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Orthogonality is a term with various meanings depending on the context. In mathematics, orthogonality is the generalization of the geometric notion of...

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension...

In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way...

In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product...

In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a table ("array") whose entries come from a fixed finite set...

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval...

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

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to linear algebra of bilinear forms. Two elements u and...

mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other...

linear algebra, an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates...

the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle...

In mathematics, Sobolev orthogonal polynomials are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives...

The proper orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational...

In mathematics, the multiple orthogonal polynomials (MOPs) are orthogonal polynomials in one variable that are orthogonal with respect to a finite family...

algebra, an orthogonal basis for an inner product space V {\displaystyle V} is a basis for V {\displaystyle V} whose vectors are mutually orthogonal. If the...

In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission used in digital modulation for encoding digital...

The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle...

the concept of orthogonality can be used. A projection P {\displaystyle P} on a Hilbert space V {\displaystyle V} is called an orthogonal projection if...

definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean...

In geometry, two circles are said to be orthogonal if their respective tangent lines at the points of intersection are perpendicular (meet at a right angle)...

Orthogonal array testing is a systematic and statistically-driven black-box testing technique employed in the field of software testing. This method is...

pair of conjugate hyperbolas, two conjugate diameters are hyperbolically orthogonal. This relationship of diameters was described by Apollonius of Perga and...

In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots...

are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting...

An orthogonal polyhedron is a polyhedron in which all edges are parallel to the axes of a Cartesian coordinate system, resulting in the orthogonal faces...

mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure...

In mathematics, the indefinite orthogonal group, O ⁡ ( p , q ) {\displaystyle \operatorname {O} (p,q)} is the Lie group of all linear transformations of...

In geometry, a set K ⊂ Rd is defined to be orthogonally convex if, for every line L that is parallel to one of standard basis vectors, the intersection...

Orthogonal is a science fiction trilogy by Australian author Greg Egan taking place in a universe where, rather than three dimensions of space and one...

whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R...

combinatorics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all...

Orthogonal Time Frequency Space (OTFS) is a 2D modulation technique that transforms the information carried in the Delay-Doppler coordinate system. The...

S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}} for k ≤ n and generates an orthogonal set S ′ = { u 1 , … , u k } {\displaystyle S'=\{\mathbf {u} _{1},\ldots...

the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are...

engineering, an orthogonal instruction set is an instruction set architecture where all instruction types can use all addressing modes. It is "orthogonal" in the...

In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the...

geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V,Q) on the associated...

to be "orthogonal" or "transparent" when it is implemented as an intrinsic property of the execution environment of a program. An orthogonal persistence...

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi...

(1992). "Orthogonal arrays for computer experiments, integration and visualization". Statistica Sinica. 2: 439–452. Ye, K.Q. (1998). "Orthogonal column...

Orthographic projection, or orthogonal projection (also analemma), is a means of representing three-dimensional objects in two dimensions. Orthographic...

be constructed with integers coordinates. Orthogonal polyhedra are polyhedra where all edges are orthogonal, parallel to all three axes of Cartesian coordinate...

mathematics, an orthogonal trajectory is a curve which intersects any curve of a given pencil of (planar) curves orthogonally. For example, the orthogonal trajectories...

the orthogonality principle is a necessary and sufficient condition for the optimality of a Bayesian estimator. Loosely stated, the orthogonality principle...

relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...

Orthogonal defect classification (ODC) turns semantic information in the software defect stream into a measurement on the process. The ideas were developed...

1. The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO(n), one example...

In the theory of orthogonal polynomials, Bochner's theorem is a characterization theorem of certain families of orthogonal polynomials as polynomial solutions...

the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2). Given...

origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called...

An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal. That is, the inverse wavelet transform is the adjoint of the wavelet...

if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the...

Ditrigonal orthogonal 22 11 41 Dihexagonal orthogonal RR 16 1 XVII Cubic orthogonal Simple cubic orthogonal Cubic orthogonal KU 24 5 9 1 Cubic orthogonal P, I...

surfaces of different pencils of a threefold orthogonal system is a curvature line. A threefold orthogonal system of surfaces consists of three pencils...

scheme like (in advanced cases) quadrature amplitude modulation (QAM) or orthogonal frequency-division multiplexing (OFDM), which typically makes it very...

edges of a rectilinear polygon. Rectilinear polygons are also known as orthogonal polygons. Other terms in use are iso-oriented, axis-aligned, and axis-oriented...

first city of the modern Greek state to develop a city plan applying the orthogonal rule by Stamatis Voulgaris, a Greek engineer of the French army, in 1829...

{\displaystyle V} into the orthogonal direct sum V = w ⊕ w ⊥ {\displaystyle V=w\oplus w^{\perp }} of w {\displaystyle w} and its orthogonal complement w ⊥ {\displaystyle...

plane is called monotone with respect to a straight line L, if every line orthogonal to L intersects the boundary of P at most twice. Similarly, a polygonal...

Orthogonal ligand-protein pairs (also known as re-engineered ligand-receptor interfaces or re-engineered enzyme-substrate interactions) are a protein-ligand...

Orthogonal frequency-division multiple access (OFDMA) is a multi-user version of the popular orthogonal frequency-division multiplexing (OFDM) digital...

way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed...

In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle...

algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety...

non-orthogonal quantum states rather than four. The protocol relies on the no-cloning theorem and the fundamental principle that non-orthogonal states...

vector resolution) of a vector a on (or onto) a non-zero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of...

Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with...

mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate...

A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular...

plane, or the special orthogonal group SO(2), and unitary group U(1). Reflective circular symmetry is isomorphic with the orthogonal group O(2). A 2-dimensional...

relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...

2013) proposed orthogonal initialization: initializing weight matrices as uniformly random (according to the Haar measure) semi-orthogonal matrices, multiplied...

{\displaystyle c_{1},c_{2}} orthogonally. The radical axis consists of all centers of circles, which intersect the given circles orthogonally. The method described...

Non-orthogonal frequency-division multiplexing (N-OFDM) is a method of encoding digital data on multiple carrier frequencies with non-orthogonal intervals...

Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry Zero-dimensional...

In linear algebra, the complete orthogonal decomposition is a matrix decomposition. It is similar to the singular value decomposition, but typically somewhat...

real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups...

Skew-Hermitian matrix Matrix decomposition Orthogonal group O(n) Special orthogonal group SO(n) Orthogonal matrix Semi-orthogonal matrix Quantum logic gate Special...

