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 unit vector n in R < sup > 3 </ sup > and an angle φ, let R ( φ, n ) represent a counterclockwise rotation about the axis through n ( with orientation determined by n ).

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 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 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 this account, historian Oscar G. Darlington asserts that Gerbert's division by 60 degrees instead of 360 allowed the lateral lines of his sphere to equal to six degrees.

Given the sphere defined by the points ( x, y, z ) such that

Given a point in the plane, draw a straight line connecting it with the north pole on the sphere.

Given a vector at the north pole, one can transport this vector along a curve by rotating the sphere in such a way that the north pole moves along the curve without axial rolling.

* Given Imaging ( NASDAQ: GIVN )-a pioneer in the sphere of capsule endoscopy.

Given that this mass is confined inside a 44 million km diameter sphere, this yields a density ten times higher than previous estimates.

Given any r on this sphere, and an angle – π < a ≤ π, the versor is on the 3-sphere of H.

Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional bloch sphere.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:

Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.

* Given a triangle OAB in which O is the center of a circle k, and points A ' and B ' inverses of A and B with respect to k, then

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given a triangle ABC, and a triangulation T of the triangle.

Given a counterfactual conditional, e. g., ' If there had been a circle on the blackboard then there would have been a triangle ', and the subsequent information ' in fact there was no triangle ', participants make the modus tollens inference ' there was no circle ' more often than they do from an indicative conditional ( Byrne and Tasso, 1999 ).

Given a point inside a triangle it is also desirable to obtain the barycentric coordinates, and at this point.

Given a triangle ABC and its three Malfatti circles, let D, E, and F be the points where two of the circles touch each other, opposite vertices A, B, and C respectively.

Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, then

Given An equilateral triangle inscribed on a circle and a point on the circle.

Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries ( Euclidean, spherical, or hyperbolic ) admits a triangle with the angles ( π / l, π / m, π / n ), and the space is tiled by reflections of the triangle.

Given any Euclidean triangle ABC and an arbitrary point P let d ( P ) = PA + PB + PC.

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle.

The has a crossing point at the origin if there is a triangle with sides a, b and c. Given the previous conditions, this means that the curve crosses the origin if and only if b < a + c.