Continuous Maps From C0 1 to C0 1

Topology, General

Taqdir Husain , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

V.G Connected and Locally Connected Spaces

For the definition of these spaces, see Section I.A. In addition to the properties of these spaces given in that section, we have the following.

A continuous map of a topological space into another carries connected sets into connected sets. A product of connected spaces is connected. If A is a connected subset of a topological space and B is any set such that $\text{A}\subset \text{B}\subset \overline{\text{A}}$ , then B is connected. Any interval of $\mathbb{R}$ , including $\mathbb{R}$ itself, is connected. Each convex subset of a real topological vector space (see Section VIII.B) is connected. An interesting characterization of connected spaces is: A topological space (X, T) is connected if and only if every continuous map f of X into a discrete space Y is constant, i.e., f(x)   =   {y}, y  ∈ Y.

A characterization of locally connected spaces is: A topological space is locally connected if and only if the components of its open sets are open. Local connectedness is preserved under closed maps.

A connected, metrizable compactum (X, T) is sometimes called a continuum. Let (X, T) be a continuum and (Y, T′) a Hausdorff space. If f: (X, T)   →   (Y, T′) is a continuous surjective map, then (Y, T′) is also a continuum. Further, if (Y, T′) is locally connected, so is (Y, T′).

A topological space each of whose components consists of a singleton is called totally disconnected. For example, the set of integers is totally disconnected.

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Complex Analysis

Joseph P.S. Kung , Chung-Chun Yang , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I.C Curves

A curve γ is a continuous map from a real interval [ a, b] to C. The curve γ is said to be closed if γ(a)   =   γ(b); it is said to be open otherwise. A simple closed curve is a closed curve with γ(t ₁)   =   γ(t ₂) if and only if t ₁  = a and t ₂  = b.

An intuitively obvious theorem about curves that turned out to be very difficult to prove is the Jordan curve theorem. This theorem is usually not necessary in complex analysis, but is useful as background.

The Jordan Curve Theorem. The image of a simple closed curve (not assumed to be differentiable) separates the extended complex plane into two regions. One region is bounded (and "inside" the curve) and the other is unbounded.

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Computational Methods for Modelling of Nonlinear Systems

In Mathematics in Science and Engineering, 2007

Example 5

In this example we exhibit a non-linear uniformly A-weak continuous map on a set which is closed and bounded but non-compact. We write

(2.101) $S=L_2\left(\left[0,1\right]\right)$

and define

(2.102) $T=\left\{y\right|y\in L_2\left(\left[0,1\right]\right)and{\int }_0^{1}y\left(t\right)dt=0\}.$

For each x ∈ S we define an associated function $\hat{x}$ : ℝ → ℝ in the following way. Let e_j: ℝ → ℝ be defined by

(2.103) $e_j\left(s\right)=\sqrt{2}sinj\pi s$

for each s ∈ ℝ and each j = 1, 2, …. For each x ∈ S define a corresponding sequence of real numbers {x_j } _j=1,2,… by setting

(2.104) $x_j={\int }_0^{1}x\left(s\right)e_j\left(s\right)ds$

and let $\hat{x}$ : ℝ → ℝ be the associated function defined by

(2.105) $\hat{x}=\underset{j=1}{\overset{\infty }{\Sigma }}{x}_j{e}_j.$

It is easily seen that

(2.106) $\underset{j=1}{\overset{\infty }{\Sigma }}{x}_j^2<\infty$

and that

(2.107) $\hat{x}\left(-s\right)=-\hat{x}\left(s\right)$

and

(2.108) $\hat{x}\left(s+1\right)=\hat{x}\left(s-1\right)$

for each s ∈ ℝ. Furthermore it is well known that

(2.109) $\hat{x}\left(s\right)=x\left(s\right)$

for almost all s ∈ [0, 1]. We say that the function $\hat{x}$ is the Fourier sine series representation for x and note that $\hat{x}$ is the odd periodic extension of period two for the function x.

For each y ∈ T we define an associated function $\stackrel{⌣}{y}$ : ℝ → ℝ in the following way. Let f_j: ℝ be defined by

(2.110) $f_j\left(t\right)=\sqrt{2}cosj\pi t$

for each t ∈ ℝ and each j = 1, 2, …. For each y ∈ T define a corresponding sequence of real numbers {y_j } _j=1,2,… by setting

