Homogeneous linear

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

If a material system experiences a continuous action, or a complete cycle of operations, of a perfectly reversible kind, the quantities of heat which it takes in at different temperatures are subject to a homogeneous linear equation, of which the coefficients are the reciprocals of these temperatures. If Qr be... 

This is a homogeneous linear differential equation of second order and its characteristic equation is... 

Now, we can solve the system of homogeneous linear equations for the unknowns and v 22 which are the non-normalized elements of v,2 and V22-... 

Illustration Effect of flow type on shear induced collisions in homogenous linear flows. The collision frequency for a general linear flow [Eq. (15)] is obtained following Smoluchowski s (1917) approach as (Bidkar and Khakhar, 1990)... 

Equation (125) applies for all values of the index k — 1,2,..., m. It is a set of m simultaneous, homogeneous, linear equations for the unknown values of the coefficients c . Following Cramer s rule (Section 7.8), a nontrivial solution exists only if the determinant of the coefficients vanishes. Thus, the secular determinant takes the form... 

If the unit matrix E is of order n, Eq. (67) represents a system of n homogeneous, linear equations in n unknowns. They are usually referred to as the secular equations. According to Cramer s rule [see (iii) of Section 7.8], nontrivial solutions exist only if the determinant of the coefficients vanishes. Thus, for the solutions of physical interest,... 

Homogeneous, linear Fokker-Planck equations are known to admit a multi-variate Gaussian PDF as a solution.33 Thus, this closure scheme ensures that a joint Gaussian velocity PDF will result for statistically stationary, homogeneous turbulent flow. 

The polymers and copolymers discussed here were all prepared by reaction of the homogeneous (linear) or heterogeneous (cross-linked) poly(vinylbenzylchloride) substrate polymer with the potassium or cesium salts of the suitably monofunctionalized donors (Reaction 1). 

The remaining part of the mode seeking procedure can formally proceed exactly in the same as for the straight waveguide starting from both innermost and outermost slices, two values of the immittance matrix in some suitably chosen radial position p are found and are then used to construct the set of homogeneous linear equations for the mode field amplitudes ... 

Mark-Houwink equation phys chem The relationship between intrinsic viscosity and molecular weight for homogeneous linear polymers. mark hau.wigk l.kwa-zhan Markovnikoff s rule org chem in an addition reaction, the additive molecule RH adds as H and R, with the R going to the carbon atom with the lesser number of hydrogen atoms bonded to it. mar kov-ns.kofs, riil ... 

Because Equation 3-lOla represents a set of homogeneous linear equations, multiplying the solution by a positive or negative factor is still a solution. Therefore, each column vector in Equation 3-lOlc and 3-lOld can be made a unit vector. Then the matrix T is obtained. With this matrix known, diffusion profiles can be calculated by solving Equation 3-99c. 

Vector XV solves the homogeneous linear system (16) if and only if it belongs to the space spanned by columns of the matrix N. There exists such a vector... 

These are two homogeneous linear equations in the two unknowns a and b they have a solution only if the determinant formed by their coefficients vanishes ... 

Accordingly, we can solve the p homogeneous linear vector equations (12.66) for the p... 

These two equations form a system of homogeneous linear equations in c, and c2. They obviously have the trivial solutions c, = c2 = 0. It is proved in the theory of homogeneous linear equations that other, nontrivial solutions can exist only if the matrix of the coefficients of the Ci s forms a determinant equal to zero (Cramer s theorem). Thus, we have the so-called secular equation ... 

MARK.-HOUW1NK EQUATION. Defines the relationship between the intrinsic viscosity and molecular weight for homogeneous linear polymers. 

He, Ne, and Ar in particular, for which the most extensive data are available, plausibly likely mixing anomalies are small (cf. Table 4.5). If it is assumed that only pressure and temperature variations and air injection, items 1-3 as listed earlier, contribute to the apparent saturation anomalies of a given water sample, then for each gas the observed A is a homogeneous linear combination of AP, AT, and An, representing pressure, temperature, and air injection. The coefficients depend only on the solubility (and temperature) and are different for each gas (Table 4.5). Thus, knowledge of AHe, ANe, and AAr permits inversion and determination of AP, AT, and Aa. Formal application of this approach is illustrated by Craig and Weiss (1971) (cf. Figure 4.3). 

Given this form of molecular orbitals, the Fock equations yield a system of homogeneous linear equations in the c/s, the Roothaan equations77... 

Fracture mechanics approach. Fracture mechanics provides the basis for many modern fatigue crack-growth studies. AK is the stress intensity range (kjnilx-kjnm) where K is the magnitude of the mathematically ideal crack-tip stress field in a homogeneous linear-elastic body and is a function of applied load and crack geometry. 

Volt — SI-derived measurement unit of the electric -> potential difference or voltage. Symbol V (named in honor of the Italian physicist Alessandro - Volta (1745— 1827)). Definition lvolt is the potential difference between two points of a homogeneous, linear conductor of constant temperature, when a current of one ampere converts one watt of power. 

In the present rheological context, lattice deformation may be regarded as arising from the transport of neutrally buoyant lattice points suspended within a macroscopically homogeneous linear shear flow. The local vector velocity field v at a general (interstitial or particle interior) point R of such a spatially periodic suspension can be shown to be of the form... 

Accordingly, we can solve the p homogeneous linear vector equations (12.66) for the p excess intensities RK) in terms of the chosen independent set of /-axis intensities R,). When each of the vector equations (12.66) is multiplied by ( L 1) and summed over A, the resulting solution is... 

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