Spectral line

From Wikipedia, the free encyclopedia

 

Continuous spectrum

Emission lines

Absorption lines

Absorption lines for air, under indirect illumination, with the direct light source not visible, so that the gas is not directly between source and detector. Here, Fraunhofer lines in sunlight and Rayleigh scattering of this sunlight is the "source." This is the spectrum of a blue sky somewhat close to the horizon, pointing east at around 3 or 4 pm (i.e., Sun in the West) on a clear day.

A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to identify atoms and molecules. These "fingerprints" can be compared to the previously collected "fingerprints" of atoms and molecules,[1] and are thus used to identify the atomic and molecular components of stars and planets which would otherwise be impossible.

Types of line spectra

Continuous spectrum of an incandescent lamp (mid) and discrete spectrum lines of a fluorescent lamp (bottom)

Spectral lines are the result of interaction between a quantum system (usually atoms, but sometimes molecules or atomic nuclei) and a single photon. When a photon has about the right amount of energy to allow a change in the energy state of the system (in the case of an atom this is usually an electron changing orbitals), the photon is absorbed. Then it will be spontaneously re-emitted, either in the same frequency as the original or in a cascade, where the sum of the energies of the photons emitted will be equal to the energy of the one absorbed (assuming the system returns to its original state).

A spectral line may be observed either as an emission line or an absorption line. Which type of line is observed depends on the type of material and its temperature relative to another emission source. An absorption line is produced when photons from a hot, broad spectrum source pass through a cold material. The intensity of light, over a narrow frequency range, is reduced due to absorption by the material and re-emission in random directions. By contrast, a bright, emission line is produced when photons from a hot material are detected in the presence of a broad spectrum from a cold source. The intensity of light, over a narrow frequency range, is increased due to emission by the material.

Spectral lines are highly atom-specific, and can be used to identify the chemical composition of any medium capable of letting light pass through it (typically gas is used). Several elements were discovered by spectroscopic means, such as helium, thallium, and cerium. Spectral lines also depend on the physical conditions of the gas, so they are widely used to determine the chemical composition of stars and other celestial bodies that cannot be analyzed by other means, as well as their physical conditions.

Mechanisms other than atom-photon interaction can produce spectral lines. Depending on the exact physical interaction (with molecules, single particles, etc.), the frequency of the involved photons will vary widely, and lines can be observed across the electromagnetic spectrum, from radio waves to gamma rays.

Nomenclature

Strong spectral lines in the visible part of the spectrum often have a unique Fraunhofer line designation, such as K for a line at 393.366 nm emerging from singly ionized Ca⁺, though some of the Fraunhofer "lines" are blends of multiple lines from several different species. In other cases the lines are designated according to the level of ionization by adding a Roman numeral to the designation of the chemical element, so that Ca⁺ also has the designation Ca II. Neutral atoms are denoted with the roman number I, singly ionized atoms with II, and so on, so that for example Fe IX (IX, roman 9) represents eight times ionized iron. More detailed designations usually include the line wavelength and may include a multiplet number (for atomic lines) or band designation (for molecular lines). Many spectral lines of atomic hydrogen also have designations within their respective series, such as the Lyman series or Balmer series. Originally all spectral lines were classified into series of Principle series, Sharp series, and Diffuse series. These series exist across atoms of all elements and the Rydberg-Ritz combination principle is a formula that predicts the pattern of lines to be found in all atoms of the elements. For this reason, the NIST spectral line database contains a column for Ritz calculated lines. These series were later associated with suborbitals.

Line broadening and shift

A spectral line extends over a range of frequencies, not a single frequency (i.e., it has a nonzero linewidth). In addition, its center may be shifted from its nominal central wavelength. There are several reasons for this broadening and shift. These reasons may be divided into two general categories – broadening due to local conditions and broadening due to extended conditions. Broadening due to local conditions is due to effects which hold in a small region around the emitting element, usually small enough to assure local thermodynamic equilibrium. Broadening due to extended conditions may result from changes to the spectral distribution of the radiation as it traverses its path to the observer. It also may result from the combining of radiation from a number of regions which are far from each other.