(2.111) $y_j={\int }_0^{1}y\left(t\right)f_j\left(t\right)dt$

and let $\stackrel{⌣}{y}$ : ℝ → ℝ be the associated function defined by

(2.112) $\stackrel{⌣}{y}=\underset{j=1}{\overset{\infty }{\Sigma }}{y}_j{f}_j.$

It is easily seen that

(2.113) $\underset{j=1}{\overset{\infty }{\Sigma }}{y}_j^2<\infty$

and that

(2.114) $\stackrel{⌣}{y}\left(-t\right)=\stackrel{⌣}{y}\left(t\right)$

and

(2.115) $\stackrel{⌣}{y}\left(t+1\right)=\stackrel{⌣}{y}\left(t-1\right)$

for each t ∈ ℝ. Furthermore it is well known that

(2.116) $\stackrel{⌣}{y}\left(t\right)=y\left(t\right)$

for almost all t ∈ [0, 1]. We say that the function $\stackrel{⌣}{y}$ is the Fourier cosine representation for y and note that $\stackrel{⌣}{y}$ is the even periodic extension of period two for the function y.

We define a non-linear operator F: S → S in the following way. First we define a linear operator A: S → T by setting

(2.117) $\begin{array}{lll}A\left[x\right]\left(t\right)& =& {\int }_0^1\left[u\left(s-t\right)-s\right]x\left(s\right)ds\\ & =& \overline{X}-X\left(t\right)\end{array}$

where u: ℝ → ℝ is the unit step function defined by

(2.118) $u\left(s\right)=\{\begin{array}{cc}0& ifs<0\\ 1& ifs>0,\end{array}$

X: [0, 1] → ℝ is the function defined by

(2.119) $X\left(t\right)={\int }_0^{t}x\left(s\right)ds$

for each t ∈ [0, 1] and $\overline{X}$ is the average value of X given by

(2.120) $\overline{X}={\int }_0^{1}X\left(t\right)dt.$

Note that if y = A[x] where x ∈ S then y ∈ T. Now for each x ∈ S we can define F[x] ∈ S by the formula

(2.121) $F\left[x\right]\left(s\right)=\frac{1}{2}{\int }_0^1\left[\hat{x}\left(s-t\right)+\hat{x}\left(s+t\right)\right]X\left(t\right)dt.$

The function F[x] is defined by a convolution integral and can be interpreted as the symmetric correlation of x and A[x]. Such operators are used frequently in the representation and analysis of non-linear systems. In Fourier series form we have

(2.122) $A[\underset{j=1}{\overset{\infty }{\Sigma }}{x}_j{e}_j]=\underset{j=1}{\overset{\infty }{\Sigma }}\frac{x_j}{\pi j}{f}_j$

and

(2.123) $F[\underset{j=1}{\overset{\infty }{\Sigma }}{x}_j{e}_j]=\underset{j=1}{\overset{\infty }{\Sigma }}\frac{x_j^2}{\sqrt{2}\pi j}{e}_j.$

We will show that F is uniformly A-weak continuous on the unit sphere

$\begin{array}{lll}S\left(0;1\right)& =& \left\{x|x\in Swith\Vert x\Vert \le 1\right\}\\ & \subseteq & S\text{.}\end{array}$

Note that the set S (0; 1) is bounded and closed but is not compact. Let ε > 0 bean arbitrary positive number. Choose N such that

(2.124) $\underset{j=N+1}{\overset{\infty }{\Sigma }}\frac{1}{j^2}<\in {\pi }^2$

and define operators T_k: S → S for each k = 1, 2, …, N by the formula

(2.125) $T_k\left(\underset{j=1}{\overset{\infty }{\Sigma }}{x}_j{e}_j\right)=x_k{e}_k$

with associated semi-norms ρ_k: S → ℝ given by

(2.126) ${\rho }_k\left(x\right)=\left|x_k\right|$

and consider the A-weak neighbourhood of zero σ ⊆ S defined by

(2.127) $\sigma =\left\{h|{\rho }_k\left(h\right)<\sqrt{\frac{3\in }{2}}foreachk=1,2,\dots ,N\right\}.$

Now for

$x+h,x\in S\left(0;1\right)andh\in \sigma$

it follows that

(2.128) $\begin{array}{lll}\Vert F\left[x+h\right]-F\left[x\right]\Vert & =& \Vert \underset{j=1}{\overset{\infty }{\Sigma }}\frac{{\left(x_j+h_j\right)}^2-x_j^2}{\sqrt{2}\pi j}{e}_j\Vert \\ & =& \underset{j=1}{\overset{\infty }{\Sigma }}\frac{{\left(2x_j+h_j\right)}^2{h}_j^2}{2{\pi }^2{j}^2}\\ & \le & \underset{j=1}{\overset{N}{\Sigma }}\frac{2h_j}{{\pi }^2{j}^2}+\underset{j=N+1}{\overset{\infty }{\Sigma }}\frac{1}{2{\pi }^2{j}^2}\\ & \le & \in .\end{array}$

Thus the uniform A-weak continuity of F on the unit sphere S (0; 1) is established.

