Discrete mathematics

Quantum biology is the study of applications of quantum mechanics and theoretical chemistry to biological objects and problems. Many biological processes involve the conversion of energy into forms that are usable for chemical transformations and are quantum mechanical in nature. Thanks for watching Make sure to subscribe and leave a comment below.

Discrete mathematics

Quantum biology is the study of applications of quantum mechanics and theoretical chemistry to biological objects and problems. Many biological processes involve the conversion of energy into forms that are usable for chemical transformations, and are quantum mechanical in nature. Such processes involve chemical reactionslight absorption, formation of excited electronic statestransfer of excitation energy, and the transfer of electrons and protons (hydrogen ions) in chemical processes, such as photosynthesisolfaction and cellular respiration.

Quantum biology may use computations to model biological interactions in light of quantum mechanical effects. Quantum biology is concerned with the influence of non-trivial quantum phenomena, which can be explained by reducing the biological process to fundamental physics, although these effects are difficult to study and can be speculative.


Quantum biology is an emerging field; most of the current research is theoretical and subject to questions that require further experimentation. Though the field has only recently received an influx of attention, it has been conceptualized by physicists throughout the 20th century. It has been suggested that quantum biology might play a critical role in the future of the medical world. Early pioneers of quantum physics saw applications of quantum mechanics in biological problems. Erwin Schrödinger‘s 1944 book What is Life? discussed applications of quantum mechanics in biology.Schrödinger introduced the idea of an “aperiodic crystal” that contained genetic information in its configuration of covalent chemical bonds. He further suggested that mutations are introduced by “quantum leaps”. Other pioneers Niels BohrPascual Jordan, and Max Delbruck argued that the quantum idea of complementarity was fundamental to the life sciences. In 1963, Per-Olov Löwdin published proton tunneling as another mechanism for DNA mutation. In his paper, he stated that there is a new field of study called “quantum biology”. In 1979, the Soviet and Ukrainian physicist Alexander Davydov published the first textbook on quantum biology entitled “Biology and Quantum Mechanics”.



Diagram of FMO complex. Light excites electrons in an antenna. The excitation then transfers through various proteins in the FMO complex to the reaction center to further photosynthesis.

Organisms that undergo photosynthesis absorb light energy through the process of electron excitation in antennae. These antennae vary among organisms. For example, bacteria use ring-like antennae, while plants use chlorophyll pigments to absorb photons. Photosynthesis creates Frenkel excitons, which provide a separation of charge that cells convert into usable chemical energy. The energy collected in reaction sites must be transferred quickly before it is lost to fluorescence or thermal vibrational motion.

Various structures, such as the FMO complex in green sulfur bacteria, are responsible for transferring energy from antennae to a reaction site. FT electron spectroscopy studies of electron absorption and transfer show an efficiency of above 99%, which cannot be explained by classical mechanical models like the diffusion model. Instead, as early as 1938, scientists theorized that quantum coherence was the mechanism for excitation energy transfer.

Scientists have recently looked for experimental evidence of this proposed energy transfer mechanism. A study published in 2007 claimed the identification of electronic  quantum coherence  at −196 °C (77 K). Another theoretical study from 2010 provided evidence that quantum coherence lives as long as 300 femtoseconds at biologically relevant temperatures (4 °C or 277 K). In that same year, experiments conducted on photosynthetic cryptophyte algae using two-dimensional photon echo spectroscopy yielded further confirmation for long-term quantum coherence.  These studies suggest that, through evolution, nature has developed a way of protecting quantum coherence to enhance the efficiency of photosynthesis. However, critical follow-up studies question the interpretation of these results. Single molecule spectroscopy now shows the quantum characteristics of photosynthesis without the interference of static disorder, and some studies use this method to assign reported signatures of electronic quantum coherence to nuclear dynamics occurring in chromophores.[14][15][16][17][18][19][20] A number of proposals emerged trying to explain unexpectedly long coherence. According to one proposal, if each site within the complex feels its own environmental noise, the electron will not remain in any local minimum due to both quantum coherence and thermal environment, but proceed to the reaction site via quantum walks. Another proposal is that the rate of quantum coherence and electron tunneling create an energy sink that moves the electron to the reaction site quickly. Other work suggested that geometric symmetries in the complex may favor efficient energy transfer to the reaction center, mirroring perfect state transfer in quantum networks.  Furthermore, experiments with artificial dye molecules cast doubts on the interpretation that quantum effects last any longer than one hundred femtoseconds. 

In 2017, the first control experiment with the original FMO protein under ambient conditions confirmed that electronic quantum effects are washed out within 60 femtoseconds, while the overall exciton transfer takes a time on the order of a few picoseconds.  In 2020 a review based on a wide collection of control experiments and theory concluded that the proposed quantum effects as long lived electronic coherences in the FMO system does not hold.  Instead, research investigating transport dynamics suggests that interactions between electronic and vibrational modes of excitation in FMO complexes require a semi-classical, semi-quantum explanation for the transfer of exciton energy. In other words, while quantum coherence dominates in the short-term, a classical description is most accurate to describe long-term behavior of the excitons.

Another process in photosynthesis that has almost 100% efficiency is charge transfer, again suggesting that quantum mechanical phenomena are at play. In 1966, a study on the photosynthetic bacterium Chromatium found that at temperatures below 100 K, cytochrome oxidation is temperature-independent, slow (on the order of milliseconds), and very low in activation energy. The authors, Don DeVault and Britton Chase, postulated that these characteristics of electron transfer are indicative of quantum tunneling, whereby electrons penetrate a potential barrier despite possessing less energy than is classically necessary.

Seth Lloyd is also notable for his contributions to this area of research.

DNA mutation

DNA acts as the instructions for making proteins throughout the body. It consists of 4 nucleotides: guanine, thymine, cytosine, and adenine. The order of these nucleotides gives the “recipe” for the different proteins.

Whenever a cell reproduces, it must copy these strands of DNA. However, sometimes throughout the process of copying the strand of DNA a mutation, or an error in the DNA code, can occur. A theory for the reasoning behind DNA mutation is explained in the Lowdin DNA mutation model. In this model, a nucleotide may spontaneously change its form through a process of quantum tunneling. Because of this, the changed nucleotide will lose its ability to pair with its original base pair and consequently changing the structure and order of the DNA strand.

Exposure to ultraviolet lights and other types of radiation can cause DNA mutation and damage. The radiations also can modify the bonds along the DNA strand in the pyrimidines and cause them to bond with themselves creating a dimer. 

In many prokaryotes and plants, these bonds are repaired to their original form by a DNA repair enzyme photolyase. As its prefix implies, photolyase is reliant on light in order to repair the strand. Photolyase works with its cofactor FADH, flavin adenine dinucleotide, while repairing the DNA. Photolyase is excited by visible light and transfers an electron to the cofactor FADH-. FADH- now in the possession of an extra electron gives the electron to the dimer to break the bond and repair the DNA. This transfer of the electron is done through the tunneling of the electron from the FADH to the dimer. Although the range of the tunneling is much larger than feasible in a vacuum, the tunneling in this scenario is said to be “superexchange-mediated tunneling,” and is possible due to the protein’s ability to boost the tunneling rates of the electron.

Vibration theory of olfaction

Olfaction, the sense of smell, can be broken down into two parts; the reception and detection of a chemical, and how that detection is sent to and processed by the brain. This process of detecting an odorant is still under question. One theory named the “shape theory of olfaction” suggests that certain olfactory receptors are triggered by certain shapes of chemicals and those receptors send a specific message to the brain. Another theory (based on quantum phenomena) suggests that the olfactory receptors detect the vibration of the molecules that reach them and the “smell” is due to different vibrational frequencies, this theory is aptly called the “vibration theory of olfaction.”

The vibration theory of olfaction, created in 1938 by Malcolm Dyson but reinvigorated by Luca Turin in 1996, proposes that the mechanism for the sense of smell is due to G-protein receptors that detect molecular vibrations due to inelastic electron tunneling, tunneling where the electron loses energy, across molecules. In this process a molecule would fill a binding site with a G-protein receptor. After the binding of the chemical to the receptor, the chemical would then act as a bridge allowing for the electron to be transferred through the protein. As the electron transfers across what would otherwise have been a barrier, it loses energy due to the vibration of the newly-bound molecule to the receptor. This results in the ability to smell the molecule.

While the vibration theory has some experimental proof of concept, there have been multiple controversial results in experiments. In some experiments, animals are able to distinguish smells between molecules of different frequencies and same structure,while other experiments show that people are unaware of distinguishing smells due to distinct molecular frequencies.


Vision relies on quantized energy in order to convert light signals to an action potential in a process called phototransduction. In phototransduction, a photon interacts with a chromophore in a light receptor. The chromophore absorbs the photon and undergoes photoisomerization. This change in structure induces a change in the structure of the photo receptor and resulting signal transduction pathways lead to a visual signal. However, the photoisomerization reaction occurs at a rapid rate, in under 200 femtoseconds, with high yield. Models suggest the use of quantum effects in shaping the ground state and excited state potentials in order to achieve this efficiency.