Broadening due to local effects

Natural broadening

The uncertainty principle relates the lifetime of an excited state (due to spontaneous radiative decay or the Auger process) with the uncertainty of its energy. A short lifetime will have a large energy uncertainty and a broad emission. This broadening effect results in an unshifted Lorentzian profile. The natural broadening can be experimentally altered only to the extent that decay rates can be artificially suppressed or enhanced.^[2]

Thermal Doppler broadening

The atoms in a gas which are emitting radiation will have a distribution of velocities. Each photon emitted will be "red"- or "blue"-shifted by the Doppler effect depending on the velocity of the atom relative to the observer. The higher the temperature of the gas, the wider the distribution of velocities in the gas. Since the spectral line is a combination of all of the emitted radiation, the higher the temperature of the gas, the broader the spectral line emitted from that gas. This broadening effect is described by a Gaussian profile and there is no associated shift.

Pressure broadening

The presence of nearby particles will affect the radiation emitted by an individual particle. There are two limiting cases by which this occurs:

• Impact pressure broadening or collisional broadening: The collision of other particles with the emitting particle interrupts the emission process, and by shortening the characteristic time for the process, increases the uncertainty in the energy emitted (as occurs in natural broadening).^[3] The duration of the collision is much shorter than the lifetime of the emission process. This effect depends on both the density and the temperature of the gas. The broadening effect is described by a Lorentzian profile and there may be an associated shift.
• Quasistatic pressure broadening: The presence of other particles shifts the energy levels in the emitting particle,^{[clarification needed]} thereby altering the frequency of the emitted radiation. The duration of the influence is much longer than the lifetime of the emission process. This effect depends on the density of the gas, but is rather insensitive to temperature. The form of the line profile is determined by the functional form of the perturbing force with respect to distance from the perturbing particle. There may also be a shift in the line center. The general expression for the lineshape resulting from quasistatic pressure broadening is a 4-parameter generalization of the Gaussian distribution known as a stable distribution.^[4]

Pressure broadening may also be classified by the nature of the perturbing force as follows:

• Linear Stark broadening occurs via the linear Stark effect, which results from the interaction of an emitter with an electric field of a charged particle at a distance ${\displaystyle r}$, causing a shift in energy that is linear in the field strength. ${\displaystyle (\Delta E\sim 1/r^{2})}$
• Resonance broadening occurs when the perturbing particle is of the same type as the emitting particle, which introduces the possibility of an energy exchange process. ${\displaystyle (\Delta E\sim 1/r^{3})}$
• Quadratic Stark broadening occurs via the quadratic Stark effect, which results from the interaction of an emitter with an electric field, causing a shift in energy that is quadratic in the field strength. ${\displaystyle (\Delta E\sim 1/r^{4})}$
• Van der Waals broadening occurs when the emitting particle is being perturbed by van der Waals forces. For the quasistatic case, a van der Waals profile^{[note 1]} is often useful in describing the profile. The energy shift as a function of distance^{[definition needed]} is given in the wings by e.g. the Lennard-Jones potential. ${\displaystyle (\Delta E\sim 1/r^{6})}$

Inhomogeneous broadening

Inhomogeneous broadening is a general term for broadening because some emitting particles are in a different local environment from others, and therefore emit at a different frequency. This term is used especially for solids, where surfaces, grain boundaries, and stoichiometry variations can create a variety of local environments for a given atom to occupy. In liquids, the effects of inhomogeneous broadening is sometimes reduced by a process called motional narrowing.

Broadening due to non-local effects

Certain types of broadening are the result of conditions over a large region of space rather than simply upon conditions that are local to the emitting particle.

