( 3 ) For anionics, these micelles appear to be roughly spherical assemblages in which the hydrocarbon tails come together so that the polar groups ( the ionized ends ) face outward towards the aqueous continuous phase.

For example, the allotropes of carbon include diamond ( where the carbon atoms are bonded together in a tetrahedral lattice arrangement ), graphite ( where the carbon atoms are bonded together in sheets of a hexagonal lattice ), graphene ( single sheets of graphite ), and fullerenes ( where the carbon atoms are bonded together in spherical, tubular, or ellipsoidal formations ).

For example, carbon can be found as diamond, which has a tetrahedral structure around each carbon atom ; graphite, which has layers of carbon atoms with a hexagonal structure stacked on top of each other ; graphene, which is a single layer of graphite that is incredibly strong ; fullerenes, which have nearly spherical shapes ; and carbon nanotubes, which are tubes with a hexagonal structure ( even these may differ from each other in electrical properties ).

For spherical bodies, when the occulting object is smaller than the star, the length ( L ) of the umbra's cone-shaped shadow is given by:

For example: some early medieval manuscripts of Macrobius include maps of the Earth, including the antipodes, zonal maps showing the Ptolemaic climates derived from the concept of a spherical Earth and a diagram showing the Earth ( labeled as globus terrae, the sphere of the Earth ) at the center of the hierarchically ordered planetary spheres.

( For example, in spherical coordinates, let radius r go to R² / r where R is the Earth's radius.

For a four-dimensional spherical uncharged black hole, the deformation of the horizon is similar to the type of deformation which describes the emission and absorption of particles on a string-theory world sheet.

For this reason spherical reference surfaces are frequently used in mapping programs.

For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles.

For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension locally minimizes surface area.

For example, the sum of the interior angles of a spherical triangle exceeds 180 degrees.

For Sweden men's national ice hockey team, the home arena is Ericsson Globe, one of the largest spherical building in the world, but it is also hosting concerts and other events.

For spherical symmetry there is no such distinction, it implies planes of reflection.

For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is referred to as a Morris-Thorne wormhole.

For purposes of spherical astronomy, which is concerned only with the directions to objects, it makes no difference whether this is actually the case, or if it is the Earth which rotates while the celestial sphere stands still.

For observers near sea level the difference between this geometrical horizon ( which assumes a perfectly flat, infinite ground plane ) and the true horizon ( which assumes a spherical Earth surface ) is imperceptible to the naked eye ( but for someone on a 1000-meter hill looking out to sea the true horizon will be about a degree below a horizontal line ).

For example, a polished spherical electrode 30 cm in diameter immersed in air at STP ( which has a breakdown voltage of about 30 kV / cm ) could be expected to develop a maximum voltage of about 450 kV.

For low values of the Reynolds number (< 1 ), which is true of most aerosol motion, Stokes ' law describes the force of resistance on a solid spherical particle in a fluid.

For example, an asymmetrical asteroid's light curve generally has more pronounced peaks, while a more spherical object's light curve will be flatter.

For example, the van der Waals interaction energy between spherical bodies of radii R < sub > 1 </ sub > and R < sub > 2 </ sub > and with smooth surfaces was approximated in 1937 by Hamaker ( using London's famous 1937 equation for the dispersion interaction energy between atoms / molecules as the starting point ) by:

For instance, a plane triangle has an angle sum of 180 °; an octant is a spherical triangle with three right angles, so its angle sum is 270 °, and its angle excess is 90 °.

For spherical and hyperbolic spatial geometries, the curvature gives a scale ( either by using the radius of curvature or its inverse ), a fact noted by Carl Friedrich Gauss in an 1824 letter to Franz Taurinus.

For a spherical mass of uniform density, the gravitational binding energy U is given by the formula

For a perfectly spherical droplet the apparent weight can be written as:

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number and the modulation index.

For more general distributions the asymptotic distribution can be expressed as a Bessel function.

For a circular aperture, the diffraction-limited image spot is known as an Airy disk ; the distance x in the single-slit diffraction formula is replaced by radial distance r and the sine is replaced by 2J < sub > 1 </ sub >, where J < sub > 1 </ sub > is a first order Bessel function.

For large r, the spherical Bessel functions decay as 1 / r, giving the radiated field above.

For a circular bleach spot of radius w and diffusion-dominated recovery, the fluorescence is described by an equation derived by Soumpasis ( which involves modified Bessel functions and )

For example, the special case the function reduces to a Bessel function:

For the Screened Poisson equation, where the parameter k is real and the fundamental solution a modified Bessel function,

For example, the solution for the time-dependent " start-up " process can be expressed in terms of Bessel functions.

For the interpretation of these significant investigations, it should be remembered that reciprocal relations exist in the hypothalamus with respect to autonomic and somatic functions which are closely associated with the emotions.

For many years he had provided music and entertainment for functions throughout the Northwest.

For the Greeks, Apollo was all the Gods in one and through the centuries he acquired different functions which could originate from different gods.

For liturgical functions other than the Mass the bishop typically wears the cope.

For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

For non-integer α, the functions J < sub > α </ sub >( x ) and J < sub >− α </ sub >( x ) are linearly independent, and are therefore the two solutions of the differential equation.

For instance, NASA has extremely rigorous software testing procedures for many operating systems and communication functions.

For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà – Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space.

For those priests over 80 who became cardinal-deacons and were not ordained to the episcopacy, this is the highest position they can normally attain in the Church hierarchy ( though all cardinals rank above bishops in rank and order of precedence, those cardinals who are not bishops do not have the right to perform the functions reserved solely to bishops, such as ordination ).

For some distributed parameter systems the vectors may be infinite-dimensional ( typically functions ).

For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point.

For example: clairsentience is one of the six human special functions mentioned or recorded in Buddhism.

For the same reason, filter functions whose critical response is at lower frequencies ( compared to the sampling frequency 1 / T ) require a higher order, more computationally intensive FIR filter.

For polynomials of degree four and smaller there are explicit formulae for the roots ( the formulae are elementary functions ).

For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components ( Fourier series ), and the transforms diverge at those frequencies.

Direction of causal flow is upward. For a multivariate function, functional decomposition generally refers to a process of identifying a set of functions such that

For crimes that are not committed in the exercise of his functions, the President of the Republic answers before the common courts, once his term of office has ended.

For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication.

For most types of hashing functions the choice of the function depends strongly on the nature of the input data, and their probability distribution in the intended application.

For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin, which is, like sin, conventionally denoted in roman type and not in italics ( note that software libraries of mathematical functions often use the name < tt > asin </ tt >):

For example, scientists developing a new viral drug to treat an infection with a pathogenic virus ( e. g. HIV-1 ) may find that a candidate drug functions to prevent viral replication in an in vitro setting ( typically cell culture ).

For this purpose the Internet Protocol defines an addressing system that has two functions: identifying hosts and providing a logical location service.