Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis.

By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.

Perpendicular-coloured
The line segments AB and CD are orthogonal to each other.

Etymology

The word comes from the Greek ὀρθός (orthos), meaning "upright"[1] , and γωνία (gonia), meaning "angle"[2]. The ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle.[3] Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle.[4]

Mathematics and physics

Orthogonality and rotation
Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle ϕ, right: in Minkowski spacetime through hyperbolic angle ϕ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line).[5]

Definitions

  • In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e., they form a right angle.
  • Two vectors, x and y, in an inner product space, V, are orthogonal if their inner product is zero.[6] This relationship is denoted .
  • Two vector subspaces, A and B, of an inner product space V, are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace of V that is orthogonal to a given subspace is its orthogonal complement.
  • Given a module M and its dual M, an element m′ of M and an element m of M are orthogonal if their natural pairing is zero, i.e. m′, m⟩ = 0. Two sets S′ ⊆ M and SM are orthogonal if each element of S′ is orthogonal to each element of S.[7]
  • A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent.

A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set.

In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. For example, the y-axis is normal to the curve y = x2 at the origin. However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics.

A vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two vectors results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ.

Euclidean vector spaces

In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (π/2 radians), or one of the vectors is zero.[8] Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension.

The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa.[9]

Note that the geometric concept two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle.

In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.[9]

Orthogonal functions

By using integral calculus, it is common to use the following to define the inner product of two functions f and g with respect to a nonnegative weight function w over an interval [a, b]:

In simple cases, w(x) = 1.

We say that functions f and g are orthogonal if their inner product (equivalently, the value of this integral) is zero:

Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to another inner product.

We write the norm with respect to this inner product as

The members of a set of functions {fi : i = 1, 2, 3, ...} are orthogonal with respect to w on the interval [a, b] if

The members of such a set of functions are orthonormal with respect to w on the interval [a, b] if

where

is the Kronecker delta. In other words, every pair of them (excluding pairing of a function with itself) is orthogonal, and the norm of each is 1. See in particular the orthogonal polynomials.

Examples

  • The vectors (1, 3, 2)T, (3, −1, 0)T, (1, 3, −5)T are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) = 0, (3)(1) + (−1)(3) + (0)(−5) = 0, and (1)(1) + (3)(3) + (2)(−5) = 0.
  • The vectors (1, 0, 1, 0, ...)T and (0, 1, 0, 1, ...)T are orthogonal to each other. The dot product of these vectors is 0. We can then make the generalization to consider the vectors in Z2n:
for some positive integer a, and for 1 ≤ ka − 1, these vectors are orthogonal, for example (1, 0, 0, 1, 0, 0, 1, 0)T, (0, 1, 0, 0, 1, 0, 0, 1)T, (0, 0, 1, 0, 0, 1, 0, 0)T are orthogonal.
  • The functions 2t + 3 and 45t2 + 9t − 17 are orthogonal with respect to a unit weight function on the interval from −1 to 1:
  • The functions 1, sin(nx), cos(nx) : n = 1, 2, 3, ... are orthogonal with respect to Riemann integration on the intervals [0, 2π], [−π, π], or any other closed interval of length 2π. This fact is a central one in Fourier series.

Orthogonal polynomials

Orthogonal states in quantum mechanics

  • In quantum mechanics, a sufficient (but not necessary) condition that two eigenstates of a Hermitian operator, and , are orthogonal is that they correspond to different eigenvalues. This means, in Dirac notation, that if and correspond to different eigenvalues. This follows from the fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by hermitian operators (in Heisenberg's formulation).

Art

In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines".

The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the Web site of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours." [1]

Computer science

Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results.[10] This usage was introduced by Van Wijngaarden in the design of Algol 68:

The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities.[11]

Orthogonality is a system design property which guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through the separation of concerns and encapsulation, and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them.

An instruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task)[12] and is designed such that instructions can use any register in any addressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes.

Communications

In communications, multiple-access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions. One such scheme is TDMA, where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").

Another scheme is orthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a, g, and n) versions of 802.11 Wi-Fi; WiMAX; ITU-T G.hn, DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of ADSL.

In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required.

Statistics, econometrics, and economics

When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated,[13] since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions). One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.

Taxonomy

In taxonomy, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.