We now consider the construction of an auxiliary operator that is defined on the entire space of input signals and which approximates the known operator F: B ⊆ H → Y on the given set B in a well defined way. This operator will be used in the proof of the main result. We suppose that B is a bounded set. The set of uniformly A-weak continuous maps F: B → Y will be denoted by C _A (B, Y).

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Elements of Probability Theory and Stochastic Processes

Eduardo Souza de Cursi , Rubens Sampaio , in Uncertainty Quantification and Stochastic Modeling with Matlab, 2015

1.17 Hilbertian structure

Stochastic processes possess Hilbertian properties analogous to those of random variables. For instance, we may consider the set $\mathcal{V}={\mathcal{E}}_P\left(\ell ,\left(a,b\right),L^2\left(\text{Ω},P\right)\right)$ , i.e. the set formed by the simple functions defined by partitions of (a, b) and taking their values in L ²(Ω, P). We have

$\mathcal{V}=\left\{X:\left(a,b\right)\times \text{Ω}\to \mathbb{R}:X\left(t,\omega \right)={\displaystyle \sum _{i=0}^{n-1}{X}_i\left(\omega \right)1_{\left(t_i,t_{i+1}\right)}\left(t\right);t\in \mathfrak{Part}\left(\left(a,b\right)\right),X_i\in L^2\left(\text{Ω},P\right)}\right\}.$

We may define a scalar product on $\mathcal{V}$ :

$\left(X,Y\right)={\displaystyle \underset{\left(a,b\right)}{\int }E\left(\mathit{XY}\right)}.$

Indeed, (•, •) is bilinear, symmetric and defined positive: when $X,Y\in \mathcal{V}$ , we have

$X\left(t,\omega \right)={\displaystyle \sum _{i=0}^{n_x-1}\overline{X_i}{1}_{\left(x_i,x_{i+1}\right)}\left(t\right)\text{and}Y\left(t,\omega \right)={\displaystyle \sum _{i=0}^{n_Y-1}\overline{Y_i}{1}_{\left(y_i,y_{i+1}\right)}\left(t\right)}}$

where $x=\left(x_0,\dots ,x_{n_x}\right)\in \mathfrak{Part}\left(\left(a,b\right)\right)$ and $y=\left(y_0,\dots ,y_{n_y}\right)\in \mathfrak{Part}\left(\left(a,b\right)\right)$ . For

$n>0\text{such that}h=\frac{b-a}{n}\le \text{min}\left\{\delta \left(x\right),\delta \left(y\right)\right\},$

we have

$\mathit{XY}={\displaystyle \sum _{i=0}^{n-1}{X}_i{Y}_i{1}_{\left(a_i,a_{i+1}\right)}\left(t\right),a_i=a+\left(i-1\right)h,X_i=X\left(a_i\right),Y_i=Y\left(a_i\right)}$

so that

$\left(X,Y\right)=h{\displaystyle \sum _{i=0}^{n-1}E\left(X_i{Y}_i\right)}.$

In particular,

${\parallel X\parallel }^2=\left(X,X\right)=h{\displaystyle \sum _{i=0}^{n-1}E\left({\left[X_i\right]}^2\right)}.$

The completion of $\mathcal{V}$ for this scalar product is a Hilbert space denoted V = L ²((a, b), L ²(Ω, P)).

In the following, we shall use often the extension principle , which consists of extending a linear continuous map defined on $\mathcal{V}$ to the set formed by all the Cauchy sequences of its elements and to the sets V such that $\mathcal{V}$ is dense on V. The extension principle is based on the following theorems:

Theorem 1.7

Let $\mathcal{V}$ be a pre-Hilbertian space of scalar product (•, •) and $I:\mathcal{V}\to \mathbb{R}$ a continuous linear application on $\mathcal{V}$ . Let

$\mathbb{V}=\left\{\tilde{\upsilon }={\left\{{\upsilon }_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}:\tilde{\upsilon }\text{is a Cauchy sequence for}\left(•,•\right)\right\},$

and

$\parallel \tilde{\upsilon }\parallel =\underset{n\to +\infty }{\text{lim}}\parallel {\upsilon }_n\parallel .$

Then

i)

$\forall \tilde{\upsilon }\in \mathbb{V}:{\left\{I\left({\upsilon }_n\right)\right\}}_{n\in \mathbb{N}}\subset \mathbb{R}$ is a Cauchy sequence;

ii)

There exists $\tilde{I}\left(\tilde{\upsilon }\right)\in \mathbb{R}$ such that $I\left({\upsilon }_n\right)\to \tilde{I}\left(\tilde{\upsilon }\right)$ when n → +∞.

