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Spectrum

Specialty Definition: Spectrum

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Specialty Definition: Eigenvector

(From Wikipedia, the free Encyclopedia)

In linear algebra, the eigenvectors (from the German eigen meaning "inherent, characteristic") of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. The scalar is then called the eigenvalue associated with the eigenvector.

In applied mathematics and physics the eigenvectors of a matrix or a differential operator often have important physical significance. In classical mechanics the eigenvectors of the governing equations typically correspond to natural modes of vibration in a body, and the eigenvalues to their frequencies. In quantum mechanics, operators correspond to observable variables, eigenvectors are also called eigenstates, and the eigenvalues of an operator represent those values of the corresponding variable that have non-zero probability of occurring.

Examples

Intuitively, for linear transformations of two-dimensional space R², eigenvectors are thus:

• rotation: no eigenvectors
• reflection: eigenvectors are perpendicular and parallel to the line of symmetry, the eigenvalues are -1 and 1, respectively
• scaling: all vectors are eigenvectors, and the eigenvalue is the scale factor
• projection onto a line: eigenvectors with eigenvalue 1 are parallel to the line, eigenvectors with eigenvalue 0 are parallel to the direction of projection

Definition

Formally, we define eigenvectors and eigenvalues as follows: If A : V -> V is a linear operator on some vector space V, v is a non-zero vector in V and c is a scalar (possibly zero) such that

then we say that v is an eigenvector of the operator A, and its associated eigenvalue is . Note that if v is an eigenvector with eigenvalue , then any non-zero multiple of v is also an eigenvector with eigenvalue . In fact, all the eigenvectors with associated eigenvalue , together with 0, form a subspace of V, the eigenspace for the eigenvalue .

Finding eigenvectors

For example, consider the matrix

which represents a linear operator R³ -> R³. One can check that

and therefore 2 is an eigenvalue of A and we have found a corresponding eigenvector.

The characteristic polynomial

An important tool for describing eigenvalues of square matrices is the characteristic polynomial: saying that c is an eigenvalue of A is equivalent to stating that the system of linear equations (A - cI) x = 0 (where I is the identity matrix) has a non-zero solution x (namely an eigenvector), and so it is equivalent to the determinant det(A - c I) being zero. The function p(c) = det(A - cI) is a polynomial in c since determinants are defined as sums of products. This is the characteristic polynomial of A; its zeros are precisely the eigenvalues of A. If A is an n-by-n matrix, then its characteristic polynomial has degree n and A can therefore have at most n eigenvalues.

Returning to the example above, if we wanted to compute all of A's eigenvalues, we could determine the characteristic polynomial first:

and because we see that the eigenvalues of A are 2, 1 and -1.

(In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Faster and more numerically stable methods are available, for instance the QR decomposition.)

Complex eigenvectors

Note that if A is a real matrix, the characteristic polynomial will have real coefficients, but not all its roots will necessarily be real. The complex eigenvalues will all be associated to complex eigenvectors.

In general, if v₁, ..., v_m are eigenvectors to different eigenvalues λ₁, ..., λ_m, then the vectors v₁, ..., v_m are necessarily linearly independent.

The spectral theorem for symmetric matrices states that, if A is a real symmetric n-by-n matrix, then all its eigenvalues are real, and there exist n linearly independent eigenvectors for A which all have length 1 and are mutually orthogonal.

Our example matrix from above is symmetric, and three mutually orthogonal eigenvectors of A are

These three vectors form a basis of R³. With respect to this basis, the linear map represented by A takes a particularly simple form: every vector x in R³ can be written uniquely as

and then we have

Infinite dimensions

Note that for infinite dimensional Hilbert spaces, eigenstates only exist in an extension of that space. For example, Dirac delta functions only exist in an extension of L²(R).

External links

• Eigenvector at Mathworld
• Earliest Known Uses of Some of the Words of Mathematics: E - see eigenvector and related terms

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Eigenvector."

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Electromagnetic spectrum

(From Wikipedia, the free Encyclopedia)

The electromagnetic spectrum describes the various types of electromagnetic radiation based on wavelength. Radio, representing wavelengths from a few feet to well over a mile, is at one end of the spectrum. Gamma ray radiation is at the other end: the wavelength of the harder types is so short, in the subatomic range, that we do not have instruments capable of directly measuring it.

Legend: γ = Gamma Rays HX = Hard X-Rays SX = Soft X-Rays EUV = Extreme ultraviolet NUV = Near ultraviolet NIR = Near infrared MIR = Moderate infrared FIR = Far infrared Radio waves: EHF = Extremely high frequency (Microwaves) SHF = Super high frequency (Microwaves) UHF = Ultrahigh frequency VHF = Very high frequency HF = High frequency MF = Medium frequency LF = Low frequency VLF = Very low frequency VF = Voice frequency ELF = Extremely low frequency

Classifications

While the classification scheme is generally accurate, in reality there is often some overlap between neighboring types of electromagnetic radiation. For example some low energy gamma-rays actually have a longer wavelength than some high energy X-rays. This is possible because "gamma-ray" is the name given to the photons generated from nuclear decay or other nuclear and subnuclear processes, whereas X-rays on the other hand are generated by electronic transitions involving highly energetic inner electrons. Therefore the distinction between gamma-ray and x-ray is related to the radiation source rather than the radiation wavelength. Generally, nuclear transitions are much more energetic than electronic transitions, so most gamma-rays are more energetic than x-rays. However, there are a few low-energy nuclear transitions (eg. the 14.4 keV nuclear transition of Fe-57) that produce gamma-rays that are less energetic than some of the higher energy X-rays.

Use of the radio frequency spectrum is regulated by governments. This is called frequency allocation.

Radio Waves

Radio is at the weak end of the spectrum, with low energy and long wavelength. It's used for transmission of data, via modulation. Television, mobile phones, wireless networking and amateur radio all use it. Radio Waves can be detected at the Ultra High Frequency (UHF), Very High Frequency (VHF), Shortwave (HF or high frequency), Medium Wave (AM), Longwave, Very Low Frequency (VLF), and Extreme Low Frequency (ELF) bandwidth.

Microwaves

Microwaves come next. They can cause entire molecules to resonate. This resonance causes water to move rapidly and enables the microwave oven to cook food. Low intensity microwave radiation is used in Wi-Fi.

Between 300 GHz and the mid-infrared, the absorption of electromagnetic radiation by molecular vibration in the Earth's atmosphere is so great that the atmosphere is effectively opaque to electromagnetic radiation, until the atmosphere becomes transparent again in the so-called infrared and optical window freqency ranges. However, there are certain wavelength ranges ("windows") within the opaque range which allow partial transmission, and can be used for astronomy.

It should be noted that the average Microwave oven is, in close range, powerful enough to cause interference with poorly shielded electromagnetic fields such as those found in mobile medical devices and cheap consumer electronics.

Infra-red Radiation

The next category is infra-red. This makes chemical bonds resonate. When a chemical bond resonates, the vibrations add internal energy to the molecule. The molecule becomes hot. The bulk substance becomes hot when its molecules' bonds are all resonating. When you touch it, you feel its warmth or you lose the tip of your finger, depending on how violent the resonance is.

Visible radiation (light)

After infra-red comes visible light. This is the range in which the sun and stars similar to it emit most of their radiation. When this is scattered or reflected by an object, we can infer the existence of the object. A person can see the light scattered from his or her room's light by his or her keyboard, so his or her brain infers that the keyboard exists.

Since current computer screens only use 3 primary colours, only the red, green and blue actually consist of single colours in the image, the rest is composite. Not drawn to scale.

Ultraviolet light

Next comes ultraviolet. This is radiation whose wavelength is shorter than the violet end of the visible spectrum. It was discovered to be useful for astronomy by a Mariner probe at Mercury, which detected UV that "had no right to be there". The dying probe was turned over to the UV team full time. The UV source turned out to be a star, but UV astronomy was born. Being very energetic, UV can break chemical bonds. Chlorine will not normally react with an alkane, but give it UV and it reacts quickly. This is because the UV breaks the bond holding chlorine atoms into molecules of Cl₂. Lone atoms are extremely reactive and will react with the otherwise almost-inert alkanes. It also makes a mess of DNA, causing cell death at best and uncontrolled cell reproduction (cancer) at worst.

X-rays

After UV come X-rays. Hard X-rays are of shorter wavelengths than soft X-rays. X-rays are used for seeing through some things and not others, as well as for high-energy physics and astronomy. Black holes and neutron stars emit x-rays, which enable us to study them.

Gamma rays

After hard X-rays come gamma rays. These are the most energetic photons, having no lower limit to their wavelength. They are useful to astronomers in the study of high-energy objects or regions and find a use with physicists thanks to their penetrative ability and their production from radioisotopes.

Note that there are no defined boundaries between the types of electromagnetic radiation. Some wavelengths have a mixture of the properties of two regions of the spectrum. For example, red light resembles infra-red radiation in that it can resonate some chemical bonds.

