As in the case of the isoelectronic nitrate ion, the symmetry can be achieved by a resonance between three structures:

and measure the parameter s from c. Assume r is to the right of c since the other case is implied by symmetry.

In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry

There are further FFT specializations for the cases of real data that have even / odd symmetry, in which case one can gain another factor of ( roughly ) two in time and memory and the DFT becomes the discrete cosine / sine transform ( s ) ( DCT / DST ).

In the case of, conjugate-symmetry reduces to symmetry, and sesquilinear reduces to bilinear.

An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

then there is a point ( 0, f )— the focus, F — such that any point P on the parabola will be equidistant from both the focus and the linear directrix, L. The linear directrix is a line perpendicular to the axis of symmetry of the parabola ( in this case parallel to the x axis ) and passes through the point ( 0 ,- f ).

In the case of a line of symmetry the latter reduces to.

The actual symmetry groups in each of these cases have two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.

There is no figure which has as full symmetry group the circle group, but for a vector field it may apply ( see the 3D case below ).

In the case of a finite automorphism group of the whole space, its order is the order of the symmetry group of the figure multiplied by the number of isomorphic versions of the figure.

which for the common case of circular symmetry where and yields

In that case, the penetration depth also varies exponentially with temperature T. If there are nodes in the energy gap as in the d symmetry HTS, electron pair can more easily be broken, the superfluid density should have a stronger temperature dependence, and the penetration depth is expected to increase as a power of T at low temperatures.

Boy's ( the case p = 3 ) and Morin's ( the case p = 2 ) surfaces are beginning of a sequence of half-way models with higher symmetry first proposed by George Francis, indexed by the even integers 2p ( for p odd, these immersions can be factor through a projective plane ).

# Attractive or repulsive electrostatic interactions between permanent charges ( in the case of molecular ions ), dipoles ( in the case of molecules without inversion center ), quadrupoles ( all molecules with symmetry lower than cubic ), and in general between permanent multipoles.

In either case, there are n axes of symmetry altogether and 2n elements in the symmetry group.

In this case, the principal axes are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken.

In the inequality case, there is no symmetry.

On the other hand, the group T < sub > h </ sub > of pyritohedral symmetry also has 24 members and a subgroup of index 3 ( this time it is a D < sub > 2h </ sub > prismatic symmetry group, see point groups in three dimensions ), but in this case the whole subgroup is a normal subgroup.

To obtain a new equivalence relation or preorder one must take the transitive closure ( reflexivity and symmetry — in the case of equivalence relations — are automatic ).

The Wightman postulate of a unique vacuum state doesn't necessarily make the Wightman axioms inappropriate for the case of spontaneous symmetry breaking because we can always restrict ourselves to a superselection sector.

The arguments advanced by those individuals and groups who oppose the system in force and who would drastically curtail or do away entirely with hospital care for the non-service-connected case, seem to be coldly impractical and out-of-step with the wishes of the general public.

Although biopsies have shown structural changes in some of the reported cases of steroid-induced weakness, this case provides the only example known to us in which necropsy afforded the opportunity for extensive study of multiple muscle groups.

In this case it is primarily a matter of conflict of racial groups rather than social-class groups.

Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test.

In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself.

In the case of phospholipids, the polar groups are considerably larger and more polar, as described below.

The Holy See accepts as valid the ordinations of the Old Catholics in communion with Utrecht, as well as the Polish National Catholic Church ( which received its orders directly from Utrecht, and was — until recently — part of that communion ); but Roman Catholicism does not recognise the orders of any group whose teaching is at variance with what they consider the core tenets of Christianity ; this is the case even though the clergy of the Independent Catholic groups may use the proper ordination ritual.

The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups.

In the case of groups, the morphisms are the group homomorphisms.

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.

The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.

While most Covenant-breakers are involved in schismatic groups, that is not always the case.

In some groups, the slap bass was utilized as band percussion in lieu of a drummer ; such was the case with Bill Haley & His Saddlemen ( the forerunner group to the Comets ), which did not use drummers on recordings and live performances until late 1952 ; prior to this the slap bass was relied on for percussion, including on recordings such as Haley's versions of Rock the Joint and Rocket 88.

The " interventions " advocated and practiced by eugenicists involved prominently the identification and classification of individuals and their families, including the poor, mentally ill, blind, deaf, developmentally disabled, promiscuous women, homosexuals and entire racial groups — such as the Roma and Jews — as " degenerate " or " unfit "; the segregation or institutionalisation of such individuals and groups, their sterilization, euthanasia, and in the extreme case of Nazi Germany, their mass extermination.

A case in point is the Third Ecumenical Council where two groups met as duly called for by the emperor, each claiming to be the legitimate council.

Here, for the case of squares inscribed in plane curves, we remove the restriction to convexity and give certain other results.

In the first case the fixed elements within each pencil are the multiple secant and the line joining the vertex, P, to the intersection of **zg and the plane of the pencil which does not lie on the multiple secant.

For if such were the case, either the plane of the two lines would meet **zg in more than K points or, alternatively, the order of the image regulus of the pencil determined by the two lines would be too high.

In at least one case, a plane was hijacked by the official pilot.

# International common strike in case of the impunity of the hijackers or in case of not letting the crew of the hijacked plane out within 48 hours ;

This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 0 specifies only the single point ( 0, 0 ).

If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).

Centrifugal force arises in the analysis of orbital motion and, more generally, of motion in a central-force field: in the case of a two-body problem, it is easy to convert to an equivalent one-body problem with force directed to or from an origin, and motion in a plane, so we consider only that.

In such a case, the adjacent planes are not straight, but instead bend around the edge of the terminating plane so that the crystal structure is perfectly ordered on either side.

Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex ; the other two ( open and unbounded ) cases are parabolas and hyperbolas.

Rectangular coordinates in the plane can be used intuitively with respect to one's current location, in which case the axis will point to the local North.

In case of plane or spatial coordinates, we typically need several datum points.

In the case of plane or spatial coordinates, datum transformation takes the form of a similarity or Helmert transformation, consisting of a rotation and scaling operation in addition to a simple translation.

In the case of plane geometry ( valid for small areas on the Earth's surface ) the solutions to both problems reduce to simple trigonometry.

This corresponds to the area of the plane covered by the polygon, or to the area of a simple polygon having the same outline as the self-intersecting one ( or, in the case of the cross-quadrilateral, the two simple triangles ).

If the areas of the two parallel faces are A < sub > 1 </ sub > and A < sub > 3 </ sub >, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A < sub > 2 </ sub >, and the height ( the distance between the two parallel faces ) is h, then the volume of the prismatoid is given by ( This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height.

An empty flight case apparently belonging to Clemente was the only personal item recovered from the plane.

The plane of the Moon's orbit around the Earth lies close to the plane of the Earth's orbit around the Sun ( the ecliptic ), rather than in the plane perpendicular to the axis of rotation of the Earth ( the equator ) as is usually the case with planetary satellites.

The Euclidean TSP is a particular case of the metric TSP, since distances in a plane obey the triangle inequality.

In this case, the polarization along one plane is intrinsically tied to the polarization along the other, and by Heisenberg ’ s principle, there is a limit to the certainty with which both states can be known.

In the case of the single-moded plane wave, is a uniform distribution.