iii)

If $\tilde{w}\in \mathbb{V}$ verifies ∥w_n − υ_n ∥ → 0 when n → +∞ then $\tilde{I}\left(\tilde{w}\right)=\tilde{I}\left(\tilde{\upsilon }\right)$ .

iv)

If $\alpha ,\beta \in \mathbb{R}$ ; $\tilde{\upsilon },\tilde{w}\in \mathbb{V}$ , $\tilde{u}=\alpha \tilde{\upsilon }+\beta \tilde{w}$ , then $\tilde{I}\left(\tilde{u}\right)=\alpha \tilde{I}\left(\tilde{\upsilon }\right)+\beta \tilde{I}\left(\tilde{w}\right)$ .

v)

Let $M\in \mathbb{R}$ verify $\left|I\left(\upsilon \right)\right|\le M\parallel \upsilon \parallel ,\forall \upsilon \in \mathcal{V}$ . Then $\left|\tilde{I}\left(\tilde{\upsilon }\right)\right|\le M\parallel \tilde{\upsilon }\parallel ,\forall \tilde{\upsilon }\in \mathbb{V}$ .■

Proof

Let us observe that ${\left\{\parallel {\upsilon }_n\parallel \right\}}_{n\in \mathbb{N}}\subset \mathbb{R}$ is a Cauchy sequence, since

$\left|\parallel {\upsilon }_m\parallel -\parallel {\upsilon }_n\parallel \right|\le \parallel {\upsilon }_m-{\upsilon }_n\parallel$

and

$m,n\ge n\left(\epsilon \right)\Rightarrow \parallel {\upsilon }_m-{\upsilon }_n\parallel \le \epsilon \Rightarrow \left|\parallel {\upsilon }_m\parallel -\parallel {\upsilon }_n\parallel \right|\le \epsilon .$

Thus, there exists $m\in \mathbb{R}$ such that ∥υ_n ∥ → m when n → +∞.

i)

${\left\{I\left({\upsilon }_n\right)\right\}}_{n\in \mathbb{N}}\subset \mathbb{R}$ is a Cauchy sequence, since

$\left|I\left({\upsilon }_m\right)-I\left({\upsilon }_n\right)\right|=\parallel I\left({\upsilon }_m-{\upsilon }_n\right)\parallel \le M\parallel {\upsilon }_m-{\upsilon }_n\parallel$

and ${\left\{{\upsilon }_n\right\}}_{n\in \mathbb{N}}$ is a Cauchy sequence.

ii)

Given that $\mathbb{R}$ is complete, there exists $m\in \mathbb{R}$ such that

$m=\underset{n\to +\infty }{\text{lim}}I\left({\upsilon }_n\right).$

The result is obtained by taking $\tilde{I}\left(\tilde{\upsilon }\right)=m$ .

iii)

If ${\left\{w_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}$ verifies ∥w_n − υ_n ∥ → 0 when n → +∞, we have

$\left|I\left(w_n\right)-I\left({\upsilon }_n\right)\right|=\parallel I\left(w_n-{\upsilon }_n\right)\parallel \le M\parallel w_n-{\upsilon }_n\parallel \to 0,$

so that

$\underset{n\to +\infty }{\text{lim}}I\left(w_n\right)=\underset{n\to +\infty }{\text{lim}}I\left({\upsilon }_n\right)$

and, as a consequence, $\tilde{I}\left(\tilde{w}\right)=\tilde{I}\left(\tilde{\upsilon }\right)$ .

iv)

We have

$I\left(u_n\right)=I\left(\alpha {\upsilon }_n+\beta w_n\right)=\alpha I\left({\upsilon }_n\right)+\beta I\left(w_n\right)\to \alpha \tilde{I}\left(\tilde{\upsilon }\right)+\beta \tilde{I}\left(\tilde{w}\right),$

so that $\tilde{I}\left(\tilde{u}\right)=\alpha \tilde{I}\left(\tilde{\upsilon }\right)+\beta \tilde{I}\left(\tilde{w}\right)$ .

v)

Since $\parallel {\upsilon }_n\parallel \to \parallel \tilde{\upsilon }\parallel$ when n → +∞ and $\left|I\left({\upsilon }_n\right)\right|\to \left|\tilde{I}\left(\tilde{\upsilon }\right)\right|$ , we have

$\left|I\left({\upsilon }_n\right)\right|\le M\parallel {\upsilon }_n\parallel ,\forall n\in \mathbb{N}\Rightarrow \left|\tilde{I}\left(\tilde{\upsilon }\right)\right|\le M\parallel \upsilon \parallel .$

