Inverse Function Theorem

Theorem B.2 (Inverse Function Theorem) Suppose that M and N are manifolds of the same dimension, and suppose that f M —> N is a differentiable function. Suppose m e M. Suppose the linear transformation dfim). TmM Tf(m)N is invertible. Then f is a local diffeomorphism at m. 

See Boothby [Bo, II.6] or Bamberg and Sternberg [BaS, p. 237] for a proof of the inverse function theorem on R". The corresponding theorem for manifolds follows by restricting to coordinate neighborhoods of m and f(m). We will use the following theorem about group actions on differentiable manifolds. 

As shown in Figure 4.1, together with the givens and an assumption of a nonzero Jacobian of I and K at the optimum, we will invoke the Inverse Function Theorem (Appendix 4.A, p. 115) to contradict the optimum of I. This step will entail the Jacobian being zero and lead to the desired conclusion. 

Since D 0,0) is not zero, the Inverse Function Theorem (Appendix 4.A, p. 115) guarantees the existence of inverse functions... 

Applying the Inverse Function Theorem as before, we establish... 

Having shown that a contraction has a unique fixed point, we consider the givens of the Inverse Function Theorem. Based on f(x), we propose an auxiliary function and prove it to be a contraction. 

The partial derivatives of F of course, are unknown. But the application of the inverse function theorem [26, 27] allows one to rewrite Equation 11.11 as Equation 11.12 ... 

The existence of a solution path x s) at least in a neighborhood of (x(0),0) can be ensured by standard existence theorems for ordinary differential equations as long as Hx has a bounded inverse in that neighborhood. For the global homotopy (3.8.1) this requirement is met if F satisfies the conditions of the Inverse Function Theorem 3.2.1. 

As with the contraction mapping approach, with two players the Theorem becomes easy to visualize. Suppose we have found best response functions X = fi x2) and X2 = /2( i) as in Figure 2,2. Find an inverse function X2 = fi xi) and construct an auxiliary function g xi) = f xi) — f2 xi) that measures the distance between two best responses. It remains to show that g x ) crosses zero only once since this would directly imply a single crossing point of fi xi) and f2 x2)- Suppose we could show that every time crosses zero, it does so Jrom below. If that is the case, we are assured there is only a single crossing it is impossible for a continuous function to cross zero more than once from below because it would also have to cross zero from above somewhere. It can be shown that the function g xi) crosses zero only from below if the slope of g xi) at the crossing point is positive as follows... 

Using the impact approximation presented in Chapter 6, they may easily be found for any rotational band even if rotational-vibrational interaction is nonlinear in J. In 1954 R W. Anderson proved as a theorem [104] that expansion of the spectral wings in inverse powers of frequency is controlled by successive odd derivatives of the correlation function at the origin. In impact approximation the lowest non-zero derivative of this type is the third and therefore asymptotics G/(co) is described by the power expansion [20]... 

We now present two theorems which can be used to find the values of the time-domain function at two extremes, t = 0 and t = °°, without having to do the inverse transform. In control, we use the final value theorem quite often. The initial value theorem is less useful. As we have seen from our very first example in Section 2.1, the problems that we solve are defined to have exclusively zero initial conditions. 

A basic theorem of quantum mechanics, which will be presented here without proof, is If a and commute, namely [a, / ] = 0, there exists an ensemble of functions that are eigenfunctions of both a and - and inversely. 

B. PARTIAL-FRACTIONS EXPANSION. The linearity theorem [Eq. (18.36)] permits us to expand the function into a sum of simple terms and invert each individually. This is completely analogous to Laplace-transformation inversion. Let F, be a ratio of polynomials in z, Mth-order in the numerator and iVth-order in the denominator. We factor the denominator into its N roots pi, P2, Ps,... 

The generalized Fisher theorems derived in this section are statements about the space variation of the vectors of the relative and absolute space-specific rates of growth. These vectors have a simple natural (biological, chemical, physical) interpretation They express the capacity of a species of type u to fill out space in genetic language, they are space-specific fitness functions. In addition, the covariance matrix of the vector of the relative space-specific rates of growth, gap, [Eq. (25)] is a Riemannian metric tensor that enters the expression of a Fisher information metric [Eqs. (24) and (26)]. These results may serve as a basis for solving inverse problems for reaction transport systems. 

In this section we studied the phenomenon of enhanced (hydrodynamic) transport, induced by population growth in reaction-diffusion systems. Based on our Fisher theorem approach, we have shown that the expressions for the emerging hydrodynamic speeds have a simple physical interpretation They are proportional to space specific fitness functions, which express the ability of a population to fill out space. Based on our approach, we came up with simple rules for solving inverse problems in geographical population genetics. 

Selected entries from Methods in Enzymology [vol, page(s)] Application in fluorescence, 240, 734, 736, 757 convolution, 240, 490-491 in NMR [discrete transform, 239, 319-322 inverse transform, 239, 208, 259 multinuclear multidimensional NMR, 239, 71-73 shift theorem, 239, 210 time-domain shape functions, 239, 208-209] FT infrared spectroscopy [iron-coordinated CO, in difference spectrum of photolyzed carbonmonoxymyo-globin, 232, 186-187 for fatty acyl ester determination in small cell samples, 233, 311-313 myoglobin conformational substrates, 232, 186-187]. 

The spirit of the Hohenberg-Kohn theorem is that the inverse statement is also true The external potential v(r) is uniquely determined by the ground-state electron density distribution, n(r). In other words, for two different external potentials vi(r) and V2(r) (except a trivial overall constant), the electron density distributions ni(r) and 2(r) must not be equal. Consequently, all aspects of the electronic structure of the system are functionals of n(r), that is, completely determined by the function (r). 

Helmholtz theorem is a direct consequence of Fourier s formalism. In order to demonstrate this claim, we shall first consider the direct and inverse transforms of the B(r) function, respectively known as... 

Next we show differentiability. Consider Figure B.l. By construction, the function is surjective. So given an arbitrary element c e S[/(V)/ there is an element A SU(V) such that ttiIA) = c. By Theorem B.3, we know that TTi is a local diffeomorphism. Hence there is a neighborhood TV of A such that TTi I// has a differentiable inverse. The inclusion function is automatically differentiable. Finally, from Theorem B.3 we know that 712 is a differentiable function. Hence the function... 

We have mentioned already that when the MSA or HNC closure is used with Eq. (71), the contact value of the density profile is some function of the inverse compressibility, dp/dp rather than the pressure recall Eq. (87). In the OZ2 theory, Eq. (87) is satisfied formally if the BGY equation is used. If the LMBW equation is used, there is no known exact theorem. However, experience has shown that Eq. (87) seems to be satisfied closely. 

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