External links

• U.S. Frequency Allocation Chart - Covering the range 3kHz-300GHz (from Department of Commerce)
• Canadian Table of Frequency Allocations
• UK frequency allocation table (from the Radiocommunications Agency, pdf format)

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Electromagnetic spectrum."

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Optical spectrum

(From Wikipedia, the free Encyclopedia)

The optical spectrum (visible light or visible spectrum) is that portion of the electromagnetic spectrum which is visible to the human eye. The optical spectrum is a composite, or mixture, of the various colors.

There are no exact bounds to the optical spectrum ; a light-adapted eye typically has a maximum sensitivity of ~555 nm (in the green). Commonly the response of the eye is considered to cover 380 nm to 780 nm although a range of 400 nm to 700 nm range is more common. The eye may, however, have some visual response at even wider wavelength ranges.

Wavelengths in the range visible to the eye occupy most of the "optical window", a range of wavelengths that are easily transmitted through the Earth's atmosphere.

Note: Ultraviolet and Infrared are often considered to be "light" but are generally not visible to the human eye.

The Optical Spectrum and Isaac Newton

It was Sir Isaac Newton in 1666 who first used the word spectrum to refer to the celebrated Phenomenon of Colours which can be extracted from sunlight, by a glass prism.

Visible Light

Visible light is electromagnetic radiation that is made of colors of light that the eye can see. This light has wavelengths that are generally expressed in nanometers.

See frequency, wavelength, Rydberg formula.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Optical spectrum."

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Political spectrum

(From Wikipedia, the free Encyclopedia)

A political spectrum is a way of comparing or visualizing different political positions, by placing them upon one or more geometric axes.

Introduction

The key assumption of such a spectrum is that people's views on many issues correlate strongly, or that one essential issue subsumes or dominates all others. For a political spectrum to exist, there must be range of beliefs. Political systems in which most people fall clearly into one group or another with almost no one in between, such as most nationalist controversies, are not well described by a political spectrum.

In a modern Islamic country, for instance, a political spectrum might be divided along the issue of the clergy's role in government. Those who believe clerics should have the power to enforce Islamic law are on one end of the spectrum, those who support a secular society are on the other; moderates fall at various points in between. In Taiwan, the political spectrum is defined in terms of Chinese reunification versus Taiwan independence.

Political spectrums can end when one group wins so throughly that there is no longer a divergence of opinions. This occurred in the 1970's in the People's Republic of China in the case between the rightists and the leftists in which the rightists won or in the late 18th century controversy between the Federalists and the Anti-federalists in the United States. However, what tends to happen in this situation is that the winners start disagreeing over new issues and a new political spectrum is created.

However, many times the political spectrum remains although the issues which define the spectrum changes. An example was the controversy over the succession of William of Orange's successor to the English throne. This helped to define the British political spectrum which exists to this day, long after the original controversy was resolved.

Left and Right

In modern Western countries, the political spectrum usually is described along left-right lines. This traditional political spectrum is defined along an axis with Conservatism ("the right") on one end, and Socialism ("the left", called Liberalism in the United States) on the other. The term left and right was also used to describe politics in China starting in the 1920's until the 1980's although the issues often were very different from the one's in Western nations.

There are various different opinions about what is actually being measured along this axis:

• Whether the state should prioritize equality (left) or liberty (right).
• Whether the government's involvement with the economy should be interventionist (left) or laissez-faire (right).
• Whether the government should be secular and separate itself from religious beliefs (left) or should take a stance of religious morality (right).
• Fair outcomes (left) versus fair processes (right)
• Whether one embraces change (left) or prefers rigorous justification for change (right). This was proposed by Eric Hoffer.
• Whether human nature and society is malleable (left) or fixed (right). This was proposed by Thomas Sowell.

Historical Origin of the Terms

The usage in Western politics of "right" and "left" to refer to political affiliation stems at least from the French National Assembly in 1789, during the French Revolution. There, the Second Estate, or nobility, sat to the right of the chamber, and the Third Estate, or common people (at the time the radicals) to the left. Thus, "right" generally meant conservative, upholding the existing social or political order, and "left" meant radical, attempting to change or overthrow the existing order. The usage may actually be earlier, from the pre-Revolutionary Estates-General, where right and left referred to supporters and opponents of the monarchy.

Alternative Spectra

Some people feel that it is not obvious how these various concepts are related. They say that it is very confusing to speak of the right or the left without indicating what exactly you are referring to. They believe that one should first establish context by defining the axes upon which different positions will be measured.

