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</style><div role="note" class="hatnote navigation-not-searchable">For the mathematics journal, see <a href="Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)"><i>Discrete Mathematics</i> (journal)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Finite math" redirects here. For the syllabus, see <a href="Finite_mathematics" title="Finite mathematics">Finite mathematics</a>.</div>

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Mathematics" title="Mathematics">Mathematics</a></th></tr><tr><td class="sidebar-above" style="padding-bottom:0.35em;">

<ul><li><a href="History_of_mathematics" title="History of mathematics">History</a></li>

<li><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Index</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Number_theory" title="Number theory">Number theory</a></li>

<li><a href="Geometry" title="Geometry">Geometry</a></li>

<li><a href="Algebra" title="Algebra">Algebra</a></li>

<li><a href="Calculus" title="Calculus">Calculus</a> and <a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Logic</a></li>

<li><a href="Set_theory" title="Set theory">Set theory</a></li>

<li><a href="Probability" title="Probability">Probability</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a> and <a href="Decision_theory" title="Decision theory">Decision theory</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)">Relationship with sciences</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Mathematical_physics" title="Mathematical physics">Physics</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li>

<li><a href="Geomathematics" title="Geomathematics">Geosciences</a></li>

<li><a href="Computational_mathematics" title="Computational mathematics">Computation</a></li>

<li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li>

<li><a href="Mathematical_linguistics" title="Mathematical linguistics">Linguistics</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Economics</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy</a></li>

<li><a href="Mathematics_education" title="Mathematics education">Education</a></li></ul></div></div></td>

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<p><b>Discrete mathematics</b> is the study of <a href="Mathematical_structures" class="mw-redirect" title="Mathematical structures">mathematical structures</a> that can be considered "discrete" (in a way analogous to <a href="Discrete_variable" class="mw-redirect" title="Discrete variable">discrete variables</a>, having a one-to-one correspondence (<a href="Bijection" title="Bijection">bijection</a>) with <a href="Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>), rather than "continuous" (analogously to <a href="Continuous_function" title="Continuous function">continuous functions</a>). Objects studied in discrete mathematics include <a href="Integer" title="Integer">integers</a>, <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>, and <a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">statements</a> in <a href="Mathematical_logic" title="Mathematical logic">logic</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> By contrast, discrete mathematics excludes topics in "continuous mathematics" such as <a href="Real_number" title="Real number">real numbers</a>, <a href="Calculus" title="Calculus">calculus</a> or <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. Discrete objects can often be <a href="Enumeration" title="Enumeration">enumerated</a> by <a href="Integers" class="mw-redirect" title="Integers">integers</a>; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with <a href="Countable_set" title="Countable set">countable sets</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (finite sets or sets with the same <a href="Cardinality" title="Cardinality">cardinality</a> as the natural numbers). However, there is no exact definition of the term "discrete mathematics".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>The set of objects studied in discrete mathematics can be finite or infinite. The term <b>finite mathematics</b> is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.

</p><p>Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of <a href="Digital_computers" class="mw-redirect" title="Digital computers">digital computers</a> 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 <a href="Computer_science" title="Computer science">computer science</a>, such as <a href="Computer_algorithm" class="mw-redirect" title="Computer algorithm">computer algorithms</a>, <a href="Programming_language" title="Programming language">programming languages</a>, <a href="Cryptography" title="Cryptography">cryptography</a>, <a href="Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a>, and <a href="Software_development" title="Software development">software development</a>. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems.

</p><p>Although the main objects of study in discrete mathematics are discrete objects, <a href="Analysis_(mathematics)" class="mw-redirect" title="Analysis (mathematics)">analytic</a> methods from "continuous" mathematics are often employed as well.

</p><p>In university curricula, discrete mathematics appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by <a href="Association_for_Computing_Machinery" title="Association for Computing Machinery">ACM</a> and <a href="Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a> into a course that is basically intended to develop <a href="Mathematical_maturity" title="Mathematical maturity">mathematical maturity</a> in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well.<sup id="cite_ref-LevasseurDoerr_6-0" class="reference"><a href="#cite_note-LevasseurDoerr-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Howson1988_7-0" class="reference"><a href="#cite_note-Howson1988-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Some high-school-level discrete mathematics textbooks have appeared as well.<sup id="cite_ref-Rosenstein_8-0" class="reference"><a href="#cite_note-Rosenstein-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> At this level, discrete mathematics is sometimes seen as a preparatory course, like <a href="Precalculus" title="Precalculus">precalculus</a> in this respect.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Fulkerson_Prize" title="Fulkerson Prize">Fulkerson Prize</a> is awarded for outstanding papers in discrete mathematics.

</p>

<div class="mw-heading mw-heading2"><h2 id="Topics">Topics</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Outline_of_discrete_mathematics" title="Outline of discrete mathematics">Outline of discrete mathematics</a></div>

<div class="mw-heading mw-heading3"><h3 id="Theoretical_computer_science">Theoretical computer science</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></div>

<p>Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on <a href="Graph_theory" title="Graph theory">graph theory</a> and <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>. Included within theoretical computer science is the study of algorithms and data structures. <a href="Computability" title="Computability">Computability</a> studies what can be computed in principle, and has close ties to logic, while complexity studies the time, space, and other resources taken by computations. <a href="Automata_theory" title="Automata theory">Automata theory</a> and <a href="Formal_language" title="Formal language">formal language</a> theory are closely related to computability. <a href="Petri_net" title="Petri net">Petri nets</a> and <a href="Process_algebra" class="mw-redirect" title="Process algebra">process algebras</a> are used to model computer systems, and methods from discrete mathematics are used in analyzing <a href="VLSI" class="mw-redirect" title="VLSI">VLSI</a> electronic circuits. <a href="Computational_geometry" title="Computational geometry">Computational geometry</a> applies algorithms to geometrical problems and representations of <a href="Geometry" title="Geometry">geometrical</a> objects, while <a href="Computer_image_analysis" class="mw-redirect" title="Computer image analysis">computer image analysis</a> applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics.

</p>

<div class="mw-heading mw-heading3"><h3 id="Information_theory">Information theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Information_theory" title="Information theory">Information theory</a></div>

<p>Information theory involves the quantification of <a href="Information" title="Information">information</a>. Closely related is <a href="Coding_theory" title="Coding theory">coding theory</a> which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: <a href="Analog_signal" title="Analog signal">analog signals</a>, <a href="Analog_coding" class="mw-redirect" title="Analog coding">analog coding</a>, <a href="Analog_encryption" class="mw-redirect" title="Analog encryption">analog encryption</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Logic">Logic</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div>

<p>Logic is the study of the principles of valid reasoning and <a href="Inference" title="Inference">inference</a>, as well as of <a href="Consistency" title="Consistency">consistency</a>, <a href="Soundness" title="Soundness">soundness</a>, and <a href="Completeness_(logic)" title="Completeness (logic)">completeness</a>. For example, in most systems of logic (but not in <a href="Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>) <a href="Peirce's_law" title="Peirce's law">Peirce's law</a> (((<i>P</i>→<i>Q</i>)→<i>P</i>)→<i>P</i>) is a theorem. For classical logic, it can be easily verified with a <a href="Truth_table" title="Truth table">truth table</a>. The study of <a href="Mathematical_proof" title="Mathematical proof">mathematical proof</a> is particularly important in logic, and has accumulated to <a href="Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a> and <a href="Formal_verification" title="Formal verification">formal verification</a> of software.

</p><p><a href="Well-formed_formula" title="Well-formed formula">Logical formulas</a> are discrete structures, as are <a href="Proof_theory" title="Proof theory">proofs</a>, which form finite <a href="Tree_structure" title="Tree structure">trees</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> or, more generally, <a href="Directed_acyclic_graph" title="Directed acyclic graph">directed acyclic graph</a> structures<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> (with each <a href="Rule_of_inference" title="Rule of inference">inference step</a> combining one or more <a href="Premise" title="Premise">premise</a> branches to give a single conclusion). The <a href="Truth_value" title="Truth value">truth values</a> of logical formulas usually form a finite set, generally restricted to two values: <i>true</i> and <i>false</i>, but logic can also be continuous-valued, e.g., <a href="Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a>. Concepts such as infinite proof trees or infinite derivation trees have also been studied,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> e.g. <a href="Infinitary_logic" title="Infinitary logic">infinitary logic</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Set_theory">Set theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Set_theory" title="Set theory">Set theory</a></div>

<p>Set theory is the branch of mathematics that studies <a href="Set_(mathematics)" title="Set (mathematics)">sets</a>, which are collections of objects, such as {blue, white, red} or the (infinite) set of all <a href="Prime_number" title="Prime number">prime numbers</a>. <a href="Partially_ordered_set" title="Partially ordered set">Partially ordered sets</a> and sets with other <a href="Relation_(mathematics)" title="Relation (mathematics)">relations</a> have applications in several areas.

</p><p>In discrete mathematics, <a href="Countable_set" title="Countable set">countable sets</a> (including <a href="Finite_set" title="Finite set">finite sets</a>) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by <a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a>'s work distinguishing between different kinds of <a href="Infinite_set" title="Infinite set">infinite set</a>, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in <a href="Descriptive_set_theory" title="Descriptive set theory">descriptive set theory</a> makes extensive use of traditional continuous mathematics.

</p>

<div class="mw-heading mw-heading3"><h3 id="Combinatorics">Combinatorics</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Combinatorics" title="Combinatorics">Combinatorics</a></div>

<p>Combinatorics studies the ways in which discrete structures can be combined or arranged.

<a href="Enumerative_combinatorics" title="Enumerative combinatorics">Enumerative combinatorics</a> concentrates on counting the number of certain combinatorial objects - e.g. the <a href="Twelvefold_way" title="Twelvefold way">twelvefold way</a> provides a unified framework for counting <a href="Permutations" class="mw-redirect" title="Permutations">permutations</a>, <a href="Combinations" class="mw-redirect" title="Combinations">combinations</a> and <a href="Partition_of_a_set" title="Partition of a set">partitions</a>.

