In mathematics , particularly linear algebra and functional analysis , the spectraltheorem is any of a number of results about linear operator s or about matrix mathematics matrices . In broad terms the spectraltheorem provides conditions under which an Operator mathematics operator or a matrix can ... s, which are as simple as one can hope to find. In more abstract language, the spectraltheorem is a statement ... of operators to which the spectraltheorem applies are self adjoint operator s or more generally normal operator s on Hilbert space s. The spectraltheorem also provides a Canonical Mathematics .... In this article we consider mainly the simplest kind of spectraltheorem, that for a self adjoint operator on a Hilbert space. However, as noted above, the spectraltheorem also holds for normal operators ... the proof. The spectraltheorem holds also for symmetric matrices on finite dimensional real ... matrices The spectraltheorem extends to a more general class of matrices. Let A be an operator on a finite ... spaces in general, the statement of the spectraltheorem for compact operator compact self adjoint ... characterization of eigenvalues. The above spectraltheorem holds for real or complex Hilbert spaces ... eigenvectors. Bounded self adjoint operators seealso Eigenfunction Self adjoint operator Spectraltheorem ... . There is also an analogous spectraltheorem for bounded normal operator s on Hilbert spaces. The only ... formulation of the spectraltheorem expresses the operator math A math as an integral of the coordinate ... of the spectraltheorem reduces to the finite dimensional spectraltheorem above, except that the operator ... operators , are unbounded. There is also a spectraltheorem for self adjoint operator ... decomposition , a generalisation of spectraltheorem to arbitrary matrices. Eigendecomposition of a matrix ... , http www.jstor.org stable 2313117 What Does the SpectralTheorem Say? , American Mathematical Monthly ..., but requires some modification for operators on infinite dimensional spaces. In general, the spectral ... more details
In mathematics and physics , the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalue s of an Operator mathematics operator . In mathematics, the spectral asymmetry arises in the study of elliptic operator s on compact manifold s, and is given a deep meaning by the Atiyah Singer index theorem . In physics, it has numerous applications, typically resulting in a fractional charge physics charge due to the asymmetry of the spectrum of a Dirac operator . For example, the vacuum expectation value of the baryon number is given by the spectral asymmetry of the Hamiltonian operator . The spectral asymmetry of the confined quark fields is an important property of the chiral bag model . Definition Given an operator with eigenvalue s math omega n math , an equal number of which are positive and negative, the spectral asymmetry may be defined as the sum math B lim t to 0 frac 1 2 sum n sgn omega n exp t omega n math where math sgn x math is the sign function . Other regularization physics regulator s, such as the zeta function regulator , may be used. The need for both a positive and negative spectrum in the definition is why the spectral asymmetry usually occurs in the study of Dirac operator s. Example As an example, consider an operator with a spectrum math omega n n theta math where n is an integer, ranging over all positive and negative values. One may show in a straightforward manner that the spectral asymmetry in this case is math B theta math . Discussion Related to the spectral asymmetry is the vacuum expectation value of the energy associated with the operator, the Casimir energy , which is given by math E lim t to 0 frac 1 2 sum n omega n exp t omega n math This sum is formally divergent, and the divergences must be accounted for and removed using standard regularization techniques. References MF Atiyah, VK Patodi and IM Singer, Spectral ... 1990 p. 109 143. Category Spectral theory ... more details
refimprove date May 2008 Mergefrom Spectral energy distribution discuss Talk Spectral energy distribution date September 2010 In statistical signal processing and physics , the spectral density , power spectral density PSD , or energy spectral density ESD , is a positive real function of a frequency .... Intuitively, the spectral density captures the frequency content of a stochastic process and helps ... wave , random vibration , or an sound wave acoustic wave . The spectral density of the wave ..., per unit frequency, known as the power spectral density PSD of the signal. Power spectral density is commonly ... UC&pg PR20&dq 22power spectral density 22 W Hz&lr &as brr 0&ei VYwvSImyA4L4sQPxxJXzAg&sig bko0DhmJwzISN6PcHszF9E3qUE ...