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Page "Analytical hierarchy" ¶ 9
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equivalent and definition
Another definition of asymmetric makes asymmetry equivalent to antisymmetry plus irreflexivity.
An equivalent definition is the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with an angular frequency of radians per day ; or that length such that, when used to describe the positions of the objects in the Solar System, the heliocentric gravitational constant ( the product GM < sub >☉</ sub >) is equal to ()< sup > 2 </ sup > AU < sup > 3 </ sup >/ d < sup > 2 </ sup >.
This definition is equivalent to the statement that a unital associative R-algebra is a monoid in R-Mod ( the monoidal category of R-modules ).
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.
For alternative ( equivalent ) formulations of the definition in terms of a bijective function or a surjective function, see the section Formal definition and properties below.
It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g ′( a ) and a function ε ( h ) that tends to zero as h tends to zero, and furthermore
These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not.
Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.
: Turing's thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i. e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.
An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.
A bit, or a group of malfunctioning physical bits ( not always the specific defective bit is known ; group definition depends on specific storage device ) is typically automatically fenced-out, taken out of use by the device, and replaced with another functioning equivalent group in the device, where the corrected bit values are restored ( if possible ).
A bit, or a group of malfunctioning physical bits ( not always the specific defective bit is known ; group definition depends on specific storage device ) is typically automatically fenced-out, taken out of use by the device, and replaced with another functioning equivalent group in the device, where the corrected bit values are restored ( if possible ).
) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.
In addition to the definition above, there is an equivalent definition.
Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane ( a focus ) and a particular circle, the directrix circle ( whose center is the other focus ).
The electric field was formally defined as the force exerted per unit charge, but the concept of potential allows for a more useful and equivalent definition: the electric field is the local gradient of the electric potential.
Boltzmann showed that this definition of entropy was equivalent to the thermodynamic entropy to within a constant number which has since been known as Boltzmann's constant.
Baslieus, a title which had long been used for Alexander the Great was already in common usage as the Greek word for the Roman emperor, but its definition and sense was " King " in Greek, essentially equivalent with the Latin Rex.
An equivalent definition of group homomorphism is: The function h: G → H is a group homomorphism if whenever we have.
This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if Y coincides with the codomain of ƒ.
In this case, the above definition can be characterized by the following equivalent statement:
In an equivalent formal definition, NP is the set of decision problems where the " yes "- instances can be decided in polynomial time by a non-deterministic Turing machine.

equivalent and analytical
* An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic ; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space.
However, equivalent weights continued to be used for many compounds for another hundred years, particularly in analytical chemistry.
When choosing primary standards in analytical chemistry, compounds with higher equivalent weights are generally more desirable because weighing errors are reduced.
In analytical chemistry, a solution of any substance which contains one equivalent per litre is known as a normal solution ( abbreviated N ), so the example sodium hydroxide solution would be 0. 0893 N. The relative uncertainty ( u < sub > r </ sub >) in the measured concentration can be estimated by assuming a Gaussian distribution of the measurement uncertainties:
The equivalent VAR ( 1 ) form is more convenient for analytical derivations and allows more compact statements.

equivalent and hierarchy
* It is Turing equivalent to the halting problem and thus at level of the arithmetical hierarchy.
History paintings were traditionally regarded as the highest form of Western painting, occupying the most prestigious place in the hierarchy of genres, and considered the equivalent to the epic in literature.
In the jurisdiction of the original decision, however, a judge should only overturn the holding of a court lower or equivalent in the hierarchy.
Suppose that you add blue, then the blue – red – black tree defined like red – black trees but with the additional constraint that no two successive nodes in the hierarchy will be blue and all blue nodes will be children of a red node, then it becomes equivalent to a B-tree whose clusters will have at most 7 values in the following colors: blue, red, blue, black, blue, red, blue ( For each cluster, there will be at most 1 black node, 2 red nodes, and 4 blue nodes ).
In the E-carrier European hierarchy, each digroup supports 15 PCM channels or their equivalent in other services.
Consequently by the 11th century, a hierarchy of minsters became apparent ; cathedral churches, or head minsters having pre-eminence within a diocese ; surviving old minsters being pre-eminent within an area broadly equivalent to an administrative hundred ; while newer lesser minsters and field churches were increasingly proliferating on local estates.
In Finland, the Elämän Sana (" the Word of life ") group, as the most " mainline " of the different branches of Laestadianism, has been prominent within the hierarchy of the Evangelical Lutheran Church of Finland: Two members have been elected bishops of Oulu, and one has served as Chaplain General ( head chaplain of the Finnish Defence Forces and the equivalent of a Major General ).
Italo Balbo had become one of the " Ras ," adopted from an Ethiopian title somewhat equivalent to a duke, of the Fascist hierarchy by 1922, establishing his local leadership in the party.
The file hierarchy is roughly equivalent to the common Unix-like hierarchy of directories, sub-directories, and files.
There are multiple equivalent definitions of the classes of the polynomial hierarchy.
Legionary centurions, the equivalent of mid-level commissioned officers, were organised in an elaborate hierarchy.
Usually their rank in the feudal hierarchy was equivalent to that of the simple sire ( dominus ), between the baron and the chevalier ; but occasionally they were great nobles with an extensive jurisdiction, as in the Low Countries ( see Burgrave ).
In many but not all North American institutions, the provost or equivalent is the second-ranking officer in the administrative hierarchy.
It ranks equivalent to an intermediate court in the judiciary hierarchy.
In the projective hierarchy, Π < sup > 1 </ sup >< sub > 1 </ sub > Wadge determinacy is equivalent to Π < sup > 1 </ sup >< sub > 1 </ sub > determinacy, as proved by Harrington.

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