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| {{redirect|Orthogonal|the trilogy of novels by [[Greg Egan]]|Orthogonal (novel)}}
| | Alyson Meagher is the name her mothers and fathers gave her but she doesn't like when individuals use her complete title. It's not a common thing but what I like doing is to climb but I don't have the time lately. My day occupation is an invoicing officer but I've currently applied for another 1. Mississippi is exactly where his house is.<br><br>Also visit my web-site real psychic [[http://Www.Rusload.de/uprofile.php?UID=413606 Recommended Browsing]] |
| {{Refimprove|date=January 2010}}
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| [[File:Perpendicular-coloured.svg|thumb|right|220px|The line segments AB and CD are orthogonal to each other.]]
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| In [[mathematics]], '''orthogonality''' is the relation of two lines at [[right angle]]s to one another ([[perpendicularity]]), and the generalization of this relation into ''n'' dimensions; and to a variety of mathematical relations thought of as describing non-overlapping, [[uncorrelated]], or [[Independence (mathematical logic)|independent]] objects of some kind. A widespread example is with vinyl records and how in the 1960s they were able to get left and right stereo signals from one single groove. By making the groove a 90-degree cut into the vinyl, wave motion on one wall was independent from any motion (or non-motion) along the other wall.
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| The concept of orthogonality has been broadly [[generalization|generalized]] in mathematics, [[science]], and [[engineering]], especially since the beginning of the 16th century. Much of it has involved the concepts of [[mathematical function]]s, [[calculus]], and [[linear algebra]].
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| ==Etymology==
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| The word comes from the [[Ancient Greek|Greek]] ''{{lang|grc|ὀρθός}}'' (''orthos''), meaning "straight", and ''{{lang|grc|γωνία}}'' (''gonia''), meaning "angle".
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| The ancient Greek ὀρθογώνιον ''orthogōnion'' (< ὀρθός ''orthos'' 'upright'<ref>Liddell and Scott, ''[[A Greek–English Lexicon]]'' [http://www.perseus.tufts.edu/hopper/morph?l=o%29rqos&la=greek#lexicon ''s.v.'' ὀρθός]</ref> + γωνία ''gōnia'' 'angle'<ref>Liddell and Scott, ''[[A Greek–English Lexicon]]'' [http://www.perseus.tufts.edu/hopper/morph?l=gwni%2Fa&la=greek#lexicon ''s.v.'' γονία]</ref>) and classical Latin ''orthogonium'' originally denoted a [[rectangle]].<ref>Liddell and Scott, ''[[A Greek–English Lexicon]]'' [http://www.perseus.tufts.edu/hopper/morph?l=o%29rqog%2Fwnion&la=greek#lexicon ''s.v.'' ὀρθογώνιον]</ref> 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.<ref>[[Oxford English Dictionary]], Third Edition, September 2004, ''s.v.'' orthogonal</ref>
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| ==Mathematics==
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| [[File:Orthogonality and rotation.svg|thumb|350px|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 [[worldline]]s of a light signal, a vector is orthogonal to itself if it lies on this line).<ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|page=58|year=1973|isbn=0-7167-0344-0}}</ref>]]
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| === Definitions ===
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| *In [[geometry]], two [[Euclidean vector]]s are '''orthogonal''' if they are [[perpendicular]], ''i.e.'', they form a [[right angle]].
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| *Curves or functions in the plane are orthogonal at an intersection if their tangent lines are perpendicular at that point.
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| *Two [[vector space|vectors]], ''x'' and ''y'', in an [[inner product space]], ''V'', are ''orthogonal'' if their [[dot product|inner product]] <math>\langle x, y \rangle</math> is zero.<ref>{{cite web|title=Wolfram MathWorld|url=http://mathworld.wolfram.com/Orthogonal.html}}</ref> This relationship is denoted <math>x \, \bot \, y</math>.
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| *Two [[Linear subspace|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]].
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| *A [[linear transformation]], ''T'' : ''V'' → ''V'', is called an ''[[orthogonal transform|orthogonal linear transformation]]'' if it preserves the inner product, and thus the [[angle]] between and the [[length]]s of vectors. That is, for all pairs of vectors ''x'' and ''y'' in the inner product space ''V'', <math>\langle Tx, Ty \rangle = \langle x, y \rangle</math>.
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| *A [[term rewriting system]] is said to be [[orthogonality (term rewriting)|orthogonal]] if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are [[confluence (term rewriting)|confluent]].
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| A set of vectors is called '''pairwise orthogonal''' if each pairing of them is orthogonal. Such a set is called an '''orthogonal set'''. Nonzero pairwise orthogonal vectors are always [[linearly independent]].
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| In certain cases, the word ''normal'' is used to mean ''orthogonal'', particularly in the geometric sense as in the [[surface normal|normal to a surface]]. For example, the ''y''-axis is normal to the curve ''y'' = ''x''<sup>2</sup> 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 vector]]s. 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]].
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| 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 space|pseudo-Euclidean plane]] uses the term [[hyperbolic orthogonality]]. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given φ.
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| ===Euclidean vector spaces===
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| In 2-D or higher-[[dimension]]al [[Euclidean space]], two vectors are orthogonal [[if and only if]] their [[dot product]] is zero, i.e. they make an angle of 90°, or π/2 [[radian]]s.<ref>{{cite book|author=Trefethen, Lloyd N. & Bau, David|title=Numerical linear algebra|publisher=SIAM|year=1997|isbn=978-0-89871-361-9|page=13|url=http://books.google.com/books?id=bj-Lu6zjWbEC&pg=PA13}}</ref> Hence orthogonality of vectors is an extension of the concept of [[perpendicular]] vectors into higher-dimensional spaces.
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| In terms of [[Euclidean subspace]]s, the "orthogonal complement" of a [[line (geometry)|line]] is the [[plane (mathematics)|plane]] perpendicular to it, and vice-versa.<ref name="R. Penrose 2007 417–419">{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books|pages=417–419| year=2007 | isbn=0-679-77631-1}}</ref>
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| Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the origin(definition of vector subspace).
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| 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.<ref name="R. Penrose 2007 417–419"/>
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| === Orthogonal functions ===
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| {{Main|Orthogonal functions}}
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| By using [[integral calculus]]. it is common to use the following to define the [[inner product]] of two [[function (mathematics)|functions]] ''f'' and ''g'':
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| :<math>\langle f, g\rangle_w = \int_a^b f(x)g(x)w(x)\,dx.</math>
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| Here we introduce a nonnegative [[weight function]] <math>w(x)</math> in the definition of this inner product. In simple cases, w(x) = 1, exactly.
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| We say that these functions are '''orthogonal''' if that inner product is zero:
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| :<math>\int_a^b f(x)g(x)w(x)\,dx = 0.</math>
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| We write the [[norm (mathematics)|norm]]s with respect to this inner product and the weight function as
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| :<math>\|f\|_w = \sqrt{\langle f, f\rangle_w}</math>
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| The members of a set of functions { ''f''<sub>''i''</sub> : ''i'' = 1, 2, 3, ... } are:
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| * ''orthogonal'' on the closed interval [a, b] if
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| :<math>\langle f_i, f_j \rangle=\int_a^b f_i(x) f_j(x) w(x)\,dx=\|f_i\|^2\delta_{i,j}=\|f_j\|^2\delta_{i,j}</math>
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| * ''orthonormal'' on the interval [a, b] if
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| :<math>\langle f_i, f_j \rangle=\int_a^b f_i(x) f_j(x) w(x)\,dx=\delta_{i,j}</math>
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| where | |
| :<math>\delta_{i,j}=\left\{\begin{matrix}1 & \mathrm{if}\ i=j \\ 0 & \mathrm{if}\ i\neq j\end{matrix}\right.</math>
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| is the "[[Kronecker delta]]" function. In other words, any two of them are orthogonal, and the norm of each is 1 in the case of the orthonormal sequence. See in particular the [[orthogonal polynomials]]. | |
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| ===Examples===
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| * The vectors (1, 3, 2)<sup>T</sup>, (3, −1, 0)<sup>T</sup>, (1/3, 1, −5/3)<sup>T</sup> are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) = 0, (3)(1/3) + (−1)(1) + (0)(−5/3) = 0, and (1)(1/3) + (3)(1) + (2)(−5/3) = 0.
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| * The vectors (1, 0, 1, 0, ...)<sup>T</sup> and (0, 1, 0, 1, ...)<sup>T</sup> are orthogonal to each other. The dot product of these vectors is 0. We can then make the generalization to consider the vectors in '''Z'''<sub>2</sub><sup>''n''</sup>:
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| ::<math>\mathbf{v}_k = \sum_{i=0\atop ai+k < n}^{n/a} \mathbf{e}_i</math>
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| :for some positive integer ''a'', and for 1 ≤ ''k'' ≤ ''a'' − 1, these vectors are orthogonal, for example (1, 0, 0, 1, 0, 0, 1, 0)<sup>T</sup>, (0, 1, 0, 0, 1, 0, 0, 1)<sup>T</sup>, (0, 0, 1, 0, 0, 1, 0, 0)<sup>T</sup> are orthogonal.
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| * Take two quadratic functions 2''t'' + 3 and 5''t''<sup>2</sup> + ''t'' − 17/9. These functions are orthogonal with respect to a unit weight function on the interval from −1 to 1. The product of these two functions is 10''t''<sup>3</sup> + 17''t''<sup>2</sup> − 7/9 ''t'' − 17/3, and now,
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| ::<math>
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| \begin{align}
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| & {} \qquad \int_{-1}^1 \left(10t^3+17t^2-{7\over 9}t-{17\over 3}\right)\,dt \\[6pt]
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| & = \left[{5\over 2}t^4 + {17\over 3}t^3-{7\over 18}t^2-{17\over 3} t \right]_{-1}^1 \\[6pt]
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| & = \left({5\over 2}(1)^4+{17\over 3}(1)^3-{7\over 18}(1)^2-{17\over 3}(1)\right)-\left({5\over 2}(-1)^4+{17\over 3}(-1)^3-{7\over 18}(-1)^2-{17\over 3}(-1)\right) \\[6pt]
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| & = {19\over 9} - {19\over 9} = 0.
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| \end{align}
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| </math>
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| * 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]].
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| ====Orthogonal polynomials====
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| * Various polynomial sequences named for [[mathematician]]s of the past are sequences of [[orthogonal polynomials]]. In particular:
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| **The [[Hermite polynomials]] are orthogonal with respect to the [[Gaussian distribution]] with zero mean value.
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| **The [[Legendre polynomials]] are orthogonal with respect to the [[uniform distribution (continuous)|uniform distribution]] on the interval [−1, 1].
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| **The [[Laguerre polynomials]] are orthogonal with respect to the [[exponential distribution]]. Somewhat more general Laguerre polynomial sequences are orthogonal with respect to [[gamma distribution]]s.
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| **The [[Chebyshev polynomials]] of the first kind are orthogonal with respect to the measure <math>1/\sqrt{1-x^2}.</math>
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| **The Chebyshev polynomials of the second kind are orthogonal with respect to the [[Wigner semicircle distribution]].
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| ====Orthogonal states in quantum mechanics====
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| * In [[quantum mechanics]], two [[eigenstates]] of a [[Hermitian operator]], <math> \psi_m </math> and <math> \psi_n </math>, are orthogonal if they correspond to different eigenvalues. This means, in [[Dirac notation]], that <math> \langle \psi_m | \psi_n \rangle = 0 </math> unless <math> \psi_m </math> and <math> \psi_n </math> correspond to the same eigenvalue. This follows from the fact that [[Schrödinger equation|Schrödinger's equation]] is a [[Sturm–Liouville theory|Sturm–Liouville]] equation (in Schrödinger's formulation) or that observables are given by hermitian operators (in Heisenberg's formulation).{{citation needed|date=February 2012}}
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| ==Art==
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| In art, the [[Perspective (graphical)|perspective]] (imaginary) lines pointing to the [[vanishing point]] are referred to as "orthogonal lines".
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| 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." [http://www.museothyssen.org/thyssen_ing/coleccion/obras_ficha_texto_print497.html]
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| ==Computer science==<!-- This section is linked from [[Motorola 68000]] -->
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| Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results.<ref>Michael L. Scott, ''Programming Language Pragmatics'', p. 228</ref> This usage was introduced by [[Adriaan van Wijngaarden|van Wijngaarten]] in the design of [[Algol 68]]:
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| <blockquote>
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| 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.<ref>1968, Adriaan van Wijngaarden et al., Revised Report on the Algorithmic Language ALGOL 68, section 0.1.2, Orthogonal design</ref>
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| </blockquote>
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| 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 [[Information Hiding#Encapsulation|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.
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| An [[instruction set]] is said to be '''[[orthogonal instruction set|orthogonal]]''' if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task)<ref>{{cite book|author=Null, Linda & Lobur, Julia|title=The essentials of computer organization and architecture|publisher=Jones & Bartlett Learning|edition=2nd|year=2006|isbn=978-0-7637-3769-6|page=257|url=http://books.google.com/books?id=QGPHAl9GE-IC&pg=PA257}}</ref> and is designed such that instructions can use any [[processor register|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.{{Citation needed|date=April 2011}}
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| ==Communications==
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| 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 function]]s. One such scheme is [[time division multiple access|TDMA]], where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").
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| 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]].
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| 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.
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| ==Statistics, econometrics, and economics==
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| When performing statistical analysis, [[Dependent and independent variables|independent variables]] that affect a particular [[Dependent and independent variables|dependent variable]] are said to be orthogonal if they are uncorrelated,<ref>{{cite book |title=Probability, Random Variables and Stochastic Processes |coauthors=Athanasios Papoulis, S. Unnikrishna Pillai |year=2002 |pages=211 |publisher=McGraw-Hill |isbn= 0-07-366011-6}}</ref> 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 linear regression|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).
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| One [[econometrics|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.
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| ==Taxonomy==
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| In [[Taxonomy (general)|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.
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| ==Combinatorics==
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| In [[combinatorics]], two ''n''×''n'' [[Latin squares]] are said to be orthogonal if their [[superimposition]] yields all possible ''n''<sup>2</sup> combinations of entries.<ref>{{cite book|author=Hedayat, A. et al|title=Orthogonal arrays: theory and applications|publisher=Springer|year=1999|isbn=978-0-387-98766-8|page=168|url=http://books.google.com/books?id=HrUYlIbI2mEC&pg=PA168}}</ref>
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| ==Chemistry==
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| In [[organic synthesis|synthetic organic chemistry]] orthogonal [[protecting group|protection]] is a strategy allowing the deprotection of [[functional group]]s independently of each other. 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.
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| ==System reliability==
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| 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.
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| ==Neuroscience==
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| In [[neuroscience]], a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.
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| ==Gaming==
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| In board games such as [[chess]] which feature a grid of squares, 'orthogonal' is commonly used to mean "in the same row or column". In this context 'orthogonal' and 'diagonal' are considered opposites.<ref>{{cite web|title=chessvariants.org chess glossary|url=http://www.chessvariants.org/misc.dir/coreglossary.html#orthogonal_direction}}</ref>
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| ==See also==
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| {{Wiktionary|orthogonal}}
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| *[[Imaginary number]]
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| *[[Isogonal]]
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| *[[Isogonal trajectory]]
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| *[[Orthogonal complement]]
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| *[[Orthogonal group]]
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| *[[Orthogonal matrix]]
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| *[[Orthogonal polynomials]]
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| *[[Orthogonalization]]
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| **[[Gram–Schmidt process]]
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| *[[Orthonormal basis]]
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| *[[Orthonormality]]
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| * Pan-orthogonality occurs in [[coquaternion]]s
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| *[[Surface normal]]
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| ==References==
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| {{Reflist}}
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| * [http://www.faqs.org/docs/artu/ch04s02.html Chapter 4 – Compactness and Orthogonality] in ''[[The Art of Unix Programming]]''
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| {{linear algebra}}
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| [[Category:Abstract algebra]]
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| [[Category:Linear algebra]]
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| [[Category:Factoring]]
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