Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron ( i. e. orbitals ) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule.

Alternatively, the objects are called the first and second coordinates, or the left and right projections of the ordered pair.

Alternatively this formula is found using spherical coordinates, with volume element

Alternatively, the area element on the sphere is given in spherical coordinates by.

Alternatively, it follows from () that if the polar coordinates of the point p are ( r < sub > 1 </ sub >, θ < sub > 1 </ sub >) and those of q are ( r < sub > 2 </ sub >, θ < sub > 2 </ sub >), then the distance between the points is

Alternatively, in polar coordinates, the gain of the transfer function is plotted as the radial coordinate, while the phase of the transfer function is plotted as the angular coordinate.

Alternatively, if a line is constructed perpendicularly between Alpha Centauri and Beta Centauri, the point where the above mentioned line and this line intersect marks the Southern Celestial Pole.

Alternatively, sparring may continue with the point noted by the judges.

Alternatively, microscopes can be classed on whether they analyse the sample via a scanning point ( confocal optical microscopes, scanning electron microscopes and scanning probe microscopes ) or analyse the sample all at once ( wide field optical microscope and transmission electron microscopes ).

Alternatively a white dwarf star may accumulate sufficient material from a stellar companion ( either through accretion or via a merger ) to raise its core temperature enough to ignite carbon fusion, at which point it undergoes runaway nuclear fusion, completely disrupting it.

Alternatively, one may point out that greater good theodicies lead us to see every conceivable state of affairs as compatible with the existence of God, and in that case the notion of God's goodness is rendered meaningless.

Alternatively ambient noise levels may be measured to provide a reference point for analyzing an intrusive sound to a given environment.

Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.

Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x < sub > 0 </ sub > in its domain

Alternatively, compass point answers are often clued as " XXX to YYY direction ", where XXX and YYY are two place names.

Alternatively, some scholars identify Ptolemy Neos Philopator with Ptolemy Memphites, a son of Ptolemy VIII and Cleopatra II who was murdered by his father about 132 / 131 BC after his mother had tried to depose Physcon and proclaim their son king ; yet others point to a number of minor co-regents – all of whom were named Ptolemy as was the tradition in the dynasty.

Alternatively, it may be that railroad crews referred to the town as " Graftin " because it was the point at which a number of branch railroad lines met ( grafted to ) the railroad's mainline.

Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S

Alternatively, in a normal coordinate system based at p, at the point p

Alternatively, with respect to a large region without airfields, e. g. an ocean, it can mean the point before which it is closer to turn around and after which it is closer to continue.

Alternatively, the MSIC has had opportunity to continuously assist their clients over a period of many years and not just at a single point of time.

Alternatively, that the corresponding map T < sup > 2 </ sup >→ S < sup > 2 </ sup > from the torus T < sup > 2 </ sup > to the sphere S < sup > 2 </ sup > ( given by the point with given azimuth and elevation ) is not a covering map at these points.

In fact, the fraction fields of the rings of regular functions on any open set will be the same, so we define, for any U, K < sub > X </ sub >( U ) to be the common fraction field of any ring of regular functions on any open affine subset of X. Alternatively, one can define the function field in this case to be the local ring of the generic point.

Alternatively, on any day during the Fringe the pedestrianised area of the High Street around St Giles ' Cathedral and the Fringe Office becomes the focal point for theatre companies to hand out flyers, perform scenes from their shows, and attempt to sell tickets.

where the integral is over a path starting at the point at infinity with argument -( 1 / 3 ) π and ending at the point at infinity with argument ( 1 / 3 ) π. Alternatively, we can use the differential equation to extend Ai ( x ) and Bi ( x ) to entire functions on the complex plane.

Alternatively, Pym may die in the retelling of the story at precisely the same point he should have died during the actual adventure.

Alternatively, a variant scoring system determines the winner via a point system — 1 point for a knock out, 2 points for a propaganda knock out, 3 points for a retaliation knock out, a variable number of points for position depending on number of players, and finally 2 points for surviving ( with the survivor not necessarily being the points winner ).

< p > Alternatively < em > emphasized </ em > < strong >< em > strong emphasized </ em > strong </ strong ></ p >

Alternatively, variable B could be collapsed out entirely, alternately sampling from p ( A | C ) and p ( C | A ) and not sampling over B at all.

Alternatively, you could compute the dimension of the nullspace of to be p = 1, and thus there are m-p = 1 generalized eigenvectors.

Alternatively, if characters ' attribute ranks are close, the weaker character can try to change the relevant attribute by changing the nature of the conflict.

Alternatively, the rock picker can feel underneath rocks at low tides for abalone.

Alternatively they can be organized by whether the raw stock is of natural, or synthetic origin, or by their starting physical phase.

Alternatively, if the boards are pre-dealt, " curtain cards " may be supplied which have each hand printed on them, so that each player can check at the beginning of the deal that he has the right cards.

Alternatively SCSI Enclosure Services can be used.

Alternatively, she can seek a surgical solution.

Alternatively, there may further be requirements for maximum terms that a government can keep power before holding an election.

Alternatively, rasterization can be done in a more complicated manner by first rendering the vertices of a face and then rendering the pixels of that face as a blending of the vertex colors.

Alternatively, diagnosis and strain identification can be made using polymerase chain reaction ( PCR ).

Alternatively, the business can be sold as a going concern with the net proceeds of the sale distributed to creditors ratably in accordance with statutory priorities.

Alternatively, dualism can mean the tendency of humans to perceive and understand the world as being divided into two overarching categories.

Alternatively, if positive pressures are required, standard glassware can not be used, energy must be used for pressurization and there is a higher chance of side reactions occurring in the distillation, such as decomposition, due to the higher temperatures required to effect boiling.

Alternatively, a single hostname may correspond to many IP addresses: this can facilitate fault tolerance and load distribution, and also allows a site to move physical locations seamlessly.

Alternatively, and especially in connection with the relational model of database management, the relation between attributes drawn from a specified set of domains can be seen as being primary.

Alternatively, the method can be used to value the company based on the value of total invested capital.

Alternatively, they can be fitted with tom heads and tuned as shallow concert toms.

Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly.

Alternatively, the image can be photographically recorded by exposing a photographic film or plate directly to the electron beam, or a high-resolution phosphor may be coupled by means of a lens optical system or a fibre optic light-guide to the sensor of a CCD ( charge-coupled device ) camera.

Alternatively entropy can be defined from a classical thermodynamics viewpoint, in which the molecular interactions are not considered and instead the system is viewed from perspective of the gross motion of very large masses of molecules and the behavior of individual molecules is averaged and obscured.

Alternatively, for an orientable surface the formula can be given in terms of the genus of a surface, g:

Alternatively, gdbserver can be used to remotely debug the program without needing to change it in any way.

Alternatively, they can be termed when they are designed for combat as opposed to iaitō training swords.

Alternatively, it can be thought of as maximizing the minimum gain ( maximin ).