Nonetheless, the right-left spectrum is so common as to be taken for granted. Many people even have a hard time conceptualizing any alternative to it. However, numerous alternatives exist, usually having been developed by people who feel their views are not fairly represented on the traditional right-left spectrum.

Perhaps the simplest alternative to the left-right spectrum was devised as a rhetorical tool during the Cold War. This was a circle which brought together the far right and left ends of the traditional spectrum, equating "extreme socialism" (i.e. the Communist Party) with "extreme conservatism" (i.e. Fascism). This nexus was particularly useful to those opposed to rapprochement with the Soviet Union.

Another alternative spectrum offered at American Federalist Journal emphasizes the degree of political control, and thus places communism and fascism [totalitarianism] at one extreme and anarchism [no government at all] at the other extreme.

Another alternative currently popular among certain environmentalists uses a single axis to measure what they consider to be the good of the Earth against the good of big business, which is seen as being the force most likely to harm the earth.

In 1998, political author Virginia Postrel, in her book The Future and Its Enemies, offered a new single axis spectrum that measures one's view of the future. On one extreme are those who allegedly fear the future and wish to control it, whom Postrel calls stasists. On the other hand are those who want the future to unfold naturally and without attempts to plan and control, for whom she uses the name dynamists.

Other axes that might merit consideration include:

• Role of the church: Clericism vs. Anti-clericism. This axis is not significant in the United States where views of the role of religion tend to get subsumed into the general left-right axis, but in Europe clericism versus anti-clericism is much less correlated with the left-right spectrum.
• Urban vs. rural: This axis is also much more significant in European politics than American.
• Foreign policy: interventionism (the nation should exert power abroad to implement its policy) vs. isolationism (the nation should keep to its own affairs)
• Market policy: socialism (government should democratize or control economic productivity) vs. laissez-faire (government should leave the market alone) vs. corporatism (government should subsidize or support existing successful businesses)
• Political violence: pacifism (political views should not be imposed by violent force) vs. militancy (violence is a legitimate or necessary means of political expression). Informally, these people are often referred to as "doves" and "hawks", respectively.
• Foreign trade: globalization (world economic markets should become integrated and interdependent) vs. autarky (the nation or polity should strive for economic independence)
• Diversity: multiculturalism (the nation should represent a diversity of cultural ideas) vs. assimilationism or nationalism (the nation should represent the dominant ethnic group)
• Participation: Positive Liberty (positive participation in the government) vs. (rule by a limited number of people)

Multi-axis models

A number of proposals have been made for a two-axis system, which combines two models of the political spectrum as axes.

The Nolan Chart

The first person to devise such a two-axis system was David Nolan, creator of the Nolan Chart. This chart shows economic freedom (taxation, free trade and free enterprise) on the x axis and personal freedom (issues like drug legalization, abortion and the draft) on the y axis. This puts liberals in the top left quadrant, libertarianss in the top right, conservativess in the bottom right, and authoritarianss (whom Nolan originally named populists in the bottom left.

This has the interesting effect that the traditional left-right spectrum forms a diagonal across the plane, with communism and fascism both in the ultra-authoritarian corner of the plane.

The Nolan Chart has been reoriented and visually represented in many forms since David Nolan first created it, with the various representations all combining an axis for economic freedom with an axis for personal freedom. It has been the inspiration for an endless array of political self-quizzes, many of which are available over the Internet.

A second, very different, two axis model was created by Jerry Pournelle. Pournelle's model has liberty (a dimension similar to the diagonal of the Nolan Chart, with those on the left seeking liberty and those on the right focusing control) perpendicular to belief in the power of one's political philosophy of choice (with those on the top believing that all the evils their ideology attempts to fight would go away if only their ideals were instituted, and those at the bottom reduced to blind, celebratory attachment to their ideology for its own sake -- the fascist who will now do anything to celebrate "greatness", the anarchist given to tossing bombs around for the fun of it).

Having three axes is a modified Nolan Chart created by the Friesian Institute. It combines the economic liberty and personal liberty axes with positive liberty, creating a cube showing the form of government crossed with the four corners of the Nolan Chart. Another three-dimensional representation is the Vosem Chart, the axes of which represent cultural issues, fiscal issues, and corporate issues.

See also:

• political models
• Nolan chart

External links

• Advocates for Self-Government
• Alternate Spectrum - American Federalist Journal
• World's Smallest Political Quiz
• The Political Compass
• The Pournelle Political Axes - All Ends of the Spectrum
• Friesian Institute

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Political spectrum."

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Power spectrum

(From Wikipedia, the free Encyclopedia)

The power spectrum is a plot of the portion of a signal's power (energy per unit time) falling within given frequency bins. As opposed to the frequency spectrum, the power spectrum does not show spatial or phase angle information.

