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Page "Spectrum (functional analysis)" ¶ 43
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complex and number
Tone systems are certainly more complex than the number of units would suggest, and often analytically more difficult than much larger consonantal systems.
By applying this general principle, a great number of complex compounds of osmium, ruthenium, iridium, and rhenium, with triphenylphosphine, triphenylarsine, and triphenylstibine have been obtained in this laboratory during the past few years.
* Argument ( complex analysis ), a function which returns the polar angle of a complex number
It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.
Alexander Grothendieck's work during the ` Golden Age ' period at IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis.
For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.
This might sound complex, but first of all the number of di-atomic molecules that can exist at the temperatures of the atomizers used in AAS is relatively small, and second, the correction is performed by the computer within a few seconds.
In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ ( n ) defined on the set of natural numbers ( i. e. positive integers ) that " expresses some arithmetical property of n ."
Among the vast number of different biomolecules, many are complex and large molecules ( called biopolymers ), which are composed of similar repeating subunits ( called monomers ).
Particularly in team competition there can be a large number of bowls on the green towards the conclusion of the end, and this gives rise to complex tactics.
The inner product of two vectors is a complex number.
A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to say that bras are linear functionals on kets, i. e. operators that input a ket and output a complex number.
( Technically, the quantum states are rays of vectors in the Hilbert space, as corresponds to the same state for any nonzero complex number c .)
On the left side, is a function mapping any point in space to a complex number ; on the right side, is a ket.
If A is a self-adjoint operator, then is always a real number ( not complex ).
* The Hermitian conjugate of a complex number is its complex conjugate.
for an arbitrary real or complex number α ( the order of the Bessel function ); the most common and important cases are for α an integer or half-integer.
where ν > − 1 / 2 and z is a complex number.
It differs from a retirement home, which is a single building or small complex, by having a number of autonomous households.
The number of ligands that react with a central metal atom can be found using the 18-electron rule, saying that the valence shells of a transition metal will collectively accommodate 18 electrons, whereas the symmetry of the resulting complex can be predicted with the crystal field theory and ligand field theory.
They are usually complex proprietary formulations containing Portland clinker and a number of other ingredients that may include limestone, hydrated lime, air entrainers, retarders, waterproofers and coloring agents.
and expresses the two parameters of the associated linear Cauchy distribution for x as a complex number:
* Complex conjugation, the involution multiplying the imaginary part of a complex number by − 1

complex and λ
The spectrum of an element x ∈ A, denoted by, consists of all those complex scalars λ such that x − λ1 is not invertible in A.
# for every λ in C and every x in A ; here, denotes the complex conjugate of λ.
For example, suppose the state space is the n-dimensional complex Hilbert space C < sup > n </ sup > and A is a Hermitian matrix with eigenvalues λ < sub > i </ sub >, with corresponding eigenvectors ψ < sub > i </ sub >.
* If the base field is C, then for all complex numbers λ, where denotes the complex conjugation of λ.
The De Bruijn – Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function H ( λ, z ), where λ is a real parameter and z is a complex variable.
The method for solving linear differential equations is similar to the method above — the " intelligent guess " ( ansatz ) for linear differential equations with constant coefficients is where λ is a complex number that is determined by substituting the guess into the differential equation.
where the complex numbers λ and μ are called the degree and order of the associated Legendre functions respectively.
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 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.
A complex number λ is then in the spectrum if this property fails to hold.
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.
Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.
That is, for any complex number λ in the spectrum, one has | λ |= 1.
Note that any matrix obtained by multiplying by a complex scalar λ determines the same transformation, so a Möbius transformation determines its matrix only up to scalar multiples.
Example: Suppose that the manifold is the circle ( thought of as R / Z ), and D is the operator d / dx − λ for some complex constant λ.

complex and is
Accidental war is so sensitive a subject that most of the people who could become directly involved in one are told just enough so they can perform their portions of incredibly complex tasks.
I am suggesting that a case-history approach to the Oedipus complex is a blind alley for a storyteller.
It is most probable that Freud and the Oedipus complex never entered his head in the writing of this story.
The board's action shows what free enterprise is up against in our complex maze of regulatory laws.
The transportation system which serves the National Forests is a complex of highways and access roads and trails under various ownerships and jurisdictions.
But there is still the sometimes complex problem of helping campers choose the best equipment for their individual needs.
Perhaps the best way to indicate the versatility of design that characterizes the use of plastics in signs and displays would be to look at what is happening in only one of the areas in this complex field -- changeable signs.
Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.
Hence the totality of singular lines is the T order complex of lines which meet Aj.
Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.
The most obvious of these is the quadratic complex of tangents to Q, each line of which is transformed into the entire pencil of lines tangent to Q at the image of the point of tangency of the given line.
We now observe that the case in which **zg is a Af curve on a quadric is impossible if the complex of singular lines consists exclusively of the lines which meet Aj.
However, if there is no additional complex of singular lines, the order of the image regulus of a pencil is precisely Af.
In societies like ours, however, its place is less clear and more complex.
This behavior is more `` veridical '' -- or true -- than other testing behavior for some types of evaluation, and so can give quick and accurate estimates of complex functioning.
Of all the possible forms of nonverbal expression, that which seems best to give release, and communicational expression, to complex and undifferentiated feelings is laughter.

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