Opacity broadening

Electromagnetic radiation emitted at a particular point in space can be reabsorbed as it travels through space. This absorption depends on wavelength. The line is broadened because the photons at the line center have a greater reabsorption probability than the photons at the line wings. Indeed, the reabsorption near the line center may be so great as to cause a self reversal in which the intensity at the center of the line is less than in the wings. This process is also sometimes called self-absorption.

Macroscopic Doppler broadening

Radiation emitted by a moving source is subject to Doppler shift due to a finite line-of-sight velocity projection. If different parts of the emitting body have different velocities (along the line of sight), the resulting line will be broadened, with the line width proportional to the width of the velocity distribution. For example, radiation emitted from a distant rotating body, such as a star, will be broadened due to the line-of-sight variations in velocity on opposite sides of the star. The greater the rate of rotation, the broader the line. Another example is an imploding plasma shell in a Z-pinch.

Radiative broadening

Radiative broadening of the spectral absorption profile occurs because the on-resonance absorption in the center of the profile is saturated at much lower intensities than the off-resonant wings.Therefore, as intensity rises, absorption in the wings rises faster than absorption in the center, leading to a broadening of the profile. Radiative broadening occurs even at very low light intensities.

Combined effects

Each of these mechanisms can act in isolation or in combination with others. Assuming each effect is independent, the observed line profile is a convolution of the line profiles of each mechanism. For example, a combination of the thermal Doppler broadening and the impact pressure broadening yields a Voigt profile.

However, the different line broadening mechanisms are not always independent. For example, the collisional effects and the motional Doppler shifts can act in a coherent manner, resulting under some conditions even in a collisional narrowing, known as the Dicke effect.

Spectral lines of chemical elements

Element	Z	Symbol	Spectral lines
hydrogen	1	H
helium	2	He
lithium	3	Li
beryllium	4	Be
boron	5	B
carbon	6	C
nitrogen	7	N
oxygen	8	O
fluorine	9	F
neon	10	Ne
sodium	11	Na
magnesium	12	Mg
aluminium	13	Al
silicon	14	Si
phosphorus	15	P
sulfur	16	S
chlorine	17	Cl
argon	18	Ar
potassium	19	K
calcium	20	Ca
scandium	21	Sc
titanium	22	Ti
vanadium	23	V
chromium	24	Cr
manganese	25	Mn
iron	26	Fe
cobalt	27	Co
nickel	28	Ni
copper	29	Cu
zinc	30	Zn
gallium	31	Ga
germanium	32	Ge
arsenic	33	As
selenium	34	Se
bromine	35	Br
krypton	36	Kr
rubidium	37	Rb
strontium	38	Sr
yttrium	39	Y
zirconium	40	Zr
niobium	41	Nb
molybdenum	42	Mo
technetium	43	Tc
ruthenium	44	Ru
rhodium	45	Rh
palladium	46	Pd
silver	47	Ag
cadmium	48	Cd
indium	49	In
tin	50	Sn
antimony	51	Sb
tellurium	52	Te
iodine	53	I
xenon	54	Xe
caesium	55	Cs
barium	56	Ba
lanthanum	57	La
cerium	58	Ce
praseodymium	59	Pr
neodymium	60	Nd
promethium	61	Pm
samarium	62	Sm
europium	63	Eu
gadolinium	64	Gd
terbium	65	Tb
dysprosium	66	Dy
holmium	67	Ho
erbium	68	Er
thulium	69	Tm
ytterbium	70	Yb
lutetium	71	Lu
hafnium	72	Hf
tantalum	73	Ta
tungsten	74	W
rhenium	75	Re
osmium	76	Os
iridium	77	Ir
platinum	78	Pt
gold	79	Au
mercury	80	Hg
thallium	81	Tl
lead	82	Pb
bismuth	83	Bi
polonium	84	Po
radon	86	Rn
radium	88	Ra
actinium	89	Ac
thorium	90	Th
protactinium	91	Pa
uranium	92	U
neptunium	93	Np
plutonium	94	Pu
americium	95	Am
curium	96	Cm
berkelium	97	Bk
californium	98	Cf
einsteinium	99	Es

Orthogonality

From Wikipedia, the free encyclopedia

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

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

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

Etymology

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

Mathematics and physics

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

Definitions

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

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

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

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

Euclidean vector spaces

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

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

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

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

Orthogonal functions

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

${\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x)g(x)w(x)\,dx.}$

In simple cases, w(x) = 1.

