO: Full form for UC is Unary Counting in Maths category
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INFO: Full form for DST is Double Spin Torus in Maths category
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In abstract algebra, the idea of an inverse element generalises the concepts of negation (sign reversal) (in relation to addition) and reciprocation (in relation to multiplication). The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.

The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'.

INFO: Full form for QI is Quadrics Intersection in Maths category
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INFO: Full form for QI is Quarter Integer in Maths category
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INFO: Full form for FD is Failure Divergence in Maths category
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INFO: Full form for FD is Finite Domain in Maths category
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In relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database. In other words, a functional dependency is a constraint between two keys.Given a relation R and sets of attributes X , Y ⊆ R {\displaystyle X,Y\subseteq R} , X is said to functionally determine Y (written X → Y) if and only if each X value in R is associated with precisely one Y value in R; R is then said to satisfy the functional dependency X → Y. Equivalently, the projection Π X , Y R {\displaystyle \Pi _{X,Y}R} is a Function, i.e. Y is a function of X. In simple words, if the values for the X attributes are known (say they are x), then the values for the Y attributes corresponding to x can be determined by looking them up in any tuple of R containing x. Customarily X is called the determinant set and Y the dependent set. A functional dependency FD: X → Y is called trivial if Y is a subset of X.

In other words, a dependency FD: X → Y means that the values of Y are determined by the values of X. Two tuples sharing the same values of X will necessarily have the same values of Y.

The determination of functional dependencies is an important part of designing databases in the relational model, and in database normalization and denormalization. A simple application of functional dependencies is Heath's theorem; it says that a relation R over an attribute set U and satisfying a functional dependency X → Y can be safely split in two relations having the lossless-join decomposition property, namely into Π X Y ( R ) ⋈ Π X Z ( R ) = R {\displaystyle \Pi _{XY}(R)\bowtie \Pi _{XZ}(R)=R} where Z = U − XY are the rest of the attributes. (Unions of attribute sets are customarily denoted by mere juxtapositions in database theory.) An important notion in this context is a candidate key, defined as a minimal set of attributes that functionally determine all of the attributes in a relation. The functional dependencies, along with the attribute domains, are selected so as to generate constraints that would exclude as much data inappropriate to the user domain from the system as possible.