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In mathematics, and especially in category theory, a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition.
In mathematical analysis, the staircase paradox is a pathological example showing that limits of curves do not necessarily preserve their length. [1] It consists of a sequence of "staircase" polygonal chains in a unit square , formed from horizontal and vertical line segments of decreasing length, so that these staircases converge uniformly to ...
In the mathematical field of model theory, the elementary diagram of a structure is the set of all sentences with parameters from the structure that are true in the structure. It is also called the complete diagram .
The properties that the logistic map exhibits in the period doubling cascade are also universal in a broader class of maps, as will be discussed later. To get an overview of the final behavior of an orbit for a certain parameter, an approximate bifurcation diagram, orbital diagram, is useful.
Diagrams and functor categories are often visualized by commutative diagrams, particularly if the index category is a finite poset category with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be ...
A category C consists of two classes, one of objects and the other of morphisms.There are two objects that are associated to every morphism, the source and the target.A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as f : X → Y or X Y the latter form being better suited for commutative diagrams.
In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2] The term map may be used to distinguish some special types of functions, such as homomorphisms.
In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the joint embedding property, among others.