Quantum vision implications

Experiments have shown that the sensor in the retina of the human eye is sensitive enough to detect a single photon. Single photon detection could lead to multiple different technologies. One area of development is in quantum communication and cryptography. The idea is to use a biometric system to measure the eye using only a small number of points across the retina with random flashes of photons that “read” the retina and identify the individual. This biometric system would only allow a certain individual with a specific retinal map to decode the message. This message can not be decoded by anyone else unless the eavesdropper were to guess the proper map or could read the retina of the intended recipient of the message.

Enzymatic activity (quantum biochemistry)

Enzymes have been postulated to use quantum tunneling in order to transfer electrons from one place to another in electron transport chains. It is possible that protein quaternary architectures may have adapted to enable sustained quantum entanglement and coherence, which are two of the limiting factors for quantum tunneling in biological entities. These architectures might account for a greater percentage of quantum energy transfer, which occurs through electron transport and proton tunneling (usually in the form of hydrogen ions, H+). Tunneling refers to the ability of a subatomic particle to travel through potential energy barriers. This ability is due, in part, to the principle of complementarity, which holds that certain substances have pairs of properties that cannot be measured separately without changing the outcome of measurement. Particles, such as electrons and protons, have wave-particle duality; they can pass through energy barriers due to their wave characteristics without violating the laws of physics. In order to quantify how quantum tunneling is used in many enzymatic activities, many biophysicists utilize the observation of hydrogen ions. When hydrogen ions are transferred, this is seen as a staple in an organelle’s primary energy processing network; in other words, quantum effects are most usually at work in proton distribution sites at distances on the order of an angstrom (1 Å).  In physics, a semiclassical (SC) approach is most useful in defining this process because of the transfer from quantum elements (e.g. particles) to macroscopic phenomena (e.g. biochemicals). Aside from hydrogen tunneling, studies also show that electron transfer between redox centers through quantum tunneling plays an important role in enzymatic activity of photosynthesis and cellular respiration (see also Mitochondria section below).[57][58] For example, electron tunneling on the order of 15–30 Å contributes to redox reactions in cellular respiration enzymes, such as complexes I, III, and IV in mitochondria.  Without quantum tunneling, organisms would not be able to convert energy quickly enough to sustain growth.  Quantum tunneling actually acts as a shortcut for particle transfer; according to quantum mathematics, a particle’s jump from in front of a barrier to the other side of a barrier occurs faster than if the barrier had never been there in the first place. (For more on the technicality of this, see Hartman effect.)


Organelles, such as mitochondria, are thought to utilize quantum tunneling in order to translate intracellular energy.  Traditionally, mitochondria are known to generate most of the cell’s energy in the form of chemical ATP. Mitochondria conversion of biomass into chemical ATP is 60-70% efficient, which is superior than the classical regime of man-made engines.[62] To achieve chemical ATP, researchers have found that a preliminary stage before chemical conversion is necessary; this step, via the quantum tunneling of electrons and hydrogen ions (H+), requires a deeper look at the quantum physics that occurs within the organelle. 

Because tunneling is a quantum mechanism, it is important to understand how this process may occur for particle transfer in a biological system. Tunneling is largely dependent upon the shape and size of a potential barrier, relative to the incoming energy of a particle. Because the incoming particle can be defined by a wave equation, its tunneling probability is dependent upon the potential barrier’s shape in an exponential way, meaning that if the barrier is akin to a very wide chasm, the incoming particle’s probability to tunnel will decrease. The potential barrier, in some sense, can come in the form of an actual biomaterial barrier. Mitochondria are encompassed by a membrane structure that is akin to the cellular membrane, on the order of ~75 Å (~7.5 nm) thick. The inner membrane of a mitochondria must be overcome to permit signals (in the form of electrons, protons, H+) to transfer from the site of emittance (internal to the mitochondria) and the site of acceptance (i.e. the electron transport chain proteins). In order to transfer particles, the membrane of the mitochondria must have the correct density of phospholipids to conduct a relevant charge distribution that attracts the particle in question. For instance, for a greater density of phospholipids, the membrane contributes to a greater conductance of protons. 

