If we cannot stop warfare in our own economic system, how can we expect to abolish it internationally??

If your house is to have a forced warm air system, cooling can be a part of it.

If you plan to add cooling later to your heating system, there are things to watch for.

If the problems of combining cooling with your heating are knotty, it may be cheaper to plan on a completely separate cooling system.

If you use one of the new year-round cooling system fluids such as `` Dowguard '' be sure to check it.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If tyrosine and a system generating hydrogen peroxide are added to a cell-free homogenate of the thyroid, large quantities of free mono-iodotyrosine can be formed ( Alexander, 1959 ).

If the house is not wired adequately for electricity or if plumbing or a central heating system must be installed, check into the cost of making these improvements.

If laborers are merely commodities competing against each other in a market place like so many bags of wheat and corn ( unsupported, by the way, by any agricultural subsidy ), then they may be pardoned for reacting with complete antagonism to a system that imposes such status upon them.

If the demands of these two sovereigns upon his duty of allegiance come into conflict, those of the United States have the paramount authority in American law ; likewise, those of the foreign land have paramount authority in their legal system.

If we use a cylindrical coordinate system with basis vectors, then the gradient of and the divergence of are given by

If under the existing system he could not assemble forces quickly enough to intercept mobile Viking raiders, the obvious answer was to have a standing field force.

If administrative appeal is available, no appeal to the judicial system may be made.

If the system be entirely behind the aperture stop, then this is itself the entrance pupil ( front stop ); if entirely in front, it is the exit pupil ( back stop ).

If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i. e. their distance from the axis.

If a system consists of several particles, the total angular momentum about a point can be obtained by adding ( or integrating ) all the angular momenta of the constituent particles.

If not, the system may either be rejected or accepted on conditions previously agreed between the sponsor and the manufacturer.

If the pressure in a system remains constant ( isobaric ), a vapor at saturation temperature will begin to condense into its liquid phase as thermal energy ( heat ) is removed.

If a newer system is attempting to achieve integration with an older system which has known flaws ( or " bugs "), then the new system may be referred to as " bugwards-compatible ".

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

" That rich island ," he wrote on 1 December 1881, " the key to the Gulf of Mexico, is, though in the hands of Spain, a part of the American commercial system … If ever ceasing to be Spanish, Cuba must necessarily become American and not fall under any other European domination.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If that set is the set of real numbers, we speak of " polynomials over the reals ".

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.

If F is a field and f and g are polynomials in F with g ≠ 0, then there exist unique polynomials q and r in F with

If the areas of the two parallel faces are A < sub > 1 </ sub > and A < sub > 3 </ sub >, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A < sub > 2 </ sub >, and the height ( the distance between the two parallel faces ) is h, then the volume of the prismatoid is given by ( This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height.

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,

If R is a given commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring, denoted R. The same holds true for several variables.

If a is algebraic over K, then there are many non-zero polynomials g ( x ) with coefficients in K such that g ( a ) = 0.

The integration functional I defined above defines a linear functional on the subspace P < sub > n </ sub > of polynomials of degree ≤ n. If x < sub > 0 </ sub >, …, x < sub > n </ sub > are n + 1 distinct points in, then there are coefficients a < sub > 0 </ sub >, …, a < sub > n </ sub > for which

If A and B are two square n × n matrices then characteristic polynomials of AB and BA coincide:

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

If k is not algebraically closed, for example the field of real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials.

If complex numbers are allowed, only 1st-degree polynomials can be irreducible.

If one allows only rational numbers, or a finite field, then some higher-degree polynomials are irreducible.

If each level of the factorization splits every polynomial into an O ( 1 ) ( constant-bounded ) number of smaller polynomials, each with an O ( 1 ) number of nonzero coefficients, then the modulo operations for that level take O ( N ) time ; since there will be a logarithmic number of levels, the overall complexity is O ( N log N ).

If a and b are rational numbers, the equation is solvable by radicals if either its left hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers l and m such that

If two diagrams have different polynomials, they represent different knots.

If the polynomials f are such that the term on the left is zero, and for, then the orthogonality relationship will hold:

If the notation He is used for these Hermite polynomials, and H for those above, then these may be characterized by

If V has finite dimension n, another way of looking at the tensor algebra is as the " algebra of polynomials over K in n non-commuting variables ".

If the indeterminates are X < sub > 1 </ sub >,…, X < sub > n </ sub >, then examples of such symmetric polynomials are