In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for the normalized prime-counting function π < sub > 0 </ sub >( x ) which is related to the prime-counting function π ( x ) by

Given a normalized continuous positive definite function f on G, one can construct a strongly continuous unitary representation of G in a natural way: Let F < sub > 0 </ sub >( G ) be the family of complex valued functions on G with finite support, i. e. h ( g )

Given that white is the most effective color in terms of facilitating the reflection of ground light, the hypothesis that straight hair could have been adaptively favorable, cannot be fully discounted in this regard.

" Given that there's a gulf of 100, 000, 000 light years, he checks his diary to find out what he had been doing that day, and finds out that it was something he wished to hide.

The Fascist government had direct rule only over the Benadir territory Given the defeat of the Dervish movement in the early 1920s and the rise of fascism in Europe, on 10 July 1925, Mussolini gave the green light to De Vecchi to start the takeover of the north-eastern sultanates.

Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.

Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them.

Given the development of the metropolitan area, further plans feature the creation of a light rail track between Gilău and Jucu that will use these modernised tracks in the city.

Given a source of white light, they produce a white shine.

Given an event, the light cone classifies all events in space + time into 5 distinct categories:

Given that the high latitudes in this period were warm and tropical, it is hypothesized that the deciduous habit evolved in response to the unusual light availability patterns, not to major seasonal variations in temperature.

Given the coarse construction of the walls, there was no second story to a typical home, and the roof would have been constructed of light wooden beams and thatch mixed with mud.

Given what later came to light, however, Wooden granted that he may have had " tunnel vision " and that he perhaps " trusted too much ".

Given a faster than light technology and weapons scaled similar to those of the Age of Sail, some of the battles described in the Honor Harrington series resemble real-life sea battles in our historical record.

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 an event, the light cone classifies all events in spacetime into 5 distinct categories:

Given that the angle of incidence is larger than the critical angle ( with respect to the normal of the surface ) and the refractive index is lower on the other side of the boundary relative to the incident ray, TIR will occur and no light will be able to pass through.

Given a mean AGB lifetime of one Myr and an outer velocity of, its maximum radius can be estimated to be roughly ( 30 light years ).

Given the very light weight of the " pin top ," the. 22 is quite capable of removing the target from the table with a single well-placed shot.

Given the large aperture and low f-number of the lens, as well as high convergence ratios and synaptic gains, the ocelli are generally considered to be far more sensitive to light than the compound eyes.

Given their underfocused nature, wide fields of view, and high light collecting ability, the ocelli are superbly adapted for measuring changes in the perceived brightness of the external world as an insect rolls or pitches around its body axis during flight.

Given the relatively recent introduction of a law technically requiring motorists to stop at a red light, it is not surprising that drivers frequently disregard traffic lights and proceed on red.

Given the complexity of its refining facilities, SPC is able to refine heavy, medium and light crudes.

Given on December 1, 2010, a translated interview with Oleg Sienko, Chief Executive Officer of the Uralvagonzavod shed some light on the project's current & future status:

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

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.