More technically, the form of the mitochondria is the matrix, with inner mitochondrial membranes (IMM) and inner membrane spaces (IMS), all housing protein sites. Mitochondria produce ATP by the oxidation of hydrogen ions from carbohydrates and fats. This process utilizes electrons in an electron transport chain (ETP). The genealogy of electron transport proceeds as follows: Electrons from NADH are transferred to NADH dehydrogenase (complex I protein), which is located in the IMM. Electrons from complex I are transferred to coenzyme Q to make CoQH2; next, electrons flow to cytochrome-containing IMM protein (complex III), which further pushes electrons to cytochrome c, where electrons flow to complex IV; complex IV is the final IMM protein complex of the ETC respiratory chain.  This final protein allows electrons to reduce oxygen from an O2 molecule to a single O, so that it can bind to the hydrogen ions to produce H2O. The energy produced from the movement of electrons through the ETC induces proton movement (known as H+ pumping) out of the mitochondria matrix into the IMS.[60] Because any charge movement creates a magnetic field, the IMS now houses a capacitance across the matrix. The capacitance is akin to potential energy, or what is known as a potential barrier. This potential energy guides ATP synthesis via complex V (ATP synthase), which conflates ADP with another P to create ATP by pushing protons (H+) back into the matrix (this process is known as oxidative phosphorylation). Finally, the outer mitochondrial membrane (OMM) houses a voltage-dependent anion channel called the VDAC.  This site is important for converting energy signals into electro-chemical outputs for ATP transfer.

Molecular solitons in proteins

Alexander Davydov developed the quantum theory of molecular solitons in order to explain the transport of energy in protein α-helices in general and the physiology of muscle contraction in particular.  He showed that the molecular solitons are able to preserve their shape through nonlinear interaction of amide I excitons and phonon deformations inside the lattice of hydrogen-bonded peptide groups. In 1979, Davydov published his complete textbook on quantum biology entitled “Biology and Quantum Mechanics” featuring quantum dynamics of proteinscell membranesbioenergeticsmuscle contraction, and electron transport in biomolecules.


Magnetoreception refers to the ability of animals to navigate using the inclination of the magnetic field of the earth  A possible explanation for magnetoreception is the entangled radical pair mechanism. The radical-pair mechanism is well-established in spin chemistry,  and was speculated to apply to magnetoreception in 1978 by Schulten et al.. The ratio between singlet and triplet pairs is changed by the interaction of entangled electron pairs with the magnetic field of the earth.  In 2000, cryptochrome was proposed as the “magnetic molecule” that could harbor magnetically sensitive radical-pairs. Cryptochrome, a flavoprotein found in the eyes of European robins and other animal species, is the only protein known to form photoinduced radical-pairs in animals. When it interacts with light particles, cryptochrome goes through a redox reaction, which yields radical pairs both during the photo-reduction and the oxidation. The function of cryptochrome is diverse across species, however, the photoinduction of radical-pairs occurs by exposure to blue light, which excites an electron in a chromophore. Magnetoreception is also possible in the dark, so the mechanism must rely more on the radical pairs generated during light-independent oxidation.

Experiments in the lab support the basic theory that radical-pair electrons can be significantly influenced by very weak magnetic fields, i.e. merely the direction of weak magnetic fields can affect radical-pair’s reactivity and therefore can “catalyze” the formation of chemical products. Whether this mechanism applies to magnetoreception and/or quantum biology, that is, whether earth’s magnetic field “catalyzes” the formation of biochemical products by the aid of radical-pairs, is undetermined for two reasons. The first is that radical-pairs may need not be entangled, the key quantum feature of the radical-pair mechanism, to play a part in these processes. There are entangled and non-entangled radical-pairs. However, researchers found evidence for the radical-pair mechanism of magnetoreception when European robins, cockroaches, and garden warblers, could no longer navigate when exposed to a radio frequency that obstructs magnetic fields  and radical-pair chemistry. To empirically suggest the involvement of entanglement, an experiment would need to be devised that could disturb entangled radical-pairs without disturbing other radical-pairs, or vice versa, which would first need to be demonstrated in a laboratory setting before being applied to in vivo radical-pairs.

Other biological applications

Other examples of quantum phenomena in biological systems include the conversion of chemical energy into motion and brownian motors in many cellular processes.

Discrete mathematics – complete mathematical induction, linear Diophantine equationsFermat’s little theoremroute inspection problem and recurrence relations

Discrete mathematics is the study of mathematical structures that can be considered “discrete” (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than “continuous” (analogously to continuous functions). Objects studied in discrete mathematics include integersgraphs, and statements in logic.[1][2][3][4] By contrast, discrete mathematics excludes topics in “continuous mathematics” such as real numberscalculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[5] (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term “discrete mathematics”.[6]

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 algorithmsprogramming languagescryptographyautomated 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.

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 ACM and MAA into a course that is basically intended to develop mathematical maturity in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well.[7][8] Some high-school-level discrete mathematics textbooks have appeared as well.[9] At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect.[10]

The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.