<a href="Analytic_combinatorics" title="Analytic combinatorics">Analytic combinatorics</a> concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from <a href="Complex_analysis" title="Complex analysis">complex analysis</a> and <a href="Probability_theory" title="Probability theory">probability theory</a>. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and <a href="Generating_functions" class="mw-redirect" title="Generating functions">generating functions</a> to describe the results, analytic combinatorics aims at obtaining <a href="Asymptotic_analysis" title="Asymptotic analysis">asymptotic formulae</a>.

<a href="Topological_combinatorics" title="Topological combinatorics">Topological combinatorics</a> concerns the use of techniques from <a href="Topology" title="Topology">topology</a> and <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>/<a href="Combinatorial_topology" title="Combinatorial topology">combinatorial topology</a> in <a href="Combinatorics" title="Combinatorics">combinatorics</a>.

Design theory is a study of <a href="Combinatorial_design" title="Combinatorial design">combinatorial designs</a>, which are collections of subsets with certain <a href="Set_intersection" class="mw-redirect" title="Set intersection">intersection</a> properties.

<a href="Partition_theory" class="mw-redirect" title="Partition theory">Partition theory</a> studies various enumeration and asymptotic problems related to <a href="Integer_partition" title="Integer partition">integer partitions</a>, and is closely related to <a href="Q-series" class="mw-redirect" title="Q-series">q-series</a>, <a href="Special_functions" title="Special functions">special functions</a> and <a href="Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a>. Originally a part of <a href="Number_theory" title="Number theory">number theory</a> and <a href="Analysis" title="Analysis">analysis</a>, partition theory is now considered a part of combinatorics or an independent field.

<a href="Order_theory" title="Order theory">Order theory</a> is the study of <a href="Partially_ordered_sets" class="mw-redirect" title="Partially ordered sets">partially ordered sets</a>, both finite and infinite.

</p>

<div class="mw-heading mw-heading3"><h3 id="Graph_theory">Graph theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Graph_theory" title="Graph theory">Graph theory</a></div>

<p>Graph theory, the study of <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> and <a href="Network_theory" title="Network theory">networks</a>, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of <a href="Topology" title="Topology">topology</a>, e.g. <a href="Knot_theory" title="Knot theory">knot theory</a>. <a href="Algebraic_graph_theory" title="Algebraic graph theory">Algebraic graph theory</a> has close links with group theory and <a href="Topological_graph_theory" title="Topological graph theory">topological graph theory</a> has close links to <a href="Topology" title="Topology">topology</a>. There are also <a href="Continuous_graph" class="mw-redirect" title="Continuous graph">continuous graphs</a>; however, for the most part, research in graph theory falls within the domain of discrete mathematics.

</p>

<div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Number_theory" title="Number theory">Number theory</a></div>

<p>Number theory is concerned with the properties of numbers in general, particularly <a href="Integer" title="Integer">integers</a>. It has applications to <a href="Cryptography" title="Cryptography">cryptography</a> and <a href="Cryptanalysis" title="Cryptanalysis">cryptanalysis</a>, particularly with regard to <a href="Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>, <a href="Diophantine_equations" class="mw-redirect" title="Diophantine equations">diophantine equations</a>, linear and quadratic congruences, prime numbers and <a href="Primality_test" title="Primality test">primality testing</a>. Other discrete aspects of number theory include <a href="Geometry_of_numbers" title="Geometry of numbers">geometry of numbers</a>. In <a href="Analytic_number_theory" title="Analytic number theory">analytic number theory</a>, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include <a href="Transcendental_number" title="Transcendental number">transcendental numbers</a>, <a href="Diophantine_approximation" title="Diophantine approximation">diophantine approximation</a>, <a href="P-adic_analysis" title="P-adic analysis">p-adic analysis</a> and <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function fields</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_structures">Algebraic structures</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Abstract_algebra" title="Abstract algebra">Abstract algebra</a></div>

<p><a href="Algebraic_structure" title="Algebraic structure">Algebraic structures</a> occur as both discrete examples and continuous examples. Discrete algebras include: <a href="Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a> used in <a href="Logic_gate" title="Logic gate">logic gates</a> and programming; <a href="Relational_algebra" title="Relational algebra">relational algebra</a> used in <a href="Databases" class="mw-redirect" title="Databases">databases</a>; discrete and finite versions of <a href="Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a> and <a href="Field_(mathematics)" title="Field (mathematics)">fields</a> are important in <a href="Algebraic_coding_theory" class="mw-redirect" title="Algebraic coding theory">algebraic coding theory</a>; discrete <a href="Semigroup" title="Semigroup">semigroups</a> and <a href="Monoid" title="Monoid">monoids</a> appear in the theory of <a href="Formal_languages" class="mw-redirect" title="Formal languages">formal languages</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Discrete_analogues_of_continuous_mathematics">Discrete analogues of continuous mathematics</h3></div>