Combinatorics

In combinatorics, two n×n Latin squares are said to be orthogonal if their superimposition yields all possible n2 combinations of entries.[14]

Chemistry and biochemistry

In synthetic organic chemistry orthogonal protection is a strategy allowing the deprotection of functional groups independently of each other. In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of the other pair. For example, DNA has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively.[15] Bioorthogonal chemistry refers to chemical reactions occurring inside living systems without reacting with naturally present cellular components. In supramolecular chemistry the notion of orthogonality refers to the possibility of two or more supramolecular, often non-covalent, interactions being compatible; reversibly forming without interference from the other.

In analytical chemistry, analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing the reliability of the measurement. This is often required as a part of a new drug application.

System reliability

In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure.

Neuroscience

In neuroscience, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.

Gaming

In board games such as chess which feature a grid of squares, 'orthogonal' is used to mean "in the same row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent".[16] In the ancient Chinese board game Go a player can capture the stones of an opponent by occupying all orthogonally-adjacent points.

Other examples

Stereo vinyl records encode both the left and right stereo channels in a single groove. The V-shaped groove in the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side.[17] A pure horizontal motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals.

See also

Notes

  1. ^ Liddell and Scott, A Greek–English Lexicon s.v. ὀρθός
  2. ^ Liddell and Scott, A Greek–English Lexicon s.v. γωνία
  3. ^ Liddell and Scott, A Greek–English Lexicon s.v. ὀρθογώνιον
  4. ^ Oxford English Dictionary, Third Edition, September 2004, s.v. orthogonal
  5. ^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. ISBN 0-7167-0344-0.
  6. ^ "Wolfram MathWorld".
  7. ^ Bourbaki, "ch. II §2.4", Algebra I, p. 234
  8. ^ Trefethen, Lloyd N. & Bau, David (1997). Numerical linear algebra. SIAM. p. 13. ISBN 978-0-89871-361-9.
  9. ^ a b R. Penrose (2007). The Road to Reality. Vintage books. pp. 417–419. ISBN 0-679-77631-1.
  10. ^ Michael L. Scott, Programming Language Pragmatics, p. 228
  11. ^ 1968, Adriaan van Wijngaarden et al., Revised Report on the Algorithmic Language ALGOL 68, section 0.1.2, Orthogonal design
  12. ^ Null, Linda & Lobur, Julia (2006). The essentials of computer organization and architecture (2nd ed.). Jones & Bartlett Learning. p. 257. ISBN 978-0-7637-3769-6.
  13. ^ Athanasios Papoulis; S. Unnikrishna Pillai (2002). Probability, Random Variables and Stochastic Processes. McGraw-Hill. p. 211. ISBN 0-07-366011-6.
  14. ^ Hedayat, A.; et al. (1999). Orthogonal arrays: theory and applications. Springer. p. 168. ISBN 978-0-387-98766-8.
  15. ^ Karver, Mark R.; Hilderbrand, Scott A. (2012). "Bioorthogonal Reaction Pairs Enable Simultaneous, Selective, Multi-Target Imaging". Angewandte Chemie International Edition. 51 (4): 920–2. doi:10.1002/anie.201104389. PMC 3304098. PMID 22162316.
  16. ^ "chessvariants.org chess glossary".
  17. ^ For an illustration, see YouTube.

References

AVR microcontrollers

AVR is a family of microcontrollers developed since 1996 by Atmel, acquired by Microchip Technology in 2016. These are modified Harvard architecture 8-bit RISC single-chip microcontrollers. AVR was one of the first microcontroller families to use on-chip flash memory for program storage, as opposed to one-time programmable ROM, EPROM, or EEPROM used by other microcontrollers at the time.

AVR microcontrollers find many applications as embedded systems. They are especially common in hobbyist and educational embedded applications, popularized by their inclusion in many of the Arduino line of open hardware development boards.

Anderson orthogonality theorem

The Anderson orthogonality theorem is a theorem in physics by the physicist P. W. Anderson.

It relates to the introduction of a magnetic impurity in a metal. When a magnetic impurity is introduced into a metal, the conduction electrons will tend to screen the potential that the impurity creates. The N-electron ground state for the system when , which corresponds to the absence of the impurity and , which corresponds to the introduction of the impurity are orthogonal in the thermodynamic limit .