The rows of W {\displaystyle W} are pairwise orthogonal. Similarly, the columns are pairwise orthogonal. Each row and each column of W {\displaystyle...

vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line...

of perpendicularity is replaced by the concept of orthogonality: two vectors v and w are orthogonal if their inner product ⟨ v , w ⟩ {\displaystyle \langle...

an orthogonal central plane which is not its completely orthogonal plane, but Clifford parallel to it. It rotates with its completely orthogonal plane...

The principle of orthogonal design (abbreviated POOD) was developed by database researchers David McGoveran and Christopher J. Date in the early 1990s...

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

nodes xi and weights wi for i = 1, ..., n. The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common...

Orthogonal Signal Correction (OSC) is a spectral preprocessing technique that removes variation from a data matrix X that is orthogonal to the response...

line through the centroid is a line of best orthogonal fit. If S ≠ 0 {\displaystyle S\neq 0} , the orthogonal regression line goes through the centroid...

The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a...

P_{n}^{(\alpha ,\beta )}(x)} are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ( 1 − x ) α ( 1 + x ) β {\displaystyle...

named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous...

both mathematics and physics. The three main examples are the Gaussian orthogonal (GOE), unitary (GUE), and symplectic (GSE) ensembles. These are classified...

glans is the urinary meatus. An erection is the stiffening expansion and orthogonal reorientation of the penis, which occurs during sexual arousal. Erections...

obtain a set of n2 triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the Latin square...

In mathematics, an orthogonal symmetric Lie algebra is a pair ( g , s ) {\displaystyle ({\mathfrak {g}},s)} consisting of a real Lie algebra g {\displaystyle...

acts as a traditional inverse of ⁠ A {\displaystyle A} ⁠ on the subspace orthogonal to the kernel. In the following discussion, the following conventions...

that is, it is one of two vertical directions at a specified location, orthogonal to a horizontal flat surface. The direction opposite of the nadir is the...

are called orthogonal if ab = ba = 0. If a is idempotent in the ring R (with unity), then so is b = 1 − a; moreover, a and b are orthogonal. An idempotent...

dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product, x is ε-orthogonal to y if | ⟨ x , y...

to be transmitted on the same carrier frequency by using waves of two orthogonal polarization states. It is used in microwave links such as satellite television...

the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group...

valuable from the perspective of an artificial intelligence. Bostrom's "orthogonality thesis" argues instead that almost any level of intelligence can be...

ensemble models Hamiltonians lacking time-reversal symmetry. The Gaussian orthogonal ensemble GOE ( n ) {\displaystyle {\text{GOE}}(n)} is described by the...

theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described...

Matrices for which this property holds are called orthogonal matrices. The group of all 3 × 3 orthogonal matrices is denoted O(3), and consists of all proper...

mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's...

being pairwise orthogonal to each other but not orthogonal to A 0 {\displaystyle A_{0}} , which is the facet opposite the orthogonal corner. For a 2-simplex...

and the names of the respective problems are axis-aligned rectangles (orthogonal range searching), simplices, halfspaces, and spheres/circles. Query types:...

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets...

are orthogonal. This means that the STBC is designed such that the vectors representing any pair of columns taken from the coding matrix is orthogonal. The...

These may not have unit length, and may also not be orthogonal. In the case that they are orthogonal at all points where the derivatives are well-defined...

invariant, the rotation angle α {\displaystyle \alpha } for a proper orthogonal 3×3 rotation matrix A {\displaystyle A} is found by α = cos − 1 ⁡ ( A...

{\displaystyle {\mbox{SF}}={\frac {\mbox{chip rate}}{\mbox{symbol rate}}}} Orthogonal variable spreading factor (OVSF) is an implementation of code-division...

{\displaystyle \sum _{i}r_{i}=\operatorname {rank} (A)} . But after an orthogonal transform, A = diag ⁡ ( I M , 0 ) {\displaystyle A=\operatorname {diag}...

This is a list of q-analogs in mathematics and related fields. Iwahori–Hecke algebra Quantum affine algebra Quantum enveloping algebra Quantum group Jackson...

) ( n ) = h n {\displaystyle \varphi (h)(n)=hn} . The orthogonal group O(n) of all orthogonal real n × n matrices (intuitively the set of all rotations...

this J {\displaystyle J} is orthogonal; writing all the groups as matrix groups fixes a J {\displaystyle J} (which is orthogonal) and ensures compatibility)...

below, the images of lines parallel to the x- and y-axes are orthogonal and provide orthogonal coordinates. If H = (EG)1⁄2, then the Gaussian curvature is...

relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}...

space, and is commonly denoted E(n) or ISO(n), for inhomogeneous special orthogonal group. The Euclidean group E(n) comprises all translations, rotations...

A scheme of using PACE to evolve orthogonal aminoacyl-tRNA synthetases. Without synthetase activity, T7 RNAP cannot be fully translated due to the presence...

direction of an incoming wave by translation via reflections from the three orthogonal sides. The three intersecting surfaces often are triangles (forming a...

subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect...

decomposed as A = Q R , {\displaystyle A=QR,} where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning Q T = Q − 1 {\displaystyle Q^{\textsf...

restore proper movement and alignment of spine and joint dysfunction. Atlas Orthogonal Technique is an upper cervical chiropractic treatment technique created...

October 1996). "The gravitational Hamiltonian in the presence of non-orthogonal boundaries". Classical and Quantum Gravity. 13 (10): 2735–2752. arXiv:gr-qc/9603050...

orthogonal central planes of the 16-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an octahedron with 3 orthogonal...

( n , R ) {\displaystyle \operatorname {GL} (n,\mathbb {R} )} is the orthogonal group O ⁡ ( n ) {\displaystyle \operatorname {O} (n)} , while "the" maximal...

vertices, edges, and faces. All three square faces surrounding a vertex are orthogonal to each other, meaning the planes are perpendicular, forming a right angle...

passes T 2 {\displaystyle T_{2}} , too, and intersects c {\displaystyle c} orthogonal: The circle with center P {\displaystyle P} and radius Π ( P ) {\displaystyle...

direction, orthogonal or diagonal, except orthogonally forward. Notation: FbWsW Left general (左将) Step: The left general can move one square orthogonally vertically...

Plugging the three eigenvalues 1 and e±iθ and their associated three orthogonal axes in a Cartesian representation into Mercer's theorem is a convenient...

In mathematics, sieved orthogonal polynomials are orthogonal polynomials whose recurrence relations are formed by sieving the recurrence relations of...

the reconstruction of a finite vector space from any subspace W and its orthogonal complement: R n = W ⊕ W ⊥ {\displaystyle \mathbb {R} ^{n}=W\oplus W^{\perp...

Orthogonality as a property of term rewriting systems (TRSs) describes where the reduction rules of the system are all left-linear, that is each variable...

statecharts introduce the new concepts of hierarchically nested states and orthogonal regions, while extending the notion of actions. UML state machines have...

Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials". dlmf.nist.gov. "DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter...

procedure known today as the matching pursuit with post-back fitting or the orthogonal matching pursuit. Petr Vaníček, a Canadian geophysicist and geodesist...

non-zero complex number) represent the same state. A system with n mutually orthogonal quantum states can be described by a Hilbert space of dimension n. Pure...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

^{2}} (generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise the elliptic curves with j ≤ 1...

agreed requirements Manual testing – Testing software without automation Orthogonal array testing – Software testing technique Pair testing – Software testing...

functions. In orthogonal collocation method, we instead use the finite-dimensional subspace spanned by the first N vectors in some orthogonal polynomial...

0\leq \phi \leq 2\pi .} Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue...

projection-valued measures, the values of F {\displaystyle F} are required to be orthogonal projections. In the discrete case, the POVM element F i {\displaystyle...

stochastic process can be represented as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function...

subgroups (including the special linear group SL(n, K), orthogonal group O(n, K), special orthogonal group SO(n, K), and symplectic group Sp(n, K)) are Lie...

Greg Egan (born 20 August 1961) is an Australian science fiction writer and mathematician, best known for his works of hard science fiction. Egan has won...

surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum...

coordinates x = A cos(φ), y = A sin(φ), can be represented as the sum of orthogonal components: [x, 0] + [0, y]. Similarly in trigonometry, the angle sum...

Daubechies wavelets, based on the work of Ingrid Daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a...

extension of the complex numbers. They are a double cover of special orthogonal group SO(8). The special unitary group SO(3) has an eight-dimensional...

+1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant...

needed to improve spatial resolution. MIMO radar systems transmit mutually orthogonal signals from multiple transmit antennas, and these waveforms can be extracted...

mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α) n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function...

stones) is captured when surrounded by the opponent's stones on all orthogonally adjacent points. The game proceeds until neither player wishes to make...

bilinear form to be symmetric and substitutes the energy minimization with orthogonality constraints determined by the same basis functions that are used to...

kernel of A, if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (since orthogonality is defined as having a dot product...

to itself by point inversion. The regular tetrahedron has two special orthogonal projections, one centered on a vertex or equivalently on a face, and one...

\operatorname {Im} (e^{-i2\pi 3t})} oscillate at the same rate but with orthogonal phase. The absolute value of the Fourier transform at +3 Hz is 0.5, which...

loop. Orthogonal trajectories of a given pencil of curves are curves which intersect all given curves orthogonally. For example, the orthogonal trajectories...

complex numbers with absolute value 1. For completeness, there are also the orthogonal and symplectic subgroups, SU ⁡ ( n ) ⊃ SO ⁡ ( n ) , SU ⁡ ( 2 n ) ⊃ Sp...

multivariate statistics. There one often encounters integrals over the orthogonal group or over the Stiefel manifold with respect to an invariant measure...

mathematics: an orthogonal Procrustes problem is a method which can be used to find out the optimal rotation and/or reflection (i.e., the optimal orthogonal linear...

coordinate system, one must designate an origin point in space, O, and two orthogonal directions: the zenith reference direction and the azimuth reference direction...

^{\mathsf {T}}A\mathbf {y} .} An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal...

space which fixes the origin. It can be written as the composition of an orthogonal transformation (an origin-preserving rigid transformation) with a uniform...

Court is separated from the Outer Court by an oblong courtyard lying orthogonal to the city's main axis. It was the home of the Emperor and his family...

equation is written in a general orthogonal coordinate system, in other words a system where the basis vectors are orthogonal. Note that this by no means limits...

instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced...

contains the 6-sphere. In six-dimensional Euclidean space, six mutually orthogonal axes may be drawn through any point. These form a group isomorphous with...

polarization states (i.e. be real). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero...

Orthogonal projections Projective symmetry [2] [4] [6] Triakis octahedron Truncated cube...

space to the real orthogonal group O ( n ) {\displaystyle \mathrm {O} (n)} at the identity matrix; formally, the special orthogonal Lie algebra. In this...

the five factors are not fully orthogonal to one another; that is, the five factors are not independent. Orthogonality is viewed as desirable by some...

intersection of curves. Orthogonal coordinates: coordinate surfaces meet at right angles Skew coordinates: coordinate surfaces are not orthogonal The log-polar...

Orthogonal projections Projective symmetry [3] [5]+ [2] Image Dual image...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

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 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...

orthogonal protecting groups X and Y Fmoc solid state peptide synthesis with orthogonal protecting groups A further important example of orthogonal protecting...

⁠ and ⁠ V {\displaystyle \mathbf {V} } ⁠ can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted ⁠ U Σ V T {\displaystyle...

Orthogonal projections Projective symmetry [3] [4]+ [2] Image Dual image...

In geometry, the great rhombihexahedron (or great rhombicube) is a nonconvex uniform polyhedron, indexed as U21. It has 18 faces (12 squares and 6 octagrams)...

Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence...

given the three phases can change independently, they are by definition orthogonal to each other. This implies a three-dimensional perspective, as shown...

determine the angle between two vectors: in particular if a and b are orthogonal the product is zero. Just as the scalar product can be formulated as the...

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude...

toroidal polyhedron. The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three...

calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry...

discover it. The term is also used to describe similar formulas for other orthogonal polynomials. Askey (2005) describes the history of the Rodrigues formula...

representations of the orthogonal group. For the real or complex case, the alternating representations are Γr = ΛrV, the representation of the orthogonal group on skew...

Orthogonal polarization spectral imaging (OPS imaging) is a method for imaging small blood vessels in tissue like the nail bed or lip. It uses a light...

the perpendicular tetrahedron. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier...

is denoted SO(4). The name comes from the fact that it is the special orthogonal group of 4 by 4 real matrices. In this article rotation means rotational...

neighborhood, in which each cell in a two-dimensional grid has the four orthogonally adjacent grid cells as neighbors, continues to be used for other cellular...

front-left to back-right and top to bottom. A 3D projection of a tesseract performing a double rotation about two orthogonal planes in 4-dimensional space....

120° to each other. Two simple roots that are not joined by a line are orthogonal. The Cartan matrix of a rank r root system is an r × r matrix whose entries...

the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal to g h k ℓ = h b 1 + k b 2 + ℓ b 3 {\displaystyle \mathbf {g} _{hk\ell...

linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly...

, Classifying space for orthogonal group BO {\displaystyle \operatorname {BO} } , Classifying space for infinite orthogonal group In chemistry and physics...

turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system. Under certain assumptions, it is possible to...

orthogonal line through T {\displaystyle T} to a chosen face is coplanar with two other orthogonal lines to the same face. The first is an orthogonal...

Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces...

transpose of a matrix. The orthogonal group has two connected components; the identity component is called the special orthogonal group S O ( n ) {\displaystyle...

horizontal fill U+25A5 ▥ Square with vertical fill U+25A6 ▦ Square with orthogonal crosshatch fill U+25A7 ▧ Square with upper left to lower right fill U+25A8...

aminoacyl tRNA synthetase are required, these are called the orthogonal set. The orthogonal set must not crosstalk with the endogenous tRNA and synthetase...

right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these "cuboids", without qualifying...

A circle that is orthogonal to two fixed circles is orthogonal to every circle in the pencil they determine. The circles orthogonal to two fixed circles...

expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses an orthonormal basis...

not an axis, of the 5-cell. Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them. The characteristic isoclinic rotation...

rotational symmetry. reflection symmetry on one orthogonal axis. reflection symmetry on two orthogonal axes. reflection symmetry on one diagonal axis....

اسماعيل; born April 27, 1944, in Cairo, Egypt) is a mathematician working on orthogonal polynomials and special functions. Ismail received his bachelor's degree...

of the orthogonal group O ( n ) {\displaystyle O(n)} . It is a product of n orthogonal reflections (reflection through the axes of any orthogonal basis);...

particularly important: The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group...

cross-loadings) There are two main types of factor rotation: orthogonal and oblique rotation. Orthogonal rotations constrain factors to be perpendicular to each...

confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles). Parabolas have only one focus, so, by convention,...

perpendicular distance may be generalized to orthogonal distance, between more abstract non-geometric orthogonal objects, as in linear algebra (e.g., principal...

second-order logic in descriptive complexity Special orthogonal group, a subset of an orthogonal group SO(2), a term used in mathematics, the group of...

In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group...

Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. There are also other systems, based on two poles...

Legendre polynomials are not mutually orthogonal in general. For example, P 1 1 {\displaystyle P_{1}^{1}} is not orthogonal to P 2 2 {\displaystyle P_{2}^{2}}...

alternately colored hexagons. The truncated cuboctahedron has two special orthogonal projections in the A2 and B2 Coxeter planes with [6] and [8] projective...

of A {\displaystyle A} on the diagonal, and V {\displaystyle V} is an orthogonal matrix whose columns are the corresponding eigenvectors of A {\displaystyle...

space of continuous functions on the interval I. If {Pn} is a sequence of orthogonal polynomials for this product, we can create the sequence of associated...

_{\mathbb {R} }\psi _{n,k}(t)^{2}\,dt=1.} The Haar functions are pairwise orthogonal, ∫ R ψ n 1 , k 1 ( t ) ψ n 2 , k 2 ( t ) d t = δ n 1 n 2 δ k 1 k 2 , {\displaystyle...

diagonalized by an orthogonal matrix. More explicitly: For every real symmetric matrix A {\displaystyle A} , there exists a real orthogonal matrix Q {\displaystyle...

Emergo match (2004) played on an orthogonal depiction of a standard checkerboard...

ionization, ESI) are generally interfaced to the TOF mass analyzer by "orthogonal extraction" in which ions introduced into the TOF mass analyzer are accelerated...

Chess problems Jumps three squares diagonally followed by one square orthogonally outwards. Atlantosaur c1✳ cWcF Fairy Chess problems (J. de A. Almay,...

has 7 prismatic forms from the lowest 5-orthotope, { }5, and upwards as orthogonal edges are constrained to be of equal length. The vertices in a prism are...

Their boundless faces and edges as elongated prisms or pyramids are orthogonal to the central planes and faces of their dual hemipolyhedra; the coincidental...

represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its...

In the theory of orthogonal functions, Lauricella's theorem provides a condition for checking the closure of a set of orthogonal functions, namely: Theorem...

three-dimensional special orthogonal group, SO(3), that is it is an orthogonal matrix with determinant 1. That it is an orthogonal matrix means that its rows...

If x and y are orthogonal and q(x)q(y) < 0, then x is hyperbolic-orthogonal to y. The standard basis of the real n-space is orthogonal. There are no orthonormal...

precisely an ordered basis. If the vectors are orthogonal, or orthonormal, the frame is called an orthogonal frame, or orthonormal frame, respectively. The...

representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely...

Consider the ground plane π, as y = c which is, for the sake of simplicity, orthogonal to the image plane. Also, consider a line L that lies in the plane π,...

transcription method that represents the entire trajectory as a single high-order orthogonal polynomial. Mesh (Grid) After transcription, the formerly continuous trajectory...

floating point. Similarly, the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix. The system Q(Rx) = b is solved...

identification orthogonalization, Volterra series must be rearranged in terms of orthogonal non-homogeneous G operators (Wiener series): y ( n ) = ∑ p H p x ( n )...

Petrie polygon for many higher-dimensional polytopes, shown in these orthogonal projections in various Coxeter planes: The number of sides in the Petrie...

i = 1 t b i θ i {\displaystyle \sum _{i=1}^{t}b_{i}\theta _{i}} , are orthogonal if ∑ i = 1 t a i b i = 0 {\displaystyle \sum _{i=1}^{t}a_{i}b_{i}=0} ...

the subgroup which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection group...

Gel permeation chromatography (GPC) is a type of size-exclusion chromatography (SEC), that separates high molecular weight or colloidal analytes on the...

have been developed, development of new reactions continues to generate orthogonal methods for labeling to allow multiple methods of labeling to be used...

forming MP, permutes the columns of M. Every permutation matrix P is orthogonal, with its inverse equal to its transpose: P − 1 = P T {\displaystyle P^{-1}=P^{\mathsf...

A tetromino is a geometric shape composed of four squares, connected orthogonally (i.e. at the edges and not the corners). Tetrominoes, like dominoes and...

several built-in pointers for the task. The AVR instruction set is more orthogonal than those of most eight-bit microcontrollers, in particular the 8051...

In the context of computer programming, magic is an informal term for abstraction; it is used to describe code that handles complex tasks while hiding...

For any orthogonal matrix A ∈ O ( d ) {\displaystyle A\in O(d)} , RA and R have the same distribution. Orthogonality: The rows of R are orthogonal to each...

form B are the kernels of B1 and B2 respectively; they are the vectors orthogonal to the whole space on the left and on the right. If V is finite-dimensional...

Orthogonal projections Projective symmetry [2] [2] [2] [2] [6] [10] Image Dual image...

⁠√8φ+7/2⁠ = ⁠√11+4√5/2⁠ ≈ 2.233. The rhombicosidodecahedron has six special orthogonal projections, centered, on a vertex, on two types of edges, and three types...

achieved by the upper convected time derivative. It may be shown that, in orthogonal coordinates, the j-th component of the convection term of the material...

Orthogonal projections Projective symmetry [2] [6] [10] Image Dual image...

Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel. The orthogonal complement of its kernel is called the initial subspace...

coincidence. A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are several minor variations of these...

where the columns are pairwise orthogonal and each has norm d. Furthermore, we have that ⁠1/d⁠E(α) belongs to the orthogonal group S O ( 3 , Q ) {\displaystyle...

symmetry (a 90° rotational symmetry, which also includes a symmetry on both orthogonal axis, 180° rotational symmetry, and diagonal symmetry) is known to exist...

(later called radio waves) can be seen by the fact that they consist of orthogonal electric (E) and magnetic (B or H) waves. The E waves consist of undulating...

Stoeckenius W (October 1981). "Walsby's square bacterium: fine structure of an orthogonal procaryote". Journal of Bacteriology. 148 (1): 352–60. doi:10.1128/JB...

g^{ab}Z_{ab}=0} ⁠. However, it is quite an important tensor since it reflects an "orthogonal decomposition" of the Ricci tensor. The following, not so trivial, property...

fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps...

sculptures of the period, the work is carved in a shallow plane, using orthogonal lines to simulate depth. Scholars have debated the location and consistency...

High-dimensional space allows many mutually orthogonal vectors. However, If vectors are instead allowed to be nearly orthogonal, the number of distinct vectors in...

A 2D orthogonal projection of a 5-cube...

on. An element called a blade may be used to represent a subspace and orthogonal projections onto that subspace. Rotations and reflections are represented...

of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental...

{\displaystyle L_{1}} distance between datapoints. Orthogonal random features uses a random orthogonal matrix instead of a random Fourier matrix. In NIPS...

respect to a circle k, then the circles k and q are orthogonal. If the circles k and q are orthogonal, then a straight line passing through the center O...

a k-dimensional space. K-dimensional is that which concerns exactly k orthogonal axes or a space of any number of dimensions. k-d trees are a useful data...

and blocking Treatment Effect size Contrast Interaction Confounding Orthogonality Blocking Covariate Nuisance variable Models and inference Linear regression...

outside the walls of the historical city, and is characterized by an almost orthogonal tract (developing from two phases of distinct urbanization). It is a fundamental...

A bridge design utilizes support elements (columns) and spanning elements (two orthogonal sets of beams)...

spread spectrum on a single carrier, whereas 802.11a, Wi-Fi 4, 5 and 6 use orthogonal frequency-division multiplexing. Channels are used half duplex and can...

group of F4, which is generated by reflections through the hyperplanes orthogonal to the F4 roots. This is a solvable group of order 1152. The rotational...

provide the best visual fit; which usually means trying to minimize the orthogonal distance to the curve (e.g., total least squares), or to otherwise include...

any rotation or reflection of R k {\displaystyle \mathbb {R} ^{k}} (any orthogonal transformation, that is, any Euclidean isometry preserving 0) to the sequence...

(besides symmetric groups: certain classes of symplectic, unitary, and orthogonal groups), but he also found 3 very large new groups. These groups are usually...

and blocking Treatment Effect size Contrast Interaction Confounding Orthogonality Blocking Covariate Nuisance variable Models and inference Linear regression...

dual, the disdyakis dodecahedron can be drawn in a number of symmetric orthogonal projective orientations. Between a polyhedron and its dual, vertices and...

their approaches, these derivations share a common topic—proving the orthogonality of the residuals and conjugacy of the search directions. These two properties...

an ideal polarizing beamsplitter these would be fully polarized, with orthogonal polarizations. For many common beam-splitting polarizers, however, only...

least squares include inverting the matrix of the normal equations and orthogonal decomposition methods. Consider the linear equation where A ∈ R m × n...

(177.31 m2) 4-bladed main-rotor and 4-bladed tail-rotor with 55° non-orthogonal blade offset Blade section: root: HH-02; tip: NACA 64A006 Performance...

\varphi =\cdots \ .} ) An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. For cardioids the following...

vector spaces, for example, orthogonal complements exist. However, these spaces may have null vectors that are orthogonal to themselves, and consequently...

relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...

cells, 18 triangular faces, 15 edges, and 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, and edges connecting all pairs...

{z}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}} They form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra...

near ideal hybridisation with orthogonal hybrid orbitals. For heavier p block elements this assumption of orthogonality cannot be justified. These deviations...

patterns is minimized if the input vectors are orthogonal to each other. Input vectors are said to be orthogonal to each other if the pairwise product of their...

Combining orthogonal velocity maps into a vector map highlights the anisotropy of the microstructure and improves contrast between grains in this large...

disturbing the signal if the two states one is trying to distinguish are not orthogonal (see no-cloning theorem); and (2) the existence of an authenticated public...

the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George W....

{h}}} , the pair ( h {\displaystyle {\mathfrak {h}}} , σ) is called an orthogonal symmetric Lie algebra of compact type. Any inner product on h {\displaystyle...

matrix that is slightly off may not be orthogonal any more and is harder to convert back to a proper orthogonal matrix. Quaternions also avoid a phenomenon...

dimension. Several techniques are used to help visualise them. Orthogonal projection Orthogonal projections can be used to show various symmetry orientations...

In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like...

Net 3D projection with 3 different rotations Skew orthogonal projections with primary triangles and squares colored...

includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues...

piece on the game moves in a characteristic pattern. Pieces move either orthogonally (that is, forward, backward, left, or right, in the direction of one...

is the adjustment of two planes or axis so that they are parallel or orthogonal (perpendicular) to each other. It is important in machine tools to ensure...

groups. It can be viewed as the group of symmetries of the integers. The orthogonal group O(2), i.e., the symmetry group of the circle, also has similar properties...

capture both diagonally and orthogonally, recovering some of Alquerque's original versatility. Turkish draughts retained orthogonal movement and developed...

may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise...

bearing is a bearing that permits rotation about a central point in two orthogonal directions (usually within a specified angular limit based on the bearing...

component of a force, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds...

_{1}\,d\varphi _{2}\cdots d\varphi _{n-1}.} The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials...

theorem, the loop spaces of the stable orthogonal group can be interpreted as reductive symmetric spaces. Orthogonal symmetric Lie algebra Relative root...

In geometry, the cubitruncated cuboctahedron or cuboctatruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16. It has 20 faces (8 hexagons...

algorithm is sometimes called partial Procrustes superimposition (see also orthogonal Procrustes problem). Let P and Q be two sets, each containing N points...

Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)...

constructed as the duoprism {4} × {4}, the product of two equal-size orthogonal squares, composed of 16 vertices. The 5-cube can be constructed as a duoprism...

refer to: Spin group, Spin(n), a particular double cover of the special orthogonal group SO(n) the corresponding spin algebra, s p i n ( n ) {\displaystyle...

arranged in orthogonal direction – for transmitting / receiving both polarization orientations. If fed with 90° phase-shifted signals, orthogonal devices...

three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions...

understanding, it is usual to consider the function of the arms in each of three orthogonal planes: Top view: The arms have to control toe/steer and lateral compliance...

pulse-position modulation (FPPM and PPM) Multiple frequency-shift keying (MFSK) Orthogonal frequency-division multiplexing (OFDM) Continuous Phase Modulation (CPM)...

perspective projection makes a duoprism look like a torus, with two sets of orthogonal cells, p-gonal and q-gonal prisms. The p-q duoprisms are identical to...

Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)...

rates. Used extensively in Wi-Fi, cable television, and LTE systems. Orthogonal Frequency Division Multiplexing (OFDM): Splits the data across multiple...

thus chosen so that the scores form an orthogonal basis. This is a major difference with PCA where orthogonality is imposed onto loadings (and not the...

follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

three measurements taken with the sensor axis set up along three mutually orthogonal directions, in a X, Y, Z configuration. As an example, it can be used...

simultaneously brought to quasi-triangular form by an orthogonal matrix. There exists an orthogonal matrix Q such that, for every Ai in the given family...

Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation...

of his generation and made fundamental contributions to the theory of orthogonal polynomials and Toeplitz matrices building on the work of his contemporary...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized...

introducing the Hamilton–Jacobi equation approach to mechanics. The orthogonality of mechanical trajectories characteristic of geometrical optics to the...

seems that humans prefer others who are unlike themselves (e.g., with an orthogonal immune system), perhaps because this will lead to a baby that has the...

diagonals and orthogonals, and the lion and lion hawk may do so. Many pieces can move any number of empty squares along a straight orthogonal or diagonal...

of A, and (b) since Av = 0, the vector v is orthogonal to every row vector of A and, hence, is orthogonal to every vector in the row space of A. The facts...

confluence with other sutures (sagitta; arrow in Latin). A section plane orthogonal to the axis of any elongated form in principle is held to be transverse...

mobility for pieces (that cannot move diagonally) compared to a standard orthogonal chessboard. For example, a rook on a hexboard usually has six natural...

follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous...

point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line t to a circle C intersects the circle at a single...

Lie algebra B 6 {\displaystyle B_{6}} via the orthogonal group and its corresponding special orthogonal Lie algebra (by symmetries shared between unordered...

reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents...

Orthomode is a contraction of orthogonal mode. Orthomode transducers serve either to combine or to separate two orthogonally polarized microwave signal paths...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

piece on the game moves in a characteristic pattern. Pieces move either orthogonally (that is, forward, backward, left, or right, in the direction of one...

parasitic effect induced by two orthogonal rotations sculling: a parasitic effect induced by an acceleration orthogonal to a rotation centrifugal accelerations...

spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, certain functions defined on...

preserved. In the classical proof of the lemma, the embedding is a random orthogonal projection. The lemma has applications in compressed sensing, manifold...

In signal processing, the Kautz filter, named after William H. Kautz, is a fixed-pole traversal filter, published in 1954. Like Laguerre filters, Kautz...

piece on the game moves in a characteristic pattern. Pieces move either orthogonally (that is, forward, backward, left, or right, in the direction of one...

science-fiction novel by Australian author Greg Egan and the first part of the Orthogonal trilogy. The novel was published by Night Shade Books on 1 July 2011 with...

mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription)...

Orthogonal curvilinear coordinate. In orthogonal mesh the grid lines are perpendicular to intersection. This is shown in Figure 2. b) Non–orthogonal coordinate...

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae...

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental...

Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

ISBN 978-0-201-61622-4. Dave Thomas, interviewed by Bill Venners (2003-10-10). "Orthogonality and the DRY Principle". Retrieved 2006-12-01. Object Oriented Software...

point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric...

microcode from the machine's Writable Control Store. The TI-990 has a fairly orthogonal instruction set. The instruction formats allows for one, two and three...

Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

minimum time for a quantum system to evolve between two distinguishable (orthogonal) states. QSL theorems are closely related to time-energy uncertainty relations...

Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moments. The wavelet...

Grand 600-cell Orthogonal projection Type Regular star 4-polytope Cells 600 {3,3} Faces 1200 {3} Edges 720 Vertices 120 Vertex figure {3,5/2} Schläfli...

relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} that generalizes the two-dimensional...

matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix...

\Gamma ^{\perp }} , where Γ ⊥ {\displaystyle \Gamma ^{\perp }} is the orthogonal subspace in Z n {\displaystyle \mathbb {Z} ^{n}} with respect to the standard...

{\displaystyle (1-x^{2})y''-xy'+n^{2}y=0,\quad (1-x^{2})y''-3xy'+n(n+2)y=0} Orthogonal polynomials Damped harmonic oscillator 2 m d 2 x d t 2 + c d x d t + k...

(punctuation, X) + – orthogonally (four possible directions) > – orthogonally forwards < – orthogonally backwards <> – orthogonally forwards and backwards...

421 142 241 Rectified 421 Rectified 142 Rectified 241 Birectified 421 Trirectified 421 Orthogonal projections in E6 Coxeter plane...

whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if A T = A − 1 . {\displaystyle \mathbf {A} ^{\text{T}}=\mathbf...

semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix...

recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theory of orthogonal polynomials by Favard (1935) and Shohat...

(qudit with radix 3), that may be in a superposition of three mutually orthogonal quantum states. The qutrit is analogous to the classical radix-3 trit...

{\frac {\hbar }{2}}.} Angular momentum uncertainty relation: For two orthogonal components of the total angular momentum operator of an object: σ J i...

called a "bug", which is a "3-point" form could have a "drop point and 2 orthogonal wheels". He wrote that the "bug" would be "easier" and "more natural"...

representations described here, the orthogonal group O ( n , C ) {\displaystyle O(n,\mathbb {C} )} and special orthogonal group S O ( n , C ) {\displaystyle...

frequency and are out of phase with each other by 90°, a condition known as orthogonality or quadrature. The transmitted signal is created by adding the two carrier...

an object are not orthogonal to the projection plane, and orthographic views in which the principal axes of the object are orthogonal to the projection...

\lambda } , the mutual orthogonality of u n ( λ ) {\displaystyle \mathbf {u} _{n}(\lambda )} is not guaranteed. However, the orthogonal set can be obtained...

February 1944) is an Italian mathematician, working on mathematical physics, orthogonal polynomials, special functions, numerical analysis, approximation theory...

traits, and motives. The interpersonal circumplex is defined by two orthogonal axes: a vertical axis (of status, dominance, power, ambitiousness, assertiveness...

angle) through any axis of a sphere form a Lie group called the special orthogonal group SO(3). (The '3' refers to the three-dimensional space of an ordinary...

prime 1813 = number of polyominoes with 26 cells, symmetric about two orthogonal axes 1814 = 1 + 6 + 36 + 216 + 1296 + 216 + 36 + 6 + 1 = sum of 4th row...

problem in computer graphics is to generate a non-zero vector in ℝ3 that is orthogonal to a given non-zero vector. There is no single continuous function that...

military access to nine military bases in the Philippines, which is orthogonally situated between Taiwan and the South China Sea, expediting the full...

necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto...

applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point x to the boundary of a set Ω in a metric space...

4-dimensional. Moreover, in this case it can be seen that they are all orthogonal to the vector n = [6, −1, 4, −4, 0] (n is an element of the kernel of...

(The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus A = U D U ∗ {\displaystyle A=UDU^{*}} and A ∗ = U D...

and blocking Treatment Effect size Contrast Interaction Confounding Orthogonality Blocking Covariate Nuisance variable Models and inference Linear regression...

A} has an eigenvector, then the spectral theorem follows by taking the orthogonal complement and arguing by induction on the dimension of the inner product...

projection, depending on the exact angle at which the view deviates from the orthogonal. A typical characteristic of orthographic pictorials is that one axis...

relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration...

Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)...

{\displaystyle a=b} , the ellipse is a circle and "conjugate" means "orthogonal".) If the standard ellipse is shifted to have center ( x ∘ , y ∘ ) {\displaystyle...

are either orthogonal or non-orthogonal, in regression, it is essential that the coefficient values assigned in contrast coding be orthogonal. Furthermore...

Shape model of Dinkinesh viewed from six orthogonal directions, with the +z direction representing Dinkinesh's rotational north pole....

| ψ j ⟩ {\displaystyle |\psi _{j}\rangle } (which are not necessarily orthogonal) are prepared with probability p j {\displaystyle p_{j}} each. This is...

orthogonal convex hull or rectilinear convex hull is the intersection of all orthogonally convex and connected supersets, where a set is orthogonally...

angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and...

)} , Σ = L L T + Ψ . {\displaystyle \Sigma =LL^{T}+\Psi .\,} For any orthogonal matrix Q {\displaystyle Q} , if we set L ′ =   L Q {\displaystyle L^{\prime...

or other types may be packed. Atomicity does not behave completely orthogonally with regard to the other ACID properties of transactions. For example...

Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism....

Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row,...

higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

delivered in a short impulse. Contact forces are often decomposed into orthogonal components, one perpendicular to the surface(s) in contact called the...

the dabaaba or dabbabah, is a fairy chess piece that jumps two squares orthogonally (the directions a rook can move), leaping over any intermediate piece...

the Toda lattice. The three-term recurrence relationship of orthogonal polynomials, orthogonal over a positive and finite Borel measure. Algorithms devised...

k-frame, and any two k-frames are related by some orthogonal transformation. In other words, the orthogonal group O(n) acts transitively on V k ( R n ) ....

function of other random variables. The polynomials are chosen to be orthogonal with respect to the joint probability distribution of these random variables...

implementation, it works by using a floating lens element that is moved orthogonally to the optical axis of the lens using electromagnets. Vibration is detected...

 Pk is a set of mutually orthogonal polynomials of degree 0, ..., k. Full details on how to obtain expressions for the orthogonal polynomials and the relationship...

both cases either orthogonally or diagonally (fQbK); this move was adopted by the TSA. In the SRZ it is described as ranging orthogonally forward or to the...

neutrinos pass through every square centimeter on the part of the Earth orthogonal to the direction of the Sun. Since neutrinos are insignificantly absorbed...

consecutive blocks (tetrads) of 4 constitute a sextet. The matrices of Co0 are orthogonal; i. e., they leave the inner product invariant. The inverse is the transpose...

interface cards. Orthogonal midplanes connect vertical cards on one side to horizontal boards on the other side. One common orthogonal midplane connects...

create sets of joints which are known as orthogonal joint sets, that intersect at nearly 90 degrees. Orthogonal joint sets quite often result in the formation...

congruence subgroups are at least 1 / 4 {\displaystyle 1/4} . Selberg's orthogonality conjecture: generalization of Mertens' theorem for functions in Selberg...

elephant in Korean janggi has the same move, but outward from an initial orthogonal step, also without the ability to jump over an intervening piece or pawn...

frequencies. MFSK is a form of M-ary orthogonal modulation, where each symbol consists of one element from an alphabet of orthogonal waveforms. M, the size of the...

representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix R T = R − 1 {\displaystyle R^{\mathsf {T}}=R^{-1}} representing...

angular shapes, while BLU bases use cool colors, industrial materials, and orthogonal shapes. During the July 2006 Electronic Arts press conference, Valve revealed...

about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal...

Wainwright, J (1986). "Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions". Class. Quantum Grav. 3 (6): 1105–1124...

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier...

gave way to, the perpendicular style from the 1320s, with straightened, orthogonal tracery topped with fan-vaulting. Perpendicular Gothic was unknown in...

and for planar point sets with disconnected orthogonal hulls, the tight span differs from the orthogonal convex hull. The definition above embeds the...

through the origin is obtained with all orthogonal matrices (i.e. with determinant 1 and −1) forming orthogonal group O(2). In the case of a determinant...

inherent characteristics of the grid plan, frequent intersections and orthogonal geometry, facilitate movement. The geometry helps with orientation and...

Shift by one quarter period orthogonal angles Shift by one half period opposite angles Shift by full periods coincident angles Period sin ⁡ ( θ ± π 2 )...

triacontahedron has three types of vertices which can be centered in orthogonally projection: The disdyakis triacontahedron, as a regular dodecahedron...

surfaces are not orthogonal, as in orthogonal coordinates. Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since...

solution of the orthogonal Procrustes problem with applications to orthogonal and oblique rotation," proposed a solution to the orthogonal Procrustes problem...

this period, the settlement was restructured according to a relatively orthogonal urban layout, wooden dwellings were replaced with stone or brick constructions...

small surface with area A (SI unit: m2) centered at a given point M and orthogonal to the motion of the charges at M. If IA (SI unit: A) is the electric...

would allow national security customers to use Claude Gov. Anthropic's orthogonal usage policy to the surveillance systems implemented at the Federal Bureau...

From left to right: Zinc complex with pyridine group and orthogonal bipyridine groups in Borromeate. Reduction to Borromeand with removal of zinc coordination...

completely orthogonal squares which do not share any vertices, or as 100 dual pairs of non-orthogonal hexagons between which all axis pairs are orthogonal, or...

Flag of Plzeň (orthogonal) Flag of Panama (orthogonal) Flag of Barcelona, Spain (orthogonal) Flag of Maryland, United States (orthogonal; the armorial...

each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space...

straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation...

assembly languages and to compiler writers. For a related concept see orthogonal instruction set which deals with the ability of any instruction to use...

{\textstyle {\vec {v}}\in V} (a vector with length 1 {\textstyle 1} ) that is orthogonal to the hyperplane. The reflection of a point x {\textstyle x} about this...

mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic...

in the coefficient estimates. The VIF equals 1 when the vector Xj is orthogonal to each column of the design matrix for the regression of Xj on the other...

principal component analysis (PCA) A statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated...

hetero bicyclic chemical compound consisting of pyridine ring fused orthogonally at any position to a pyrimidine ring. List of pyridopyrimidines "pyridopyrimidine...

Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry Zero-dimensional...

longitudinal to the axis of the long bone, meaning that the fracture itself is orthogonal to that axis. The word "torus" originates from the Latin word "protuberance...

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about...

involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors...

Two vectors are orthogonal if ⟨u, v⟩ = 0. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given...

dependent and independent variables have equal variance, this results in orthogonal regression in which the degree of imperfection of the fit is measured...

reconstruction of data in non-orthogonal (oblique) planes, which help in the visualization of organs which are not in orthogonal planes. It is better suited...

specified in a Cartesian coordinate system, defined in terms of mutually orthogonal coordinate axes. Any arbitrary direction can be represented numerically...

rate of 1 metre per second per second. In mechanical engineering, force orthogonal to the main load (such as in parallel helical gears) is referred to as...

non-orthogonal filter bank with linear phase (1988), Ingrid Daubechies' orthogonal wavelets with compact support (1988), Stéphane Mallat's non-orthogonal...

the Gaussian matrix ensembles. The three main examples are the circular orthogonal ensemble (COE) on symmetric unitary matrices, the circular unitary ensemble...

sesquilinear form φ over a module M and a subspace (submodule) W of M, the orthogonal complement of W with respect to φ is W ⊥ = { v ∈ M ∣ φ ( v , w ) = 0 ...

this city – 22,000 acres. Do you lay it out like an American city, rigid orthogonal from side to side? Being more sensitive in 1966–7, the designers decided...

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976) and used in the proof...

less trusted world (the Normal world). This world switch is generally orthogonal to all other capabilities of the processor, thus each world can operate...

the player is to stick with the initial choice, or change to another "orthogonal" option. The latter strategy turns out to double the chances, just as...

(October 2018). "Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation" (PDF). IEEE Transactions on Wireless...

known as the orthogonal group. Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group, or...

neither simple nor semisimple. Another counter-example are the special orthogonal groups in even dimension. These have the matrix − I {\displaystyle -I}...

Structured Query Language (SQL) (pronounced /ˌɛsˌkjuˈɛl/ S-Q-L; or alternatively as /ˈsiːkwəl/ "sequel") is a domain-specific language used to manage data...

Dichronauts describes the cosmological inverse situation to Egan's earlier Orthogonal trilogy comprising The Clockwork Rocket (2011), The Eternal Flame (2012)...

to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse...

settlements, often preserving the original grid layout around two main orthogonal axes (such as Turin, Italy); and cities heavily damaged during World War...

of the orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(n)...

three-dimensional Minkowski space. Equivalently, it is isomorphic to the indefinite orthogonal group SO+(1,2). It follows that SL(2, R) is isomorphic to the spin group...

Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)...

in orthogonal basis functions (OBFs) that work by using the dipole model. OBF decomposition works by expanding the measured field into an orthogonal basis...

In mathematics, the Konhauser polynomials, introduced by Joseph Konhauser, are biorthogonal polynomials for the distribution function of the Laguerre polynomials...

diagonally and one square orthogonally "outward" (not ending adjacent to its starting square), or one square orthogonally followed by one square diagonally...

{\displaystyle ds^{2}=g_{ij}dq^{i}dq^{j}=dx^{2}+dy^{2}+dz^{2}} For all orthogonal coordinates the metric tensor is given by: [ g i j ] = ( h 1 2 0 0 0 h...

Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

field acting on the charged particle. This is equivalent to saying that the charged particle's velocity orthogonal to the wave magnetic field is zero....

basis in an inner-product space Orthogonal polynomials Fourier analysis and Fourier series Harmonic analysis Orthogonal wavelet Biorthogonal wavelet Radial...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

have built major fortified Hippodamian cities (rectangular walls with an orthogonal network of streets) in Sogdiana, such as Bukhara and Panjikent, as they...

In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

g(x, x) associated with the metric tensor applied to each vector of any orthogonal basis produces n real values. By Sylvester's law of inertia, the number...

automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation. When studying representations of groups...

piece on the game moves in a characteristic pattern. Pieces move either orthogonally (that is, forward, backward, left, or right, in the direction of one...

mechanics, mixing angles are the angles between two sets of (complex-valued) orthogonal basis vectors, or states, usually the eigenbases of two quantum mechanical...

Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic...

Orthogonal projection of the Coxeter–Todd lattice onto two dimensions...

piece on the game moves in a characteristic pattern. Pieces move either orthogonally (that is, forward, backward, left, or right, in the direction of one...

analysis (PCA), which computes orthogonal modes that lack predetermined temporal behaviors. Because its modes are not orthogonal, DMD-based representations...