Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis.

Given that the eigenstates of the Hamiltonian always from a set of orthonormal basis, both the cases of m

Given a C < sup > n + 1 </ sup >- curve γ in R < sup > n </ sup > which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors

Given an orthonormal basis

Given an orthonormal basis, any pure state of a two-level quantum system can be written as a superposition of the basis vectors

Given the frame field, one can also define a metric by conceiving of the frame field as an orthonormal vector field.

If V is finite-dimensional, then V * has the same dimension as V. Given a basis

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given the complexity of NLP problems, it is often difficult to predict performance only on the basis of glass-box evaluation, but this type of evaluation is more informative with respect to error analysis or future developments of a system.

Given Poseidon's connection with horses as well as the sea, and the landlocked situation of the likely Indo-European homeland, Nobuo Komita has proposed that Poseidon was originally an aristocratic Indo-European horse-god who was then assimilated to Near Eastern aquatic deities when the basis of the Greek livelihood shifted from the land to the sea, or a god of fresh waters who was assigned a secondary role as god of the sea, where he overwhelmed the original Aegean sea deities such as Proteus and Nereus.

Given the medieval medicine's main theory of humorism, spices and herbs were indispensable to balance " humors " in food, a daily basis for good health at a time of recurrent pandemics.

The isomorphism is not canonical ; it depends on a choice of basis in V. Given a basis ( e < sub > 1 </ sub >, ..., e < sub > n </ sub >) of V and an automorphism T in GL ( V ), we have

Given that there is no constitutional office of Deputy Prime Minister, with the position being recreated on a case by case basis, the person who holds the post has no official residence.

Given a basis f = ( X < sub > 1 </ sub >,..., X < sub > n </ sub >) of V, there is a unique reciprocal basis f < sup >#</ sup > = ( Y < sup > 1 </ sup >,..., Y < sup > n </ sup >) of V determined by requiring that

Given the Protestant belief of justification through faith alone, versus on the basis of merit, most Protestant Christians consider themselves saved without having to keep the commandments of the Mosaic Law as a whole.

Given that it is still legal to discriminate against individuals on the basis of gender identity within New Zealand, the transgender community often finds that many of its younger members require survival sex for food, shelter and rest.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Given a set, a free module on is a free module with basis.

For a finite group G, the left regular representation λ ( over a field K ) is a linear representation on the K-vector space V whose basis is the elements of G. Given g ∈ G, λ ( g ) is the linear map determined by its action on the basis by left translation by g, i. e.

Given the proximity of Boarstall to the king's palace at Brill it would appear that this legend certainly has some basis in fact.

Given the Likert Scale's ordinal basis, summarizing the central tendency of responses from a Likert scale by using either the median or the mode is best, with ‘ spread ’ measured by quartiles or percentiles.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given a probability space, a filtration is a weakly increasing collection of sigma-algebras on,, indexed by some totally ordered set T, and bounded above by.

Given the extremely disadvantaged position in which the English were, Vernon ordered a retreat but not before throwing some 300 bombs on the city.

Given an ordered sequence of real numbers: the first difference is defined as

Given the distance that these troops had to march in order to make contact with the enemy, it was clear to the two Corps commanders that they would be unable to attack at 4: 00 as ordered.

Given intelligence that the rebels had been stockpiling weapons at Concord, Massachusetts, he ordered a troop of British regulars to march there on the night of 18 April to confiscate them.

Given a non-empty totally ordered set R with the following four properties:

Given a partially ordered class P, we can form a category with P's elements as objects and a single arrow going from one object ( element ) to another if the first is less than or equal to the second.

Given 25, 000 men, he was ordered to clear Confederate obstacles on the Mississippi River.

Given two partially ordered sets A and B, the lexicographical order on the Cartesian product A × B is defined as

Given points a, b, c, and d on a line, if it is known that the points are ordered as ( a, b, c ) and ( b, c, d ), then it is also true that ( a, b, d ).

#: Given three ordered periods,,, the price index for periods and times the price index for periods and should be equivalent to the price index for periods and.

Given that the documentary had aroused much tension among the Party ’ s top leaders, the Ministry of Radio, Film and Television ordered CCTV to revise the original version of the film before it could be broadcast again.

Given an ordered choice of linearly independent eigenvectors, especially an eigenbasis, they can be indexed by eigenvalues, i. e. using a double index, with x < sub > i, j </ sub > being the j < sup > th </ sup > eigenvector for the i < sup > th </ sup > eigenvalue.

Given its inner-city roots, chicken and food is quite often ordered a la carte.

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given a simple, binary component feed, analytical methods such as the McCabe-Thiele method or the Fenske equation can be used.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given these connotations, dandyism can be seen as a political protestation against the rise of levelling egalitarian principles, often including nostalgic adherence to feudal or pre-industrial values, such as the ideals of " the perfect gentleman " or " the autonomous aristocrat ", though paradoxically, the dandy required an audience, as Susann Schmid observed in examining the " successfully marketed lives " of Oscar Wilde and Lord Byron, who exemplify the dandy's roles in the public sphere, both as writers and as personae providing sources of gossip and scandal.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given the Greek fleet's unpreparedness resulting from the premature outbreak of the war, such an early Ottoman attack might well have been able to achieve a crucial victory.

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

Given the toleration of Hinduism, such imperial leeway was not shocking.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given the need for such fine regulation of Ca < sup > 2 +</ sup > signaling, it is perhaps unsurprising that dysregulated mitochondrial Ca < sup > 2 +</ sup > has been implicated in several neurodegenerative diseases, while the catalogue of tumor suppressors includes a few that are enriched at the MAM.

Given its continental origin as a fragment of Zealandia, unlike many of the islands of the Pacific such as the Hawaiian chain, New Caledonia is not of geographically recent volcanic provenance.

* Named entity recognition ( NER ): Given a stream of text, determine which items in the text map to proper names, such as people or places, and what the type of each such name is ( e. g. person, location, organization ).

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.

Given X such that

Given that the authors were the primary developers of the Unix operating system, it is natural that QED had a strong influence on the classic UNIX text editors ed, sed and their descendants such as ex and sam.

Given that in such a case pits the freedom of an innocent child against the freedom of the evil-doer, it is not clear why God would not intervene for the sake of the child.