If matrix support is polar ( e. g. paper, silica etc.

Formulae connecting a tangential angle, the angle anchored at the ellipse's center ( called also the polar angle from the ellipse center ), and the parametric angle t < ref > If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, the angle is the geocentric latitude, and parametric angle t is a parametric ( or reduced ) latitude of auxiliary circle </ ref > are:

If instead we use polar coordinates with the origin at one focus, with the angular coordinate still measured from the major axis, the ellipse's equation is

If Earth were smaller, a thinner atmosphere would allow temperature extremes, thus preventing the accumulation of water except in polar ice caps ( as on Mars ).

If B is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function ( of integer order ) of the radial component.

If the polar vertex is " pulled down " 90 °, so that the vertex is on the equator, or transverse diameter, then it becomes the transverse graticule, upon which all spherical trigonometry is ultimately based ( if the longitudinal vertex is between the poles and equator, then it is considered an oblique graticule ).

If eaten in one meal, 30 to 90 grams of polar bear liver is enough to kill a human being, or to make even sled dogs very ill.

If z is a complex number, written in polar form as

If a curve is defined in polar coordinates as, then its curvature is

There exists a link between sudden stratospheric warmings and the quasi-biennial oscillation: If the QBO is in its easterly ( westward ) phase, the atmospheric waveguide is modified in such a way that upward-propagating Rossby waves are focused on the polar vortex, intensifying their interaction with the mean flow.

If the result is between 0. 4 and 1. 7 then, generally, the bond is polar covalent.

If a complex number is represented in polar form z = re < sup > iθ </ sup >, then the logarithm of z is

If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition.

If C < sub > 1 </ sub > and C < sub > 2 </ sub > are given in polar coordinates by and respectively, then the equation describes the cissoid of C < sub > 1 </ sub > and C < sub > 2 </ sub > relative to the origin.

If the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ.

If dinosaurs found a similar haven in which they outlived the rest of their kind, then we think polar Gondwana, including southeastern Australia, is a likely place to look for it.

If the metabolites of phase I reactions are sufficiently polar, they may be readily excreted at this point.

If the body lacks cylinder ( axial ) symmetry, a last rotation around its z-axis ( which has polar coordinates and ) is necessary to specify its orientation completely.

If a chemical is very nonpolar it will not dissolve at all in a very polar solvent.

If the filter paper has small pores, long chained non polar liquids such as oil may clog the filter paper.

If the polar distance of the Sun is equal to the observer's latitude, the shadow path of a gnomon's tip on a sundial will be a parabola ; at higher latitudes it will be an ellipse and lower, a hyperbola.

If the introduced surface had an ionic or polar nature, there would be water molecules standing upright on 1 ( along the axis of an orbital for ionic bond ) or 2 ( along a resultant polarity axis ) of the 4 sp3 orbitals.

If the mobile phase is changed to a more polar solvent or mixture of solvents, it is more capable of dispelling solutes from the silica binding places and all compounds on the TLC plate will move higher up the plate.

If a circle with radius a is rotated about an axis at distance b from its center, then the equation of the resulting hippopede in polar coordinates

If F ( r ) represents gravity, it is a negative term proportional to 1 / r < sup > 2 </ sup >, so the net acceleration in r in the rotating frame depends on a difference of reciprocal square and reciprocal cube terms, which are in balance in a circular orbit but otherwise typically not.

If chances of meeting another reciprocal altruist are high enough or the game is repeated for a long enough amount of time, this form of altruism can evolve within a population.

If z is replaced by the negative reciprocal of its complex conjugate, then the functions g < sub > 1 </ sub >, g < sub > 2 </ sub >, and g < sub > 3 </ sub > of z are left unchanged.

If all firms have an equal share the reciprocal of the index shows the number of firms in the industry.

If the event being measured repeats itself with sufficient stability and the frequency is considerably lower than that of the clock oscillator being used, the resolution of the measurement can be greatly improved by measuring the time required for an entire number of cycles, rather than counting the number of entire cycles observed for a pre-set duration ( often referred to as the reciprocal technique ).