Atomicity (database systems)

In database systems, atomicity (; from Ancient Greek: ἄτομος, translit. átomos, lit. 'undividable') is one of the ACID (Atomicity, Consistency, Isolation, Durability) transaction properties. An atomic transaction is an indivisible and irreducible series of database operations such that either all occur, or nothing occurs. A guarantee of atomicity prevents updates to the database occurring only partially, which can cause greater problems than rejecting the whole series outright. As a consequence, the transaction cannot be observed to be in progress by another database client. At one moment in time, it has not yet happened, and at the next it has already occurred in whole (or nothing happened if the transaction was cancelled in progress).

An example of an atomic transaction is a monetary transfer from bank account A to account B. It consists of two operations, withdrawing the money from account A and saving it to account B. Performing these operations in an atomic transaction ensures that the database remains in a consistent state, that is, money is neither lost nor created if either of those two operations fail.

Bilinear form

In mathematics, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:

B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v)

B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v)The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.

When K is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.

Character theory

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

Galerkin method

In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem. In principle, it is the equivalent of applying the method of variation of parameters to a function space, by converting the equation to a weak formulation. Typically one then applies some constraints on the function space to characterize the space with a finite set of basis functions.

The approach is usually credited to Boris Galerkin but the method was discovered by Walther Ritz, to whom Galerkin refers. Often when referring to a Galerkin method, one also gives the name along with typical approximation methods used, such as Bubnov–Galerkin method (after Ivan Bubnov), Petrov–Galerkin method (after Georgii I. Petrov) or Ritz–Galerkin method (after Walther Ritz).

Examples of Galerkin methods are:

the Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method,

the boundary element method for solving integral equations,

Krylov subspace methods.

Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α)n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.

Hyperbolic orthogonality

In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity.

Orthogonal instruction set

In computer engineering, an orthogonal instruction set is an instruction set architecture where all instruction types can use all addressing modes. It is "orthogonal" in the sense that the instruction type and the addressing mode vary independently. An orthogonal instruction set does not impose a limitation that requires a certain instruction to use a specific register.

Orthogonality (programming)

In computer programming, orthogonality means that operations change just one thing without affecting others. The term is most-frequently used regarding assembly instruction sets, as orthogonal instruction set.

Orthogonality in a programming language means that a relatively small set of primitive constructs can be combined in a relatively small number of ways to build the control and data structures of the language. It is associated with simplicity; the more orthogonal the design, the fewer exceptions. This makes it easier to learn, read and write programs in a programming language. The meaning of an orthogonal feature is independent of context; the key parameters are symmetry and consistency (for example, a pointer is an orthogonal concept).

An example from IBM Mainframe and VAX highlights this concept. An IBM mainframe has two different instructions for adding the contents of a register to a memory cell (or another register). These statements are shown below:

A Reg1, memory_cell

AR Reg1, Reg2

In the first case, the contents of Reg1 are added to the contents of a memory cell; the result is stored in Reg1. In the second case, the contents of Reg1 are added to the contents of another register (Reg2) and the result is stored in Reg1.

In contrast to the above set of statements, VAX has only one statement for addition:

ADDL operand1, operand2

In this case the two operands (operand1 and operand2) can be registers, memory cells, or a combination of both; the instruction adds the contents of operand1 to the contents of operand2, storing the result in operand1.

VAX’s instruction for addition is more orthogonal than the instructions provided by IBM; hence, it is easier for the programmer to remember (and use) the one provided by VAX.

The design of C language may be examined from the perspective of orthogonality. The C language is somewhat inconsistent in its treatment of concepts and language structure, making it difficult for the user to learn (and use) the language. Examples of exceptions follow:

Structures (but not arrays) may be returned from a function.

An array can be returned if it is inside a structure.

A member of a structure can be any data type (except void, or the structure of the same type).

An array element can be any data type (except void). Everything is passed by value (except arrays).

Void can be used as a type in a structure, but a variable of this type cannot be declared in a function.Though this concept was first applied to programming language, orthogonality has since become recognized as a valuable feature in the design of APIs and even user interfaces. There, too, having a small set of composable primitive operations without surprising cross-linkages is valuable. leading to systems that are easier to explain and less frustrating to use.

Orthogonality (term rewriting)

Orthogonality as a property of term rewriting systems describes where the reduction rules of the system are all left-linear, that is each variable occurs only once on the left hand side of each reduction rule, and there is no overlap between them.

Orthogonal term rewriting systems have the consequent property that all reducible expressions (redexes) within a term are completely disjoint -- that is, the redexes share no common function symbol.

For example, the term rewriting system with reduction rules

is orthogonal -- it is easy to observe that each reduction rule is left-linear, and the left hand side of each reduction rule shares no function symbol in common, so there is no overlap.

Orthogonal term rewriting systems are confluent.

Orthogonality principle

In statistics and signal processing, the orthogonality principle is a necessary and sufficient condition for the optimality of a Bayesian estimator. Loosely stated, the orthogonality principle says that the error vector of the optimal estimator (in a mean square error sense) is orthogonal to any possible estimator. The orthogonality principle is most commonly stated for linear estimators, but more general formulations are possible. Since the principle is a necessary and sufficient condition for optimality, it can be used to find the minimum mean square error estimator.

Perpendicular

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.

A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.

Perpendicularity easily extends to segments and rays. For example, a line segment is perpendicular to a line segment if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, means line segment AB is perpendicular to line segment CD. For information regarding the perpendicular symbol see Up tack.

A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.

Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle (90 degrees).

Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal.

Quadrature amplitude modulation

Quadrature amplitude modulation (QAM) is the name of a family of digital modulation methods and a related family of analog modulation methods widely used in modern telecommunications to transmit information. It conveys two analog message signals, or two digital bit streams, by changing (modulating) the amplitudes of two carrier waves, using the amplitude-shift keying (ASK) digital modulation scheme or amplitude modulation (AM) analog modulation scheme. The two carrier waves of the same frequency are out of phase with each other by 90°, a condition known as orthogonality and as quadrature. Being the same frequency, the modulated carriers add together, but can be coherently separated (demodulated) because of their orthogonality property. Another key property is that the modulations are low-frequency/low-bandwidth waveforms compared to the carrier frequency, which is known as the narrowband assumption.

Phase modulation (analog PM) and phase-shift keying (digital PSK) can be regarded as a special case of QAM, where the magnitude of the modulating signal is a constant, but its sign changes between positive and negative. This can also be extended to frequency modulation (FM) and frequency-shift keying (FSK), for these can be regarded as a special case of phase modulation.

QAM is used extensively as a modulation scheme for digital telecommunication systems, such as in 802.11 Wi-Fi standards. Arbitrarily high spectral efficiencies can be achieved with QAM by setting a suitable constellation size, limited only by the noise level and linearity of the communications channel. QAM is being used in optical fiber systems as bit rates increase; QAM16 and QAM64 can be optically emulated with a 3-path interferometer.

Racah polynomials

In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients.

The Racah polynomials were first defined by Wilson (1978) and are given by

Askey & Wilson (1979) introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by

They are sometimes given with changes of variables as

Right angle

In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.

Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry.

Schur orthogonality relations

In mathematics, the Schur orthogonality relations, which is proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups.

They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

Sesquilinear form

In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of what a vector is.

A motivating special case is a sesquilinear form on a complex vector space, V. This is a map V × V → C that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism.

An application in projective geometry requires that the scalars come from a division ring (skewfield), K, and this means that the "vectors" should be replaced by elements of a K-module. In a very general setting, sesquilinear forms can be defined over R-modules for arbitrary rings R.

Up tack

The up tack or falsum (⊥, \bot in LaTeX, U+22A5 in Unicode) is a constant symbol used to represent:

The bottom element in lattice theory

The bottom type in type theory

A logical constant denoting "false" in logic

Mixed radix decoding in the APL programming languageThe glyph of the up tack appears as an upside-down tee symbol, and as such is sometimes called eet (the word "tee" in reverse). Tee plays a complementary or dual role in many of these theories.

The similar-looking perpendicular symbol (⟂, \perp in LaTeX, U+27C2 in Unicode) is a binary relation symbol used to represent:

Perpendicularity of lines in geometry

Orthogonality in linear algebra

Independence of random variables in probability theory

Comparability in order theory

Coprimality in number theory

Basic concepts
Vector algebra
Multilinear algebra
Matrices
Algebraic constructions
Numerical

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