Circle Theorem

For a general Jacobian matiix pertaining to C components and N theoretical trays, as shown by Distefano [Am. Jnst. Chem. Eng. J., 14, 946 (1968)]. Gerschgorin s circle theorem (Varga, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N.J., 1962) may be employed to obtain bounds on the maximum and minimum absolute eigenvalues. Accord-... 

Figure 2.7 The Gershgorin s circle theorem. The three eigenvalues of the matrix A are located within the Gershgorin s circles.

Proof. Since A has nonnegative off-diagonal entries and is irreducible, Theorem A.5 asserts that 5(A) is a simple eigenvalue of A, larger than the real parts of all other eigenvalues. The inequality hypothesis and the Gershgorin circle theorem (Theorem A.l) together imply that 5(A) < 0. If 5(A) < 0, then the final assertion of the lemma follows from Theorem A.12. If 5(A) = 0, then Theorem A.5 implies that there exists x > 0 such that Ax = 0. We can assume that Xy < 1 for all j and that x, = 1 for a nonempty subset / of indices. If J is the complementary set of indices then J is non-empty by our assumptions on the row sums of A. For iel we have... 

The next result is also general and, while the conditions are not often met, it is an important tool when it can be applied. The theorem is called the Gerschgorin circle theorem. An excellent general reference on matrices is Lancaster and Tismenetsky [LT], and most of the results here are quoted from that source. Another important source is Berman and Plem-mons [BP], particularly for special results on nonnegative matrices. 

Flows in anisotropic media can be similarly treated that is, first renormalize x and y so that the resulting equations take on an isotropic, homogeneous form. Then, the results of this section apply directly. The foregoing development for pressure and streamfunction provides the first of two powerful uses of complex variables. The second, introduced in Chapter 5, called conformal mapping, potentially transforms simple, trivial flows into exact flow solutions past complicated shapes. Before focusing on these applications, we present a powerful tool known as the Circle Theorem, used by aerodynamicists to transform seemingly artificial flows past circles into real flows past airfoils. 

Using the Circle Theorem. First, we consider a uniform flow alone. In the z-plane, we would like to have the complex uniform velocity (see Equation 4-77) at angle of attack a,... 

To model the flow at upstream and downstream infinity in this plane, a source is placed at -e, while a sink is placed at +e in the plane. Then, the complex potential is obtained by superposing this flow with the image defined by the Circle Theorem of Discussion 4-7. The end result is... 

Discussion 4-7. Circle Theorem Exact solutions to Laplace s equation, 66 Discussion 4-8. Generalized streamline tracing and volume flow rate computations, 68... 

The author and cowoikers have introduced a number of theorems from the analytic theory of polynomial equations and perturbation theory for the purpose of gaming irrsight irrto the distribution of eigenvalues by simply knowing the Lanczos parameters. These theorerrrs, which include the Gershgorin circle theorems, enable one to constract spectral domains in the complex plane to which the eigenvalues of L) and are confined. It has been... 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

According to Cauchy s theorem, plus the integral over the large half-circle Cm... 

Tangent-Secant Theorem— f given an angle with its vertex on a circle, formed by a secant ray and a tangent ray, then the measure of the angle is half the measure of the intercepted arc. 

Two-Secant Power Theorem— f given a circle C and an exterior point Q, let L be a secant line through Q, intersecting C at points R and S, and let be another secant line through Q, intersecting C at U and T, then... 

Two-Chord Power Theorem—RS and TU are chords of the same circle, intersecting at Q then... 

According to a general theorem, the three properties combined imply that the unit circle is a singular curve for q(x). This fact can be established without involving the general theorem, by making better use of the continued fraction (8 ). A proof is outlined below. 

On and inside the unit circle z = 1, we apply Bouch6 s theorem to the denominator of the expression on the right whose two zeros are... 

To determine the unknown probabilities in the numerator of the right hand side of (5-21) we apply Rouch6 s theorem to the denominator. This leads to the condition that requires the vanishing of the numerator at the zero a2 which lies inside the unit circle, which yields ... 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

To use the DFT properly for evaluating normal surface deformation, the linear convolution appearing in Eq (27) has to be transformed to the circular convolution. This requires a pretreatment for the influence coefficient Kj and pressure pj so that the convolution theorem for circle convolution can be applied. The pretreatment can be performed in two steps ... 

Here we derive Avrami s theorem for a simple case [4]. Consider an area A that is partially covered by N circles each of area a, where a A. The circles overlap so that the area that is actually covered is smaller than the extended area Na. Show that the probability that a particular point in not covered by any circle is ... 

We have closed the contour C by means of a semicircle of infinite radius in the lower half plane. The integrand is zero on this half circle because r is positive. In the closed contour thus obtained, we have applied the residue theorem. The only contributions to be retained come from the poles at z 0 indeed, the poles of the operators Yg/ -n) are situated in the lower half plane and give terms affected by an exponential which vanishes for r —> oo. 

This theorem has important implications in the box model theory. It states that every eigenvalue of A x , possibly complex, lies in the complex plane inside at least one of the circles centered at the diagonal entry a and with a radius equal to the sum Z a0- (i j) of all the off-diagonal elements of the ith row. 

We can apply the Z — P = iV theorem of Chap. 13 to this new problem. We pick a contour that goes completely around the area in the z plane that is outside the unit circle, as shown in Fig. 19.9. We then plot in the A plane and look at the encirclements of the ( — 1, 0) point to find N. 

Theorem 2. Let us assume that for the discrete linear system (23) there exists a matrix Kd such that Ad + BdKd has all its eigenvalues inside the unit circle in the complex plane and Sd has all its eigenvalues in the unit circle. Then the state feedback regulator problem is solvable if and only if the Francis equations... 

It is based on Denjoy s theorem, and ris the rotation number. This algorithm, implemented by Chan (1983) computes invariant circles with irrational rotation numbers. We may, of course, discretize and solve for the whole invariant surface and not just for a section of it. Instead of having to integrate the system equations, we will then be solving for a much larger number of unknowns resulting from the additional dimension we had suppressed in the shooting approach we used. 

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