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spectrum and any
An artist is a person engaged in one or more of any of a broad spectrum of activities related to creating art, practicing the arts and / or demonstrating an art.
Black is the color of objects that do not emit or reflect light in any part of the visible spectrum ; they absorb all such frequencies of light.
The spectrum of any element x is a closed subset of the closed disc in C with radius || x || and center 0, and thus is compact.
* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.
For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of C as spectrum.
* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).
Spectroscopic studies revealed absorption lines in the Jovian spectrum due to diatomic sulfur ( S < sub > 2 </ sub >) and carbon disulfide ( CS < sub > 2 </ sub >), the first detection of either in Jupiter, and only the second detection of S < sub > 2 </ sub > in any astronomical object.
These are capable of the most severe types of molecular damage, which can happen in biology to any type of biomolecule, including mutation and cancer, and often at great depths from the skin, since the higher end of the X-ray spectrum, and all of the gamma ray spectrum, are penetrating to matter.
Fascists have commonly opposed having a firm association with any section of the left-right spectrum, considering it inadequate to describe their beliefs, though fascism's goal to promote the rule of people deemed innately superior while seeking to purge society of people deemed innately inferior is identified as a prominent far-right theme.
Individuals were categorized according to their so-called " rejection spectrum " which allowed doctors to counter any immune system responses to the new organs, allowing transplants to " take " for life.
We can also show that the possible values of the observable A in any state must belong to the spectrum of A.
In linguistics the term orthography is often used to refer to any method of writing a language, without judgment as to right and wrong, with a scientific understanding that orthographic standardization exists on a spectrum of strength of convention.
Pacifism covers a spectrum of views ranging from the belief that international disputes can and should be peacefully resolved ; to calls for the abolition of the institutions of the military and war ; to opposition to any organization of society through governmental force ( anarchist or libertarian pacifism ); to rejection of the use of physical violence to obtain political, economic or social goals ; to opposition to violence under any circumstance, including defense of self and others.
Pacifism covers a spectrum of views, including the belief that international disputes can and should be peacefully resolved, calls for the abolition of the institutions of the military and war, opposition to any organization of society through governmental force ( anarchist or libertarian pacifism ), rejection of the use of physical violence to obtain political, economic or social goals, the obliteration of force except in cases where it is absolutely necessary to advance the cause of peace, and opposition to violence under any circumstance, even defense of self and others.
If the surface has any transparent or translucent properties, it refracts a portion of the light beam into itself in a different direction while absorbing some ( or all ) of the spectrum ( and possibly altering the color ).
Fundamental strings exist in 9 dimensions and the strings can vibrate in any direction, meaning that the spectrum of vibrational modes is much richer.
As with any religious movement, a theological spectrum exists within Adventism comparable to the fundamentalist-moderate-liberal spectrum in the wider Christian church and in other religions.
In general, any particular instrument will operate over a small portion of this total range because of the different techniques used to measure different portions of the spectrum.

spectrum and bounded
When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.
In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices.
Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator.
In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.
In this case a complex number λ is said to be in the spectrum of such an operator T: D → X ( where D is dense in X ) if there is no bounded inverse ( λI − T )< sup >− 1 </ sup >: X → D.
The spectrum of a bounded linear operator T acting on a Banach space X is the set of complex numbers λ such that λI − T does not have an inverse that is a bounded linear operator.
The spectrum of a bounded operator T is always a closed, bounded and non-empty subset of the complex plane.
The boundedness of the spectrum follows from the Neumann series expansion in λ ; the spectrum σ ( T ) is bounded by || T ||.
So the spectrum includes the set of approximate eigenvalues, which are those λ such that is not bounded below ; equivalently, it is the set of λ for which there is a sequence of unit vectors x < sub > 1 </ sub >, x < sub > 2 </ sub >, ... for which
It can be shown that, in general, the approximate point spectrum of a bounded multiplication operator is its spectrum.
One can classify the spectrum in exactly the same way as in the bounded case.
For λ to be in the resolvent ( i. e. not in the spectrum ), as in the bounded case λI − T must be bijective, since it must have a two-sided inverse.
Therefore, as in the bounded case, a complex number λ lies in the spectrum of a closed operator T if and only if λI − T is not bijective.
* Positive element of a C *- algebra ( such as a bounded linear operator ) whose spectrum consists of positive real numbers
In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ (·).
The Koszul complex is essential in defining the joint spectrum of a tuple of bounded linear operators in a Banach space.

spectrum and symmetric
Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all ( although the spectrum of any self-adjoint operator is nonempty ).
* the spectrum of a real valued process is symmetric:, or in other words, it is an even function
* The spectrum of a graph is symmetric if and only if it's a bipartite graph.
An undirected graph has a symmetric adjacency matrix and therefore has real eigenvalues ( the multiset of which is called the graph's spectrum ) and a complete set of orthonormal eigenvectors.

spectrum and operator
Once the operator took his finger off, the player would speed up until its tachometer was back in phase with the master, and as this happened, the phasing effect would appear to slide up the frequency spectrum.
* Spectrum of an operator, in functional analysis ( a generalisation of the spectrum of a matrix )
Once the operator took his finger off, the player would speed up until its tachometer was back in phase with the master, and as this happened, the phasing effect would appear to slide up the frequency spectrum.
In simpler instruments the absorption is determined one wavelength at a time and then compiled into a spectrum by the operator.
One of the key results of loop quantum gravity is quantization of areas: the operator of the area of a two-dimensional surface should have a discrete spectrum.
In the absence of such control or alternative arrangements such as a privatized electromagnetic spectrum, chaos might result if, for example, airlines didn't have specific frequencies to work under and an amateur radio operator were interfering with the pilot's ability to land an aircraft.
Residual spectrum of a normal operator is empty.
This notion can be extended to the spectrum of an operator in the infinite-dimensional case.
The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues.
However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues.
On the other hand 0 is in the spectrum because the operator R − 0 ( i. e. R itself ) is not invertible: it is not surjective since any vector with non-zero first component is not in its range.

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