The most common way of generating a power spectrum is by using a Fourier transform and taking the magnitude of the complex coefficients. Other techniques such as the maximum entropy method can also be used to calculate the power spectrum.

In acoustics, the spectral centroid of a sound is the midpoint of its spectral energy distribution, i.e. the frequency that divides the distribution into two parts of equal energy.

Information about the spectrum during time is called the spectrogram.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Power spectrum."

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Spectroscopy

(From Wikipedia, the free Encyclopedia)

Originally a spectrum was what was observed when white light was dispersed through a prism. Soon the term referred to a plot of light intensity as a function of frequency or wavelength. Planck later realized that frequency represents electromagnetic energy.

E = h ν

Today the term has been generalized even further to include any data that are the result of a study where an energy (or frequency) is systematically varied

The recording and studying of spectrum of energy levels in atoms or molecules in the physical sciences is called spectroscopy. A device for recording a spectrum is a spectrometer or spectrophotometer. The latter term is used when an optical spectrum is recorded by the device.

Types of spectroscopy

Energy of photons

• Circular dichroism
• electromagnetic spectroscopy - The study electomagnetic radiation spectra given off or absorbed by atoms or molecules changing energy levels.
• Atomic absorption spectroscopy
• infra-red spectroscopy - The study of spectra showing infra-red radiation absorbed by atoms or molecules making them vibrate.
• UV/VIS spectroscopy
• Magnetic circular dichroism
• Mossbauer spectroscopy - Measures the absorption of gamma-rays by atoms bound in a solid as a function of gamma-ray energy.
• Nuclear magnetic resonance - Measures the resonant absorption of RF radiation by nuclei in a strong magnetic field. Absorption peaks correspond to transitions in the nuclear spin states of the nuclei.
• Electron spin resonance spectroscopy
• Raman spectroscopy The study of spectra caused by the scattering and change in frequency of light due to the transition between vibrational/rotational energy levels in molecules.
• Stark spectroscopy
• Fluorescence spectroscopy
• X-ray fluorescence
• Stellar Spectroscopy

kinetic energy of electrons

• X-ray photoelectron spectroscopy
• Electron energy loss spectroscopy
• Auger electron spectroscopy - The analysis of the energies of the stimulated emission of Auger electrons

kinetic energy of ions or molecules

• mass spectrometry - The study of the mass of molecules or atoms, measured by how much they bend as they are exposed to a magnetic field

vibrational energy

• Acoustic spectroscopy
• Dynamic mechanical spectroscopy
• Dielectric spectroscopy

Other Topics

• Rotational spectroscopy
• Vibrational spectroscopy

• Fourier transform spectroscopy - An efficient method for collecting various spectra. Frequently applied to infra-red spectroscopy (FTIR) and Nuclear Magnetic Resonance (NMR) Spectroscopy.
• Force spectroscopy - An AFM-based analytical technique

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Spectroscopy."

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Spectrum

(From Wikipedia, the free Encyclopedia)

A spectrum is a usually 2-dimensional plot, of a compound signal, depicting the components by another measure. Sometimes, it is used to refer to the compound signal itself, such as the "spectrum of visible light", a reference to those electromagnetic waves which are visible to the human eye.

The spectrogram is the result of calculating the frequency spectrum of windowed frames of the signal.

There are many specific meanings of spectrum:

In Sports

• The Spectrum was the sports arena in Philadelphia where the Philadelphia Flyers Hockey team and the Philadelphia 76ers Basketball team played untill the First Union Center was built; it is still located, used largely by the Philadelphia Phantoms, in the sports complex on Broad Street in Philadelphia.

In Mathematics

• The frequency spectrum is the result of a frequency transform of a mathematical function into the frequency domain.
• The spectrum of an operator has to do with the invertibility of an operator in function spaces.
• The spectrum of a matrix is basically the spectrum of an operator where the matrix is considered as operator. Precisely, it is the set of the matrix's eigenvalues.
• The spectrum of a ring is the set of prime ideals of a ring.
• A (strange) construction, similar to the frequency spectrum, is the cepstrum of the quefrency.

In Music

• Spectra is one of the determinants of the timbre or quality of a sound. It is the relative strength of pitches called harmonics and partials (collectively overtones) at various frequencies usually above the fundamental frequency, which is the actual note named (eg. an A).