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

${\displaystyle \langle f,g\rangle _{w}=0.}$

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

We write the norm with respect to this inner product as

${\displaystyle \|f\|_{w}={\sqrt {\langle f,f\rangle _{w}}}}$

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

${\displaystyle \langle f_{i},f_{j}\rangle _{w}=0\quad i\neq j.}$

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

${\displaystyle \langle f_{i},f_{j}\rangle _{w}=\delta _{i,j},}$

where

${\displaystyle \delta _{i,j}=\left\{{\begin{matrix}1,&&i=j\\0,&&i\neq j\end{matrix}}\right.}$

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

Examples

• The vectors (1, 3, 2)^T, (3, −1, 0)^T, (1, 3, −5)^T are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) = 0, (3)(1) + (−1)(3) + (0)(−5) = 0, and (1)(1) + (3)(3) + (2)(−5) = 0.
• The vectors (1, 0, 1, 0, ...)^T and (0, 1, 0, 1, ...)^T are orthogonal to each other. The dot product of these vectors is 0. We can then make the generalization to consider the vectors in Z₂ⁿ:

${\displaystyle \mathbf {v} _{k}=\sum _{i=0 \atop ai+k$

for some positive integer a, and for 1 ≤ k ≤ a − 1, these vectors are orthogonal, for example (1, 0, 0, 1, 0, 0, 1, 0)^T, (0, 1, 0, 0, 1, 0, 0, 1)^T, (0, 0, 1, 0, 0, 1, 0, 0)^T are orthogonal.

• The functions 2t + 3 and 45t² + 9t − 17 are orthogonal with respect to a unit weight function on the interval from −1 to 1:
  ${\displaystyle \int _{-1}^{1}\left(2t+3\right)\left(45t^{2}+9t-17\right)\,dt=0}$
• The functions 1, sin(nx), cos(nx) : n = 1, 2, 3, ... are orthogonal with respect to Riemann integration on the intervals [0, 2π], [−π, π], or any other closed interval of length 2π. This fact is a central one in Fourier series.

Orthogonal polynomials

• Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials. In particular:
  • The Hermite polynomials are orthogonal with respect to the Gaussian distribution with zero mean value.
  • The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval [−1, 1].
  • The Laguerre polynomials are orthogonal with respect to the exponential distribution. Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions.
  • The Chebyshev polynomials of the first kind are orthogonal with respect to the measure ${\displaystyle 1/{\sqrt {1-x^{2}}}.}$
  • The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.

Orthogonal states in quantum mechanics

• In quantum mechanics, a sufficient (but not necessary) condition that two eigenstates of a Hermitian operator, ${\displaystyle \psi _{m}}$ and ${\displaystyle \psi _{n}}$, are orthogonal is that they correspond to different eigenvalues. This means, in Dirac notation, that ${\displaystyle \langle \psi _{m}|\psi _{n}\rangle =0}$ if ${\displaystyle \psi _{m}}$ and ${\displaystyle \psi _{n}}$ correspond to different eigenvalues. This follows from the fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by hermitian operators (in Heisenberg's formulation).^{[citation needed]}

Art

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

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

Computer science

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

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

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

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

Communications

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

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

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

Statistics, econometrics, and economics

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

Taxonomy

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

Combinatorics

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

Chemistry and biochemistry

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

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

System reliability

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

Neuroscience

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

Gaming

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

Other examples

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

Topological group

From Wikipedia, the free encyclopedia

 

The real numbers form a topological group under addition

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics.