<p>There are many concepts and theories in continuous mathematics which have discrete versions, such as <a href="Discrete_calculus" title="Discrete calculus">discrete calculus</a>, <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transforms</a>, <a href="Discrete_geometry" title="Discrete geometry">discrete geometry</a>, <a href="Discrete_logarithm" title="Discrete logarithm">discrete logarithms</a>, <a href="Discrete_differential_geometry" title="Discrete differential geometry">discrete differential geometry</a>, <a href="Discrete_exterior_calculus" title="Discrete exterior calculus">discrete exterior calculus</a>, <a href="Discrete_Morse_theory" title="Discrete Morse theory">discrete Morse theory</a>, <a href="Discrete_optimization" title="Discrete optimization">discrete optimization</a>, <a href="Discrete_probability_theory" class="mw-redirect" title="Discrete probability theory">discrete probability theory</a>, <a href="Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete probability distribution</a>, <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a>, <a href="Discrete_dynamical_system" class="mw-redirect" title="Discrete dynamical system">discrete dynamical systems</a>, and <a href="Shapley%E2%80%93Folkman_lemma#Probability_and_measure_theory" title="Shapley–Folkman lemma">discrete vector measures</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Calculus_of_finite_differences,_discrete_analysis,_and_discrete_calculus">Calculus of finite differences, discrete analysis, and discrete calculus</h4></div>

<p>In <a href="Discrete_calculus" title="Discrete calculus">discrete calculus</a> and the <a href="Calculus_of_finite_differences" class="mw-redirect" title="Calculus of finite differences">calculus of finite differences</a>, a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> defined on an interval of the <a href="Integer" title="Integer">integers</a> is usually called a <a href="Sequence" title="Sequence">sequence</a>. A sequence could be a finite sequence from a data source or an infinite sequence from a <a href="Discrete_dynamical_system" class="mw-redirect" title="Discrete dynamical system">discrete dynamical system</a>. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a> or <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a>. Difference equations are similar to <a href="Differential_equation" title="Differential equation">differential equations</a>, but replace <a href="Derivative" title="Derivative">differentiation</a> by taking the difference between adjacent terms; they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance, where there are <a href="Integral_transforms" class="mw-redirect" title="Integral transforms">integral transforms</a> in <a href="Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> for studying continuous functions or analogue signals, there are <a href="Discrete_transform" title="Discrete transform">discrete transforms</a> for discrete functions or digital signals. As well as <a href="Discrete_metric_space" class="mw-redirect" title="Discrete metric space">discrete metric spaces</a>, there are more general <a href="Discrete_topological_space" class="mw-redirect" title="Discrete topological space">discrete topological spaces</a>, <a href="Finite_metric_space" class="mw-redirect" title="Finite metric space">finite metric spaces</a>, <a href="Finite_topological_space" title="Finite topological space">finite topological spaces</a>.

</p><p>The <a href="Time_scale_calculus" class="mw-redirect" title="Time scale calculus">time scale calculus</a> is a unification of the theory of <a href="Difference_equations" class="mw-redirect" title="Difference equations">difference equations</a> with that of <a href="Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>, which has applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modeling such a situation is the notion of <a href="Hybrid_system" title="Hybrid system">hybrid dynamical systems</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Discrete_geometry">Discrete geometry</h4></div>

<p><a href="Discrete_geometry" title="Discrete geometry">Discrete geometry</a> and combinatorial geometry are about combinatorial properties of <i>discrete collections</i> of geometrical objects. A long-standing topic in discrete geometry is <a href="Tessellation" title="Tessellation">tiling of the plane</a>.

</p><p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the concept of a curve can be extended to discrete geometries by taking the <a href="Spectrum_of_a_ring" title="Spectrum of a ring">spectra</a> of <a href="Polynomial_ring" title="Polynomial ring">polynomial rings</a> over <a href="Finite_field" title="Finite field">finite fields</a> to be models of the <a href="Affine_space" title="Affine space">affine spaces</a> over that field, and letting <a href="Algebraic_variety" title="Algebraic variety">subvarieties</a> or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, every point of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}}</annotation>

</semantics>

</math></span><img src="./147df15a1780a002602cd26438adbec315699e2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.429ex; height:3.176ex;" alt="{\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> a field can be studied either as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}</annotation>

</semantics>

</math></span><img src="./80cc3220b7c86e6d0862f1bcf3fbac3ffc0191a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.076ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}" loading="lazy"></span>, a point, or as the spectrum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} K[x]_{(x-c)}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} K[x]_{(x-c)}}</annotation>

</semantics>

</math></span><img src="./76e6e66e203e1805ec5b90fa25d9f0c817f28dd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.169ex; height:3.176ex;" alt="{\displaystyle \operatorname {Spec} K[x]_{(x-c)}}" loading="lazy"></span> of the <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">local ring at (x-c)</a>, a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of <a href="Tangent_space" title="Tangent space">tangent space</a> called the <a href="Zariski_tangent_space" title="Zariski tangent space">Zariski tangent space</a>, making many features of calculus applicable even in finite settings.

</p>

<div class="mw-heading mw-heading4"><h4 id="Discrete_modelling">Discrete modelling</h4></div>

<p>In <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>, <a href="Discrete_modelling" title="Discrete modelling">discrete modelling</a> is the discrete analogue of <a href="Continuous_modelling" title="Continuous modelling">continuous modelling</a>. In discrete modelling, discrete formulae are fit to <a href="Data" title="Data">data</a>. A common method in this form of modelling is to use <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>. <a href="Discretization" title="Discretization">Discretization</a> concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. <a href="Numerical_analysis" title="Numerical analysis">Numerical analysis</a> provides an important example.

