Let F and G Be Continuous Mappings

Handbook of Mathematical Fluid Dynamics

George R. Sell , in Handbook of Mathematical Fluid Dynamics, 2007

Definition

A continuous mapping ϕ: (− ∞, T ₁) → $\mathcal{V}$ ¹ is said to be a negative continuation of the solution S(θ, t)v ₀ provided that ϕ satisfies: (1) ϕ(0) = v ₀, (2) T ₁ = T ₀(θ, v ₀), and (3) for all τ ∈ (−∞, T ₁), ϕ satisfies

(4.16) $\begin{array}{cc}S\left(\theta \cdot \tau ,t\right)\varphi \left(\tau \right)=\varphi \left(\tau +t\right),& \text{for\hspace{0.17em}all}t\in \left[0,T_1-\tau \right)\cdot \end{array}$

A global solution through the point (θ, v ₀) ∈ M is a continuous mapping ϕ: ℝ → $\mathcal{V}$ ¹ such that: (1) ϕ(0) = v ₀ and (2) ϕ satisfies

$\begin{array}{cc}S\left(\theta \cdot \tau ,t\right)\varphi \left(\tau \right)=\varphi \left(\tau +t\right),& \text{for\hspace{0.17em}all}t\in ℝ\text{and\hspace{0.17em}all}t\in \left[0,\infty \right)\cdot \end{array}$

It is important to note that, when a negative continuation of a solution S(θ, t)v ₀ exists, it need not be unique. This lack of uniqueness is a major complication that arises in infinite dimensional dynamical systems, such as the dynamics generated by solutions of partial differential equations. Nevertheless, it is convenient to adopt a notational convention here. For τ ≤ 0, we set S(θ, τ)v ₀:= ϕ(τ), where ϕ is some negative continuation of S(θ, t)v ₀. In this way, (4.16) reads

$\begin{array}{cc}S\left(\theta \cdot \tau ,t\right)S\left(\theta ,\tau \right)v_0=S\left(\theta ,\tau +t\right)v_0,& \text{for\hspace{0.17em}all}t\in \left[0,T_1-\tau \right)\end{array}\cdot$

Since a global solution, S(θ, τ)ϕ(0) = S(θ, τ)v ₀ = ϕ(τ) is defined for all τ ∈ ℝ, it satisfies:

(4.17) $\begin{array}{cc}S\left(\theta \cdot \tau ,t\right)S\left(\theta ,\tau \right)v_0=S\left(\theta ,\tau +t\right)v_0,& \text{for\hspace{0.17em}all}\tau \in ℝ\text{and\hspace{0.17em}all}t\ge 0\end{array}\cdot$

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Modern General Topology

In North-Holland Mathematical Library, 1985

Theorem VI.11

Let A be a set with power μ where μ is an infinite cardinal number. Then a topological space X with weight ≤ μ is metrizable if and only if it is homeomorphic with a subspace of the product space of countably many star-spaces with index set A.

Proof

It suffices to prove only the necessity. First pursuing the same discussion and using the same notation as in the previous proof, we define F _α and $U_{\alpha },\alpha \in A={\displaystyle {\cup }_{n=1}^{\infty }{A}_n}.$ Then for each α we define a continuous function f _α such that

$\begin{array}{cccc}{f}_{\alpha }(F_{\alpha })=1,& f_{\alpha }(X-U_{\alpha })=0& \text{and}& 0\le f_{\alpha }\end{array}\le 1.$

Now, for each A_n , we construct a star-space S(A_n ) and note that S(A_n ) is the sum of the unit segments I _α, α ∈ A_n , whose zeros are identified. We put

Now, for each natural number n, we define a mapping f_n of X into S(A_n ) by

$f_n(p)=\{\begin{array}{ll}{f}_{\alpha }(p)\in I_{\alpha }\hfill & \text{if}p\in U_{\alpha },\hfill \\ 0\hfill & \text{if}p\notin \cup \left\{U_{\alpha }\right|\alpha \in A_n\}.\hfill \end{array}$

Since ${\mathcal{U}}_n=\left\{U_{\alpha }|\alpha \in A_n\right\}$ is discrete, this uniquely defines a mapping f_n

over X. Moreover, it follows from the discreteness of ${\mathcal{U}}_n$ that f_n is continuous. Hence the mapping

is a continuous mapping of X into P. We can also prove in the same way as for Theorem VI.10 that f is a homeomorphism. Thus X is homeomorphic with a subspace of P. Since each S(A_n ) is a subspace of S(A), X is homeomorphic with a subspace of the product space of countably many copies of the star-space S(A). ¹

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Nonlinearity and Functional Analysis

In Pure and Applied Mathematics, 1977

3.1A The contraction mapping principle

Given a continuous mapping A of a set S into itself, one attempts to locate the fixed points of A by defining a sequence (x ₀, Ax ₀, A ² x ₀, …, A ⁿx ₀, …,) for x ₀ ∈ S and by seeking conditions on S and A that ensure the convergence of this sequence. A simple answer is the (3.1.1) Contraction Mapping Theorem Denote by $S\left(\overline{x},\rho \right)$ the sphere of radius ρ and center $\overline{x}$ of a Banach space X. Suppose A maps $S\left(\overline{x},\rho \right)$ into itself and satisfies the condition that for any x, y ∈ $S\left(\overline{x},\rho \right)$ ,

(3.1.2) $\left|\right|Ax-Ay\left|\right|\le K\left|\right|x-y\left|\right|,$

where K is an absolute constant less than 1. Then A has one and only one fixed point x _∞ in $S\left(\overline{x},\rho \right)$ , and x _∞ is the limit of the sequence x_n = A ⁿx ₀ (n = 0, 1, 2, …) for any choice of x ₀ in $S\left(\overline{x},\rho \right)$ .

Proof: First we show that x_n = A ⁿx ₀ is a Cauchy sequence for any x ₀ ∈ $S\left(\overline{x},\rho \right)$ . Indeed, for any integers n and p, comparison with the geometric series Kⁿ + K ⁿ⁺¹ + · · · yields

$\begin{array}{c}\Vert x_{n+p}-x_n\Vert =\Vert A^{n+1p}{x}_0-A^n{x}_0\Vert \le \sum _{j=n}^{n+p-1}\Vert A^{j+1}{x}_0-A^j{x}_0\Vert \\ \le \sum _{j=n}^{n+p-1}{K}^j\Vert A{x}_0-x_0\Vert \le \frac{K^n}{1-K}\Vert A{x}_0-x_0\Vert .\end{array}$

Hence as $n\to \infty ,\left|\right|x_{n+p}-x_n\left|\right|\to 0$ independently of p, so that {x_n } is indeed a Cauchy sequence in $S\left(\overline{x},\rho \right)$ . Since $S\left(\overline{x},\rho \right)$ is complete, x_n → x _∞ (say) with $x_{\infty }\in S\left(\overline{x},\rho \right)$ . Hence by the continuity of A

(3.1.3) $A{x}_{\infty }=\underset{n\to \infty }{lim}A{x}_n=\underset{n\to \infty }{lim}{x}_{n+1}=x_{\infty },$

i.e., x _∞ is a fixed point; and it is unique since if y _∞ were another fixed point, then (3.1.3) would imply

$\left|\right|x_{\infty }-y_{\infty }\left|\right|=\left|\right|A{x}_{\infty }-A{y}_{\infty }\left|\right|\le K\left|\right|x_{\infty }-y_{\infty }\left|\right|,$

which is possible only if x _∞ = y _∞

Of the many interesting extensions of (3.1.1) the following one is quite useful when the map A depends on a parameter β.

(3.1.4) Corollary Suppose A(x, β) is a continuous mapping of $S\left(\overline{x},\rho \right)$ × B → $S\left(\overline{x},\rho \right)$ for some metric space B, and furthermore that A satisfies (3.1.2) for each β ∈ B. Then the mapping g: B → x_β (the unique fixed point of x = A(x, β)) is a continuous mapping of B into X.

Proof: Let β_n → β_∞ in B. Then $g\left({\beta }_n\right)=x_{{\beta }_n}=A\left(x_{{\beta }_n},{\beta }_n\right)$ , and similarly for β = β_∞. Hence

$\begin{array}{l}\left|\right|g\left({\beta }_n\right)-g\left({\beta }_{\infty }\right)\left|\right|=\left|\right|A\left(x_{{\beta }_n},{\beta }_n\right)-A\left(x_{{\beta }_{\infty }},{\beta }_{\infty }\right)\left|\right|\\ \le \left|\right|A\left(x_{{\beta }_n},{\beta }_n\right)-A\left(x_{{\beta }_{\infty }},{\beta }_n\right)\left|\right|+\left|\right|A\left(x_{{\beta }_{\infty }},{\beta }_n\right)-A\left(x_{{\beta }_{\infty }},{\beta }_{\infty }\right)\left|\right|\\ \le K\left|\right|x_{{\beta }_n}-x_{{\beta }_{\infty }}\left|\right|+\left|\right|A\left(x_{{\beta }_{\infty }},{\beta }_n\right)-A\left(x_{{\beta }_{\infty }},{\beta }_{\infty }\right)\left|\right|\end{array}$

so that

$\left|\right|g\left({\beta }_n\right)-g\left({\beta }_{\infty }\right)\left|\right|\le \frac{1}{1-K}\left|\right|A\left(x_{{\beta }_{\infty }},{\beta }_n\right)-A\left(x_{{\beta }_{\infty }},{\beta }_{\infty }\right)\left|\right|.$

Since A is continuous in β, the right-hand side above tends to 0; and the result follows.

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Modern General Topology

In North-Holland Mathematical Library, 1985

Example II.12

The concept of continuous mapping is a generalization of that of real-valued continuous function. In fact a real-valued continuous function with n real variables is a continuous mapping of E ⁿ into E ¹ if it is defined for every value of the variables. Every monotone continuous function with a real variable gives an example of a topological mapping. For example, the mapping f in Example I.7 is a topological mapping of the open interval (-1, 1) onto E ¹. Thus (-1, 1) and E ¹ are homeomorphic.

Let us consider an Euclidean plane $E^2=\left\{\left(x,y,z\right)|z=0\right\}$ and a sphere $S^2=\left\{\left(x,y,z\right)|x^2+y^2+{\left(z-1\right)}^2=1\right\}$ in E ³. To each point p ∈ S ² we assign the intersection f(p) of E ² and the straight line connecting p and p ₀ = (0, 0, 2). Then it is easily seen that f is a topological mapping of S − {p ₀} onto E ². Thus they are homeomorphic. On the other hand, E ² and E ¹ are not homeomorphic, because E ¹ minus one point can be decomposed into two disjoint, non-empty, open sets while this is not the case for E ² . Two homeomorphic topological spaces have the same topological properties and hence in topology, they are often regarded as the same space.

Among various types of mappings appearing in topology, continuous mappings including topological mappings are the most important ones, but there are many other interesting conditions for mappings. Here we shall give two of them, leaving some others to the later chapters. Let f be a mapping of a topological space X onto a topological space Y. If every closed set of X is mapped by f onto a closed set of Y, i.e., if the image f(F) of each closed set F of X is a closed set of Y, then we call f a closed mapping. If every open set of X is mapped by f onto an open set of Y, then f is called an open mapping. In view of (ii) (or (i)) of B), we can say that a one-to-one mapping f of X onto Y is a homeomorphism if and only if f is a closed (or open) continuous mapping.

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TOPOLOGICAL AND METRIC SPACES

L.V. KANTOROVICH , G.P. AKILOV , in Functional Analysis (Second Edition), 1982

PROPERTY 5.

A necessary and sufficient condition for f to be continuous is that, for every x ∈ X and every net {x _α} (α ∈ A)such that $x_{\alpha }\underset{\text{A}}{\to }x$ , we have f (x _α) → f (x).

Let us prove the sufficiency. Let Φ ⊂ Y be a closed set and let F = f ⁻¹(Φ). Consider a net {x _α} (α ∈ A) of elements of F converging to x ∈ X. By assumption, $f\left(x_{\alpha }\right)\underset{\text{A}}{\to }f\left(x\right)$ and f (x _α) ∈ Φ (α ∈ A), so we have also f (x) ∈ Φ; consequently, x ∈ F. Thus F is closed, and so f is continuous.

Assume that f is a continuous mapping. Take any x ∈ X and a net {x _α} converging to x. Let U be a neighbourhood of y = f (x). There exists a neighbourhood V of x such that f (V) ⊂ U. Further, there exists α _V ∈ A such that x _α ∈ V for α ⩾ α _V . For such an α, we have f (x _α) ∈ U, whence it follows that $f\left(x_{\alpha }\right)\underset{\text{A}}{\to }f\left(x\right)$ . ^*

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DISTRIBUTED ALGORITHMS WITH DYNAMICAL RANDOM TRANSITIONS

NADINE GUILLOTIN-PLANTARD , RENÉ SCHOTT , in Dynamic Random Walks, 2006

THEOREM 8.1

Let us assume that for every j, l ∈ {1,…, d}, f_j ∈ C ₂(S), f_j f_l ∈ C ₂(S) and ${\displaystyle {\int }_E{f}_{j}d\mu =\frac{1}{2d}}.$ Then, the sequence of processes

${(\frac{1}{m}{Y}_m([m^{2}t]))}_m$

converges in D to a d-dimensional reflected and absorbed Brownian motion (W(t))_t with zero mean and covariance matrix At evolving inside the bounded domain

$D=\{x=(x_1,\dots ,x_d)\in {ℝ}^d;0\le x_i\le 1,i=1,\dots ,d;\underset{i=1}{\overset{d}{\Sigma }}{x}_i\le 1\}$

with boundary conditions: reflections on the axes [0,1] and absorption on the sloping side

$H_a=\{x=(x_1,\dots ,x_d)\in D;\underset{i=1}{\overset{d}{\Sigma }}{x}_i=1\}.$

Let us denote T_m = inf {k ≥ 1;Y_m (k) ∈ Γ _a }and Z_m = Y_m (T_m ). Then,

$\frac{T_m}{m^2}\stackrel{ℒ}{\to }T$

and

$\frac{Z_m}{m}\stackrel{ℒ}{\to }Z$

where T = {t > 0;W(t) ∈ H_a } and Z = W(T).

Proof:

The theorem follows from Theorem 2.2 and the continuous mapping theorem (see [ 15], corollary 1, p. 31) by remarking that before absorption the process (Y_m (m))_m can be written as

$S_m+\underset{0\le i\le m}{\max }(-S_i)$

and that the hitting time T is a continuous functional of the limiting process W(.).

We give a method to find the transition densities of (W(t))_t before the hitting time T for any dimension, in the case when $A=\frac{1}{d}$ I_d where I_d is the identity matrix. In particular, we show that the characteristics of the process (W(t))_t before the hitting time can be obtained via a geometric transformation from the d–dimensional Brownian motion (W^R(t))_t , centered, with covariance matrix $\frac{t}{d}{I}_d,$ evolving inside the hypercube [0, 1]^d with normal reflections on the sides.

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Nonlinearity and Functional Analysis

In Pure and Applied Mathematics, 1977

5.2C Removal of the coerciveness restriction

Under certain circumstances, the result (5.2.3) can be substantially improved by replacing the coerciveness hypothesis (f(x), (x)||x||⁻¹ → ∞ as ||x|| → ∞ with less restrictive conditions. Indeed, an operator f ∈ B(X, X ^*) satisfying Condition (G) may well map f onto a proper subset of X ^*, and thus fail to be surjective.

The coerciveness hypothesis just mentioned was used in (5.2.3) to prove (a) the solvability of the Galerkin approximations (5.2.1) _n , and (b) an a priori estimate for the solutions of these approximate equations. Thus the improvement we now state will imply both (a) and (b) in some cases.

(5.2.14) Theorem Suppose f is a bounded, continuous mapping of a real separable reflexive Banach space into itself, satisfying the following conditions:

(I)

f is an odd mapping, i.e., f(− x) = −f(x) for all x ∈ X.

(II)

Condition (G') If x_n → x weakly in X, f(x_n ) → y weakly in X ^*, and (f(x_n ), x_n ) → (y, x), then x_n → x strongly.

Then, if ||f(x)|| ⩾ α for $x\in \partial {\Sigma }_R=\{z||z||=R\}$ , the equation f(x) = g has a solution in Σ _R for all g with ||g|| < α.

Proof: Again the basic idea is to use the hypotheses of the theorem to ensure both the existence of the solutions of the Galerkin approximations and a priori bounds for the solutions so obtained. For then, since Condition (G') implies Condition (G), our previous arguments imply that a subsequence of the solutions of the Galerkin approximations will converge to a solution of f(x) = g.

To show that the Galerkin approximations (5.2.1) _n have a solution x_n on ${\Sigma }_R\cap X_n$ for n sufficiently large, we first note that if g ∈ X, ||g|| < α, and f(x) ≠ tg for any t ∈ [0, 1] any y ∈ [0, 1] and $x\in \partial {\Sigma }_R$ , then for n sufficiently large, there are constants β and N > 0 for which $||P_n^{*}\left(f\left(z\right)-tg\right)||\ge \beta$ for t ∈ [0, 1] and $z\in \partial {\Sigma }_R\cap X_n$ for all n ⩾ N. Indeed, otherwise there would be sequences $\left\{P_{n_k}^{*}\right\},\left\{z_k\right\}$ , and {t_k } such that $z_k\in {\Sigma }_R\cap X_n,t_k\in [0,1]$ , and $||P_{n_k}^{*}\left[f\left(z_k\right)-t_{k}g\right]||\to 0$ as k → ∞. Thus, after passing to appropriate subsequences, we may suppose that z_k → z ₀ weakly, t_k → t ₀, and P_kf(z_k ) → t ₀ g strongly in X ^*. Hence, for any $w\in X\left(P_{n_k}^{*}f\left(z_k\right),w\right)=\left(f\left(z_k\right),P_{n_k}w\right)\to \left(t_{0}g,w\right)$ Also

$\begin{array}{cc}|(f\left(z_k\right)-t_{0}g,& P_{k}w-w)\end{array}|\le ||f\left(z_k\right)-t_{0}g||||P_{k}w-w||\to 0.$

Consequently, expanding $\left(f\left(z_k\right)-t_{0}g,P_{k}w-w\right)$ , we find $f\left(z_k\right)\to t_{0}g$ weakly in X ^*. Thus Condition (G') implies that z_k → z ₀ strongly and so f(z ₀) = t ₀ g. Consequently $||z_0||=R,f\left(z_0\right)=t_{0}g$ and $||f\left(z_0\right)||\le ||g||$ , which contradicts the hypothesis that $||f\left(z\right)||\ge \alpha >||g||$ for $z\in \partial {\Sigma }_R\cap X_n$ .

This result shows that for n ⩾ N, first, the mappings $P_n^{*}\left(f\left(z\right)-g\right)$ and P_n ^* f(z) are homotopic on $\partial {\Sigma }_R\cap X_n$ , and secondly, with g = 0 that P_n ^* f(z) ≠ 0 on $\partial {\Sigma }_R\cap X_n$ . Thus, by (1.6.3), the Brouwer degree $d\left(P_n^{*}f\left(z\right)\right),0,{\Sigma }_R\cap X_n$ is an odd integer and hence not zero. Consequently, by the homotopy invariance of degree, $d\left(P_n^{*}f\left(z\right)-g\right),0,{\Sigma }_R\cap X_n\ne 0$ . This means that for $n\ge N,P_n^{*}\left(f\left(z\right)-g\right)=0$ has a solution z_n in ${\Sigma }_R\cap X_n$ . Thus the Galerkin approximations (5.2.1) _n have a solution {z_n } for n ⩾ N, and these solutions automatically satisfy the a priori bound ||z_n || < R. Thus the theorem is established.

As in Section 5.2A, we now state hypotheses implying Condition (G').

(5.2.15) Let f be a bounded continuous operator mapping X into X ^*. Then f satisfies Condition (G′) if there is a completely continuous operator R : X → X ^* such that with P = f − R

(5.2.16) $\begin{array}{ccc}\left(Px-Pz,x-z\right)+f\left(x-z\right)\ge c\left(||x-z||\right)& \text{for}& x,z\in X.\end{array}$

where f is weakly upper semicontinuous and satisfies f(0) = 0, and c(r) is real-valued, positive, and continuous, and c(r) → 0 if and only if r → 0.

Proof: If x_n → x, f(x_n ) → y, and (f(x_n ), x_n ) → (y, x), then a short computation shows that (Px_n − Px, x_n − x) → 0. Also, if x_n → x weakly, then $\overline{lim}f\left(x_n-x\right)\le 0$ . Thus (5.2.16) implies that ${lim}_{n\to \infty }c\left(||x_n-x||\right)=0$ . Since c is continuous and c(β) = 0 if and only if β = 0, x_n → x strongly in X.

For the class of quasilinear elliptic operators discussed in (5.2.3), we prove the following condition analogous to Condition (G′).

(5.2.17) Theorem Suppose A is a quasilinear operator satisfying the ellipticity condition (5.2.13), and in addition:

(a)

the associated abstract operator $\mathcal{a}$ : ${\stackrel{°}{W}}_{m,p}(\Omega )\to W_{-m,q}(\Omega )$ satisfies the conditions mentioned in Section 5.2B together with the hypothesis (*) if u_n → u weakly, then (P(u_n, u_n ) – P(u, u_n ), u_n ) → 0;

(b)

for fixed y there are integrable functions c ₀(y) > 0 and c ₁(y) such that

$\sum _{|\alpha |=m}{A}_{\alpha }\left(x,y,z\right)z_{\alpha }\ge c_0\left(y\right){|z|}^p-c_1\left(y\right).$

Then the abstract operator $\mathcal{a}$ associated with A by duality satisfies Condition (G′).

Proof: First we observe that since $\mathcal{a}$ satisfies hypothesis (a), by the hypothesis of Condition (G′), $\mathcal{a}$ u_n → $\mathcal{a}$ u strongly in ${\stackrel{\circ }{W}}_{m,p}(\Omega )$ .

To show that u_n → u strongly in ${\stackrel{°}{W}}_{m,p}(\Omega )$ , we shall prove that for |α| = m, (i) the integrals $f_{\Omega }|D^m{u}_n{|}^p$ are equiabsolutely continuous and that (ii) $D^{\alpha }{u}_n\to D^{\alpha }u$ in measure. The desired strong convergence then follows by Vitali's theorem. Now the result (ii) follows immediately from our assumptions, by virtue of (5.2.13). On the other hand, to prove (i) we use hypotheses (b) and (*) as follows:

By hypothesis (5.2.5) and the fact that ( $\mathcal{a}$ u_n , u_n ) → ( $\mathcal{a}$ u, u), we deduce (after a short computation) that (Pu_n, u_n ) → (Pu, u). Then hypothesis (*) implies

(5.2.18) $\left(P\left(u,u_n\right),u_n\right)\to \left(Pu,u\right).$

Now by virtue of the definition and hypothesis (b),

$\left(P\left(u,u_n\right),u_n\right)=\sum _{\begin{array}{l}|\alpha |=m\\ |\gamma |<m\end{array}}{\int }_{\Omega }{A}_{\alpha }\left(x,D^{\gamma }u,D^m{u}_n\right)D^{\alpha }{u}_n$

(5.1.18') $c_1\left(D^{\gamma }u\right)+{\sum _{\begin{array}{l}|\alpha |=m\\ |\gamma |<m\end{array}}{A}_{\alpha }\left(x,D^{\gamma }u,D^m{u}_n\right)D^{\alpha }{u}_n\ge c_0\left(D^{\gamma }u\right)|D^m{u}_n|}^p.$

Using the facts concerning equiabsolutely continuous integrals, (5.2.13), the fact that $D^{\alpha }{u}_n\to D^{\alpha }u$ in measure, and the positivity of the expression Σ_{|α| = m} A _α(x, y, z)z _α implies that the functions Σ_{|α| = m} A_α (x, D ^γ u) D ^m u _n D ^α u _n have equiabsolutely continuous integrals over Ω. But in that case, the inequality (5.2.18′) implies the same equiabsolutely continuous property for the integrals |D ^α u_n |p for |α| = m. Thus u_n → u strongly in ${\stackrel{°}{W}}_m,p(\Omega )$ and the result is proven.

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Non-linear Registration

J. Ashburner , K. Friston , in Statistical Parametric Mapping, 2007

Deformation models

At its simplest, image registration involves estimating a smooth, continuous mapping between the points in one image, and those in another. This mapping allows one image to be re-sampled so that it is warped (deformed) to match another ( Figures 5.2 and 5.3). There are many ways of modelling such mappings, but these fit into two broad categories of parameterization (Miller et al., 1997).

FIGURE 5.2. This figure illustrates a hypothetical deformation field that maps between points in one image and those in another. This is a continuous function over the domain of the image.

FIGURE 5.3. This figure illustrates a deformation field that brings the top left image into alignment with the bottom left image. At the top right is the image with the deformation field overlayed, and at the bottom right is this image after it has been warped. Note that in this example, the deformation wraps around at the boundaries.

•

The small deformation framework does not necessarily preserve topology ³ – although if the deformations are relatively small, then it may still be preserved.

•

The large deformation framework generates deformations (diffeomorphisms) that have a number of elegant mathematical properties, such as enforcing the preservation of topology.

Both of these approaches require some model of a smooth vector field. Such models will be illustrated with the simpler small deformation framework, before briefly introducing the principles that underlie the large deformation framework.

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Handbook of Algebra

S. MacLane , I. Moerdijk , in Handbook of Algebra, 1996

3.7 Points

Motivated by the correspondence (3.1 (i)) between continuous mappings X → Y between topological spaces and morphisms of topoi Sh(X) → Sh(Y), together with the observation that Sets is the category Sh(1) of sheaves on the one-point space, one defines a point of a topos ε to be a geometric morphism p: Sets → ε. A topos ε is said to have enough points if all the inverse image functors p ^*: ε → Sets of points p are collectively faithful; or in other words, if for any two distinct parallel arrows f, g: A → B in ε there exists a point p: Sets → ε so that p ^*(f) and p ^*(g): p ^*(A) → p ^*(B) are still distinct.

For a topos ε, having enough points is a useful property, because it implies that any statement expressible in terms of colimits and finite limits and true in Sets will be true in ε. (For general topoi, Barr's Theorem (5.3 below) provides a similar useful result.)

Call a site (ℂ, J) of finite type if ℂ has pullbacks and every covering family in J is finite. "Deligne's Theorem" states that any topos Sh(ℂ, J) of sheaves on such a site of finite type has enough points.

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Nonlinearity and Functional Analysis

In Pure and Applied Mathematics, 1977

1.6A Mappings between Euclidean spaces

Let Ω denote an open set in ℝ ^N , and f a smooth mapping (of class C^p say): Ω →ℝ ^M. Then one attempts to determine the mapping properties of f by studying the derivative of f, f′(x) (i.e., the N × M matrix (D_jf_i ) where f = (f ₁, …, f_N )). Thus, if at a point x ₀, rank(f′(x ₀)) = M, f maps a small neighborhood of x ₀ onto a small neighborhood of f(x ₀). Such a point x ₀ is called regular with respect to f. The complement in Ω, i.e., the set $\mathcal{C}$ = { x | x ∈ Ω, rank f'(x) < M}, is called the critical set and a point x ∈ Ω is called a critical point. The set $\mathcal{C}$ is closed in Ω since if x_n → x in Ω, rank f'(x_n ) ≥ rank f(x). The following additional results concerning the set C are important.

(1.6.1) Let Ω be an open subset of ℝ ^N. Then:

(I) Sard's Theorem If f(x) is a p-times continuously differentiable mapping of Ω into ℝ ^m , then the critical values f( $\mathcal{C}$ ) have measure zero in ℝ ^m , provided N − m + 1 ≤ p.

(II) A. Morse's Theorem If F(x) is a N-times continuously differentiable real-valued function defined on Ω, and $\mathcal{C}$ denotes the critical points of F(x), F( $\mathcal{C}$ ) has measure zero in ℝ¹.

These two results have numerous applications in analysis, and in Chapter 3 we shall discuss infinite-dimensional analogues of these facts.

For complex analytic mappings f defined on a bounded domain Ω of ℂ ^N into ℂ ^M , many additional mapping properties of f are known. Thus

(1.6.2) (i) For N = M, then z ₀ is a singular point of f if and only if f is not one-to-one near z ₀.

(ii) If z ₀ is a point on the set S = {z | f(z) = p) in a small neighborhood U of z ₀, S ∩ U consists of a finite number of irreducible components (V_i ) each of which is either a point or contains an analytic (nontrivial) curve. Moreover, if V_i ≠ V_j, V_i contains an analytic curve not contained in V_j.

(iii) If S = {z | f(z) = p} is compact, S consists of a finite number of points.

Sard's theorem can be used to define the degree of a continuous mapping f: Ω → ℝ ^R. This integer provides an "algebraic" count of the number of solutions of the equation f(x) = p in Ω for p ∈ ℝ ^N provided f(x) ≠ p on ∂Ω. Our definition is given in three parts:

(i)

Suppose f is a C ¹ mapping of Ω → ℝ ^N and the rank of f'(x) is N, so that whenever f(x ₀) = p, the Jacobian determinant of f at x ₀ |J_f (x ₀)| ≠ 0. Then we define the degree of f at p relative to Ω as

$d\left(f,p,\Omega \right)=\sum _{f\left(x\right)=p}sgn\left|J_f\left(x\right)\right|.$

This sum is finite by virtue of the compactness of $\overline{\Omega }$ , and the inverse function theorem.

(ii)

Suppose now that f is known to be a C ¹ mapping of Ω → ℝ ^N. Then by (1.6.1(i)) we can find a sequence of regular points {p_n } (with respect to (f, Ω)) such that p_n → p in ℝ ^N. We then define the degree of f at p as

$d\left(f,p,\Omega \right)=\underset{n\to \infty }{lim}d\left(f,p_n,\Omega \right).$

(iii)

Finally, if f is only known to be continuous in Ω, there is a sequence of C ¹ mappings f_n → f uniformly on Ω, and we set

$d\left(f,p,\Omega \right)=\underset{n\to \infty }{lim}\left(f_n,p,\Omega \right).$

The function d(f, p, Ω) can be shown to be well defined in (ii) and (iii), and the limits exist and are independent of the approximating sequences. The basic properties of the degree function follow readily from this definition. For a bounded domain D ⊂ ℝ ^N , they are

(1.6.3) (i) (boundary value dependence) d(f, p, D,) is uniquely determined by the action of f(x) on ∂D.

(ii) (homotopy invariance) Suppose H(x, t) = p has no solution x ∈ ∂D for any f ∈ [0, 1], then d(H(x, t),p D) is constant independent of t ∈ [0, 1] provided H(x, t) is a continuous function of x and t.

(iii) (continuity) d(f, p, D) is a continuous function of f ∈ C( $\overline{D}$ ) (with respect to uniform convergence) and of p ∈ D.

(iv) d(f, p, D) = d(f, p', D) for p and p' in the same component of ℝ ^N − f(∂D).

(v) (domain decomposition) If {D_i } is a finite collection of disjoint open subsets of D and f(x) ≠ p for $x\in \left(\overline{D}-U_i{D}_i\right)$ , then d(f, p, D) = Σ_id(f, p, D_i ).

(vi) (Cartesian product formula) If $p\in D\subset {ℝ}^n,p\prime \in D\prime \subset {ℝ}^m$ and $f:D\to {ℝ}^n,g:D\prime \to {ℝ}^m$ , then d((f, g), (p, p'), D × D') = d(f, p, D). d(g, p', D') provided the right-hand side is defined.

(vii) If f(x) ≠ p in $\overline{D}$ , then d(f, p, D) = 0.

(viii) (odd mappings) Let D be a symmetric domain about the origin, and f(– x) = – f(x) on ∂D, with f: D → ℝ ⁿ , and f(x) ≠ 0 on ∂D, then d(f, 0, D)) is an odd integer.

(ix) If d(f, p, D) ≠ 0, then the equation f(x) = p has solutions in D.

(x) Let f be a complex analytic mapping of a neighborhood U of the origin of ℂ into itself with f(0) = 0. If the origin is an isolated zero of f, and the Jacobian determinant det |J_f (0)| = 0, then d(f, 0, U') ≥ 2 for any sufficiently small open neighborhood U' of the origin.

The degree of a mapping just described is most useful in establishing qualitative properties of continuous mappings. As a simple example, we prove

(1.6.4) Brouwer Fixed Point Theorem Let f be a continuous mapping of the unit ball σ = (x| |x| ≤ 1) in ℝ ^N into itself. Then f has at least one fixed point in σ.

Proof: We show f has a fixed point either on the boundary of σ or in the interior. Suppose f has no fixed point on the boundary ∂σ of σ, then the degree $\delta =d\left(x-f\left(x\right),0,\left|x\right|<1\right)$ is defined. In this case we show δ = 1, by use of the homotopy h(x, t) = x − tf(x), t ∈ [0, 1], joining the identity mapping to x − f(x). Since f maps σ into itself, |f(x)| ≤ 1; and so the equation h(x, t) = 0 has no solutions on the boundary of σ, ∂σ. Indeed, if the equation had solutions on ∂σ, t = 1 and f would, of necessity, have a fixed point on ∂σ. Thus by the homotopy invariance property of degree mentioned in (1.6.3), δ = d(x, 0, |x| < 1) = 1 (by definition). Thus f has a fixed point in σ by (1.6.3(ix)) and so the result is established.

In Chapter 5, we shall investigate the extensions of the degree to classes of mappings between Banach spaces. Moreover, this extension is used there to solve many problems of analysis.

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