Formal definition

A topological group, G, is a topological space which is also a group such that the group operations of product:

${\displaystyle G\times G\to G:(x,y)\mapsto xy}$

and taking inverses:

${\displaystyle G\to G:x\mapsto x^{-1}}$

are continuous. Here G × G is viewed as a topological space with the product topology.

Although not part of this definition, many authors^[1] require that the topology on G be Hausdorff; it is equivalent to assume that the singleton containing the identity element 1 is a closed subset of G. The reasons, and some equivalent conditions, are discussed below. In any case, any topological group can be made Hausdorff by taking an appropriate canonical quotient.

In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.

Homomorphisms

A homomorphism of topological groups means a continuous group homomorphism G ${\displaystyle \to }$ H. An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.

Topological groups, together with their homomorphisms, form a category.

Examples

Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups.

The real numbers, R with the usual topology form a topological group under addition. More generally, Euclidean n-space Rⁿ is a topological group under addition. Some other examples of abelian topological groups are the circle group S¹, or the torus (S¹)ⁿ for any natural number n.

The classical groups are important examples of non-abelian topological groups. For instance, the general linear group GL(n,R) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL(n,R) as a subspace of Euclidean space R^n×n. Another classical group is the orthogonal group O(n), the group of all linear maps from Rⁿ to itself that preserve the length of all vectors. The orthogonal group is compact as a topological space. Much of Euclidean geometry can be viewed as studying the structure of the orthogonal group, or the closely related group O(n) ⋉ Rⁿ of isometries of Rⁿ.

The groups mentioned so far are all Lie groups, meaning that they are smooth manifolds in such a way that the group operations are smooth, not just continuous. Lie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about Lie algebras and then solved.

An example of a topological group which is not a Lie group is the additive group Q of rational numbers, with the topology inherited from R. This is a countable space, and it does not have the discrete topology. An important example for number theory is the group Z_p of p-adic integers, for a prime number p, meaning the inverse limit of the finite groups Z/pⁿ as n goes to infinity. The group Z_p is well behaved in that it is compact (in fact, homeomorphic to the Cantor set), but it differs from (real) Lie groups in that it is totally disconnected. More generally, there is a theory of p-adic Lie groups, including compact groups such as GL(n,Z_p) as well as locally compact groups such as GL(n,Q_p), where Q_p is the locally compact field of p-adic numbers.

Some topological groups can be viewed as infinite dimensional Lie groups; this phrase is best understood informally, to include several different families of examples. For example, a topological vector space, such as a Banach space or Hilbert space, is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, are loop groups, Kac–Moody groups, diffeomorphism groups, homeomorphism groups, and gauge groups.

In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication. For example, the group of invertible bounded operators on a Hilbert space arises this way.

Properties

The inversion operation on a topological group G is a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G.

Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.^[2] If G is not abelian, then these two need not coincide. The uniform structures allow one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

As a uniform space, every topological group is completely regular. It follows that if the identity element is closed in a topological group G, then G is T₂ (Hausdorff), even T_3½ (Tychonoff). If G is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group G/K, where K is the closure of the identity.^[3] This is equivalent to taking the Kolmogorov quotient of G.

The Birkhoff–Kakutani theorem states that the following three conditions on a topological group G are equivalent:^[4]

• The identity element 1 is closed in G, and there is a countable basis of neighborhoods for 1 in G.
• G is metrizable (as a topological space).
• There is a left-invariant metric on G that induces the given topology on G. (A metric on G is called left-invariant if for each point a in G, the map x ↦ ax is an isometry from G to itself.)

Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G, the set of left cosets G/H with the quotient topology is called a homogeneous space for G. The quotient map q : G → G/H is always open. For example, for a positive integer n, the sphere Sⁿ is a homogeneous space for the rotation group SO(n+1) in Rⁿ⁺¹, with Sⁿ = SO(n+1)/SO(n). A homogeneous space G/H is Hausdorff if and only if H is closed in G.^[5] Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.