If a lease exists at the sole discretion of the landlord, the law of the jurisdiction may imply that the tenant is granted, by operation of law, a reciprocal right to terminate the lease at will.

If a landlord can terminate the tenancy at will, a tenant by operation of law is also granted a reciprocal right to terminate at will.

If the exponent here is given in NAF form a digit value 1 implies a multiplication by the base, and a digit value-1 by its reciprocal.

If the index is calculated from the most recent time period to the earlier time period, it should be the reciprocal of the index found going from the earlier period to the more recent.

If this is then compared at progressive offsets with the known letter-sequence of an Enigma rotor, quite a few possibilities are discounted due to violating either the " reciprocal " property or the " no-self-ciphering " property of the Enigma machine:

( If one is obligated to partake of a festive meal that day, such as if he is the father of an infant on the day of circumcision, this opinion requires him to undertake a reciprocal fast at the soonest opportunity.

If we reciprocate such a polyhedron about its intersphere, the dual polyhedron will share the same edge-tangency points and so must also be canonical ; it is the canonical dual, and the two together form a canonical dual compound.

If they act otherwise, let them be subject to the canonical penalties.

If he has presumed to do this in the past or shall presume to do so in the future, his action shall be null and he shall be subject to the canonical penalties.

If pedobaptists accept this text as canonical, they can still point out that the second clause mentions believing, but not baptism.

If a diagram F: J → C has a limit in C, denoted by lim F, there is a canonical isomorphism

If it does, however, it is unique in a strong sense: given another direct limit X ′ there exists a unique isomorphism X ′ → X commuting with the canonical morphisms.

The decree proceeded to affirm, after listing the books of the Bible according to the Roman Catholic canon, that " If anyone receive not, as sacred and canonical, the said books entire with all their parts, as they have been used to be read in the Catholic Church, and as they are contained in the old Latin Vulgate edition, and knowingly and deliberately condemn the traditions aforesaid ; let him be anathema.

If R is a ring and S is a subset, consider all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit.

From a Catholic canonical point of view, provisions of the joint synodal decree are fully consistent with the provisions of canon 671 of the 1991 Code of Canons of the Eastern Churches, which states: " If necessity requires it or genuine spiritual advantage suggests it and provided that the danger of error or indifferentism is avoided, it is permitted for Catholic Christian faithful, for whom it is physically or morally impossible to approach a Catholic minister, to receive the sacraments of penance, the Eucharist and anointing of the sick from non-Catholic ministers, in whose Churches these sacraments are valid.

If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL ( n, C ).

If one fail to do this, the problem is that ( for example ) instead of evolving a given three-dimensional manifold into one of Thurston's canonical forms, we might just shrink its size.

If R is a quotient of by a homogeneous ideal, then the canonical surjection induces the closed immersion

If is the twisting sheaf of Serre on, we let denote the pullback of to ; that is, for the canonical map

If C has a zero object 0, given two objects X and Y in C, there are canonical morphisms f: 0 → X and g: Y → 0.

If η ( θ ) = θ, then the exponential family is said to be in canonical form.

If we have two sets of variables, and, and there are correlations among the variables, then canonical correlation analysis will enable us to find linear combinations of the's and the's which have maximum correlation with each other.

If K < sub > v </ sub > is a non-archimedean local field, there is a canonical isomorphism inv < sub > v </ sub >: Br ( K < sub > v </ sub >) → Q / Z constructed in local class field theory.

If is the canonical vector-valued 1-form on the frame bundle,

If the transformation is canonical, these two must be equal, resulting in the equations

If we accept Helmut Koester's theory that the canonical Mark is a revision of Secret Mark, another explanation is possible: namely, that the ancient editor who deleted an earlier encounter of Jesus with such a young man in a cloth, then added this incident also involving a young man during Jesus ' arrest.

If Secret Mark is dependent on all four canonical gospels, then it was written after John and could not have been used as a source by John's author.

If what is portrayed in Secret Mark is indeed a baptism, then the placement of this story within the canonical Mark is highly significant.