■

Theorem 1.8

Let $\mathcal{V}$ be a pre-Hilbertian space for the scalar product (•, •) and $I:\mathcal{V}\to \mathbb{R}$ a continuous linear application on $\mathcal{V}$ . If V is a linear space such that $\mathcal{V}$ is dense on V, then I may be extended to V, i.e. there exists $I_V:V\to \mathbb{R}$ linear continuous which coincides with I on $\mathcal{V}$ . In addition, if $M\in \mathbb{R}$ verifies $\left|I\left(\upsilon \right)\right|\le M\parallel \upsilon \parallel ,\forall \upsilon \in \mathcal{V}$ , then |I_V (υ| ≤ M ∥υ∥, ∀ υ ∈ V.■

Proof

Let us recall that a linear map $I:\mathcal{V}\to \mathbb{R}$ is continuous if and only if there exists $M\in \mathbb{R}$ such that $\left|I\left(\upsilon \right)\right|\le M\parallel \upsilon \parallel ,\forall \upsilon \in \mathcal{V}$ . Given that $\mathcal{V}$ is dense on V: for any υ ∈ V, there exists $\tilde{\upsilon }={\left\{{\upsilon }_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}$ such that ∥υ_n − υ∥ → 0 when n → +∞. Let us consider

$I_V\left(\upsilon \right)=\underset{n\to +\infty }{\text{lim}}I\left({\upsilon }_n\right).$

$\tilde{\upsilon }$ is a sequence of Cauchy (since it converges). Thus, $I_V\left(\upsilon \right)=\tilde{I}\left(\tilde{\upsilon }\right)$ . It yields from the preceding theorem that the limit exists and is well-defined: if $\tilde{w}={\left\{w_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}$ verifies ∥w_n − υ∥ → 0 when n → +∞ then ${\left\{w_n\right\}}_{n\in \mathbb{N}}$ is a sequence of Cauchy (given that it converges) and ∥w_n − υ_n ∥ → 0 when n → +∞ so that $\tilde{I}\left(\tilde{\upsilon }\right)=\tilde{I}\left(\tilde{w}\right)$ and, as a consequence,

$\underset{n\to +\infty }{\text{lim}}I\left({\upsilon }_n\right)=\underset{n\to +\infty }{\text{lim}}I\left(w_n\right).$

$I_V:V\to \mathbb{R}$ is linear: let $\alpha ,\beta \in \mathbb{R}$ ; υ, w ∈ V; u = αυ + βw; $\tilde{\upsilon }={\left\{{\upsilon }_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}$ , $\tilde{w}={\left\{w_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}$ , ∥υ_n − υ∥ → 0 and ∥w_n − w∥ → 0 when n → +∞. Then $\tilde{u}={\left\{u_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}$ given by u_n = αυ_n + βw_n verifies $\tilde{I}\left(\tilde{u}\right)=\alpha \tilde{I}\left(\tilde{\upsilon }\right)+\beta \tilde{I}\left(\tilde{w}\right)$ . Thus, I_V (w) = αI_V (u)+ βI_V (υ). $I_V:V\to \mathbb{R}$ is continuous : if $\tilde{\upsilon }={\left\{{\upsilon }_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}$ verifies ∥υ_n − υ∥ → 0 when n → +∞, then ∥υ_n ∥ → ∥υ∥, so that $\parallel \tilde{\upsilon }\parallel =\parallel \upsilon \parallel$ and we have $\left|\tilde{I}\left(\tilde{\upsilon }\right)\right|\le M\parallel \upsilon \parallel \Rightarrow \left|I_V\left(\upsilon \right)\right|\le M\parallel \upsilon \parallel$ .■

Corollary 1.14

Let $\mathcal{V}$ be a pre-hilbertian space for the scalar product (•, •) and $I:\mathcal{V}\to \mathbb{R}$ a continuous linear map on $\mathcal{V}$ . If V is the completion of $\mathcal{V}$ for (•, •), then I extends to V.■

Proof

It is enough to notice that $\mathbb{V}\left\{{\left\{{\upsilon }_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{V}:{\left\{{\upsilon }_n\right\}}_{n\in \mathbb{N}}\right.$ is a Cauchy sequence (•, •)} is dense on V.■

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Cellular Automata

Jean-Paul Allouche , ... Gencho Skordev , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IV Cellular Automata as Global Maps

IV.A Cellular Automata as Special Maps on the Set of Configurations

As we have seen above, a cellular automaton can be regarded as a map on the set of configurations and its time evolution can be regarded as the orbit of the initial configuration under this map. We will see in this section that cellular automata are exactly the maps on the set of configurations which are continuous and homogeneous.