In Physics

• The electromagnetic spectrum is the power spectrum of an electromagnetic signal - see: electromagnetic spectrum, spectroscopy
• The optical spectrum is the electromagnetic spectrum of visual light
• The power spectrum is the distribution of the energy of a function in the frequency domain, which is actually the same as the magnitude of the frequency spectrum. See spectroscopy.

In Pharmacology

• The spectrum of activity of an antibiotic evaluates how wide a range of infections can be treated.

In Politics

• There is a political spectrum which is said to go from left to right.

In Psychology

• There exists the concept of a bipolar spectrum.

In Telecommunication

• The term spread spectrum is used for certain kinds of signal transmission.

Other meanings

• Spectrum is the name of the defence force of Earth in the science fiction television series Captain Scarlet.

• The ZX Spectrum was also the name of a 1980s personal computer see ZX Spectrum.

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Spectrum."

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Spectrum of a ring

(From Wikipedia, the free Encyclopedia)

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with a topology, the Zariski topology, and with a structure sheaf, turning it into a locally ringed space.

Zariski topology

Spec(R) can be turned into a topological space as follows: a subset V of Spec(R) is closed if and only if there exists a subset I of R such that V consists of all those prime ideals in R that contain I. This is called the Zariski topology on Spec(R).

Spec(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. Spec(R) is always a T₀ space, however.

Sheaves and schemes

To every open set U of Spec(R), one may assign a commutative ring R_U in the following way: let S be the complement of the union of all the prime ideals in U. Then S is a multiplicative set and we define R_U as the ring of quotients of R with respect to S. This endows Spec(R) with a sheaf of rings O. If P is an element of Spec(R), then the stalk O_P at P of this sheaf is equal to the localization of R at P, which is a local ring. Thus Spec(R) is a locally ringed space.

Every sheaf of rings of this form is called an affine scheme. General schemes are obtained by "gluing together" several affine schemes.

Functoriality

It is useful to use the language of category theory and observe that Spec is a functor. Every ring homomorphism f : R → S induces a continuous map Spec(f) : Spec(S) → Spec(R) (since the preimage of any prime ideal in S is a prime ideal in R). In this way, Spec can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover for every prime P the homomorphism f descends to homomorphisms

O_f ^-1(P) → O_P,

of local rings. Thus Spec even defines a contravariant functor from the category of commutative rings to the category of locally ringed spaces. In fact it is the universal such functor and this can be used to define the functor Spec up to natural isomorphism.

The functor Spec yields a contravariant equivalence between the category of commutative rings and the category of affine schemes.

Example

A special but quite typical case of an affine scheme is obtained as follows. Take a field K and n variables, x₁,...,x_n. Given m polynomials, p₁,...,p_m in these variables over K, there is a functor F from the category of commutative K-algebras to sets characterized by F(A)={(x₁,...,x_n) in Aⁿ|p₁=...=p_m=0}. Then F is represented by Spec(B) where B is the quotient of K[x₁,...,x_n] by the ideal I generated by the p_j.

Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kⁿ (where K is an algebraically closed field) which are defined as the common zeros of a set of polynomials in n variables. If A is such an algebraic set, one considers the commutative ring R of all polynomial functions A → K. The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals or R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets).

The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points of A are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of A, i.e. the maximal ideals in R, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets).

One can thus view the topological space Spec(R) as an "enrichment" of the topological space A (with Zariski topology): for every subvariety of A, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the "generic point" for the subvariety. Furthermore, the sheaf on Spec(R) and the sheaf of polynomial functions on A are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemess.

External links:

• Kevin R. Coombes: The Spectrum of a Ring, http://odin.mdacc.tmc.edu/~krc/agathos/spec.html

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Spectrum of a ring."

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Spread spectrum

(From Wikipedia, the free Encyclopedia)

Spread-spectrum telecommunications is a technique in which a signal is transmitted in a bandwidth considerably greater than the frequency content of the original information.

Note: Frequency hopping, direct sequence PN spreading, time scrambling, chirp, and combinations of these techniques are forms of spread spectrum. Ultra Wideband (UWB) is another modulation technique that accomplishes much the same purpose, based on transmitting short duration pulses.

Frequency-hopped spread spectrum was invented by actress Hedy Lamarr and musician George Antheil. U.S patent 2,292,387 was awarded in 1942. They proposed using punched paper rolls (like those in player pianos familiar to George Antheil) to coordinate the frequency shifts of sender and receiver[1].

Spread-spectrum telecommunications is a signal structuring technique that employs direct sequence, frequency hopping or a hybrid of these, which can be used for multiple access and/or multiple functions. This technique decreases the potential interference to other receivers while achieving privacy and increasing the immunity of spread spectrum receivers to noise and interference. Spread spectrum generally makes use of a sequential noise-like signal structure to spread the normally narrowband information signal over a relatively wide band of frequencies. The receiver correlates the signals to retrieve the original information signal.

