For non-integer α, Y < sub > α </ sub >( x ) is related to J < sub > α </ sub >( x ) by:

For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator.

) For example, Aldebaran is designated α Tauri ( pronounced Alpha Tauri ), which means " Alpha of the Bull ".

For integer order α = n, J < sub > n </ sub > is often defined via a Laurent series for a generating function:

For example, if A < sub > α </ sub >

For example, the " Little Dipper " asterism of the constellation Ursa Minor has ten stars with Bayer designation, α UMi to π UMi.

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

* Prediction: For every state in S ( k ) of the form ( X → α • Y β, j ) ( where j is the origin position as above ), add ( Y → • γ, k ) to S ( k ) for every production in the grammar with Y on the left-hand side ( Y → γ ).

* Completion: For every state in S ( k ) of the form ( X → γ •, j ), find states in S ( j ) of the form ( Y → α • X β, i ) and add ( Y → α X • β, i ) to S ( k ).

For anisotropic multicomponent diffusion coefficients one needs 4-index quantities, for example,, where i, j are related to the components and α, β = 1, 2, 3 correspond to the space coordinates.

For most voltage-gated ion channels, the pore-forming subunit ( s ) are called the α subunit, while the auxiliary subunits are denoted β, γ, and so on.

For instance, if the result of the measurement is, < var > α </ var > is changed to 1 ( up to phase ) and < var > β </ var > is changed to 0.

For instance, were α to change by 4 %, stellar fusion would not produce carbon, so that carbon-based life would be impossible.

For example, the components of α Canis Majoris ( Sirius ) are α Canis Majoris A and α Canis Majoris B ( Sirius A and Sirius B ); the components of 44 Boötis are 44 Boötis A and 44 Boötis B ; the components of ADS 16402 are ADS 16402A and ADS 16402B ; and so on.

For example, a smooth curve α ( t ): 1 → M has tangent vector α ′( t < sub > 0 </ sub >) in the tangent space TM ( α ( t < sub > 0 </ sub >)) at any point t < sub > 0 </ sub > ∈ ( 0, 1 ), and each such vector has length ‖ α ′( t < sub > 0 </ sub >)‖, where ‖·‖ denotes the norm induced by the inner product on TM ( α ( t < sub > 0 </ sub >)).

For the interpretation of these significant investigations, it should be remembered that reciprocal relations exist in the hypothalamus with respect to autonomic and somatic functions which are closely associated with the emotions.

For many years he had provided music and entertainment for functions throughout the Northwest.

For the Greeks, Apollo was all the Gods in one and through the centuries he acquired different functions which could originate from different gods.

For liturgical functions other than the Mass the bishop typically wears the cope.

For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

For the spherical Bessel functions the orthogonality relation is:

For instance, NASA has extremely rigorous software testing procedures for many operating systems and communication functions.

For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà – Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space.

For those priests over 80 who became cardinal-deacons and were not ordained to the episcopacy, this is the highest position they can normally attain in the Church hierarchy ( though all cardinals rank above bishops in rank and order of precedence, those cardinals who are not bishops do not have the right to perform the functions reserved solely to bishops, such as ordination ).

For some distributed parameter systems the vectors may be infinite-dimensional ( typically functions ).

For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point.

For example: clairsentience is one of the six human special functions mentioned or recorded in Buddhism.

For the same reason, filter functions whose critical response is at lower frequencies ( compared to the sampling frequency 1 / T ) require a higher order, more computationally intensive FIR filter.

For polynomials of degree four and smaller there are explicit formulae for the roots ( the formulae are elementary functions ).

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number and the modulation index.

For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components ( Fourier series ), and the transforms diverge at those frequencies.

Direction of causal flow is upward. For a multivariate function, functional decomposition generally refers to a process of identifying a set of functions such that

For crimes that are not committed in the exercise of his functions, the President of the Republic answers before the common courts, once his term of office has ended.

For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication.

For most types of hashing functions the choice of the function depends strongly on the nature of the input data, and their probability distribution in the intended application.

For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin, which is, like sin, conventionally denoted in roman type and not in italics ( note that software libraries of mathematical functions often use the name < tt > asin </ tt >):

For example, scientists developing a new viral drug to treat an infection with a pathogenic virus ( e. g. HIV-1 ) may find that a candidate drug functions to prevent viral replication in an in vitro setting ( typically cell culture ).

For this purpose the Internet Protocol defines an addressing system that has two functions: identifying hosts and providing a logical location service.

For example, in the authoritative modern monograph about the creed, J. N. D.

For more information on Stone and his work, see " Arthur J.

( For one particle, J = L + S.

* Hudson, J. Francis Esther: For Such a Time as This.

For example, J. M. W. Topp collected data annually between 1993 – 2003 and found that on the average three new plant species arrived each year, mainly on Diego Garcia.

For example Major General Charles J. Dunlap, Jr. says that United States Air Force culture includes an egalitarianism bred from officers as warriors who work with small groups of enlisted airmen either as the service crew or onboard crew of their aircraft.

Diagram: For categories C and J, a diagram of type J in C is a covariant functor.

( Category theoretical ) presheaf: For categories C and J, a J-presheaf on C is a contravariant functor.

Limit functor: For a fixed index category J, if every functor J → C has a limit ( for instance if C is complete ), then the limit functor C < sup > J </ sup >→ C assigns to each functor its limit.

For example, J. B. Clark saw the co-ordinating function in production and distribution as being served by entrepreneurs ; Frank Knight introduced managers who co-ordinate using their own money ( financial capital ) and the financial capital of others.

For example, the H. 263v2 added a deblocking filter via Annex J.

For instance, J. Scott Armstrong challenged Al Gore to a climate-related bet in 2007 that focused on year-to-year variation in temperatures but not on betting over longer term changes in global average temperatures.

For the five 6. 4 meganewton sea-level thrust F-1 rocket engines of the Saturn V, burning together, the reaction generated roughly 1. 62 × 10 < sup > 11 </ sup > watts ( J / s ) ( 162 gigawatt ) or 217 million horsepower.

* 1975 – For the first time, ever, the Zapruder film of the assassination of John F. Kennedy is shown in motion to a national TV audience by Robert J. Groden and Dick Gregory.

For example, the charm quark was first seen in the J / Psi meson () in 1974, and the bottom quark in the upsilon meson () in 1977.

For example, J. G. A.

For this work, Alan J. Heeger, Alan G. MacDiarmid, and Hideki Shirakawa were jointly awarded the Nobel Prize in Chemistry in 2000.

For example, assume that a Texas hold ' em board looks like this after the third round: 5 ♠ < font color = red > K ♦ 7 ♦</ font > J ♠, and that a player is holding < font color = red > A ♦ 10 ♦</ font >.