We restrict our attention to cellular automata Γ defined on Z ⁿ with values in some finite set A. Recall that the set of configurations is the set ${\text{A}}^{{\text{Z}}^{\text{n}}}$ of all maps from Z ⁿ to A. This set is equipped with the following metric: the distance d(C ₁, C ₂) between two configurations C ₁ and C ₂ is 1/k if C ₁ and C ₂ agree on all points of Z ⁿ inside the cube:

${\text{I}}_{\text{k}}=\left\{\left({\text{a}}_1,\dots ,{\text{a}}_{\text{n}}\right):\left|{\text{a}}_{\text{i}}\right|<\text{k}\right\}$

and they disagree on at least one point (a ₁,…,a _n ) such that max{∣a _i ∣:1   ≤ i  ≤ n}   = k. The topology associated with this metric is the product topology on ${\text{A}}^{{\text{Z}}^{\text{n}}}$ . This is a compact metric space; that is, all sequences of configurations have an accumulation point. In fact, ${\text{A}}^{{\text{Z}}^{\text{n}}}$ is homeomorphic to the ternary Cantor set. The shifts on the space ${\text{A}}^{{\text{Z}}^{\text{n}}}$ are the translations ${\tau }_{\left({\text{a}}_1,\mathrm{\dots },{\text{a}}_{\text{n}}\right)}$ defined by:

${\tau }_{\left({\text{a}}_1,\mathrm{\dots },{\text{a}}_{\text{n}}\right)}\left(\text{C}\right)\left({\text{b}}_1,\mathrm{\dots },{\text{b}}_{\text{n}}\right):=\text{C}\left({\text{a}}_1+{\text{b}}_1,\mathrm{\dots },{\text{a}}_{\text{n}}+{\text{b}}_{\text{n}}\right)$

A map F on the set of configurations ${\text{A}}^{{\text{Z}}^{\text{n}}}$ is said to commute with all shifts, if

$\text{F}\circ {\tau }_{\left({\text{a}}_1,\mathrm{\dots },{\text{a}}_{\text{n}}\right)}={\tau }_{\left({\text{a}}_1,\mathrm{\dots },{\text{a}}_{\text{n}}\right)}\circ \text{F}$

for all (a ₁,…,a _n )   ∈ Z ⁿ . This property is equivalent to saying that F commutes with all shifts for which all a _i but one are equal to 0, and the remaining nonzero a _i is equal to 1.

The following fundamental theorem can be found in the seminal 1969 paper of Hedlund.

Theorem.

The cellular automata defined on Z ⁿ with values in the finite set A are exactly the continuous maps of ${\text{A}}^{{\text{Z}}^{\text{n}}}$ that commute with all shifts.

The proof of this result uses essentially the compactness of the set ${\text{A}}^{{\text{Z}}^{\text{n}}}$ .

IV.B Invertibility of Global Maps Defined by Cellular Automata

An intriguing feature of cellular automata is that while the set of values A is finite and the local maps are usually simple, the global map extended from the local maps can be quite complicated. The global map is said to be injective if for any two distinct configurations C ₁ and C ₂, F(C ₁) ≠ F(C ₂). The map F is said to be onto or surjective if every configuration is the image under the global map of some "predecessor" configuration (sometimes called its father). A garden of Eden is a configuration without any predecessors for the global map. There exists a garden of Eden if and only if the global map F is not surjective.

The following unexpected result was given by Hedlund for cellular automata on Z ⁿ .

Theorem.

A cellular automaton on Z ⁿ with values in the finite set A which is injective is necessarily also surjective. (Hence, it is bijective and a homeomorphism of the space ${\text{A}}^{{\text{Z}}^{\text{n}}}$ .)

Surjective cellular automata also have a special structure. The following result was proved independently by Gleason (1988) and Hedlund (1969).

Theorem.

Each configuration of a surjective cellular automaton on Z ⁿ with values in the finite set A has only finitely many predecessors. The number of predecessors might vary from one configuration to another but is uniformly bounded.

In fact, if a cellular automaton on Z ⁿ with values in the finite set A is not surjective, then Hedlund proved that there exists a configuration with uncountably many predecessors.

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Computational Methods for Modelling of Nonlinear Systems

In Mathematics in Science and Engineering, 2007

Theorem 21

Let X and Y be separable Banach spaces. Let K ⊆ X be a compact set and F: X → Y a continuous map. For any given numbers δ > 0 and τ > 0 and for all x ∈ K and all x′ ∈ X with

$\Vert x^{\prime }-x\Vert \le \delta$

there exists an operator

$S=WZQ{G}_m:X\to Y$

defined by finite arithmetic such that

$\Vert F\left(x\right)-S\left(x^{\prime }\right)\Vert \le \frac{1}{2}\omega \left[F\right]\left(2\delta \right)+\tau .$

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Distributed Parameter Systems

N.U. Ahmed , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.B Stability, Identification, and Controllability

We present here some simple results on stability and some comments on the remaining topics.