Source: from Federal Standard 1037C and from the NTIA Manual of Regulations and Procedures for Federal Radio Frequency Management and from MIL-STD-188 and from the National Information Systems Security Glossary

See also: open spectrum

Spread-spectrum clock generation

Spread-spectrum clock generation (SSCG) is used in the design of synchronous digital systems, especially those containing microprocessors, to reduce the spectral density of the electromagnetic interference (EMI) that these systems generate. A synchronous digital system is one that is driven by a clock signal that, because of its periodic nature, has an unavoidably narrow frequency spectrum. In fact, a perfect clock signal would have all its energy concentrated at a single frequency and its harmonics, and would therefore radiate energy with an infinite spectral density. Practical synchronous digital systems radiate electromagnetic energy in a number of narrow bands at the clock frequency and its harmonics, resulting in a frequency spectrum that, at certain frequencies, can exceed the regulatory limits for electromagnetic interference (e.g. those of the FCC in the United States, JEITA in Japan and the IEC in Europe).

To avoid this problem, which is of great commercial importance to manufacturers, spread-spectrum clocking is used. This consists of modulating the frequency of the clock signal by either a regular function such as a triangular wave, or by a pseudo-random function. This method distributes the energy of the clock signal over a wider frequency range, and so reduces its peak spectral density. The technique therefore reshapes the system's electromagnetic emissions to make them comply with the electromagnetic compatibility (EMC) regulations. It is a popular technique because it can be used to gain regulatory approval with only a simple modification to the equipment.

It is important to note that this method does not reduce the peak electrical or magnetic field strength emitted by the system, nor the total energy, and therefore does not make the system any less likely to interfere with sensitive equipment such as TV and radio receivers. It works because the EMI receivers used by EMC testing laboratories divide the electromagnetic spectrum into frequency bands approximately 120 kHz wide. If the system under test were to radiate all of its energy at one frequency, then this energy would fall into a single frequency band of the receiver, which would register a large peak at that frequency. Spread-spectrum clocking distributes the energy so that it falls into a large number of the receiver's frequency bands, without putting enough energy into any one band to exceed the statutory limits.

See also :electromagnetic compatibility (EMC) and electromagnetic interference (EMI), optimum utilisation of spectrum.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Spread spectrum."

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ZX Spectrum

(From Wikipedia, the free Encyclopedia)

The Sinclair ZX Spectrum was a small home computer released in the United Kingdom in 1982 by Sinclair Research. Based on a Zilog Z80 CPU running at 3.54 MHz, the Spectrum came with either 16KB or 48KB of RAM (an expansion pack was also available to upgrade the former). Video output was to a TV, for a simple colour graphic display. The rubber keyboard (on top of a membrane, similar to calculator keys) was marked with Sinclair BASIC programming language keywords, so that pressing, say, "G" when in programming mode would insert the BASIC command GOTO. Programs and data were stored using a normal cassette recorder.

The Spectrum's video display, although rudimentary by today's standards, was perfect at the time for display on portable TV sets, and didn't present a much of a barrier to game development. The text mode display was 32 columns × 24 rows, with a choice of 8 colours in either normal or bright mode, which gave 16 shades. The graphics resolution was 256×192 with the same colour limitations. The Spectrum had an interesting method of handling colour; the colour attributes were held in a 32×24 grid, separate from the text or graphical data, but was still limited to only two colours in any given character cell. This led to what was called colour clash or attribute clash with some bizarre effects in arcade style games.

Retailing for £125125 for the 16KB and £175 for the 48KB model, the Spectrum was the first mainstream audience home computer in the UK, similar in significance to the Commodore 64 in the USA (the C64 also being the main rival to the Spectrum in the UK market). A slightly modified version of the Spectrum, in a silver colored case with hard plastic keys, was marketed in the USA by Timex as the TS2068. It had an extra 8K extension ROM and a cartridge port, as well as two joystick ports and a AY-3-8912 sound chip with extra Sinclair BASIC commands to support these devices (STICK and SOUND, respectively).

Several peripherals for the Spectrum were marketed by Sinclair: the printer was already on the market, as the Spectrum had retained the protocol for the ZX81's printer. The Interface 1 added a standard RS-232 serial port, a proprietary format local area networking port, and the ability to connect up to eight ZX Microdrives – somewhat unreliable but speedy tape-loop storage devices that was later used in a revised version on the Sinclair QL (the QL's Microdrive data storage format was electrically compatible but logically incompatible with the Spectrum's). Sinclair also released the Interface 2 which added two joystick ports and a ROM cartridge port.

