Since the Somalis were one main ethnic group, they could only elect a single candidate and the two main fractions of that tribe in Djibouti, the Issa and the Gadabuursi each strove to win.

Since the sum of the mole fractions is equal to one,

Since they were historically referred to as " Suolun people " and spoke Tungus rather than Mongolic language, they may have derived their origins from one or more fractions of the Xianbei or other ethnic groups subjugated by the Xianbei.

Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement.

Since cutting the edge of a target ring will result in scoring the higher score, fractions of an inch are important.

D for every x in F, then D is said to be a valuation ring for the field F or a place of F. Since F is in this case indeed the field of fractions of D, a valuation ring for a field is a valuation ring.

Since it can accurately measure fractions of nanometers, it could help standardize the future nanotechnology industry.

Since the boule is a single crystal, the atoms in the boule are precisely aligned, to within small fractions of a nanometer or an angstrom, over the entire boule.

Since both Nour and Moussa fractions were using ( and still are ) the same name and insignia ( ex: Ghad El-Thawra website ), it was often difficult to tell them apart.

Since these systems can be in equilibrium with other phases, many systems, especially those with high volume fractions of both the two imiscible phases, can be easily destabilised by anything that changes this equilibrium e. g. high or low temperature or addition of surface tension modifying agents.

Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable.

Since only the relative weights are relevant, any weighted mean can be expressed using coefficients that sum to one.

Since the topology on R < nowiki ></ nowiki > X < nowiki ></ nowiki > is the ( X )- adic topology and R < nowiki ></ nowiki > X < nowiki ></ nowiki > is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients ( so that they belong to the ideal ( X )): f ( 0 ), f ( X < sup > 2 </ sup >− X ) and f ( ( 1 − X )< sup >− 1 </ sup > − 1 ) are all well defined for any formal power series f ∈ R < nowiki ></ nowiki > X < nowiki ></ nowiki >.

Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients.

Since only finitely many coefficients a < sub > i </ sub > and b < sub > j </ sub > are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.

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

Since the generating function for is, the generating function for the binomial coefficients is:

Since this tends to be a rather large area compared to the projected frontal area, the resulting drag coefficients tend to be low: much lower than for a car with the same drag and frontal area, and at the same speed.

Since B is also a matrix with polynomials in t as entries, one can for each i collect the coefficients of in each entry to form a matrix B < sub > i </ sub > of numbers, such that one has

Since the filter has no resistive elements, there is no dissipation and the magnitudes of the two reflection coefficients must be equal,

Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms.

Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example.

Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic, for an irreducible polynomial to not have distinct roots its coefficients must lie in a field of prime characteristic.

Since only finitely many coefficients a < sub > i </ sub > and b < sub > j </ sub > are non-zero, all sums in effect have only finitely many terms, and hence represent polynomials from K.

In the case that V is a non-singular algebraic curve and i = 1, H < sup > 1 </ sup > is a free Z < sub > l </ sub >- module of rank 2g, dual to the Tate module of the Jacobian variety of V, where g is the genus of V. Since the first Betti number of a Riemann surface of genus g is 2g, this is isomorphic to the usual singular cohomology with Z < sub > ℓ </ sub > coefficients for complex algebraic curves.

Since the Fourier coefficients of the product of two quantities is the convolution of the Fourier coefficients of each one separately, the correspondence with Fourier series allowed Heisenberg to deduce the rule by which

Since f < sup > 0 </ sup >< sub > k </ sub > is a constant it does not contribute to the error on f. Therefore, the propagation of error follows the linear case, above, but replacing the linear coefficients, A < sub > ik </ sub > and A < sub > jk </ sub > by the partial derivatives, and.

Since only the relative phase between the coefficients of the two basis vectors has any physical meaning, we can take the coefficient of to be real and non-negative.

Since this ODE has smooth coefficients we have that solutions exist for all and are unique, given and, for all.

Since a given vector v is a finite linear combination of basis elements, the only nonzero entries of the coordinate vector for v will be the nonzero coefficients of the linear combination representing v. Thus the coordinate vector for v is zero except in finitely many entries.

More abstractly and generally, we note that the two quantities asserted to be equal count the subsets of size k and n − k, respectively, of any n-element set S. There is a simple bijection between the two families F < sub > k </ sub > and F < sub > n − k </ sub > of subsets of S: it associates every k-element subset with its complement, which contains precisely the remaining n − k elements of S. Since F < sub > k </ sub > and F < sub > n − k </ sub > have the same number of elements, the corresponding binomial coefficients must be equal.

Since the quadratic form is a scalar, so is its expectation.

Since the complex roots of a real polynomial are in conjugate pairs, the irreducible polynomials over the field of real numbers are the linear polynomials and the quadratic polynomials with no real roots.

Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated.

( Since Q is quadratic and L is linear, and are constants, so these are just numbers.

Since the quadratic term of this cubic polynomial is zero, the roots are related by the equation

Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute k < sub > B </ sub > T to the average kinetic energy in thermal equilibrium.

Since every quadratic residue modulo N has four square roots, the probability that the receiver learns m is 1 / 2.

Since larger space classes are not affected by quadratic increases, the nondeterministic and deterministic classes are known to be equal, so that for example we have PSPACE = NPSPACE.

Since this is a quadratic irrational, the continued fraction must be periodic ( unless n is square, in which case the factorization is obvious ).

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 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: