functions as real combinations of spherical harmonics Y < sub > lm </ sub >( θ, φ ) ( where l and m are quantum numbers ).

On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ.

where θ is the true angle SEE ′, is the apparent angle S ′ EE ′, and is the relative speed between the presumed fixed frame of reference ( such as heliocentric ) and the observer's one.

where θ < sub > r, p </ sub > is the angle between r and p measured from r to p ; an important distinction because without it, the sign of the cross product would be meaningless.

Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis of a Cartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±, where b = a × tan ( θ ) and where θ is the angle between the transverse axis and either asymptote.

The contact angle, θ < sub > C </ sub >, is the angle formed by a liquid at the three-phase boundary where the liquid, gas, and solid intersect.

: where E is the energy, τ is magnitude of the torque, and θ is the angle moved ( in radians ).

where μ is the location parameter, θ represents additional parameters, and is a function of the additional parameters.

: where θ is the angle of inclination of the normal, and p is the length of the normal.

where A and θ < sub > 0 </ sub > are arbitrary constants.

where θ < sub > 1 </ sub > and θ < sub > 2 </ sub > are the angles of incidence of a ray crossing the interface between two media with refractive indices n < sub > 1 </ sub > and n < sub > 2 </ sub >.

where R is the distance to the particle, θ is the scattering angle, n is the refractive index of the particle, and d is the diameter of the particle.

The standard convention ( r, θ, φ ) conflicts with the usual notation for the two-dimensional polar coordinates, where θ is often used for the azimuth.

where r is the distance from the axis of rotation to the particle, F is the magnitude of the force applied, and θ is the angle between the position and force vectors.

where E is the energy, τ is magnitude of the torque, and θ is the angle moved ( in radians ).

where W is work, τ is torque, and θ < sub > 1 </ sub > and θ < sub > 2 </ sub > represent ( respectively ) the initial and final angular positions of the body.

where ( r, θ, φ ) correspond to a spherical coordinate system.

where the numerical aperture is defined as for θ being the half-angle of the cone of rays accepted by the microscope objective.

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

where A ∈ N ( i. e., A is a single nonterminal ), α, β ∈ ( N U Σ )* ( i. e., α and β are strings of nonterminals and terminals ) and γ ∈ ( N U Σ )< sup >+</ sup > ( i. e., γ is a nonempty string of nonterminals and terminals ).

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x, y ∈ N then x · y ∈ N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N = N *, where the superscript * is the Kleene star: the set is closed under composition or concatenation of its elements.

Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E ( n ) ( the isometry group of R < sup > n </ sup >), where two subgroups H < sub > 1 </ sub >, H < sub > 2 </ sub > of a group G are conjugate, if there exists g ∈ G such that H < sub > 1 </ sub >= g < sup >− 1 </ sup > H < sub > 2 </ sub > g.

Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.

where we have used the symbol e < sub >( v, w )</ sub > to emphasize that these are taken to be linearly independent by definition for distinct ( v, w ) ∈ V × W.

A subbase for the weak topology is the collection of sets of the form φ < sup >- 1 </ sup >( U ) where φ ∈ X < sup >*</ sup > and U is an open subset of the base field K. In other words, a subset of X is open in the weak topology if and only if it can be written as a union of ( possibly infinitely many ) sets, each of which is an intersection of finitely many sets of the form φ < sup >- 1 </ sup >( U ).

where p, q ∈ C and γ ∈ R are constants and i is the imaginary unit.

The several variable case can be further generalised by taking non-commuting variables X < sub > i </ sub > for i ∈ I, where I is an index set and then a monomial X < sup > α </ sup > is any word in the X < sub > I </ sub >; a formal power series in X < sub > I </ sub > with coefficients in a ring R is determined by any mapping from the set of monomials X < sup > α </ sup > to a corresponding coefficient c < sub > α </ sub >, and is denoted.

Any quadratic polynomial over the complex numbers ( polynomials of the form where,, and ∈ ) can be factored into an expression with the form using the quadratic formula.

where t ∈ R, V ∈ TM and denotes the geodesic with initial data.

