Linear differential operator

A particularly useful property of linear differential equations may be explained by comparing an equation and its derivative in operator form,... 

In addition, the time dependence of the solution, meaning the exponential function, arises from the left hand side of Eq. (2-2), the linear differential operator. In fact, we may recall that the left hand side of (2-2) gives rise to the so-called characteristic equation (or characteristic polynomial). 

Lanczos, C. (1961). Linear Differential Operators. London Van Nostrand. 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

Proof. Acting by the linear differential operator 0 on matrix (30) yields an equality whose right-hand side can be decomposed into the sum of seven terms having the same structure ... 

The quantities in parentheses are linear differential operators which can, however, be handled as if they were algebraic quantities. The determinant of the coefficients of A, B, and E is thus... 

Various voltammetric waveforms can be employed during the stripping step, including linear scan, differential pulse, square-wave, staircase, or alternating-current operations. The differential pulse and square-wave modes are usually performed at the hanging mercury drop electrode, while linear scan stripping is usually performed in connection with the mercury film electrode. 

In general a differential equation can be expressed by an operator equation.3,9 With the first decomposition, the original deterministic non-linear differential equation can be written in the Adomian s general form as ... 

From a mathematical point of view, a random evolution is an operator 0(t,t) that is improved at both t and T times. The linear differential equation is Eq. (4.90) ... 

If we consider that V(X(t)) is a first order linear differential operator like V(X) =... 

Special situations exist for which this procedure simplifies considerably. If the intermediary under consideration is not a chain carrier but is merely produced and consumed through unimportant side reactions, then the burning velocity and the composition profiles of all other species in the flame are virtually unaffected by the presence of this intermediary. The structure of the flame (excluding the X profile) can therefore be determined completely by setting = 0 in the flame equations. After this structure is determined, a, b and the coefficients of the linear differential operator Si X ) are known functions of t. Therefore, equation (90) reduces to a linear nonhomogeneous differential equation with known variable coefficients,... 

Laplace transforms can be used to transform this system of linear differential equations in the time domain into a system of linear equations in the Laplace domain. From the table of Laplace operations (Appendix I) we obtain... 

If the spectrum of linear differential operator contains a pair of pure imaginary eigenvalues then there Is a possibility of interference of two kinds of oscillations - the oscillations of catalytic reaction and the oscillations of catalyst activity- It is not possible to use the singular perturbation method for dynamic behavior investigation- This is done by the following procedure ... 

The external potential V(x) becomes the linear differential operator in k,... 

Let a linear differential operator D act on a function tp to produce a function p. tp z) is the function sought. 

Fortunately, process control problems are most usually concerned with maintaining operating variables constant at particular values. Most disturbances to the process involve only small excursions of the process variables about their normal operating points with the result that the system behaves linearly regardless of how nonlinear the descriptive equations may be. Thus Eq. (1) is a nonlinear differential equation since both Cp and U are functions of 80 but for small changes in 8 average values of CP and U may be regarded as constants, and the equation becomes the simplest kind of first order linear differential equation. 

In this equation we have followed the NMR convention and set the constant fi = 1. This is equivalent to measuring energies in angular frequency units. Employing a suitable set of base functions of the Hilbert space, this equation can be converted into a set of linear differential equations for the matrix elements of p. In the case of a single parr of tunnel levels the Hamiltonian of the two levels with their tunnel splitting can be treated as a two-level system, employing fictitious spin 1/2 operators, describable by the Hamiltonian H... 

Operator algebra can be used to solve some differential equations. A linear differential equation with constant coefficients can be written in operator notation... 

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