</p>

<div class="mw-heading mw-heading2"><h2 id="Challenges">Challenges</h2></div>

<p>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 <a href="Four_color_theorem" title="Four color theorem">four color theorem</a>, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).<sup id="cite_ref-4colors_15-1" class="reference"><a href="#cite_note-4colors-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p><p>In <a href="Mathematical_logic" title="Mathematical logic">logic</a>, the <a href="Hilbert's_second_problem" title="Hilbert's second problem">second problem</a> on <a href="David_Hilbert" title="David Hilbert">David Hilbert</a>'s list of open <a href="Hilbert's_problems" title="Hilbert's problems">problems</a> presented in 1900 was to prove that the <a href="Axioms" class="mw-redirect" title="Axioms">axioms</a> of <a href="Arithmetic" title="Arithmetic">arithmetic</a> are <a href="Consistent" class="mw-redirect" title="Consistent">consistent</a>. <a href="G%C3%B6del's_second_incompleteness_theorem" class="mw-redirect" title="Gödel's second incompleteness theorem">Gödel's second incompleteness theorem</a>, proved in 1931, showed that this was not possible – at least not within arithmetic itself. <a href="Hilbert's_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a> was to determine whether a given polynomial <a href="Diophantine_equation" title="Diophantine equation">Diophantine equation</a> with integer coefficients has an integer solution. In 1970, <a href="Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a> proved that this <a href="Matiyasevich's_theorem" class="mw-redirect" title="Matiyasevich's theorem">could not be done</a>.

</p><p>The need to <a href="Cryptanalysis" title="Cryptanalysis">break</a> German codes in <a href="World_War_II" title="World War II">World War II</a> led to advances in <a href="Cryptography" title="Cryptography">cryptography</a> and <a href="Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>, with the <a href="Colossus_computer" title="Colossus computer">first programmable digital electronic computer</a> being developed at England's <a href="Bletchley_Park" title="Bletchley Park">Bletchley Park</a> with the guidance of <a href="Alan_Turing" title="Alan Turing">Alan Turing</a> and his seminal work, <a href="On_Computable_Numbers" class="mw-redirect" title="On Computable Numbers">On Computable Numbers</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The <a href="Cold_War" title="Cold War">Cold War</a> meant that cryptography remained important, with fundamental advances such as <a href="Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a> being developed in the following decades. The <a href="Telecommunications_industry" title="Telecommunications industry">telecommunications industry</a> has also motivated advances in discrete mathematics, particularly in graph theory and <a href="Information_theory" title="Information theory">information theory</a>. <a href="Formal_verification" title="Formal verification">Formal verification</a> of statements in logic has been necessary for <a href="Software_development" title="Software development">software development</a> of <a href="Safety-critical_system" title="Safety-critical system">safety-critical systems</a>, and advances in <a href="Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a> have been driven by this need.

</p><p><a href="Computational_geometry" title="Computational geometry">Computational geometry</a> has been an important part of the <a href="Computer_graphics_(computer_science)" title="Computer graphics (computer science)">computer graphics</a> incorporated into modern <a href="Video_game" title="Video game">video games</a> and <a href="Computer-aided_design" title="Computer-aided design">computer-aided design</a> tools.

</p><p>Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and <a href="Combinatorics" title="Combinatorics">combinatorics</a>, are important in addressing the challenging <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a> problems associated with understanding the <a href="Phylogenetic_tree" title="Phylogenetic tree">tree of life</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p><p>Currently, one of the most famous open problems in theoretical computer science is the <a href="P_%3D_NP_problem" class="mw-redirect" title="P = NP problem">P = NP problem</a>, which involves the relationship between the <a href="Complexity_class" title="Complexity class">complexity classes</a> <a href="P_(complexity)" title="P (complexity)">P</a> and <a href="NP_(complexity)" title="NP (complexity)">NP</a>. The <a href="Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> has offered a $1 million <a href="USD" class="mw-redirect" title="USD">USD</a> prize for the first correct proof, along with prizes for <a href="Millennium_Prize_Problems" title="Millennium Prize Problems">six other mathematical problems</a>.<sup id="cite_ref-CMI_Millennium_Prize_Problems_18-0" class="reference"><a href="#cite_note-CMI_Millennium_Prize_Problems-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

</p>

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<ul><li><a href="Outline_of_discrete_mathematics" title="Outline of discrete mathematics">Outline of discrete mathematics</a></li>

<li><a href="Cyberchase" title="Cyberchase">Cyberchase</a>, a show that teaches discrete mathematics to children</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Richard_Johnsonbaugh" title="Richard Johnsonbaugh">Richard Johnsonbaugh</a>, <i>Discrete Mathematics</i>, Prentice Hall, 2008.</span>

</li>

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</style><cite id="CITEREFFranklin2017" class="citation journal cs1"><a href="James_Franklin_(philosopher)" title="James Franklin (philosopher)">Franklin, James</a> (2017). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/16561/1/Discrete%20and%20Continuous.pdf">"Discrete and continuous: a fundamental dichotomy in mathematics"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Humanistic Mathematics</i>. <b>7</b> (2): <span class="nowrap">355–</span>378. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.5642%2Fjhummath.201702.18">10.5642/jhummath.201702.18</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6945363">6945363</a><span class="reference-accessdate">. Retrieved <span class="nowrap">30 June</span> 2021</span>.</cite></span>

