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 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 a system consists of several particles, the total angular momentum about a point can be obtained by adding ( or integrating ) all the angular momenta of the constituent particles.

If a diatomic molecule consists of two atoms of the same element, such as H < sub > 2 </ sub > and O < sub > 2 </ sub >, then it is said to be homonuclear, but otherwise it is heteronuclear.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

* If we think of as the ring of real numbers, then the direct product again consists of.

* If is the set of all cars, and is the equivalence relation " has the same color as ", then one particular equivalence class consists of all green cars.

If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S then an isomorphism from X to Y is a bijective function such that:

If the alphabet consists of alternative symbols, each symbol represents a message consisting of N bits.

If either party consists of two or more members, all will come to attention, but only the most senior member of the party will offer ( or return ) the physical or verbal salute.

If the diad consists of two identically oriented units, the diad is called a meso diad reflecting similar features as a meso compound.

If the diad consists of units oriented in opposition, the diad is called a racemo diad as in a racemic compound.

If the array of weights consists of N weights spaced evenly over the length L

If absent or deficient of natural tissue, alternatives can be cultured sheets of epithelial cells in vitro or synthetic compounds, such as integra, which consists of silicone and bovine tendon collagen with glycosaminoglycans.

If one assumes that all of the universe consists of protons and neutrons, the density of the universe is such that much of the currently observed deuterium would have been burned into helium-4.

If ( E, M ) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense.

* If K is a field and we consider the K-vector space K < sup > n </ sup >, then the endomorphism ring of K < sup > n </ sup > which consists of all K-linear maps from K < sup > n </ sup > to K < sup > n </ sup >.

If the body consists of a homogeneous material with sufficiently known physical properties, the thermal mass is simply the mass of material present times the specific heat capacity of that material.

If the support of a distribution consists of points, with corresponding probabilities, then the probability mass function of the distribution over can be written, using the Kronecker delta, as

If ecological sanitation is practiced municipal wastewater consists only of greywater, which can be recycled for gardening.

If a thermodynamical system consists of N particles, then a point in the 6N-dimensional phase space describes the dynamical state of every particle in that system, as each particle is associated with three position variables and three momentum variables.

If virtue consists in being useful to our fellow citizens, perhaps there were few more virtuous men than Mr. Craig.

If the universe of discourse consists of all functions from the real line R to itself, then the solutions for f are all functions whose only values are one and zero.

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.