Since we require the process to be adiabatic, the following equation needs to be true

Since this is a second-order differential equation, there must be two linearly independent solutions.

Since the separation of variables in this case involves dividing by y, we must check if the constant function y = 0 is a solution of the original equation.

Since the firearm is also a variable in the accuracy equation, careful tuning of the load to a particular firearm can yield significant

Since the left-hand side of this equation is a series of non-negative numbers, and that it converges to

Since E < sub > 2 </ sub >-E < sub > 1 </ sub > ≫ kT, it follows that the argument of the exponential in the equation above is a large negative number, and as such N < sub > 2 </ sub >/ N < sub > 1 </ sub > is vanishingly small ; i. e., there are almost no atoms in the excited state.

Since the Laplace operator appears in the heat equation, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition.

Since E and P are defined separately, this equation can be used to define D. The physical interpretation of D is not as clear as E ( effectively the field applied to the material ) or P ( induced field due to the dipoles in the material ), but still serves as a convenient mathematical simplification, since Maxwell's equations can be simplified in terms of free charges and currents.

Since temperature in principle also depends on altitude, one needs a second equation to determine the pressure as a function of height, as discussed in the next section.

Since the heat exchanged is related to the entropy change by, the equation governing the temperature as a function of height for a thoroughly mixed atmosphere is

Since the frequency range of the typical noise experiment ( e. g. 1 Hz – 1 kHz ) is low compared with typical microscopic " attempt frequencies " ( e. g. 10 < sup > 14 </ sup > Hz ), the exponential factors in the Arrhenius equation for the rates are large.

Since the previous equation does not uniquely define, one common approach is to separate each step of the process in two sub-steps ; the proposal and the acceptance-refusal.

Since these satisfy the Schrödinger equation-the solutions to Schrödinger's equation for a given situation will not only be the plane waves used to obtain it, but any wavefunctions which satisfy the Schrödinger's equation prescribed by the system, in addition to the relevant boundary conditions.

( Since activity coefficients tend to unity at low concentrations, activities in the Nernst equation are frequently replaced by simple concentrations.

Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Since the 1970s, the equation has been changed to introduce the role of inflationary expectations ( or the expected inflation rate, gP < sup > ex </ sup >).

Since the wavenumber k, is the inverse of wavelength ( k = 1 / λ ) the previous equation can be rewritten as

Since velocity changes instantly in this formalism, the Wiener equation is not suitable for short time scales.

Since equations ( 1 ) through ( 4 ) above cannot be satisfied simultaneously when the hidden variable, λ, takes the same value in each equation, GHSZ proceed by allowing λ to take different values in each equation.

Since an nth degree polynomial equation can only have n distinct roots, this implies that the powers of a primitive root z, z < sup > 2 </ sup >, ... z < sup > n − 1 </ sup >, z < sup > n </ sup >

Since the gas and liquid volumes are functions of < i > P < sub > V </ sub ></ i > and T only, this equation is then solved numerically to obtain < i > P < sub > V </ sub ></ i > as a function of temperature ( and number of particles N ), which may then be used to determine the gas and liquid volumes.

Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are real valued as well as those that have complex values.

Since momentum of a body is defined as its mass multiplied by its velocity, we can rewrite the above equation as:

Since this holds true when performing addition on any real numbers, we say that " addition of real numbers is an associative operation.

Since the rule was reinstated in its present form, Gil Hodges of the Dodgers holds the record for most sacrifice flies in one season with 19, in 1954 ; Eddie Murray holds the record for most sacrifice flies in a career with 128.

Since pure consequentialism holds that an action is to be judged solely by its result, most consequentialist theories hold that a deliberate action is no different from a deliberate decision not to act.

Since 2008 a take-over is in progress: the Schaeffler Group from Herzogenaurach ( Bavaria ) holds the majority of the stock but were required due to the financial crisis to deposit the options as securities at banks.

Since there is only one 0-tuple, the so-called empty tuple ( ), there are only two zero-place relations: the one that always holds, and the one that never holds.

Since all contact between the wrestlers must cease if any part of the body is touching, or underneath, the ropes, many wrestlers will attempt to break submission holds by deliberately grabbing the bottom ropes.

The transfer principle states that true first order statements about R are also valid in * R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals ; since R is a real closed field, so is * R. Since for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś ' theorem of 1955.

Since a limited nuclear war is a viable option for a NUTS theorist, the power to unleash such attacks holds a great deal of appeal.

Discussing the role of fortune-telling in society, Ronald H. Isaacs, an American rabbi and author, opined, “ Since time immemorial humans have longed to learn that which the future holds for them.

* Since 2004 the Colombian town of Sutamarchán holds a similar event on 15 June when a surplus of tomatoes is harvested.

Since 2008 they have played at the Stade de l ' Épopée, a stadium which holds about 12, 000 spectators.

Since 1998, every year the city of Port Hueneme holds a theme-based beach festival at 550 E. Surfside Dr., Port Hueneme, Ca 93041.

Since no actual device holds perfectly equal and opposite charges on each of the two " plates ", it is the mutual capacitance that is reported on capacitors.

Since this holds for any region Ω, the integrand must be zero

Since Christian tradition holds that Jesus was born at night ( based in Luke 2: 6-8 ), Midnight Mass is celebrated on Christmas Eve, traditionally at midnight, in commemoration of his birth.

Since this holds for all smooth regions V, it can be shown that this implies

Since then, Lillestrøm holds an annual 4 day city fair, which is entirely free of charge.

Since 1980 he has been a Senior Fellow of the Hoover Institution at Stanford University, where he holds a fellowship named after Rose and Milton Friedman, his mentor.

Since a defense is raised by the defendant in a direct attempt to avoid what would otherwise result in liability, the defendant typically holds the burden of proof.

Since the Local Government Act 1972, the Duchy holds and exerts the right to appoint Sheriffs and Lords Lieutenant in the ceremonial counties of Greater Manchester, Merseyside and Lancashire, which include areas from the historic county boundaries of Cheshire and Yorkshire.

Since February 2007, Boston College has submetered electricity in 24 of its residence halls and since 2008 the college holds an annual energy competition between residential halls.

Since its reintroduction, the European Grand Prix is usually held in a country that also holds a national Grand Prix in that same year.

Since the hospital closed its doors, a popular local legend holds that a stained glass dome in the Hall Rotunda was salvaged from the Temple of Music at Buffalo's Pan-American Exposition.