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<li id="cite_note-Howson1988-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Howson1988_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGeoffrey_Howson1988" class="citation book cs1">Geoffrey Howson, Albert, ed. (1988). <i>Mathematics as a Service Subject</i>. Cambridge University Press. pp. <span class="nowrap">77–</span>78. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-35395-3</bdi>.</cite></span>

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<li id="cite_note-Rosenstein-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rosenstein_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRosenstein" class="citation book cs1">Rosenstein, Joseph G. <i>Discrete Mathematics in the Schools</i>. American Mathematical Society. p. 323. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-8578-9</bdi>.</cite></span>

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<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFBrotherstonBornatCalcagno2008" class="citation journal cs1">Brotherston, J.; Bornat, R.; Calcagno, C. (January 2008). "Cyclic proofs of program termination in separation logic". <i>ACM SIGPLAN Notices</i>. <b>43</b> (1): <span class="nowrap">101–</span>112. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1328897.1328453">10.1145/1328897.1328453</a>.</cite></span>

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<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFMoharThomassen2001" class="citation book cs1"><a href="Bojan_Mohar" title="Bojan Mohar">Mohar, Bojan</a>; <a href="Carsten_Thomassen_(mathematician)" title="Carsten Thomassen (mathematician)">Thomassen, Carsten</a> (2001). <a rel="nofollow" class="external text" href="https://www.press.jhu.edu/books/title/1675/graphs-surfaces"><i>Graphs on Surfaces</i></a>. Johns Hopkins University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8018-6689-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/45102952">45102952</a>.</cite></span>

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<li id="cite_note-4colors-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-4colors_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-4colors_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFWilson2002" class="citation book cs1"><a href="Robin_Wilson_(mathematician)" title="Robin Wilson (mathematician)">Wilson, Robin</a> (2002). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fourcolorssuffic00wils"><i>Four Colors Suffice</i></a></span>. London: Penguin Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-11533-7</bdi>.</cite></span>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li><cite id="CITEREFBiggs2002" class="citation book cs1"><a href="Norman_L._Biggs" title="Norman L. Biggs">Biggs, Norman L.</a> (2002). <i>Discrete Mathematics</i>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-850717-8</bdi>.</cite></li>

<li><cite id="CITEREFDwyer2010" class="citation book cs1">Dwyer, John (2010). <i>An Introduction to Discrete Mathematics for Business & Computing</i>. Algana Pub. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-907934-00-1</bdi>.</cite></li>

<li><cite id="CITEREFEpp2010" class="citation book cs1"><a href="Susanna_S._Epp" title="Susanna S. Epp">Epp, Susanna S.</a> (2010-08-04). <i>Discrete Mathematics With Applications</i>. Thomson Brooks/Cole. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-495-39132-6</bdi>.</cite></li>

<li><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation book cs1"><a href="Ronald_Graham" title="Ronald Graham">Graham, Ronald</a>; <a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a>; <a href="Oren_Patashnik" title="Oren Patashnik">Patashnik, Oren</a> (1994). <a href="Concrete_Mathematics" title="Concrete Mathematics"><i>Concrete Mathematics</i></a> (2nd ed.). Addison–Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-55802-5</bdi>.</cite></li>

<li><cite id="CITEREFGrimaldi2004" class="citation book cs1"><a href="Ralph_P._Grimaldi" class="mw-redirect" title="Ralph P. Grimaldi">Grimaldi, Ralph P.</a> (2004). <i>Discrete and Combinatorial Mathematics: An Applied Introduction</i>. Addison Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-72634-3</bdi>.</cite></li>

<li><cite id="CITEREFKnuth2011" class="citation book cs1 cs1-prop-long-vol">Knuth, Donald E. (2011). <a href="The_Art_of_Computer_Programming" title="The Art of Computer Programming"><i>The Art of Computer Programming</i></a>. Vol. 1–4a Boxed Set. Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-321-75104-1</bdi>.</cite></li>

<li><cite id="CITEREFMatoušekNešetřil1998" class="citation book cs1"><a href="Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)" title="Jiří Matoušek (mathematician)">Matoušek, Jiří</a>; <a href="Jaroslav_Ne%C5%A1et%C5%99il" title="Jaroslav Nešetřil">Nešetřil, Jaroslav</a> (1998). <i>Discrete Mathematics</i>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-850208-1</bdi>.</cite></li>

<li><cite id="CITEREFObrenic2003" class="citation book cs1">Obrenic, Bojana (2003). <i>Practice Problems in Discrete Mathematics</i>. Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-045803-2</bdi>.</cite></li>

<li><cite id="CITEREFRosenMichaels2000" class="citation book cs1">Rosen, Kenneth H.; Michaels, John G. (2000). <i>Hand Book of Discrete and Combinatorial Mathematics</i>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8493-0149-0</bdi>.</cite></li>