We consider the semilinear system,

(116) $\frac{\text{d}\phi }{\text{d}\text{t}}=\text{A}\phi +\text{f}\left(\phi \right)$

in a Hilbert space (H,∥ċ∥) and assume that f is weakly nonlinear in the sense that (a) f(0)   =   0, and (b) ∥f(ξ)∥   = o(∥ξ∥), where

(117) $\underset{\parallel \xi \parallel \to 0}{lim}\left\{\frac{\text{o}\left(\parallel \xi \parallel \right)}{\parallel \xi \parallel }\right\}=0$

Theorem 20.

If the linear system $\stackrel{.}{\phi }=\text{A}\phi$ is asymptotically stable in the Lyapunov sense and f is a weakly nonlinear continuous map from H to H, then the nonlinear system (116) is locally asymptotically stable near the zero state.

The proof is based on Theorem 4.

For finite-dimensional systems, Lyapunov stability theory is most popular in that stability or instability of a system is characterized by a scalar-valued function known as the Lyapunov function. For ∞-dimensional systems a straight forward extension is possible only if strong solutions exist.

Consider the evolution equation (116) in a Hilbert space H, and suppose that A is the generator of a strongly continuous semigroup in H and f is a continuous map in H bounded on bounded sets. We assume that Eq. (116) has strong solutions in the sense that $\stackrel{.}{\phi }\left(\text{t}\right)=\text{A}\phi \left(\text{t}\right)+\text{f}\left(\phi \left(\text{t}\right)\right)$ holds a.e., and ϕ(t)   ∈ D(A) whenever ϕ₀  ∈ D(A). Let T _t , t  ≥   0, denote the corresponding nonlinear semigroup in H so that ϕ(t)   = T _t (ϕ₀), t  ≥   0. Without loss of generality we may consider f(0)   =   0 (if necessary after proper translation in H) and study the question of stability of the zero state. Let Ω be a nonempty open connected set in H containing the origin and define Ω _D   ≡   Ω   ∩ D(A), and B _a (D)   ≡   {ξ   ∈ H:∣ξ∣ _H   < a}   ∩ D(A) for each a  >   0. The system is said to be stable in the region Ω _D if, for each ball B _R (D)   ⊂   Ω _D , there exists a ball B _r (D)   ⊂ B _R (D) such that T _t (ϕ₀)   ∈ B _R (D) for all t  ≥   0 whenever ϕ₀  ∈ B _r (D). The zero state is said to be asymptotically stable if lim _t→∞ T _t (ϕ₀)   =   0 whenever ϕ₀  ∈   Ω _D .

A function V:Ω _D   →   [0, ∞] is said to be positive definite if it satisfies the properties:

a

V(x)   >   0 for x  ∈   Ω _D ⧹{0}, V(0)   =   0.

b

V is continuous on Ω _D and bounded on bounded sets.

c

V is Gateaux differentiable on Ω _D in the direction of H, in the sense that, for each x  ∈   Ω _D and h  ∈ H,

$\underset{\varepsilon \to 0}{lim}\left\{\frac{\text{V}\left(\text{x}+\varepsilon \text{h}\right)-\text{V}\left(\text{x}\right)}{\varepsilon }\right\}\equiv \text{V}\text{'}\left(\text{x},\text{h}\right)$

exists, and for each h  ∈ H, x  → V′(x, h) is continuous.

The following result is the ∞-dimensional analog of the classical Lyapunov stability theory.

Theorem 21.

Suppose the system $\stackrel{.}{\phi }=\text{A}\phi +\text{f}\left(\phi \right)$ has strong solutions for each ϕ₀  ∈ D(A) and there exists a positive definite function V on Ω _D such that along any trajectory ϕ(t), t  ≥   0, starting from ϕ₀  ∈   Ω _D ,

(118) $\begin{array}{cc}\text{V}\text{'}\left(\phi \left(\text{t}\right),\text{A}\phi \left(\text{t}\right)+\text{f}\left(\phi \left(\text{t}\right)\right)\right)\le 0& \left(<0\right)\end{array}$

for all t  ≥   0. Then the system is stable (asymptotically stable) in the region Ω _D .

If the system admits only mild solutions, Theorem 21 must be modified by using positive definite functions which have Gateaux derivatives in the directions {h} in spaces larger than H.

We conclude this section with a result for systems governed by monotone nonlinear operators as in Eqs. (107) and (112).

