If **ya is the multiple secant of **zg which passes through Af and **yb is the simple secant of **zg which passes through Af, and if Af are the points in which **ya meets **zg, and if Af is the image of Af on the generator **yb, it follows that the image of the line Af is Af.

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

If a sheet is for multiple days it shows a conversion table to convert from weekday to date and back.

If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

If multiple transiting planets exist in one system, then this method can be used to confirm their existence.

If this sounds indulgent, it allows simultaneity, absolute space and time and a deterministic universe ( along with decoherence theory ) whilst the status-quo permits time travel / causality paradoxes, subjectivity in the measurement process and multiple universes.

If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different.

If a and b are not both zero, the greatest common divisor of a and b can be computed by using least common multiple ( lcm ) of a and b:

If we are considering multiple operations on a set, then all operations must be preserved for a function to be considered as a homomorphism.

In arithmetic and number theory, the least common multiple ( also called the lowest common multiple or smallest common multiple ) of two integers a and b, usually denoted by LCM ( a, b ), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM ( a, b ) is defined to be zero.

On the other hand, LDAP does not define transactions of multiple operations: If you read an entry and then modify it, another client may have updated the entry in the meantime.

# If p is an odd prime, then any prime q that divides 2 < sup > p </ sup > − 1 must be 1 plus a multiple of 2p.

If it is desirable to encode more text than will fit in an encoded-word of 75 characters, multiple encoded-words ( separated by CRLF SPACE ) may be used.

If the hardware provides multiple rings or CPU modes, the microkernel is the only software executing at the most privileged level ( generally referred to as supervisor or kernel mode ).

In addition, multiple continuities are visited in the comic book series What If, What The --?!

If a text ( such as this one ) discusses multiple bases, and if ambiguity exists, the base ( itself represented in base 10 ) is added in subscript to the right of the number, like this: number < sub > base </ sub >.

If a language uses multiple writing systems, it may have distinct orthographies, as is the case with Kurdish, Uyghur, Serbian, Inuktitut and Turkish.

If multiple vendors agree to implement the same functionality using the same API, a shared extension may be released, using the identifier EXT.

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

If a marriage includes multiple husbands and wives, it can be called group marriage.

If T is a good symmetry of the dynamics, then

If solutions that differ only by symmetry operations ( rotations and reflections ) of the board are counted as one, the puzzle has 12 unique ( or fundamental ) solutions.

If Lorentz symmetry can cease to be a fundamental symmetry at Planck scale or at some other fundamental scale, it is conceivable that particles with a critical speed different from the speed of light be the ultimate constituents of matter.

If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

If inversion through the center of symmetry in a molecule results in the same phases for the molecular orbital, then the MO is said to have gerade, g, symmetry.

If inversion through the center of symmetry in a molecule results in a phase change for the molecular orbital, then the MO is said to have ungerade, u, symmetry.

* If the molecule has some symmetry, the degenerate atomic orbitals ( with the same atomic energy ) are grouped in linear combinations ( called symmetry-adapted atomic orbitals ( SO )), which belong to the representation of the symmetry group, so the wave functions that describe the group are known as symmetry-adapted linear combinations ( SALC ).

If a point X is located on a parabola which has focal length and if is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from and as follows, assuming they are all expressed in the same units.

If one considers a version of QCD with N < sub > f </ sub > flavors of massless quarks, then there is a global ( chiral ) flavor symmetry group SU < sub > L </ sub >( N < sub > f </ sub >) × SU < sub > R </ sub >( N < sub > f </ sub >) × U < sub > B </ sub >( 1 ) × U < sub > A </ sub >( 1 ).

If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in wider contexts ; see below.

If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axis aligns to the field.

If, in addition to treating the queens as identical, rotations and reflections of the board were allowed, we would have only 12 distinct solutions up to symmetry, signifying that two arrangements that are symmetrical to each other are considered equivalent.

If the symmetry resolution phases that are usually found at the start of the game could somehow be put off for later, then one may readily see how similar they are to the castling maneuvers in Chess.

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.

If the symmetry is specially d < sub > x < sup > 2 </ sup >- y < sup > 2 </ sup ></ sub > then the penetration depth should vary linearly with T at low temperatures.

If a pattern is found that is not known in the diffraction data libraries an attempt can be made to index the pattern, i. e. to identify the symmetry and the size of the unit cell.

If one combines this with the symmetry of the stress – energy tensor, one can show that angular momentum is also conserved,

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

If n is even there are n / 2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices.

If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.

If it is assumed that the formula given by Lodge of cosec Af applies, the pressure difference along the major axes can be calculated from the angle of inclination of the major axis, and from this the interfacial tension can be calculated.

# If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively ( like altitude on a contour map ), then as the eye moves towards the axes ( towards the south-west ), it sees the density of isotherms stay constant, but it sees the density of adiabats grow.

If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes.

# If all the particles are identical ( same energy barrier and same magnetic moment ), their easy axes are all oriented parallel to the applied field and the temperature is low enough ( T < sub > B </ sub > < T ≲ KV /( 10 k < sub > B </ sub >)), then the magnetization of the assembly is.

# If all the particles are identical and the temperature is high enough ( T ≳ KV / k < sub > B </ sub >), then, irrespective of the orientations of the easy axes:

If we let Δx and Δy be the distances ( along the x and y axes, respectively ) between two points on a curve, then the slope given by the above definition,

If the axis of polarization of the incident wave is chosen so that it makes a 45 ° with the fast and slow axes of the waveplate, then E < sub > f </ sub > = E < sub > s </ sub > ≡ E, and the resulting wave upon exiting the waveplate is

If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal ( due to the spectral theorem ) and represent the directions of the axes of the ellipsoid: the lengths of the semiaxes are given by the eigenvalues.

Prior to any measurements being made, the polarizations of the photons are indeterminate ; If a measurement is made on one of the photons using a two-channel polarizer aligned with the axes of the coordinate system, the photon assumes either horizontal or vertical polarization, with 50 % probability for each orientation, and the other two photons immediately assume the identical polarization.

If you ignore the effects of gravity, then there are ten basic ways of doing such shifts: translation through time, translation through any of the three dimensions of space, rotation ( by a fixed angle ) around any of the three spatial axes, or a boost in any of the three spatial directions.

If that pendulum can move about two axes ( in a circular or elliptical shape ), due to the principle of superposition, the motion of a rod connected to the bottom of the pendulum along one axes will be described by the equation

If the crystals of both minerals are well formed so that the directions of the crystallographic axes are clear then the epitaxic relationship can be deduced just by a visual inspection.

If the followers are native honey-hunters, when they reach the hive they incapacitate the adult bees with smoke and open the hive with axes or pangas ( machetes ).

If both axes are used simultaneously, this type of dolly shot is known as a compound move.

If the imaging system produces an inverted image, we may simply regard the image plane coordinate axes as being reversed from the object plane axes.

If the linkage has four hinged joints with axes angled to intersect in a single point, then the links move on concentric spheres and the assembly is called a spherical four-bar linkage.

If the crystal symmetry is monoclinic or triclinic, even the angles between these axes are subject to thermal changes.