<li><cite id="CITEREFRosen2007" class="citation book cs1">Rosen, Kenneth H. (2007). <i>Discrete Mathematics: And Its Applications</i>. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-288008-3</bdi>.</cite></li>

<li><cite id="CITEREFSimpson2002" class="citation book cs1"><a href="Andrew_Clive_Simpson" title="Andrew Clive Simpson">Simpson, Andrew</a> (2002). <i>Discrete Mathematics by Example</i>. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-709840-7</bdi>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Discrete_Mathematics" class="extiw external" title="wikibooks:Discrete Mathematics">Discrete Mathematics</a></b></i></div></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Discrete_mathematics" class="extiw external" title="commons:Category:Discrete mathematics">Discrete mathematics</a></span>.</div></div>

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<ul><li><a rel="nofollow" class="external text" href="http://archives.math.utk.edu/topics/discreteMath.html">Discrete mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110829184228/http://archives.math.utk.edu/topics/discreteMath.html">Archived</a> 2011-08-29 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> at the utk.edu Mathematics Archives, providing links to syllabi, tutorials, programs, etc.</li>

<li><a rel="nofollow" class="external text" href="http://www.iowacentral.edu/industrial_technology/electrical_technologies/index.asp">Iowa Central: Electrical Technologies Program</a> Discrete mathematics for <a href="Electrical_engineering" title="Electrical engineering">Electrical engineering</a>.</li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas1069" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Major_mathematics_areas1069" style="font-size:114%;margin:0 4em">Major <a href="Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_mathematics" title="History of mathematics">History</a>

<ul><li><a href="Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li>

<li><a href="Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li>

<li><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li>

<li><a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Category_theory" title="Category theory">Category theory</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li>

<li><a href="Set_theory" title="Set theory">Set theory</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_algebra" title="Abstract algebra">Abstract</a></li>

<li><a href="Commutative_algebra" title="Commutative algebra">Commutative</a></li>

<li><a href="Elementary_algebra" title="Elementary algebra">Elementary</a></li>

<li><a href="Group_theory" title="Group theory">Group theory</a></li>

<li><a href="Linear_algebra" title="Linear algebra">Linear</a></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li>

<li><a href="Universal_algebra" title="Universal algebra">Universal</a></li>

<li><a href="Homological_algebra" title="Homological algebra">Homological</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Calculus" title="Calculus">Calculus</a></li>

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Complex_analysis" title="Complex analysis">Complex analysis</a></li>

<li><a href="Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li>

<li><a href="Differential_equation" title="Differential equation">Differential equations</a></li>

<li><a href="Functional_analysis" title="Functional analysis">Functional analysis</a></li>

<li><a href="Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Combinatorics" title="Combinatorics">Combinatorics</a></li>

<li><a href="Graph_theory" title="Graph theory">Graph theory</a></li>

<li><a href="Order_theory" title="Order theory">Order theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li>

<li><a href="Analytic_geometry" title="Analytic geometry">Analytic</a></li>

<li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential</a></li>

<li><a href="Discrete_geometry" title="Discrete geometry">Discrete</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li>

<li><a href="Finite_geometry" title="Finite geometry">Finite</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic" title="Arithmetic">Arithmetic</a></li>

<li><a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li>

<li><a href="Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li>

<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_topology" title="General topology">General</a></li>

<li><a href="Algebraic_topology" title="Algebraic topology">Algebraic</a></li>

<li><a href="Differential_topology" title="Differential topology">Differential</a></li>

<li><a href="Geometric_topology" title="Geometric topology">Geometric</a></li>

<li><a href="Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li>

<li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li>

<li><a href="Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li>

<li><a href="Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Systems_science" title="Systems science">Systems science</a>

<ul><li><a href="Control_theory" title="Control theory">Control theory</a></li>

<li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer_science" title="Computer science">Computer science</a></li>

<li><a href="Theory_of_computation" title="Theory of computation">Theory of computation</a></li>

<li><a href="Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li>

<li><a href="Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li>

<li><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a></li>

<li><a href="Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a>

<ul><li><a href="List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li>

<li><a href="Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li>

<li><a href="List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li>

<li><a href="Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li>

<li><a href="Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li>

<li><a href="Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> <b>Category</b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw external" title="commons:Category:Mathematics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Industrial_and_applied_mathematics506" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Industrial_and_applied_mathematics506" style="font-size:114%;margin:0 4em"><a href="Applied_mathematics" title="Applied mathematics">Industrial and applied mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algorithm" title="Algorithm">Algorithms</a>

<ul><li><a href="Algorithm_design" class="mw-redirect" title="Algorithm design">design</a></li>

<li><a href="Analysis_of_algorithms" title="Analysis of algorithms">analysis</a></li></ul></li>

<li><a href="Automata_theory" title="Automata theory">Automata theory</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Coding_theory" title="Coding theory">Coding theory</a></li>

<li><a href="Computational_geometry" title="Computational geometry">Computational geometry</a></li>

<li><a href="Constraint_satisfaction_problem" title="Constraint satisfaction problem">Constraint satisfaction</a>

<ul><li><a href="Constraint_programming" title="Constraint programming">Constraint programming</a></li></ul></li>

<li><a href="Logic_in_computer_science" title="Logic in computer science">Computational logic</a></li>