&pg PA352&dq 22power spectral density 22 22energy spectral density 22&lr &as brr 3&ei i3IvSLL6H4 ... force g sup 2 sup Hz sup 1 sup are sometimes used for acceleration spectral density. ref cite book ... page 83 url http books.google.com books?id xPIW3AI9tdAC&pg PA83&dq acceleration spectral density ... varies in time. Definition Energy spectral density Anchor energy spectral density The energy spectral density describes how the Energy signal processing energy or variance of a signal or a time ... integrable signal, the spectral density math Phi omega math of the signal is the square of the magnitude ... with values math f n math , over an infinite number of elements, we still have an energy spectral density ... of defined values is finite, the sequence does not have an energy spectral density per se ... spectrum, or it can be extended with zeros and a spectral density can be computed as in the infinite sequence case. The continuous and discrete spectral densities are often denoted with the same symbols .... Power spectral density The above definitions of energy spectral density require that the Fourier ... integrable. In this case one would need to use the Plancherel theorem . An often more useful alternative is the power spectral density PSD , which describes how the power physics power of a signal ... more details
In noncommutative geometry and related branches of mathematics and mathematical physics , a spectral triple is a set of data which encodes geometric phenomenon in an analytic way. The definition typically involves a Hilbert space , an algebra ring theory algebra of operators on it and an unbounded self adjoint operator, endowed with supplemental structures. It was conceived by Alain Connes who was motivated by the Atiyah Singer index theorem and sought its extension to noncommutative spaces. Some authors refer to this notion as unbounded K homology K cycles or as unbounded Fredholm modules . Motivation A motivating example of spectral triple is given by the algebra of functions on a compact spin ... the smooth structure, which is Connes reconstruction theorem, but I think this result is too recent ... part of the index theorem , while the geometric local side involves the Dixmier trace and commutators ... of the index theorem . Extensions of the index theorem can be considered in cases, typically when one .... Definition An odd spectral triple is a triple A, H, D consisting of a Hilbert space H, an algebra ... D satisfying nowiki a, D &infin nowiki for any a &isin A. An even spectral triple is an odd spectral ... to this grading. One could also say that an even spectral triple is given by a quartet A, H, D, &gamma ... &gamma &gamma D. A finitely summable spectral triple is a spectral triple A, H, D such that a.D ... H . When this condition is satisfied, the triple A, H, D is said to be p summable . A spectral triple ... connes94 Let &delta T denote the commutator of D with an operator T on H. A spectral triple is said ... of the iterates &delta sup n sup of &delta . When a spectral triple A, H, D is p summable ... the dimension spectrum of A, H, D . A real spectral triple is a spectral triple A, H, D accompanied ... Given a spectral triple A, H, D , one can apply several important operations to it. The most fundamental ... from K sub 1 sub A to Z . When the spectral triple is finitely summable, one may write the above ... more details
Spectral efficiency , spectrum efficiency or bandwidth efficiency refers to the information rate that can ... protocol, and sometimes by the media access control the channel access protocol . Link spectral efficiency The link spectral efficiency of a digital communication system is measured in bit second ... or a data link . Alternatively, the spectral efficiency may be measured in in bit symbol , which is equivalent ... modulation rate or line code pulse rate. Link spectral efficiency is typically used to analyse the efficiency ... of 3,400 300 3,100 hertz. The spectral efficiency or modulation efficiency is 56,000 3,100 18.1 ... can not exceed 2 B symbols s in view to avoid intersymbol interference . Thus, the spectral efficiency ... 4 bit symbol or bpcu. Since QAM is a form of double sideband passband transmission, the spectral ... signal before passband filtering can be considered as baseband transmission, the spectral efficiency ..., a higher spectral efficiency is achieved if we consider the smaller passband bandwidth. If a forward error correction code is used, the spectral efficiency is reduced from the uncoded modulation ... that the encoder input bit rate is one half the encoder output rate, the spectral efficiency is 50 of the modulation efficiency. In exchange for this reduction in spectral efficiency, FEC usually ... . An upper bound for the spectral efficiency possible without bit error s in a channel with a certain SNR, if ideal error coding and modulation is assumed, is given by the Shannon Hartley theorem . Example 7 If the SNR is 1 times expressed as a ratio, corresponding to 0 decibel , the link spectral efficiency ... is not already efficiently compressed. The link spectral efficiency of a wireless telephony link ..., the link spectral efficiency can be somewhat misleading, as larger values are not necessarily more efficient in their overall use of radio spectrum. In a wireless network, high link spectral ... spreading and forward error correction reduce the spectral efficiency in bit s Hz but substantially ... more details