Every open subgroup H is also closed in G, since the complement of H is the open set given by the union of open sets gH for g in G \ H.

If H is a normal subgroup of G, then the quotient group G/H becomes a topological group when given the quotient topology. It is Hausdorff if and only if H is closed in G. For example, the quotient group R/Z is isomorphic to the circle group S¹.

If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup of G, the closure of H is normal in G.

In any topological group, the identity component (i.e., the connected component containing the identity element) is a closed normal subgroup. If C is the identity component and a is any point of G, then the left coset aC is the component of G containing a. So the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. It follows that the quotient group G/C is totally disconnected.^[6]

The isomorphism theorems from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups. The theorems are valid if one places certain restrictions on the maps involved. For example, the first isomorphism theorem states that if f : G → H is a homomorphism, then the homomorphism from G/ker(f) to im(f) is an isomorphism if and only if the map f is open onto its image.^[7]

Hilbert's fifth problem

There are several strong results on the relation between topological groups and Lie groups. First, every continuous homomorphism of Lie groups G → H is smooth. It follows that a topological group has a unique structure of a Lie group if one exists. Also, Cartan's theorem says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth submanifold.

Hilbert's fifth problem asked whether a topological group G which is a topological manifold must be a Lie group. In other words, does G have the structure of a smooth manifold, making the group operations smooth? The answer is yes, by Gleason, Montgomery, and Zippin.^[8] In fact, G has a real analytic structure. Using the smooth structure, one can define the Lie algebra of G, an object of linear algebra which determines a connected group G up to covering spaces. As a result, the solution to Hilbert's fifth problem reduces the classification of topological groups that are topological manifolds to an algebraic problem, albeit a complicated problem in general.

The theorem also has consequences for broader classes of topological groups. First, every compact group (understood to be Hausdorff) is an inverse limit of compact Lie groups. (One important case is an inverse limit of finite groups, called a profinite group. For example, the group Z_p of p-adic integers and the absolute Galois group of a field are profinite groups.) Furthermore, every connected locally compact group is an inverse limit of connected Lie groups.^[9] At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group.^[10] (For example, the locally compact group GL(n,Q_p) contains the compact open subgroup GL(n,Z_p), which is the inverse limit of the finite groups GL(n,Z/p^r) as r goes to infinity.)

Representations of compact or locally compact groups

An action of a topological group G on a topological space X is a group action of G on X such that the corresponding function G × X → X is continuous. Likewise, a representation of a topological group G on a real or complex topological vector space V is a continuous action of G on V such that for each g in G, the map v ↦ gv from V to itself is linear.

Group actions and representation theory are particularly well understood for compact groups, generalizing what happens for finite groups. For example, every finite-dimensional (real or complex) representation of a compact group is a direct sum of irreducible representations. An infinite-dimensional unitary representation of a compact group can be decomposed as a Hilbert-space direct sum of irreducible representations, which are all finite-dimensional; this is part of the Peter–Weyl theorem.^[11] For example, the theory of Fourier series describes the decomposition of the unitary representation of the circle group S¹ on the complex Hilbert space L²(S¹). The irreducible representations of S¹ are all 1-dimensional, of the form z ↦ zⁿ for integers n (where S¹ is viewed as a subgroup of the multiplicative group C*). Each of these representations occurs with multiplicity 1 in L²(S¹).

The irreducible representations of all compact connected Lie groups have been classified. In particular, the character of each irreducible representation is given by the Weyl character formula.

