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

If the polar reciprocal of the canonical form in the sphere is congruent to the original, then the figure is self-dual.

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 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 the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

If v < sub > n </ sub > is the number of vertices of degree n and D is the maximum degree of any vertex,

If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable.

If there is an edge from a vertex x to a vertex y, then the element is 1 ( or in general the number of xy edges ), otherwise it is 0.

If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other.

If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex ( and either one of the lower ones ).

If loops are permitted, then the graph with a single vertex and one loop is terminal.

If at any step a vertex has ( visited ) neighbours with the same label as itself, then the graph is not bipartite.

If n is odd each axis of symmetry connects the midpoint of one side to the opposite vertex.

For example, given a graph G =( V, E ), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then the path p < sub > 1 </ sub > from u to w and p < sub > 2 </ sub > from w to v are indeed the shortest paths between the corresponding vertices ( by the simple cut-and-paste argument described in CLRS ).

If G is a connected graph with infinitely many vertices such that every vertex has finite degree ( that is, each vertex is adjacent to only finitely many other vertices ) then G contains an infinitely long simple path, that is, a path with no repeated vertices.

If one angle is a right angle, the orthocenter coincides with the vertex of the right angle.

If I needed a bit of colour, I'd add it in the vertex.

If there are no negative-weight cycles, then every shortest path visits each vertex at most once, so at step 3 no further improvements can be made.

If, when a vertex is colored, there exists an edge connecting it to a previously-colored vertex with the same color, then this edge together with the paths in the breadth-first search tree connecting its two endpoints to their lowest common ancestor forms an odd cycle.

If the edges of a graph are thought of as lines drawn from one vertex to another ( as they are usually depicted in illustrations ), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.