Given the definition of morpheme as " the smallest meaningful unit " Nanosyntax aims to account for idioms where it is often an entire syntactic tree which contributes " the smallest meaningful unit.

Given that the unified atomic mass unit is one twelfth the mass of one atom of carbon-12, meaning the mass of such an atom is 12 u, it follows that there are N < sub > A </ sub > atoms of carbon-12 in 0. 012 kg of carbon-12.

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval ( 0, 1 ), the variate

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Given ω = ( xθ, yθ, zθ ), with v = ( x, y, z ) a unit vector, the correct skew-symmetric matrix form of ω is

Given the ferocity of the Russian defense, everyone was aware that such a move would cost the lives of thousands of Guardsmen, but it was thought that the presence of this prestigious unit would bolster the morale of the entire army for a final decisive push.

Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is " above " the plane.

Given *- representations π, π ' each with unit norm cyclic vectors ξ ∈ H, ξ ' ∈ K such that their respective associated states coincide, then π, π ' are unitarily equivalent representations.

Given a unit sphere, a " triangle " on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere.

Given this diversity, it is difficult to generalize widely about the folklore of India as a unit.

Given an incident direction from the surface to the light source and the surface normal direction the specularly reflected direction ( all unit vectors ) is:

Given the variable half-life of flecainide and the characteristic QRS prolongation on ECG elicited in flecainide toxicity, especially at rapid heart rates, starting flecainide or changing the level of the drug is done under telemetry monitoring ( preferably in a hospital telemetry unit ) until a steady state plasma level has been achieved, typically three to five days after the dose has been increased.

Given a unit vector u = ( u < sub > x </ sub >, u < sub > y </ sub >, u < sub > z </ sub >), where u < sub > x </ sub >< sup > 2 </ sup > + u < sub > y </ sub >< sup > 2 </ sup > + u < sub > z </ sub >< sup > 2 </ sup > = 1, the matrix for a rotation by an angle of θ about an axis in the direction of u is

The chords are of considerable historical importance because, along with the sides of the triangle and tetragon ( square ), they enable the generation of a table of half chords ( effectively sine values ) Given the unit circle with sector ABO subtending an arc of, we may write − a relationship expressed in words by Copernicus:

Given a rotation axis z represented by a unit vector k

Given that LRAC is an average quantity, one must not confuse it with the long-run marginal cost curve, which is the cost of one more unit.

Given the time required for bonding and cohesion to build in standard military units, it is questionable if six-month rotations are long enough to generate high unit cohesiveness in an international, joint-services force such as the NRF:

→ Given a fixed stock of labor, the impact on output of the last unit of capital accumulated will always be less than the one before.

Given a highly controlled moral teaching unit at a young age, he is able to re-program it by instinct to become his personal assistant and, eventually, partner in crime.

Given a timelike unit vector field ( which can be interpreted as a family or congruence of nonintersecting world lines, not necessarily geodesics ), Raychaudhuri's equation can be written

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

) Given a smooth Φ < sup > t </ sup >, an autonomous vector field can be derived from it.

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

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 a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

Given the dimensions of the ellipsoid, the conversion from lat / lon / height-above-ellipsoid coordinates to X-Y-Z is straightforward — calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid.

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

Given subspaces U and W of a vector space V, then their intersection U ∩ W :=

Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given a vector x ∈ V and y * ∈ W *, then the tensor product y * ⊗ x corresponds to the map A: W → V given by

Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix.

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

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 a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

Given a grammar in GNF and a derivable string in the grammar with length n, any top-down parser will halt at depth n.

Given objects A < sub > 1 </ sub >,..., A < sub > n </ sub > in C, their biproduct is an object A < sub > 1 </ sub > ⊕ ··· ⊕ A < sub > n </ sub > together with morphisms

* Aggregate signature-a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users.

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

Given some training data, a set of n points of the form

Given a codeword, there are roughly 2 < sup > n H ( p )</ sup > typical output sequences.