In mathematics , the spectral radius of a matrix mathematics matrix or a bounded linear operator is the supremum ... number complex eigenvalues of a matrix A C sup n n sup . Then its spectral radius A is defined as math rho A max i lambda i math The following lemma shows a simple yet useful upper bound for the spectral radius of a matrix Lemma Let A C sup n n sup be a complex valued matrix, A its spectral ... math rho A leq A k 1 k , , square math The spectral radius is closely related to the behaviour of the convergence of the power sequence of a matrix namely, the following theorem holds Theorem Let A C sup n n sup be a complex valued matrix and A its spectral radius then math lim k to infty A k ... 0 right math From the Jordan normal form theorem, we know that for any complex valued matrix math .... math square math Theorem Gelfand s formula, 1941 For any matrix norm , we have math rho A lim k to infty A k 1 k . math In other words, the Gelfand s formula shows how the spectral radius of A gives ..., math rho tilde A frac rho A rho A epsilon 1 math and, by the previous theorem, math lim ... rho check A frac rho A rho A epsilon 1 math and so, by the previous theorem, math check A k math ... 1 k rho A . , , square math Gelfand s formula leads directly to a bound on the spectral radius of a product ... definition the left lower bound with the spectral radius itself and write more precisely math ... by definition, its spectral radius is A 10. In the following table, the values of math A k 1 k math ... spectral radius coincides with its numerical radius . An example of such an operator is a normal operator . Graphs The spectral radius of a finite graph mathematics graph is defined to be the spectral ... v sum u,v in E G f u math . The spectral radius of G is defined to be the spectral radius of the bounded linear operator math gamma math . See also Spectral gap The Joint spectral radius is a generalization of the spectral radius to sets of matrices. Category Spectral theory Category Articles containing ... more details
Merge Spectral element method date October 2009 Spectral methods are a class of techniques used in applied ... s PDEs , often involving the use of the Fast Fourier Transform . Where applicable, spectral methods .... Spectral methods were developed in a long series of papers by Steven Orszag starting in 1969 including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral ... problems, and spectral iteration methods for fast solution of steady state problems. PDEs describe ... solutions to these PDEs. In a typical case, spectral methods take advantage of this fact ... ODEs. The spectral method and the finite element method are closely related and built on the same ideas the main difference between them is that the spectral method approximates the solution as linear ... as a linear combination of piecewise functions that are nonzero on small subdomains. Because of this, the spectral ... . This is part of why the spectral method works best when the solution is Smooth function smooth . In fact there are no known three dimensional single domain spectral shock capturing results. ref name CHQZ http books.google.com books?id 7COgEw5 EBQC pp 235, Spectral Methods evolution to complex geometries ... as the grid parameter h decreases to zero is sometimes called a spectral element method . The implementation of the spectral method is normally accomplished either with collocation method collocation or a Galerkin method Galerkin approach. Examples of spectral methods A concrete, linear example ... has a continuous second derivative. By the uniqueness theorem for Fourier expansions, we must then equate ... example We wish to solve the forced, transient, nonlinear Burgers equation using a spectral approach ... with the spectral element method One can show that if math g math is infinitely differentiable ... than math Ch n math for all sufficiently small values of math h math . We say that the spectral method is of order math n math , for every n 0. Because a spectral element method is a finite element ... more details
