where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

where k has the same p, and opposite σ and sign of the energy.

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

This involves an auxiliary regression, wherein the residuals obtained from estimating the model of interest are regressed on ( a ) the original regressors and ( b ) k lags of the residuals, where k is the order of the test.

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with ; such expressions are then still called binomial coefficients.

Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects ; more formally, the number of k-element subsets ( or k-combinations ) of an n-element set.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

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

The same number however occurs in many other mathematical contexts, where it is denoted by ( often read as " n choose k "); notably it occurs as coefficient in the binomial formula, hence its name binomial coefficient.

where i, j, and k are the unit vectors for the x -, y -, and z-axes, respectively.

where e < sub > k </ sub > are the coordinate vector fields.

That is a non-deterministic Turing machine with a tape of only kn cells, where n is the size of the input and k is a constant associated with the machine.

where E < sub > a </ sub > is the activation energy and k < sub > B </ sub > is the Boltzmann constant.

where and h and k are the distances between the successive pentagons on the virus surface for each axis ( see figure on right ).

After completing high school Doppler studied philosophy in Salzburg and mathematics and physics at the k. k. Polytechnisches Institut ( now Vienna University of Technology ) where he worked as an assistant since 1829.

The term " discothèque " was coined in Europe to describe clubs where there was no live music played ( a. k. a. disk-only events ).

where m is the mass and k is the spring constant that represents a measure of spring stiffness.

where a is the radius of the circular aperture, k is equal to 2π / λ and J < sub > 1 </ sub > is a Bessel function.

But California is where we're goin ''.

He thought of the jungles below him, and of the wild, strange, untracked beauty there and he promised himself that someday he would return, on foot perhaps, to hunt in this last corner of the world where man is sometimes himself the hunted, and animals the lords.

Now I wish to enter the American market, where the competition is very strong.

from downstream, where the water level is much lower, it is a high, elaborately facaded pavilion.

The answers derived by these means may determine not only the temporal organization of the dance but also its spatial design, special slips designating the location on the stage where the movement is to be performed.

Even in domains where detailed and predictive understanding is still lacking, but where some explanations are possible, as with lightning and weather and earthquakes, the appropriate kind of human action has been more adequately indicated.

Robert Penn Warren puts it this way in `` Brother To Dragons '': `` The recognition of complicity is the beginning of innocence '', where innocence, I think, means about the same thing as redemption.

A dominant motive is the poet's longing for his homeland and its boyhood associations: `` Not men-folk, but the fields where I would stray, The stones where as a child I used to play ''.

In a society where everything is for sale, Marlowe is the only man who cannot be bought.

This is done for simplicity of commands and to bring the hidden redundancy up to where misunderstanding has almost zero possibility.

As to benefits to employees, it is notorious for its callous disregard except where it depends on them for services.

`` The Moral Creed '' and `` The Will To Risk '' live happily together, if we do not examine where the line is to be drawn.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Ptolemy's problem is to forecast where, against the inverted bowl of night, some particular light will be found at future times.

She was exposing herself to temptation which it is best to avoid where it can consistently be done.

Such performance is a great tribute to American scientists and engineers, who in the past five years have had to telescope time and technology to develop these long-range ballistic missiles, where America had none before.

but it is perennially restructured in situations where anchors are dragged or lost.

`` History has this in common with every other science: that the historian is not allowed to claim any single piece of knowledge, except where he can justify his claim by exhibiting to himself in the first place, and secondly to any one else who is both able and willing to follow his demonstration, the grounds upon which it is based.

Finally we may note that the idea appears in educational theory where its influence is at present widespread.

Several species of reptiles have temperature-dependent sex determination, where incubation temperature of eggs determines the sex of the individual.

The theory predicts a linear dependence of Af on Af, where J is the experimentally determined Curie-Weiss constant.

Experiment where p is not constant.

Because the range of possible values for K < sub > a </ sub > spans many orders of magnitude, a more manageable constant, pK < sub > a </ sub > is more frequently used, where pK < sub > a </ sub >

We have a language where is a constant symbol and is a unary function and the following axioms:

where is generally taken to be the duration of the time-history, or the number of points if is constant.

where is the specific heat at constant pressure, is the specific heat at constant volume, and is the wave speed.

where is a constant.

where E < sub > 0 </ sub > is an infinite negative constant.

where < big ></ big > is the number of degrees of freedom divided by two, R is the universal gas constant and n is the number of moles in the system ( a constant ).

For a vector with linear addressing, the element with index i is located at the address B + c · i, where B is a fixed base address and c a fixed constant, sometimes called the address increment or stride.

This means that in most real-world setups where machines are added a few at a time, only one or two tries are needed before the address effectively become constant.

In transition regions, where this pressure dependent dissociation is incomplete, both the differential, constant pressure heat capacity and beta ( the volume / pressure differential ratio ) will greatly increase.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

For any system, the following restrictions on measurement results apply, where is the reduced Planck constant and is any direction vector such as x, y, or z:

Detailed analysis shows that the complexity class is unchanged by allowing error as high as 1 / 2 − n < sup >− c </ sup > on the one hand, or requiring error as small as 2 < sup >− n < sup > c </ sup ></ sup > on the other hand, where c is any positive constant, and n is the length of input.

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.

where is the difference in density of particles separated by a height difference of, is Boltzmann's constant ( namely, the ratio of the universal gas constant,, to Avogadro's number, ), and is the absolute temperature.

where A < sub > α </ sub > and B < sub > α </ sub > are any two solutions of Bessel's equation, and C < sub > α </ sub > is a constant independent of x ( which depends on α and on the particular Bessel functions considered ).

where Ω is the constant rate of rotation of the water.

What was even worse was his constant discrimination against the central and northern regions of Chad, where the southern Chadian administrators came to be perceived as arrogant and incompetent.