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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).
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]
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.
NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time. For instance, the Boolean satisfiability problem is NP-complete by the Cook–Levin theorem , so any instance of any problem in NP can be transformed mechanically into a Boolean satisfiability problem in polynomial time.
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.
It asks for the maximum number of clauses which can be satisfied by any assignment. It has efficient approximation algorithms, but is NP-hard to solve exactly. Worse still, it is APX-complete, meaning there is no polynomial-time approximation scheme (PTAS) for this problem unless P=NP.
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 ...
de Bruijn's theorem (discrete geometry) Descartes's theorem on total angular defect ; Erdős–Anning theorem (discrete geometry) Erdős–Nagy theorem (discrete geometry) Erdős–Szekeres theorem (discrete geometry) Fáry's theorem (graph theory) Fenchel's duality theorem (convex analysis) Fenchel–Moreau theorem (mathematical analysis)