In telecommunications , spectral component is any of the wave s that range outside the interval of frequency frequencies assigned to a signal. Any waveform can be disassembled into its spectral components by Fourier analysis or Fourier transformation . The length of a pulse thereby is determined by its complex spectral components, which include not just their relative intensities, but also the relative positions spectral phase of these spectral components. See also Spectral width Bandwidth limited pulse Category Telecommunications engineering Telecomm term stub ... more details
the spectraltheorem establishes conditions under which an operator can be expressed in simple ... radius , Spectrum of an operator , Spectraltheorem Self adjoint operator , functional calculus Functions ...In mathematics , spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue ... A short course on spectral theory author William Arveson chapter Chapter 1 spectral theory and Banach ... ref The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. ref name Sadovnichi cite book ... space the spectral theory of operators url http books.google.com books?id SR1QkG6OkVEC&pg PA181 ... The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic form s in infinitely many variables. The original spectraltheorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite dimensional setting. The later discovery in quantum mechanics that spectral theory ... three main ways to formulate spectral theory, all of which retain their usefulness. After Hilbert s initial formulation, the later development of abstract Hilbert space and the spectral theory of a single ... . The Fourier transform on the real line is in one sense the spectral theory of derivative differentiation ... properties, and this is carried out by means of Pontryagin duality . One can also study the spectral ... spectral properties similar to that of Matrix mathematics matrices . Physical background The background ... pse maths research analysis research.html title Research at the analysis group ref Cquote Spectral ... Mawhin, p.4 and pp. 10 11. ref The later discovery in quantum mechanics that spectral theory could ... s spectral theory. A definition of spectrum main Spectrum functional analysis Consider a bounded ... of T in the complex plane. ref name Lorch Cite book title Spectral Theory author Edgar Raymond Lorch ... more details
orphan date December 2010 unreferenced date December 2010 Spectral purity is a quantification of the monochromaticity of a given light sample. This is a particularly important parameter in areas like laser operation and time measurement. Spectral purity is given by math Spectral purity frac triangle v v math Spectral purity is easier to achieve in devices that generate visible light visible and ultraviolet light, since higher frequency light results in greater spectral purity. Category Spectroscopy ... more details
A spectral envelope is a curve in the frequency amplitude plane, derived from a Fourier magnitude spectrum. It describes one point in time one window, to be precise . nofootnotes date December 2010 one source date December 2010 In remote sensing using a spectrometer , the spectral envelope of a feature is the boundary of its electromagnetic spectrum spectral properties, as defined by the range of brightness levels in each of the spectral bands of interest. References Swartz, Diemo 1998 . Spectral Envelopes . http recherche.ircam.fr anasyn schwarz da specenv 3 3Spectral Envelopes.html . http www.unesco.org csi pub source rs15.htm Category Spectroscopy ... more details
Unreferenced stub auto yes date December 2009 Mergeto Full Spectral Imaging date July 2010 Spectral imaging is a branch of spectroscopy in which a complete spectrum or some spectral information such as the Doppler shift or Zeeman splitting of a spectral line is collected at every location in an image plane . Applications include astronomy , solar physics , analysis of Plasma physics plasmas in nuclear fusion experiments, planetology , and Earth remote sensing . This is often referred to as hyperspectral imaging . See also Dopplergraph Imaging spectrometer DEFAULTSORT Spectral Imaging Category Astronomical spectroscopy Physics stub ... more details
Infobox VG image File Spectral Gene cover.jpg 250px developer Idea Factory publisher Idea Factory designer engine series Spectral released vgrelease JP December 13, 2007 genre Tactical role playing game SRPG modes Single player video game Single player ratings platforms PlayStation 2 Spectral Gene is an Tactical role playing game SRPG for the PlayStation 2 , and is the next installment in Idea Factory s Spectral series. It was developed and published for Japan by Idea Factory and was released on December 13, 2007. It is unknown if it will ever be published outside of Japan. ref http www.gamefaqs.com console ps2 data 941627.html Spectral Gene Info Spectral Gene Information Spectral Gene Release Date Bot generated title ref References Reflist External links ja icon http www.ideaf.co.jp spectral gene index.html The official website of Spectral Gene Category 2007 video games Category Tactical role playing video games Category PlayStation 2 games Category PlayStation 2 only games Category Video games developed in Japan Category Japan exclusive video games tactical rpg videogame stub ... more details
