Explicitly, suppose a quantum system may be found in state with probability p < sub > 1 </ sub >, or it may be found in state with probability p < sub > 2 </ sub >, or it may be found in state with probability p < sub > 3 </ sub >, and so on.

Explicitly, a closed operator N is said to be normal if

Explicitly, let T be a tensor field of type ( p, q ).

Explicitly, k ( X ) consists of zero and f / g, with f, g homogeneous of the same degree, inside the field of fractions of R. For each x in X, let be the subring consisting of zero and f / g with g ( x ) ≠ 0 ; it is a local ring.

Explicitly incorporated into the concept of a Xiaokang society is the idea that economic growth needs to be balanced with sometimes conflicting goals of social equality and environmental protection.

Explicitly, let C and D be ( locally small ) categories and let F: C → D be a functor from C to D. The functor F induces a function

Explicitly, the initial topology may be described as the topology generated by sets of the form, where is an open set in.

Explicitly, a reflection has order 2 in O ( V ), r < sup > 2 </ sup > = 1, so the square of the preimage of a reflection ( which has determinant one ) must be in the kernel of Spin < sub >±</ sub >( V ) → SO ( V ), so, and either choice determines a pin group ( since all reflections are conjugate by an element of SO ( V ), which is connected, all reflections must square to the same value ).

Explicitly, the final topology may be described as follows: a subset U of X is open if and only if is open in Y < sub > i </ sub > for each i ∈ I.

Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n. This result is known as the Erdős – Ginzburg – Ziv theorem after its discoverers: it may be deduced from the Cauchy-Davenport theorem.

Explicitly stating that a bridge does not exist is not a solution to this problem, as there are many other exceptional conditions that should be excluded ( such as the presence of a rope for fastening the cannibals, the presence of a larger boat nearby, etc.

* Superscalar execution, VLIW, and the closely related Explicitly Parallel Instruction Computing concepts, in which multiple execution units are used to execute multiple instructions in parallel.

Explicitly, it is used to define the degree of a polynomial and the notion of homogeneous polynomial, as well as for graded monomial orderings used in formulating and computing Gröbner bases.

* Explicitly parallel instruction computing, a design philosophy based on VLIW used in Itanium CPUs

Explicitly, this means that for every arbitrary collection

Explicitly, this means that for every arbitrary collection

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

Explicitly, for each vector space, require that it come with the data of an isomorphism to its dual, In other words, take as objects vector spaces with a nondegenerate bilinear form This defines an infranatural isomorphism ( isomorphism for each object ).

An isometric embedding is a smooth embedding f: M → N which preserves the metric in the sense that g is equal to the pullback of h by f, i. e. g = f * h. Explicitly, for any two tangent vectors

Explicitly, for a particle in one dimension with position x, mass m and momentum p, and potential energy V which generally varies with position and time t:

Explicitly, for a positive real number m, define the subset B < sub > m </ sub > of K by

Explicitly contrasted against a social cohesion based definition for social groups is the social identity perspective, which draws on insights made in social identity theory.

Explicitly, given a Lagrangian subspace ( as defined below ), then a choice of basis ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >) defines a dual basis for a complement, by

Explicitly excluded are propaganda for war, incitement to violence and hate speech.

Explicitly, for a skew-symmetric matrix A,

Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i < sub > 1 </ sub >: X → P and i < sub > 2 </ sub >: Y → P for which the following diagram commutes:

Explicitly, the score for is the gradient of the log-likelihood with respect to.

Explicitly this means that for every positive constant ε there exists a constant N such that

Explicitly, the formulae for the barycentric co-ordinates of are:

Explicitly, the modules of the form for prime ideals p ( including p = 0, which yields R ) are indecomposable.

Explicitly, the pullback of the morphisms f and g consists of an object P and two morphisms p < sub > 1 </ sub >: P → X and p < sub > 2 </ sub >: P → Y for which the diagram

Explicitly, IOLTA applies only to funds that are " nominal in amount or held for a short period of time " so larger amounts of money held for single clients are exempt from the IOLTA program.

Explicitly, this means that for every sequence ( x < sub > n </ sub >) of points from the domain of T such that x < sub > n </ sub > converge to some x and Tx < sub > n </ sub > converge to some y, it holds that x belongs to the domain of T and.

Explicitly or implicitly, he projects himself everywhere in his writings.

Explicitly recognising two hundred years ' of ambivalence, the act states that it intended " To give statutory recognition to the existence of the position of Prime Minister, and to the historic link between the Premiership and the office of First Lord of the Treasury, by providing in respect to that position and office a salary of ..." The Act made a distinction between the " position " ( Prime Minister ) and the " office " ( First Lord of the Treasury ), emphasising the unique political character of the former.

* Anti-Semitism: Explicitly referenced in recent decades is the comparison between anti-mutant sentiment and anti-Semitism.

* Explicitly states that the federal government will rely primarily on private enterprise to achieve the four goals.

Explicitly if lightly linked to The Fionavar Tapestry, featuring two of its main characters as supporting roles.

Explicitly violent and / or sexual content and social satire are trademarks of both his drama and science fiction films.

Explicitly following G. E.

# „ Organism-centred, multi-scale topological ecology “ ( e. g. John A. Wiens ): Explicitly rejecting views expounded by Troll, Zonneveld, Naveh, Forman & Godron, etc., landscape and landscape ecology are defined independently of human perceptions, interests, and modifications of nature.

Explicitly, this means the following.