: An automaton reads a finite string of symbols a < sub > 1 </ sub >, a < sub > 2 </ sub >,...., a < sub > n </ sub >, where a < sub > i </ sub > ∈ Σ, which is called an input word.

: A sequence of states q < sub > 0 </ sub >, q < sub > 1 </ sub >, q < sub > 2 </ sub >,...., q < sub > n </ sub >, where q < sub > i </ sub > ∈ Q such that q < sub > 0 </ sub > is the start state and q < sub > i </ sub > = δ ( q < sub > i-1 </ sub >, a < sub > i </ sub >) for 0 < i ≤ n, is a run of the automaton on an input word w

For example, a smooth curve α ( t ): 1 → M has tangent vector α ′( t < sub > 0 </ sub >) in the tangent space TM ( α ( t < sub > 0 </ sub >)) at any point t < sub > 0 </ sub > ∈ ( 0, 1 ), and each such vector has length ‖ α ′( t < sub > 0 </ sub >)‖, where ‖·‖ denotes the norm induced by the inner product on TM ( α ( t < sub > 0 </ sub >)).

where represent the coordinates of w, v ∈ V in some particular F < sub > q² </ sub >- basis of the n-dimensional space V.

where ε is the augmentation map, is a free ZG-resolution of Z ( where Z is equipped with the trivial ZG-module structure, g m = m for every g ∈ G and every m ∈ Z ).

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

We have just observed that we can write Af where D is diagonalizable and N is nilpotent, and where D and N not only commute but are polynomials in T.

From Fig. 6 the relationship between these parameters can readily be derived and the cutting force is Af where **yl is the shear strength of the coating and is a parameter of the coatings material, W is the width of the removed coating and T is its thickness.

In an adiabatic irreversible process, dQ = 0 is not equal to TdS ( TdS > 0 ) where Q is thermal energy, T is temperature, and S is entropy.

where T is an absolute temperature.

where k < sub > B </ sub > is Boltzmann's constant, T is the temperature, R is the resistance, and is the bandwidth of the frequency.

auxiliary regression is TR < sup > 2 </ sup >, where T is the sample size and R < sup > 2 </ sup > is the coefficient of determination.

* AT & T Mobility-In areas where AT & T Mobility previously had D-AMPS operating on 1900 MHz frequencies, no analog AMPS network existed, and the D-AMPS network on the 1900 MHz frequency was shut down in mid-2007.

where is the Boltzmann constant, T is temperature ( assumed to be a well-defined quantity ), is the degeneracy ( meaning, the number of levels having energy ; sometimes, the more general ' states ' are used instead of levels, to avoid using degeneracy in the equation ), N is the total number of particles and Z ( T ) is the partition function.

While this is intended as a clever con trick, the machine, surprisingly, works, sending Blackadder and Baldrick back to the time of the dinosaurs, where they manage to cause the extinction of the dinosaurs, through the use of Baldrick's best, worst and only pair of underpants as a weapon against a hungry T. Rex.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

where A, B, S and T are active masses and k < sub >+</ sub > and k < sub >−</ sub > are rate constants.

Badlands near Drumheller, Alberta, where erosion has exposed the K – T boundary

( V, T, P, S ), where N / V is the Non-terminal Variable, and Σ / T is the Terminal ) is context-sensitive if all rules in P are of the form

Isolation of T. cruzi can occur by inoculation into mice, by culture in specialized media ( for example, NNN, LIT ); and by xenodiagnosis, where uninfected Reduviidae bugs are fed on the patient's blood, and their gut contents examined for parasites.

In both zones, Chagas occurs almost exclusively in rural areas, where triatomines breed and feed on the over 150 species from 24 families of domestic and wild mammals, as well as humans, that are the natural reservoirs of T. cruzi.

The tension at c is tangent to the curve at c and is therefore horizontal, and it pulls the section to the left so it may be written (− T < sub > 0 </ sub >, 0 ) where T < sub > 0 </ sub > is the magnitude of the force.