<li><a href="Cryptography" title="Cryptography">Cryptography</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Computational_statistics" title="Computational statistics">Statistics</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematicalsoftware51" scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_software" title="Mathematical software">Mathematical<br>software</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">Arbitrary-precision arithmetic</a></li>

<li><a href="List_of_finite_element_software_packages" title="List of finite element software packages">Finite element analysis</a></li>

<li><a href="Tensor_software" title="Tensor software">Tensor software</a></li>

<li><a href="List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li>

<li><a href="List_of_optimization_software" title="List of optimization software">Optimization software</a></li>

<li><a href="List_of_statistical_software" title="List of statistical software">Statistical software</a></li>

<li><a href="List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li>

<li><a href="List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li>

<li><a href="Solver" title="Solver">Solvers</a></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer_algebra" title="Computer algebra">Computer algebra</a></li>

<li><a href="Computational_number_theory" title="Computational number theory">Computational number theory</a></li>

<li><a href="Combinatorics" title="Combinatorics">Combinatorics</a></li>

<li><a href="Graph_theory" title="Graph theory">Graph theory</a></li>

<li><a href="Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_theory" title="Approximation theory">Approximation theory</a></li>

<li><a href="Clifford_analysis" title="Clifford analysis">Clifford analysis</a>

<ul><li><a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li>

<li><a href="Differential_equation" title="Differential equation">Differential equations</a>

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a></li></ul></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential geometry</a>

<ul><li><a href="Differential_form" title="Differential form">Differential forms</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul></li>

<li><a href="Dynamical_system" title="Dynamical system">Dynamical systems</a>

<ul><li><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></li>

<li><a href="Control_theory" title="Control theory">Control theory</a></li></ul></li>

<li><a href="Functional_analysis" title="Functional analysis">Functional analysis</a>

<ul><li><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></li>

<li><a href="Operator_theory" title="Operator theory">Operator theory</a></li></ul></li>

<li><a href="Harmonic_analysis_(mathematics)" class="mw-redirect" title="Harmonic analysis (mathematics)">Harmonic analysis</a>

<ul><li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li></ul></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a>

<ul><li><a href="Exterior_algebra" title="Exterior algebra">Exterior</a></li>

<li><a href="Geometric_algebra" title="Geometric algebra">Geometric</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li>

<li><a href="Vector_calculus#Vector_algebra" title="Vector calculus">Vector</a></li></ul></li>

<li><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a>

<ul><li><a href="Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li>

<li><a href="Vector_calculus" title="Vector calculus">Vector</a></li></ul></li>

<li><a href="Numerical_analysis" title="Numerical analysis">Numerical analysis</a>

<ul><li><a href="Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li>

<li><a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li>

<li><a href="Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li>

<li><a href="Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Variational calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Probability_theory" title="Probability theory">Probability theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Probability_distribution" title="Probability distribution">Distributions</a> (<a href="Random_variable" title="Random variable">random variables</a>)</li>

<li><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="Stochastic_calculus" title="Stochastic calculus">analysis</a></li>

<li><a href="Functional_integration" title="Functional integration">Path integral</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_physics" title="Mathematical physics">Mathematical<br>physics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a>

<ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li>

<li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li>

<li><a href="Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">Field theory</a>

<ul><li><a href="Classical_field_theory" title="Classical field theory">Classical</a></li>

<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal</a></li>

<li><a href="Effective_field_theory" title="Effective field theory">Effective</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge</a></li>

<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum</a></li>

<li><a href="Statistical_field_theory" title="Statistical field theory">Statistical</a></li>

<li><a href="Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li>

<li><a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a>

<ul><li><a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li>

<li><a href="Potential_theory" title="Potential theory">Potential theory</a></li>

<li><a href="String_theory" title="String theory">String theory</a>

<ul><li><a href="Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li>

<li><a href="Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li>

<li><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a>

<ul><li><a href="Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li>

<li><a href="Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraicstructures48" scope="row" class="navbox-group" style="width:1%"><a href="Algebraic_structure" title="Algebraic structure">Algebraic<br>structures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li>

<li><a href="Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li>

<li><a href="Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li>

<li><a href="Quantum_group" title="Quantum group">Quantum group</a></li>

<li><a href="Renormalization_group" title="Renormalization group">Renormalization group</a></li>

<li><a href="Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li>

<li><a href="Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li>

<li><a href="Superalgebra" title="Superalgebra">Superalgebra</a></li>

<li><a href="Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Decision_theory" title="Decision theory">Decision sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a></li>

<li><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a></li>

<li><a href="Social_choice_theory" title="Social choice theory">Social choice theory</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li>

<li><a href="Mathematical_psychology" title="Mathematical psychology">Psychology</a></li>

<li><a href="Mathematical_sociology" title="Mathematical sociology">Sociology</a></li>

<li>"<a href="The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mathematics" title="Mathematics">Mathematics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>

<ul><li><a href="Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li>

<li><a href="Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li>

<li><a href="International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li>

<li>European Community on Computational Methods in Applied Sciences</li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><b>Category</b></li>

<li><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a href="List_of_mathematics_topics" class="mw-redirect" title="List of mathematics topics">topics list</a></li></ul>

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