More generally, locally compact groups have a rich theory of harmonic analysis, because they admit a natural notion of measure and integral, given by the Haar measure. Every unitary representation of a locally compact group can be described as a direct integral of irreducible unitary representations. (The decomposition is essentially unique if G is of Type I, which includes the most important examples such as abelian groups and semisimple Lie groups.^[12]) A basic example is the Fourier transform, which decomposes the action of the additive group R on the Hilbert space L²(R) as a direct integral of the irreducible unitary representations of R. The irreducible unitary representations of R are all 1-dimensional, of the form x ↦ e^2πiax for a ∈ R.

The irreducible unitary representations of a locally compact group may be infinite-dimensional. A major goal of representation theory, related to the Langlands classification of admissible representations, is to find the unitary dual (the space of all irreducible unitary representations) for the semisimple Lie groups. The unitary dual is known in many cases such as SL(2,R), but not all.

For a locally compact abelian group G, every irreducible unitary representation has dimension 1. In this case, the unitary dual ${\displaystyle {\hat {G}}}$ is a group, in fact another locally compact abelian group. Pontryagin duality states that for a locally compact abelian group G, the dual of ${\displaystyle {\hat {G}}}$ is the original group G. For example, the dual group of the integers Z is the circle group S¹, while the group R of real numbers is isomorphic to its own dual.

Every locally compact group G has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of G (the Gelfand–Raikov theorem). By contrast, representation theory for topological groups that are not locally compact has so far been developed only in special situations, and it may not be reasonable to expect a general theory. For example, there are many abelian Banach–Lie groups for which every representation on Hilbert space is trivial.^[13]

Homotopy theory of topological groups

Topological groups are special among all topological spaces, even in terms of their homotopy type. One basic point is that a topological group G determines a path-connected topological space, the classifying space BG (which classifies principal G-bundles over topological spaces, under mild hypotheses). The group G is isomorphic in the homotopy category to the loop space of BG; that implies various restrictions on the homotopy type of G.^[14] Some of these restrictions hold in the broader context of H-spaces.

For example, the fundamental group of a topological group G is abelian. (More generally, the Whitehead product on the homotopy groups of G is zero.) Also, for any field k, the cohomology ring H*(G,k) has the structure of a Hopf algebra. In view of structure theorems on Hopf algebras by Heinz Hopf and Armand Borel, this puts strong restrictions on the possible cohomology rings of topological groups. In particular, if G is a path-connected topological group whose rational cohomology ring H*(G,Q) is finite-dimensional in each degree, then this ring must be a free graded-commutative algebra over Q, that is, the tensor product of a polynomial ring on generators of even degree with an exterior algebra on generators of odd degree.^[15]

In particular, for a connected Lie group G, the rational cohomology ring of G is an exterior algebra on generators of odd degree. Moreover, a connected Lie group G has a maximal compact subgroup K, which is unique up to conjugation, and the inclusion of K into G is a homotopy equivalence. So describing the homotopy types of Lie groups reduces to the case of compact Lie groups. For example, the maximal compact subgroup of SL(2,R) is the circle group SO(2), and the homogeneous space SL(2,R)/SO(2) can be identified with the hyperbolic plane. Since the hyperbolic plane is contractible, the inclusion of the circle group into SL(2,R) is a homotopy equivalence.

Finally, compact connected Lie groups have been classified by Wilhelm Killing, Élie Cartan, and Hermann Weyl. As a result, there is an essentially complete description of the possible homotopy types of Lie groups. For example, a compact connected Lie group of dimension at most 3 is either a torus, the group SU(2) (diffeomorphic to the 3-sphere S³), or its quotient group SU(2)/{±1} ≅ SO(3) (diffeomorphic to RP³).

Generalizations

Various generalizations of topological groups can be obtained by weakening the continuity conditions:^[16]

• A semitopological group is a group G with a topology such that for each c in G the two functions G → G defined by ${\displaystyle x\mapsto xc}$ and ${\displaystyle x\mapsto cx}$ are continuous.
• A quasitopological group is a semitopological group in which the function mapping elements to their inverses is also continuous.
• A paratopological group is a group with a topology such that the group operation is continuous.