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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).
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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]
To Create His Geometric Artwork, M.C. Escher Had to Learn Math the Hard Way. Fourier Transforms: The Math That Made Color TV Possible. The Game of Trees is a Mad Math Theory That Is Impossible to ...
Quadratic programming (NP-hard in some cases, P if convex) Subset sum problem [3]: SP13 Variations on the Traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric.
Kawasaki's theorem (mathematics of paper folding) Kelvin's circulation theorem ; Kempf–Ness theorem (algebraic geometry) Kepler conjecture (discrete geometry) Kharitonov's theorem (control theory) Khinchin's theorem (probability) Killing–Hopf theorem (Riemannian geometry) Kinoshita–Lee–Nauenberg theorem (quantum field theory)
A simple example of an NP-hard problem is the subset sum problem. Informally, if H is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP-hard (unless ...
The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. [45] Discrete mathematics includes: [14] Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints.