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

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

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

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

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...

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

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

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...

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

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)...

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

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...

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

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...

In computational complexity theory, the Orthogonal Vectors (OV) problem is a fundamental problem that is used to prove conditional lower bounds under the...

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...

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

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

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...

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

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

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

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

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

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...

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...

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

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...

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

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...

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

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

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

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

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

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

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

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...

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...

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

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

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

In chemistry, orthogonality is when two or more protecting groups in a chemical compound can be removed under conditions that do not affect the other(s)...

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

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

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

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...

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...

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

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...

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

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...

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

{\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...

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...

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...

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

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

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...

Ditrigonal orthogonal 11 41 16 1 Dihexagonal orthogonal RR XVII Cubic orthogonal Simple cubic orthogonal 5 9 1 Cubic orthogonal KU 96 5 Cubic orthogonal P, I...

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

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...

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

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

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...

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...

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

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

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

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

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...

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

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

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

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

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...

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

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

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

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

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...

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

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...

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

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

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

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

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

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

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...

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

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...

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

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...

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

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

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

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

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

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

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...

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...

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

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

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

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

spaces. Additional geometric visualizations of orthogonal diagonalization, including reflection and orthogonal projection matrices, are available at Wikimedia...

q-Jacobi polynomials Pn(x;a,b,c;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. The polynomials are given in terms...

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

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)...

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...

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

(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...

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

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...

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

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

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...

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...

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

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...

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

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...

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

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

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

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...

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...

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

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} ...

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

) ( 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...

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

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

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

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

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

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,...

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...

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

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

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...

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...

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...

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...

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...

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

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...

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

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...

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...

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

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...

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...

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

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...

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

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

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

was On Orthogonal Arrays of Strength Four and Their Applications. She later published this work, "the first significant progress on orthogonal arrays...

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

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...

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...

^{\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...

In mathematics, sieved Pollaczek polynomials are a family of sieved orthogonal polynomials, introduced by Ismail (1985). Their recurrence relations are...

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...

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

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

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...

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...

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

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

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

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...

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

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

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...

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

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

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...

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

Stieltjes–Wigert polynomials P(α) n(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by Moak 1981. Koekoek...

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...

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

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

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...

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

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...

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

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

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

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

have symmetry types governed by the compact classical groups (unitary, orthogonal, or symplectic), and that the distributions of their low-lying zeros should...

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...

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...

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

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...

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

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

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

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...

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

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

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

⁠√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...

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...

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...

Codazzi, de Lamé, de Bour. Polar surfaces. The Elements of a Ternary Orthogonal Substitution. The Motion of an Articulated Quadrilateral; Robert Stephenson's...

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...

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...

14, 1929 – February 11, 2026) was an American mathematician working on orthogonal polynomials who introduced Al-Salam–Chihara polynomials, Brenke–Chihara...

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

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...

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...

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...

In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987...

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)}...

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...

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

cos β − cos γ Tetragonal orthogonal (K) KG a ≠ b = c ≠ d α = β = γ = δ = ε = ζ = 90° (P) P, S, I, Z, G Hexagonal orthogonal (R) R, RS a ≠ b = c ≠ d α...

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

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

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

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

_{\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...

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...

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

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

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

the sequence or the size of the object. As Cartesian moments are non-orthogonal, so are Cartesian velocity moments, so different moments can be closely...

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...

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...

engineering, and molecular evolution. Liu is best known for developing orthogonal DNA replication, a genetic architecture that achieves durable continuous...

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...

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

Object–relationship modeling Object–role modeling Physical data model Principle of orthogonal design (POOD) Relational database Relational model Semantic Web Three-schema...

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...

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

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...

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

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

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...

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...

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

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)...

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

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...

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...

described by a symmetric three-by-three matrix. This symmetric matrix can be orthogonally diagonalized by a set of mutually perpendicular principal axes. Torques...

thin films discovered that it was possible to move magnetic signals in orthogonal directions within the film. This seminal work led to a patent application...

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...

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}}...

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...

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

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...

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...

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

is one of the five simple scheme representing a man upside down with orthogonal arms often behind or slightly arched at the end of the limbs and the junction...

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

be mutually orthogonal. However, if two eigenvectors belong to two different eigenvalues, it may be impossible for them to be orthogonal to each other...

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...

p\times p} orthogonal matrix, then H U H T ∼ B ( a , b ) . {\displaystyle HUH^{T}\sim B(a,b).} Also, if H {\displaystyle H} is a random orthogonal p × p {\displaystyle...

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

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

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...

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

Orthogonal projections Centered by Vertex Edge 5–6 Edge 6–6 Face Hexagon Face Pentagon Image Projective symmetry [2] [2] [2] [6] [10]...

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

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...

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...

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...

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

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...

quantum q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky...

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

mathematician who listed her research interests as "special functions, orthogonal polynomials and approximation theory". She was an emeritus professor at...

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

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

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...

generated by the superposition of two counter propagating beams of light with orthogonal polarizations. This creates a gradient where the polarization varies in...

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...

Ruled surface made of lines orthogonal to an axis...

(n-1)} expressing these as discrete subgroups (point groups). The special orthogonal group has a 2-fold cover by the spin group Spin(n) → SO(n), and restricting...

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

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

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);...

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

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

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...

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

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...

In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is...

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...

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...

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...

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....

The Apollonian circles, two orthogonal families of circles...

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

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 ⁠ V {\displaystyle \mathbf {V} } ⁠ can be found which are real (orthogonal) matrices; a real-valued SVD is often denoted ⁠ U Σ V T {\displaystyle...

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

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

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...

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

three battens in the three orthogonal directions. A Rietveld joint is an overlapping joint of three battens in the three orthogonal directions. It was a prominent...

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

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...

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

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

contributions to real analysis, Fourier analysis, interpolation theory, orthogonal series, complex analysis, and probability theory. Concepts named after...

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

point r → {\displaystyle {\vec {r}}} . In the figures on the right, an orthogonal metal strip with 3 nodes and 2 cells, and the corresponding PEEC circuit...

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...

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...

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

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

the Stieltjes polynomials En are polynomials associated to a family of orthogonal polynomials Pn. They are unrelated to the Stieltjes polynomial solutions...

hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line...

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

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

{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...

specializations of the original art-gallery theorem. For instance, for orthogonal polygons, those whose edges/walls meet at right angles, only ⌊ n / 4 ⌋...

\|R_{n}\|} decreases monotonically. As at each step, the residual is orthogonal to the selected filter, the energy conservation equation is satisfied...

Paraboloidal coordinates are three-dimensional orthogonal coordinates ( μ , ν , λ ) {\displaystyle (\mu ,\nu ,\lambda )} that generalize two-dimensional...

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...

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...

by Lesky for one of three finite sequences of orthogonal polynomials y. Since they form an orthogonal subset of Routh polynomials it seems consistent...

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

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

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...

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...

Small stellated 120-cell Orthogonal projection Type Schläfli-Hess polytope Cells 120 {5/2,5} Faces 720 {5/2} Edges 1200 Vertices 120 Vertex figure {5,3}...

perpendicularity is the relation of conjugate diameters of a circle, so hyperbolic orthogonality is the relation of conjugate diameters of rectangular hyperbolas. The...

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

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

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

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

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

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

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...

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

pieces move orthogonally (forward, sideways i.e. left and right). It especially resembles Turkish draughts in that pieces only move orthogonally. The game...

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...

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...

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

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....

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...

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

In mathematics, the Bateman polynomials are a family Fn of orthogonal polynomials introduced by Harry Bateman (1933). The Bateman–Pasternack polynomials...

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

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)...

global standard for the air interface of 5G networks. It is based on orthogonal frequency-division multiplexing (OFDM), as is the 4G (fourth generation)...

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...

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...

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...

exploration. Common triaxial seismometers provide signal outputs in three orthogonal axes oriented towards east–west (E), north–south (N) and up-down (Z),...

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

galaxies) should be hypersurface orthogonal. Meaning, they should form a 3-bundle of non-intersecting geodesics orthogonal to a series of spacelike hypersurfaces...

spanned by a twisted annulus. They have a common center point in 4-dimensional Euclidean space, and could lie in completely orthogonal rotation planes....

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

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...

walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function...

for Excellence in Scientific Research in Basic Sciences (2007) Scientific career Fields Approximation theory, orthogonal polynomials, spectral methods...

more strategic and better balanced than the original. Because of the new orthogonal decisions required by the 'merchandise tokens', I definitely wouldn't...

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

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...

and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials. Brenke (1945) introduced sequences of Brenke polynomials...

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

bisector of the cone) and selects the nearest neighbor with respect to orthogonal projections to that ray. The resulting graph exhibits several good spanner...

geometry Sphere S n {\displaystyle S^{n}} Orthogonal group O ( n + 1 ) {\displaystyle \mathrm {O} (n+1)} Orthogonal group O ( n ) {\displaystyle \mathrm {O}...

Dual snub 24-cell Orthogonal projection of dual snub 24-cell Type 4-polytope Cells 96, each with three kites and nine isosceles triangles Faces 432 Edges...

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

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...

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...

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

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

posed by Grace Wahba in 1965, seeks to find a rotation matrix (special orthogonal matrix) between two coordinate systems from a set of (weighted) vector...

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...

Icosahedral bipyramid Orthogonal projection:   Base icosahedron edges (30)   Base icosahedron vertices (12)   Apex vertices (2)   Connecting edges (24)...

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...

\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...

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

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...

functionality by hydroformylation, olefination, and then hydrogenation. Orthogonal tandem catalysis is a "one-pot reaction in which sequential catalytic...

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

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...

stereographic projection, centered on one pentagrammic crossed-antiprism Orthogonal projection, with vertices colored by overlaps, red, orange, yellow, green...

on a certain spline interpolation formula. Though these wavelets are orthogonal, they do not have compact supports. There is a certain class of wavelets...

}} B and Bj are orthogonal, since j is orthogonal to B. If a is in B, then j a = a* j, since by orthogonal 0 = 2(j, a*) = ja − a*j. The formula...

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

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

orthonormal frame of a vector space is an ordered basis consisting of orthogonal unit vectors (an orthonormal basis). An affine frame of an affine space...

space, each pseudovector/bivector can be defined as the sum of three orthogonal components parallel to the standard basis pseudovectors/bivectors: A =...

The Plancherel–Rotach asymptotics are asymptotic results for orthogonal polynomials. They are named after the Swiss mathematicians Michel Plancherel and...

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...

Phosphorus monoxide (PO), an unstable radical inorganic compound Projective orthogonal group, an action in projective geometry and linear algebra Per os or peroral...

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...

Coxeter notation Order Abstract structure Rotation points #valence Diagrams Orthogonal Stereographic T (332) [3,3]+ 12 A4 43 32 Th (3*2) [4,3+] 24 A4 × C2 43...

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

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

finite fields, and adele rings.) Hamiltonian mechanics Metaplectic group Orthogonal group Paramodular group Projective unitary group Representations of classical...

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

Multiprocessing Cognitive Neuromorphic Instruction set architectures Types Orthogonal instruction set CISC RISC Application-specific EDGE TRIPS VLIW EPIC MISC...

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...

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)...

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...

q_{k}(i)} , play a prominent role in the theory. They satisfy well-defined orthogonality relations. The p-numbers are the eigenvalues of the adjacency matrix...

attain a specific target for) the response variable(s) of interest. Orthogonality The property that allows individual effects of the k-factors to be estimated...

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...

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

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

Tetrahedral bipyramid Orthogonal projection. 4 red vertices and 6 blue edges make central tetrahedron. 2 yellow vertices are bipyramid apexes. Type Polyhedral...

Heckman–Opdam polynomials (sometimes called Jacobi polynomials) Pλ(k) are orthogonal polynomials in several variables associated to root systems. They were...

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...

methods, including dual check data computations (parity and Reed–Solomon), orthogonal dual parity check data and diagonal parity, have been used to implement...

Orthogonal projection of the 16-dimensional Barnes–Wall lattice B W 16 {\displaystyle BW_{16}} onto 2 dimensions....

q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Hahn (1949). Roelof...

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

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...

a modular representation in terms of 7 × 7 {\displaystyle 7\times 7} orthogonal matrices in the field of eleven elements, with generators given by Y =...

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...

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...

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...

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...

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

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...

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

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

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)...

6-demicube = Runcic 6-cube = Runcicantic 6-cube = Orthogonal projections in D6 Coxeter plane...

are three coordinate axes—usually labeled x, y, and z—with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in...

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

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...

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...

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

hypotheses—one of which he attributed to Ampère—by which the non-interference of orthogonally polarized beams could be explained if polarized light waves were partly...

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...

coefficient for ri has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let D i j = δ i j ( r i , r i )...

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

product. A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group O(n,R);...

The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer. As a type of a continuous wavelet, it has been applied in a number of cases, such as...

In mathematics, sieved Jacobi polynomials are a family of sieved orthogonal polynomials, introduced by Askey (1984). Their recurrence relations are a modified...

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...

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

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

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

Semigroup Vector space Affine representation Character theory Great orthogonality theorem Maschke's theorem Monstrous moonshine Projective representation...

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

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...

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...

mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky...

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...

called sphero-conal or sphero-conical coordinates, are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their...

resuscitation techniques (kappo). The above categorization of techniques is orthogonal to the categorization of domains of combat, which include tachi-waza (立技;...

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

square. Opposition is a particularly important concept in endgames. One orthogonal square separation is direct opposition; one diagonal square is diagonal...

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...

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

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

can be written as the product of a special unitary matrix (or special orthogonal matrix in the real case) and a positive definite Hermitian matrix (or...

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

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...

matrix HS upper Hessenberg matrix OP (real) orthogonal matrix, packed storage matrix OR (real) orthogonal matrix PB symmetric matrix or Hermitian matrix...

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

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...

scale known as color temperature, as well as along a green–magenta axis orthogonal to the color temperature axis. Without color-correction gels, a scene...

Sobolev. His dissertation was a generalization of the work The method of orthogonal projection in potential theory by Hermann Weyl, which he read with no...

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...

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

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

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....

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)...

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)...

space BSO ⁡ ( n ) {\displaystyle \operatorname {BSO} (n)} for the special orthogonal group SO ⁡ ( n ) {\displaystyle \operatorname {SO} (n)} is the base space...

functions and the heat kernel measure on an infinite dimensional complex orthogonal group, was supervised by Leonard Gross. Gordina held a post-doctoral appointment...

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...

of a tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form...

(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...

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...

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...

space orthogonally. Pikeman 4 Pikemen may move up to 12 spaces orthogonally or 1 space diagonally. Squire 1 The squire moves 1 space orthogonally followed...

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

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...

b. Paley, R. E. A. C. (1932), "A Remarkable Series of Orthogonal Functions I, II", Proceedings of the London Mathematical Society, Second Series, 34 (4):...

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

in fibrous red phosphorus, phosphorus chains lie parallel instead of orthogonal, unlike violet phosphorus. Fibrous red phosphorus, similar to red phosphorus...

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...

on the left side "free software", on the right side "proprietary software". On both sides, and therefore mostly orthogonal, "free download" (Freeware)....

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...

Trirectified 9-orthoplex Quadrirectified 9-cube Trirectified 9-cube Birectified 9-cube Rectified 9-cube 9-cube Orthogonal projections in BC9 Coxeter plane...