Linear-function Sentence Examples

For these only will the symbolic product be replaceable by a linear function of products of real coefficients.

Simspon concluded that for a given wind velocity dissipation is practically a linear function of ionization.

The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).

The vapour tension may approximate to a linear function of the composition, and the curve will then be practically a straight line.

If we assume that s is a linear function of 0, s= so(I +aO), the adiabatic equation takes the form, s 0 log e OW +aso(0 - Oo) +R loge(v/vo) =o

Other favourite types' of equation for approximate work are (I) p=RO/v±f(v), which makes p a linear function of 0 at constant volume, as in van der Waal's equation; (2) v=RO/p+f(p), which makes v a linear function of 0 at constant pressure.

The simplest case is that in which u is constant or is a linear function of x, i.e.

Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.

This gives an average value of the conductivity over the range, but it is better to observe the temperatures at three distances, and to assume k to be a linear function of the temperature, in which case the solution of the equation is still very simple, namely, 0+Ze6 2 =a log r+b, (3) where e is the temperature-coefficient of the conductivity.

In its essential nature a set is a linear function of any number of " distinct " units of the same species.

Perry (Steam Engine, p. 580), assuming a characteristic equation similar to Zeuner's (which makes v a linear function of the temperature at constant pressure, and S independent of the pressure), calculates S as a function of the temperature to satisfy Regnault's formula (10) for the total heat.

Under these conditions both S and s may be regarded as approximately constant, so that L is a linear function of the temperature.

A formula of the same type was given by Athenase Dupre (Theorie de chaleur, p. 96, Paris, 1869), on the assumption that the latent heat was a linear function of the temperature, taking the instance of Regnault's formula (io) for steam.

In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.

It further appears that a determinant is a linear function' of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs.

To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.

According to this formula, the Peltier effect is a linear function of the temperature.