In astronomy , the spectral index of a source is a measure of the dependence of radiative flux density on frequency . Given frequency math nu math and radiative flux math S math , the spectral index math alpha math is given implicitly by math S propto nu alpha. math Note that if flux does not follow a power law in frequency, the spectral index itself is a function of frequency. Rearranging the above, we see that the spectral index is given by math alpha frac partial log S partial log nu . math Spectral index is also sometimes defined in terms of wavelength math lambda math . In this case, the spectral index math alpha math is given implicitly by math S propto lambda n, math and at a given frequency, spectral index may be calculated by taking the derivative math alpha frac partial log S partial log lambda . math The opposite sign convention is sometimes employed, ref Burke, B.F., Graham Smith, F. 2009 . An Introduction to Radio Astronomy, 3rd Ed. , Cambridge University Press, Cambridge, UK, ISBN 9780521878081, page 132. ref in which the spectral index is given by math S propto nu alpha . math The spectral index of a source can hint at its properties. For example, using the positive sign convention, a spectral index of 0 to 2 at radio frequencies indicates thermal emission , while a steep negative spectral index typically indicates synchrotron emission . Spectral Index of Thermal emission At radio frequencies, where the Rayleigh Jeans law is a good approximation to the spectrum of thermal radiation, intensity is given by math B nu T frac 2 nu 2 k T c 2 . math Taking the logarithm ... partial log B nu T partial log nu 2. math Using the positive sign convention, the spectral index of thermal radiation is thus math alpha approx 2 math in the Raleigh Jeans regime. The spectral index ... Jeans regime, the radio spectral index is defined as ref cite web title Radio Spectral Index publisher ... stub es ndice espectral fr Indice spectral it Indice spettrale ... more details
In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalue s of a matrix or operator alternately, it is sometimes taken as the smallest non zero eigenvalue. Various theorems relate this difference to other properties of the system. See Expander graph discrete case Poincar inequality continuous case See also Spectral radius Eigengap disambig Category Mathematical disambiguation Category Spectral theory ... more details
Spectral analysis or Spectrum analysis may refer to Spectrum analysis in chemistry and physics, a method of analyzing the chemical properties of matter from bands in their visible spectrum Spectrum analyzer in signal processing, a device or algorithm that identifies a frequency domain representation of a time domain signal, typically by means of Fourier transform Spectral theory , in mathematics, a theory that extends eigenvalues and eigenvectors to linear operators on Hilbert space, and more generally to the elements of a Banach algebra Spectral analysis in statistics, a procedure that decomposes a time series into a spectrum of cycles of different lengths. Spectral analysis is also known as frequency domain analysis In nuclear and particle physics, gamma spectroscopy , and high energy astronomy, the analysis of the output of a pulse height analyzer for characteristic features such as spectral line , edges, and various physical processes producing continuum shapes disambig ... more details
In mathematics , the spectral abscissa of a matrix mathematics matrix or a bounded linear operator is the supremum among the real part of the elements in its spectrum of a matrix spectrum , sometines denoted as math eta A math Matrices Let sub 1 sub , ..., sub s sub be the real number real or complex number complex eigenvalues of a matrix A C sup n n sup . Then its spectral abscissa is defined as math eta A max i rm Re lambda i , math It is often used as a measure of stability in control theory, where a continuous system is stable iF all its eigenvalues are located in the left half plane, i.e. math eta A 0 math See also Spectral radius DEFAULTSORT Spectral Abscissa Category Spectral theory Category Matrix theory mathematics stub ... more details
Unreferenced stub auto yes date December 2009 File Spectrum of blue sky.svg thumb 300px right Spectrum of blue sky Spectral Signatures are the specific combination of reflected and absorbed electromagnetic radiation electromagnetic EM radiation at varying wavelengths which can uniquely identify an object. The spectral signature of stars indicates the spectrum according to the EM spectrum. The spectral signature of an object is a function of the incidental EM wavelength and material interaction with that section of the electromagnetic spectrum . The measurements can be made with various instruments, including a task specific spectrometer , although the most common method is separation of the Red, Green, Blue and Near Infrared portion of the EM spectrum as acquired by digital cameras. Calibration of spectral signatures under specific illumination are collected in order to apply an empirical correction to airborne or satellite imagery digital images. When using a spectrometer, you will look through it at a tube of gas with high voltages passing through it. In the Spectrometer, you will be able to see specific rays of colour falling on graduations on the inside. Each substance will have its own unique falling of rays. Most remote sensing applications process digital images to extract spectral signature at each pixel and use such signature to divide the image in groups of similar pixels segmentation using different approaches, and, as last step, assign a class to each group classification by comparing with known spectral signatures. Depending on pixel resolution, a pixel can represent many spectral signature mixed together that is why a lot of remote sensing analysis is done to unmix mixtures . Ultimately correct matching of spectral signature recorded by image pixel with spectral signature of existing elements leads to accurate classification in remote sensing. DEFAULTSORT Spectral Signature Category Spectroscopy Sci stub ca Signatura espectral es Firma espectral id Cir ... more details
In telecommunication s, spectral width is the wavelength interval over which the magnitude of all spectral component s is equal to or greater than a specified fraction of the magnitude of the component having the maximum value. In optical Telecommunication communications applications, the usual method of specifying spectral width is the full width at half maximum . This is the same convention used in Bandwidth signal processing bandwidth , defined as the frequency range where power drops by less than half at most &minus 3 dB . The FWHM method may be difficult to apply when the spectrum has a complex shape. Another method of specifying spectral width is a special case of Root mean square root mean square deviation where the independent variable is wavelength, , and f is a suitable radiometric quantity. The relative spectral width , , is frequently used where is obtained according to note 1, and is the center wavelength. See also Spectral linewidth in optics References Refimprove date January 2009 references FS1037C MS188 Category Telecommunications terms telecomm term stub fr Largeur de bande ... more details
Multiple issues orphan April 2010 technical September 2010 Spectral skewing is the term used to describe the phenomenon of the changes in relative intensities of mass spectral peaks due to the changes in concentration of the analyte in the ion source as the mass spectrum is scanned this situation occurs routinely as Chromatography chromatographic components Elution elute into a continuous ion source. Spectral skewing is not observed in ion trap Quadrupole ion trap quadrupole this has been seen also in Quadrupole mass spectrometer QMS or magnetic or Time of flight mass spectrometry time of flight TOF mass analyzer s because potentially all ion s formed in operational cycle a snapshot in time of the instrument are available for detection. References Watson, J. THrock, Sparkman,O David.Introduction to Mass Spectrometry.John Wiley & Sons, Inc. 4th Edition,2007. Page 113 DEFAULTSORT Spectral Skewing Category Mass spectrometry ... more details
Unreferenced date December 2009 The spectral linewidth characterizes the width of a spectral line , such as in the emission spectrum electromagnetic emission spectrum of an atom, or the frequency spectrum of an acoustic or electronic system. For example, the emission of an atom usually has a very thin spectral linewidth, as only transitions between discrete energy levels are allowed, leading to emission of photons with a certain energy. Several definitions are used to quantify the spectral linewidth, e.g. the full width at half maximum FWHM . While the spectral width of a resonator in electronics depends on the parameters of the components, and therefore can be easily adjusted over a wide range, linewidths are typically more difficult to adjust in physics. For example, even a resting atom which does not interact with its environment has a non zero linewidth , called the natural linewidth also called the decay width , which is a consequence of the Fourier transform Localization property Fourier transform limit classical description and the Heisenberg uncertainty principle quantum mechanical description . According to the uncertainty principle the uncertainty in energy, E , of a transition is inversely proportional to the lifetime, t of the excited state math Delta E Delta t gtrapprox frac hbar 2 . math In practice lines are further broadened by effects such as Doppler broadening . See also Bandwidth signal processing , a more generic term for a range of electromagnetic frequencies Q factor and linewidths in electronics, acoustics and optics Spectral line s in optics. This article also includes a list of sources of linewidth broadening. Spectral width in telecommunications Oscillator linewidth Laser linewidth DEFAULTSORT Spectral Linewidth Category Optics de Linienbreite lb Spektral Linnebreet tr Spektral izgi geni li i uk ... more details
Multiple issues orphan August 2008 unreferenced June 2008 context February 2009 Spectral mutability is a desired capability Clarify date August 2008 where RF system s have the appropriate mix of both manual and automatic flexibility and agility to affect all aspects of their operating parameters e.g. waveform , Bandwidth signal processing bandwidth , power, frequency , direction and maximize efficiency of spectral use and maintain optimum compatibility between systems. Category Radio technology electromagnetism stub ... more details
In mathematics , a spectral space is a topological space which is homeomorphic to the Spectrum of a ring spectrum of a commutative ring . Definition Let X be a topological space and let K sup math circ math sup X be the set of all Compact space quasi compact and Open set open subsets of X . Then X is said to be spectral if it satisfies all of the following conditions X is quasi compact and Kolmogorov space T sub 0 sub . K sup math circ math sup X is a basis of open subsets of X . K sup math circ math sup X is Closure mathematics closed under under finite intersections. X is Sober space sober , i.e. every nonempty Hyperconnected space irreducible Closed set closed subset of X has a necessarily unique generic point . Equivalent descriptions Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral X is homeomorphic to a projective limit of finite Kolmogorov space T sub 0 sub space s. X is homeomorphic to the duality theory for distributive ... in the sense of topology this indeed is only another name . Properties Let X be a spectral space and let ... sublattice of subsets of X . Every closed Subspace topology subspace of X is spectral. An arbitrary ... sup X is again spectral. X is Kolmogorov space T sub 0 sub by definition, but in general not T1 space T sub 1 sub . In fact a spectral space is T sub 1 sub if an only if it is Hausdorff space Hausdorff ... lattices and Heyting algebras. Mathematical Structures in Computer Science , 20. ref Spectral maps A spectral map f X Y between spectral spaces X and Y is a continuous map such that the preimage of every open and quasi compact subset of Y under f is again quasi compact. The category of spectral spaces which has spectral maps as morphisms is Equivalence of categories anti equivalent to the category ... 1982 ref In this anti equivalence, a spectral space X corresponds to the lattice K sup math circ math ... locales pages 62 69 year 1982 . Footnotes reflist DEFAULTSORT Spectral Space Category General topology ... more details
Orphan date February 2009 Infobox Company company name Spectral Genomics, Inc. company type Private, acquired by PerkinElmer in 2006 company logo Image SpectralGenomics.jpg 200px PerkinElme logo foundation 2000 location Houston, TX key people Robert Johnson Ph.D., CEO, and President br Keith Steward Ph.D., Director Informatics, br Brandon Perthuis, Director Sales industry Health Sciences products Genetic Screening tools homepage http www.spectralgenomics.com Spectral Genomics, Inc. is a technology technology spin off company from Baylor College of Medicine , selling Array comparative genomic hybridization aCGH microarrays and related software. History The company was founded in February 2000 by BCM technologies . Spectral licensed technology invented by its founders Alan Bradley, Ph.D., Wei wen Cai, Ph.D.. The company raised 3.0 million in the first financing round in August 2001. In March 2004 the company raised additional 9.4 million in its second financing round. In March 2005, GE Healthcare became the exclusive distributor for Spectral Genomics s products outside of North America. Spectral Genomics was acquired by PerkinElmer in May 2006, ending GE s distribution agreement. External links http www.spectralgenomics.com Corporate website Category Technology companies Category Microarrays ... more details
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