There were also a plethora of third-party hardware addons. These more well-known of these included the Kempston joystick interface, the Currah Microspeech unit (speech synthesis), and the Multiface (snapshot and disassembly tool), from Romantic Robot. There were numerous disk drive interfaces, including the Opus Discovery and the Disciple/Plus D from Miles Gordon Technology. During the mid-80s, the company Micronet800 launched a service allowing users to connect their ZX Spectrums to a network known as Prestel. This service had some similarities to the Internet, but was proprietary and fee-based.

A number of current leading games developers and development companies began their careers on the ZX Spectrum, including Peter Molyneux (ex-Bullfrog Games), Shiny Entertainment, and Ultimate Play The game (now known as Rare, Inc, maker of many famous titles for the Super NES and Nintendo 64 game consoles).

Successor models of the basic Spectrum included the ZX Spectrum+, with an improved keyboard, and the ZX Spectrum 128, with the improved keyboard, three-channel sound, 128KB of RAM, and RGB monitor output. After Amstrad's buyout of Sinclair Research in 1986, two more versions were released: the ZX Spectrum +2 (with a built-in cassette recorder, like the Amstrad CPC 464) and the ZX Spectrum +3 (with a built-in 3-inch floppy disk drive, like the Amstrad CPC 6128). Many Spectrum clones were produced, especially in Eastern Europe and South America. Some of them are still being produced such as Didaktik and the Sprinter from Peters Plus Ltd. A Russian clone of the ZX Spectrum is the Pentagon.

See also: Sinclair ZX80, ZX81, SAM Coupe, ZX Spectrum Demos, History of computing hardware II.

External links

• World of Spectrum
• Jasper - An online spectrum emulator written in Java

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "ZX Spectrum."

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Abbreviations & Acronyms: Spectrum

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Commercial Usage: Spectrum

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Image Slideshow: Spectrum

Photos: Spectrum More pictures...

Illustrations: Spectrum More pictures...

Computer Images: Spectrum More pictures...

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Photo Album: Spectrum

Thumbnail	Description & Credit	Thumbnail	Description & Credit
	Photodynamic therapy (PDT) is a procedure to treat cancer. Patients are injected with a photosensitizer which is a light sensitive drug selectively retained by cancer cells. When exposed to laser light, the photosensitizer in the cancer cells produces a toxic reaction which destroys the tumor. This photo shows an argon-ion laser, the first component of the argon pumped-dye laser (630nm red). This argon-ion laser emits blue-green light at 488/514 nm, and is used to excite a dye in the second component, the dye laser head, where the wavelength is changed to 630nm red. Different photosensitizers absorb light at different wavelengths. Some absorb light most efficiently in the blue light region of the spectrum around 400 nanometers(nm) with lesser absorption in the green and red light range. However, red light at 630 nm penetrates deeper into the tumor tissue (3-8 mm) than green or blue light. For this reason, the majority of PDT work has used 630 nm light. See artwork: GA-17. Credit: Unknown photographer/artist.		Arc with reflection higher in the sky - note reversal of spectrum. Credit: Paths Less Taken - NOAA at the Ends of the Earth.
	Arc with spectacular reflection higher in the sky - note reversal of spectrum. Credit: Paths Less Taken - NOAA at the Ends of the Earth.		Rainbow with reflection - note reversal of spectrum. Credit: National Severe Storms Laboratory (NSSL).
	Figure 45. Quartz spectrograph, meant to photographically meausure the spectrum of various materials under analysis. This instrument was constructed by the Paris firm of Jobin and Yvon in 1901. Several of these instruments were made by the engineer Amedee Jobin. Credit: Sailing for Science - the NOAA Fleet Then and Now.
Source: pictures compiled by the editor from various references; see picture credits.

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Digital Photo Gallery: Spectrum
 

"Spectrum Glasgow" by Craig Young Commentary: "This is Spectrum house which is a glass building with a silver bulge fitted to one end. Not sure why though."

Source: photographs selected by the editor, with permission from the photographers.

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Familiar Quotations: Spectrum

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Non-Fiction Usage: Spectrum

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Usage Frequency: Spectrum

Source: compiled by the editor from several corpora; see credits.

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Usage in Company Names: Spectrum

Source: compiled by the editor from Icon Group International, Inc.

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Expression: Spectrum

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Frequency of Internet Keywords: Spectrum