Grand challenges, past and present[edit]


Much research in graph theory was motivated by attempts to prove that all maps, like this one, can be colored using only four colors so that no areas of the same color share an edge. Kenneth Appel and Wolfgang Haken proved this in 1976.[11]

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 until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).[11]

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 consistentGödel’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 with the guidance of Alan Turing and his seminal work, On Computable Numbers.[12] 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 theoryFormal 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.

Computational geometry has been an important part of the computer graphics incorporated into modern video games and computer-aided design tools.

Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.[13]

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 USD prize for the first correct proof, along with prizes for six other mathematical problems.[14]

Topics in discrete mathematics[edit]

Theoretical computer science[edit]


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 mathematical logic. Included within theoretical computer science is the study of algorithms and data structures. Computability 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. 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 various continuous computational topics.

Information theory[edit]


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.

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 signalsanalog codinganalog encryption.


Logic is the study of the principles of valid reasoning and inference, as well as of consistencysoundness, and completeness. For example, in most systems of logic (but not in intuitionistic logicPeirce’s law (((PQ)→P)→P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

Logical formulas are discrete structures, as are proofs, which form finite trees[15] or, more generally, directed acyclic graph structures[16][17] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation trees have also been studied,[18] e.g. infinitary logic.

Set theory[edit]

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 numbersPartially 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, 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 descriptive set theory makes extensive use of traditional continuous mathematics.


Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects – e.g. the twelvefold way provides a unified framework for counting permutationscombinations and partitionsAnalytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-seriesspecial functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finite and infinite.

Graph theory[edit]


Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group A4.

Graph theory, the study of graphs and networks, 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.[19] 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 topology, e.g. knot theoryAlgebraic graph theory has close links with group theory. There are also continuous graphs; however, for the most part, research in graph theory falls within the domain of discrete mathematics.


Discrete probability theory deals with events that occur in countable sample spaces. For example, count observations such as the numbers of birds in flocks comprise only natural number values {0, 1, 2, …}. On the other hand, continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal. Discrete probability distributions can be used to approximate continuous ones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards, calculating the probability of events is basically enumerative combinatorics.

Number theory[edit]


The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of prime numbers.

Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography and cryptanalysis, particularly with regard to modular arithmeticdiophantine equations, linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theory include geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbersdiophantine approximationp-adic analysis and function fields.

Algebraic structures[edit]

Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groupsrings and fields 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[edit]

function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system. 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 recurrence relation or difference equation. Difference equations are similar to differential equations, but replace differentiation 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 integral transforms in harmonic analysis for studying continuous functions or analogue 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 and finite topological spaces.



Computational geometry applies computer algorithms to representations of geometrical objects.

Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry applies algorithms to geometrical problems.


Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of “continuous deformation” of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topologytopological graph theorytopological combinatoricscomputational topologydiscrete topological spacefinite topological spacetopology (chemistry).

Operations research[edit]


PERT charts like this provide a project management technique based on graph theory.

Operations research provides techniques for solving practical problems in engineering, business, and other fields — problems such as allocating resources to maximize profit, and scheduling project activities to minimize risk. Operations research techniques include linear programming and other areas of optimizationqueuing theoryscheduling theory, and network theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingalesprocess optimization, and continuous and hybrid control theory.

Game theory, decision theory, utility theory, social choice theory[edit]

Cooperate-1, -1−10, 0
Defect0, -10-5, -5
Payoff matrix for the Prisoner’s dilemma, a common example in game theory. One player chooses a row, the other a column; the resulting pair gives their payoffs

Decision theory is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision.

Utility theory is about measures of the relative economic satisfaction from, or desirability of, consumption of various goods and services.

Social choice theory is about voting. A more puzzle-based approach to voting is ballot theory.

Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex. There are even continuous games, see differential game. Topics include auction theory and fair division.


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[edit]

There are many concepts in continuous mathematics which have discrete versions, such as discrete calculusdiscrete probability distributionsdiscrete Fourier transformsdiscrete geometrydiscrete logarithmsdiscrete differential geometrydiscrete exterior calculusdiscrete Morse theorydifference equationsdiscrete dynamical systems, and discrete vector measures.

In applied mathematicsdiscrete 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 relation.

In algebraic geometry, the concept of a curve can be extended to discrete geometries by taking the spectra of polynomial rings over finite fields to be models of the affine spaces over that field, and letting subvarieties 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 {\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}} for {\ displaystyle K} a field can be studied either as {\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}, a point, or as the spectrum {\displaystyle \operatorname {Spec} K[x]_{(x-c)}} of the local ring at (x-c), a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of tangent space called the Zariski tangent space, making many features of calculus applicable even in finite settings.

Hybrid discrete and continuous mathematics[edit]

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. Another way of modeling such a situation is the notion of hybrid dynamical systems.

See also[edit]

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