If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.

If the author wished, he could apply for a second 14 ‑ year monopoly grant, but after that the work entered the public domain, so it could be used and built upon by others.

If the author has been dead more than 70 years, the work is in the public domain in most, but not all, countries.

If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.

If the name given in the delegation is a subdomain of the domain for which the delegation is being provided, there is a circular dependency.

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.

If the domain consists of the non-negative numbers, then the function is injective and invertible.

If ƒ is an invertible function with domain X and range Y, then

If one assumed that magnetopause was just a boundary between a magnetic field in a vacuum and a plasma with a weak magnetic field embedded in it, then the magnetopause would be defined by electrons and ions penetrating one gyroradius into the magnetic field domain.

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R ( more generally one can take domain and range to be the same unital associative algebra over R ).

If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely

If M is a free module over a principal ideal domain R, then every submodule of M is again free.

If Φ is a unital positive map, then for every normal element a in its domain, we have Φ ( a * a ) ≥ Φ ( a *) Φ ( a ) and Φ ( a * a ) ≥ Φ ( a ) Φ ( a *).

If the fully qualified domain name of any name server for a zone appears within that zone, the zone administrator provides IP addresses for that name server, which are installed in the parent zone as glue records ; otherwise, the delegation consists of the list of NS records for that zone.

If in addition all of the elements in the codomain are in fact mapped to by some element of the domain, then the function is said to be bijective ( see figures ).

* If the domain X = ∅ or X has only one element, the function is always injective.

The Gaussian integers form a principal ideal domain with units 1, − 1, i, and − i. If x is a Gaussian integer, the four numbers x, ix, − x, and − ix are called the associates of x.

If the waveform is digitally created directly in the time domain using a non-bandlimited form, such as y = x-floor ( x ), infinite harmonics are sampled and the resulting tone contains aliasing distortion.

If a membrane is stretched over a curve C that forms the boundary of a domain D in the plane, its vibrations are governed by the wave equation

If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇ f which takes the point a to the vector ∇ f ( a ).

If anyone violates the truce of God and after the third admonition does not make satisfaction, he shall be anathematized.

If someone under a geis violates the associated taboo, the infractor will suffer dishonor or even death.

If a young person violates a condition during the supervision period, the provincial director may place the offender back into custody.

If we can assume that the spacetime inside the Cauchy horizon violates AWEC, then the horizon becomes stable and frequency boosting effects would be canceled out by the tendency of the spacetime to act as a divergent lens.

If one violates land borders, or works somebody else ’ s land, or puts a fence on somebody else ’ s land, then he ’ ll pay 12 grivnas to the knyaz.

If landlord violates either, the tenant can terminate the lease and move out, or stay on the premises, while continuing to pay rent, and sue the landlord for damages ( or withhold rent and use breach of implied warranty of habitability as a defense when the landlord attempts to collect rent ).

If a service violates any other constraint, it cannot strictly be considered RESTful.

If the party against whom the judgment is rendered violates the terms of the consent decree, the judgment is as binding as any other, and the non-breaching party may seek enforcement through a contempt action.

If the listing of the credits violates right under the Lanham Act, the fact that the violation is pursuant to a contract with a third party is no defense .”

If this is not the case, the image needs to be deleted since it violates Wikipedia's image policy.

If an aircraft violates or is on a trajectory that will violate Restricted or Prohibited airspace, it will be warned of military interception on 121. 5 MHz.

If this node violates the α-height-balance property ( defined above ), a rebalance is required.

If you buy and sell a stock before paying for it, you are free riding, which violates the credit extension provisions of the Federal Reserve Board.

If the Secretary of Agriculture has any reason to suspect that any licensed dealer, carrier, or operator violates any provision of the AWA, then their license may be suspended temporarily, but not to exceed 21 days, until a hearing is held.

If a system exists that violates the bound, i. e. by having too much entropy, Bekenstein argued that it would be possible to violate the second law of thermodynamics by lowering it into a black hole.

If the hired Company violates the contract, the Guild takes it out of their hide and out of the bond they were paid by their employer, for the violation of contracted.

If the defender violates this rule, a technical foul is assessed against him, and the opposing team is granted one free throw and subsequent possession of the basketball.

If a childe violates this, the law demands that he be destroyed, and his sire may be punished too.

If a message is marked as infected, as spam, or violates administrator-defined restrictions, it will not be passed back to the GWIA for delivery to the intended recipient.