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Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic.
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic [1] – do not vary smoothly in this way, but have distinct, separated values. [2]
The fundamental group can be defined for discrete structures too. In particular, consider a connected graph G = (V, E), with a designated vertex v 0 in V. The loops in G are the cycles that start and end at v 0. [4] Let T be a spanning tree of G. Every simple loop in G contains exactly one edge in E \ T; every loop in G is a concatenation of ...
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Essence of combinatorics ... Discrete Optimization;
Schematic illustration of a combinatorial species structure on five elements by using a Labelle diagram. Any species consists of individual combinatorial structures built on the elements of some finite set: for example, a combinatorial graph is a structure of edges among a given set of vertices, and the species of graphs includes all graphs on all finite sets.
In differential geometry, a G-structure on an n-manifold M, for a given structure group [1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.
In mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G. Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian. Other properties: