Discrete mathematics
From Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Discrete_mathematics
Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.
For the mathematics journal, see Discrete Mathematics (journal).
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] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of the integers, including rational numbers but not real numbers). However, there is no exact, universally agreed, definition of the term "discrete mathematics."[4] Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
Grand challenges, past and present
Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored with only four colors. Kenneth Appel and Wolfgang Haken finally proved this in 1976.[5]
The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved till 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).[5]
In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Kurt Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.
The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory. Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.
Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million US prize for the first correct proof, along with prizes for six other mathematical problems.[6]
Topics in discrete mathematics
The ASCII codes for the word "Wikipedia", given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms.
Discrete mathematics occurs in all the branches of mathematics listed below:
Logic
Main article: Mathematical logic
Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. As a simple example, in most systems of logic, Peirce's law (((P→Q)→P)→P) is true, and this can be easily verified with a truth table. The study of mathematical proofs is particularly important in logic, and has applications to automated theorem proving and software development.
Logical formulae are discrete structures, as are proofs, which form finite trees[7] or, more generally, directed acyclic graph structures[8][9] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulae usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued e.g. fuzzy logic.
Set theory
Main article: Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas.
In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, and further development of the theory of infinite sets is outside the scope of discrete mathematics.
Algebra
Main article: Abstract algebra
Algebraic structures such as groups, rings and fields occur as both discrete examples and continuous examples. Discrete and finite versions are important in algebraic coding theory. Discrete semigroups and monoids appear in the theory of formal languages.
Calculus of finite differences, discrete calculus or discrete analysis
Main article: finite difference
A discrete function f(n) is usually called a sequence a(n). A sequence could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. A discrete function could be defined explicitly by a list, or by a formula for f(n) or it could be given implicitly by a recurrence relation or difference equation. A difference equation is the discrete equivalent of a differential equation and can be used to aproximate the latter or studied in its own right. Every question and method about differential equations has a discrete equivalent for difference equations. For instance where there are integral transforms in harmonic analysis for studing continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces.
Geometry
Main article: discrete geometry
Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. A major topic in discrete geometry is tiling of the plane.
Topology
See combinatorial topology, topological graph theory, topological combinatorics, compuational topology.
Number theory
Main article: Number theory
Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography, cryptanalysis, and cryptology, particularly with regard to prime numbers and primality testing. In analytic number theory, techniques from continuous mathematics are also used. Discrete mathematics is concerned just with elementary number theory and excludes topics such as transcendental numbers, diophantine approximation, p-adic analysis and function fields.
Combinatorics
Main article: Combinatorics
Combinatorics studies the way in which discrete structures can be combined or arranged, and includes topics such as design theory, enumerative combinatorics, counting, combinatorial geometry, combinatorial topology. Graph theory, the study of networks, is an important part of combinatorics, with many practical applications. In analytic combinatorics and algebraic graph theory, techniques from continuous mathematics are also used, and algebraic graph theory also has close links with group theory. Infinitary combinatorics also includes continuous topics such as continuous graphs.
Information theory
Main article: Information Theory
Computational geometry applies computer algorithms to representations of geometrical objects.
Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding theory, analog encryption.
Theoretical computer science
Main article: Theoretical computer science
Complexity studies the time taken by algorithms, such as this sorting routine.
Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic. Included within theoretical computer science is the study of algorithms for computing mathematical results. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of continuous computational topics such as analog computation, continuous computability such as computable analysis, continuous complexity such as information-based complexity, and continous systems and models of computation such as analog VLSI, analog automata, differential petri nets, hybrid process algebra.
Operations research
Main article: Operations research
PERT charts like this provide a business management technique based on graph theory.
Operations research provides techniques for solving practical problems in business and other fields — problems such as allocating resources to maximize profit, or scheduling project activities to minimize risk. Operations research techniques include linear programming, queuing theory, and a continuously growing list of others. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.
Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex, although the study of differential games can be considered to be part of control theory rather than discrete mathematics.
Discretization
Main article: Discretization
Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example.
Discrete analogues of continuous mathematics
There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, difference equations, and discrete dynamical systems.
In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relations.
Hybrid discrete and continuous mathematics
The time scale calculus is a unification of the theory of difference equations with that of differential equations, which has applications to fields requiring simultaneous modelling of discrete and continuous data.
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Tuesday 23 March 2010
Discrete mathematics
Discrete mathematics
From Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Discrete_mathematics
Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.
For the mathematics journal, see Discrete Mathematics (journal).
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] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of the integers, including rational numbers but not real numbers). However, there is no exact, universally agreed, definition of the term "discrete mathematics."[4] Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
Grand challenges, past and present
Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored with only four colors. Kenneth Appel and Wolfgang Haken finally proved this in 1976.[5]
The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved till 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).[5]
In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Kurt Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.
The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory. Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.
Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million US prize for the first correct proof, along with prizes for six other mathematical problems.[6]
Topics in discrete mathematics
The ASCII codes for the word "Wikipedia", given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms.
Discrete mathematics occurs in all the branches of mathematics listed below:
Logic
Main article: Mathematical logic
Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. As a simple example, in most systems of logic, Peirce's law (((P→Q)→P)→P) is true, and this can be easily verified with a truth table. The study of mathematical proofs is particularly important in logic, and has applications to automated theorem proving and software development.
Logical formulae are discrete structures, as are proofs, which form finite trees[7] or, more generally, directed acyclic graph structures[8][9] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulae usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued e.g. fuzzy logic.
Set theory
Main article: Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas.
In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, and further development of the theory of infinite sets is outside the scope of discrete mathematics.
Algebra
Main article: Abstract algebra
Algebraic structures such as groups, rings and fields occur as both discrete examples and continuous examples. Discrete and finite versions are important in algebraic coding theory. Discrete semigroups and monoids appear in the theory of formal languages.
Calculus of finite differences, discrete calculus or discrete analysis
Main article: finite difference
A discrete function f(n) is usually called a sequence a(n). A sequence could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. A discrete function could be defined explicitly by a list, or by a formula for f(n) or it could be given implicitly by a recurrence relation or difference equation. A difference equation is the discrete equivalent of a differential equation and can be used to aproximate the latter or studied in its own right. Every question and method about differential equations has a discrete equivalent for difference equations. For instance where there are integral transforms in harmonic analysis for studing continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces.
Geometry
Main article: discrete geometry
Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. A major topic in discrete geometry is tiling of the plane.
Topology
See combinatorial topology, topological graph theory, topological combinatorics, compuational topology.
Number theory
Main article: Number theory
Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography, cryptanalysis, and cryptology, particularly with regard to prime numbers and primality testing. In analytic number theory, techniques from continuous mathematics are also used. Discrete mathematics is concerned just with elementary number theory and excludes topics such as transcendental numbers, diophantine approximation, p-adic analysis and function fields.
Combinatorics
Main article: Combinatorics
Combinatorics studies the way in which discrete structures can be combined or arranged, and includes topics such as design theory, enumerative combinatorics, counting, combinatorial geometry, combinatorial topology. Graph theory, the study of networks, is an important part of combinatorics, with many practical applications. In analytic combinatorics and algebraic graph theory, techniques from continuous mathematics are also used, and algebraic graph theory also has close links with group theory. Infinitary combinatorics also includes continuous topics such as continuous graphs.
Information theory
Main article: Information Theory
Computational geometry applies computer algorithms to representations of geometrical objects.
Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding theory, analog encryption.
Theoretical computer science
Main article: Theoretical computer science
Complexity studies the time taken by algorithms, such as this sorting routine.
Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic. Included within theoretical computer science is the study of algorithms for computing mathematical results. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of continuous computational topics such as analog computation, continuous computability such as computable analysis, continuous complexity such as information-based complexity, and continous systems and models of computation such as analog VLSI, analog automata, differential petri nets, hybrid process algebra.
Operations research
Main article: Operations research
PERT charts like this provide a business management technique based on graph theory.
Operations research provides techniques for solving practical problems in business and other fields — problems such as allocating resources to maximize profit, or scheduling project activities to minimize risk. Operations research techniques include linear programming, queuing theory, and a continuously growing list of others. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.
Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex, although the study of differential games can be considered to be part of control theory rather than discrete mathematics.
Discretization
Main article: Discretization
Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example.
Discrete analogues of continuous mathematics
There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, difference equations, and discrete dynamical systems.
In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relations.
Hybrid discrete and continuous mathematics
The time scale calculus is a unification of the theory of difference equations with that of differential equations, which has applications to fields requiring simultaneous modelling of discrete and continuous data.
From Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Discrete_mathematics
Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.
For the mathematics journal, see Discrete Mathematics (journal).
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] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of the integers, including rational numbers but not real numbers). However, there is no exact, universally agreed, definition of the term "discrete mathematics."[4] Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
Grand challenges, past and present
Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored with only four colors. Kenneth Appel and Wolfgang Haken finally proved this in 1976.[5]
The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved till 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).[5]
In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Kurt Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.
The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory. Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.
Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million US prize for the first correct proof, along with prizes for six other mathematical problems.[6]
Topics in discrete mathematics
The ASCII codes for the word "Wikipedia", given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms.
Discrete mathematics occurs in all the branches of mathematics listed below:
Logic
Main article: Mathematical logic
Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. As a simple example, in most systems of logic, Peirce's law (((P→Q)→P)→P) is true, and this can be easily verified with a truth table. The study of mathematical proofs is particularly important in logic, and has applications to automated theorem proving and software development.
Logical formulae are discrete structures, as are proofs, which form finite trees[7] or, more generally, directed acyclic graph structures[8][9] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulae usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued e.g. fuzzy logic.
Set theory
Main article: Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas.
In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, and further development of the theory of infinite sets is outside the scope of discrete mathematics.
Algebra
Main article: Abstract algebra
Algebraic structures such as groups, rings and fields occur as both discrete examples and continuous examples. Discrete and finite versions are important in algebraic coding theory. Discrete semigroups and monoids appear in the theory of formal languages.
Calculus of finite differences, discrete calculus or discrete analysis
Main article: finite difference
A discrete function f(n) is usually called a sequence a(n). A sequence could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. A discrete function could be defined explicitly by a list, or by a formula for f(n) or it could be given implicitly by a recurrence relation or difference equation. A difference equation is the discrete equivalent of a differential equation and can be used to aproximate the latter or studied in its own right. Every question and method about differential equations has a discrete equivalent for difference equations. For instance where there are integral transforms in harmonic analysis for studing continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces.
Geometry
Main article: discrete geometry
Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. A major topic in discrete geometry is tiling of the plane.
Topology
See combinatorial topology, topological graph theory, topological combinatorics, compuational topology.
Number theory
Main article: Number theory
Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography, cryptanalysis, and cryptology, particularly with regard to prime numbers and primality testing. In analytic number theory, techniques from continuous mathematics are also used. Discrete mathematics is concerned just with elementary number theory and excludes topics such as transcendental numbers, diophantine approximation, p-adic analysis and function fields.
Combinatorics
Main article: Combinatorics
Combinatorics studies the way in which discrete structures can be combined or arranged, and includes topics such as design theory, enumerative combinatorics, counting, combinatorial geometry, combinatorial topology. Graph theory, the study of networks, is an important part of combinatorics, with many practical applications. In analytic combinatorics and algebraic graph theory, techniques from continuous mathematics are also used, and algebraic graph theory also has close links with group theory. Infinitary combinatorics also includes continuous topics such as continuous graphs.
Information theory
Main article: Information Theory
Computational geometry applies computer algorithms to representations of geometrical objects.
Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding theory, analog encryption.
Theoretical computer science
Main article: Theoretical computer science
Complexity studies the time taken by algorithms, such as this sorting routine.
Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic. Included within theoretical computer science is the study of algorithms for computing mathematical results. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of continuous computational topics such as analog computation, continuous computability such as computable analysis, continuous complexity such as information-based complexity, and continous systems and models of computation such as analog VLSI, analog automata, differential petri nets, hybrid process algebra.
Operations research
Main article: Operations research
PERT charts like this provide a business management technique based on graph theory.
Operations research provides techniques for solving practical problems in business and other fields — problems such as allocating resources to maximize profit, or scheduling project activities to minimize risk. Operations research techniques include linear programming, queuing theory, and a continuously growing list of others. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.
Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex, although the study of differential games can be considered to be part of control theory rather than discrete mathematics.
Discretization
Main article: Discretization
Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example.
Discrete analogues of continuous mathematics
There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, difference equations, and discrete dynamical systems.
In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relations.
Hybrid discrete and continuous mathematics
The time scale calculus is a unification of the theory of difference equations with that of differential equations, which has applications to fields requiring simultaneous modelling of discrete and continuous data.