Theorem 22.

Consider system (112) with the operators A and f satisfying the assumptions of Theorem 19 for all t  ≥   0, and suppose h ₁  ∈ L ₁(0, ∞; R) and the injection V  ⊂ H is continuous. Then, the system is globally asymptotically stable with respect to the origin in H.

The questions of identification of parameters appearing in any of the system equations treated above can be dealt with in a similar way as in the linear case. In fact, an identification problem may be considered as a special case of a control problem with controls appearing in the system coefficients. Such classes of problems have been covered well in the literature. However, controllability questions for the general systems are more difficult and almost nothing is known.

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Computational Methods for Modelling of Nonlinear Systems

In Mathematics in Science and Engineering, 2007

Theorem 26

Let X, Y be Banach spaces with the Grothendieck property of approximation, let K ⊆ X be a compact set and F: K → Y a continuous map. Let the operator Z ^*: ℝ ^m → ℝ ⁿ be defined by

(5.5) $Z*=Z_{c*}.$

Then for some fixed ɛ > 0 and for all x, x′ ∈ X with ||x′ − x|| < ɛ the operator S ^* = S _{c ^*} : X → Y_n in the form

(5.6) $S*=WZ*Q{G}_m$

satisfies the equality

(5.7) $\underset{x\in K}{sup}\left|\right|F(x)-S*(x\prime )\left|\right|=\underset{S\in S}{inf}\left\{\underset{x\in K}{sup}\left|\right|F(x)-S(x\prime )\left|\right|\right\}.$

Let us now extend the application of the approach presented in the preceding Chapters to the best approximation of non-linear dynamical systems when the system is completely described by a finite number of real parameters.

To this end, let us consider the approximation of operator F: KΓ → Y in the form

$\hat{S}=WZ{\hat{V}}_{\xi },$

where KΓ and Γ are the same as in Section 1.6.6. (check the section number!!!)

We suppose that X and Y are Banach spaces, and that by analogy with (5.1)–(5.4), Z = Z_c where

(5.8) $Z_c(\gamma )=(g_1(c_1;\gamma ),g_2(c_2;\gamma ),\dots ,g_n(c_n;\gamma ))$

and

(5.9) $g_k(c_k;\gamma )=\underset{s=0}{\overset{p}{\Sigma }}{c}_{k,s}{r}_s(\gamma )$

where $p=\left(p_1,p_2,\mathrm{\dots },p_m\right)\in Z_{+}^m$ is fixed and where

(5.10) $r_s(\gamma )={\gamma }^8={\gamma }_1^{s_1}{\gamma }_2^{s_2}\dots {\gamma }_m^{s_m}.$

The neighbourhoods of zero ξ, θ, ζ ⊊ ℝ ^m can be chosen to be closed and bounded. Fix ξ, θ, ζ and the method of calculation of the parameters and introduce the class Ŝ of operators given by

(5.11) $\hat{S}=\left\{\hat{S}|\hat{S}:K_{\Gamma +\theta }\to Y\text{and}\hat{S}={\hat{S}}_c=W{Z}_{c}Q{\hat{V}}_{\xi }\right\}.$

Thus the operator Ŝ_c is completely defined by the coefficients {c_{k, s} }. We suppose the map

$R_{\zeta }:\Gamma +\zeta \to {ℝ}^n$

is written in the form

(5.12) $R_{\xi }(\gamma )=(f_1(\gamma ),f_2(\gamma ),\dots ,f_n(\gamma ))$

and let $\left\{g_k\left(c_k^{*};\gamma \right)\right\}$ denote the functions which best approximate the given functions {f_k (γ)} on the closed and bounded interval Γ + ζ ⊊ ℝ ^m .

We have the following theorem.

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Liquid Crystals (Physics)

Paul Ukleja , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IV.B Other Commercial Applications

The pitch of a cholesteric liquid crystal, and thus its colored appearance, is sensitive to such things as temperature, pressure, electric and magnetic fields, and impurities. Cholesterics are used to create continuous maps of temperatures on various surfaces, for instance, to locate circuit board or welding faults and to detect radiation and carcinoma of the breast. In such applications, a coating of liquid crystal can be painted on the area. The range of temperatures to which the coating responds can be widely varied by choice of the liquid crystals used.

Ultrasonic waves have been detected with cholesteric liquid crystals in which the pitch is altered by local heating or by the direct effect of high-intensity waves. In another application the ultrasonic waves directly cause a change from one stable director configuration into another. Such detectors may be usable in sonar devices.

The development of Kevlar, a high-strength polymer competitive with steel on a weigh-per-strength basis, has stimulated the study of liquid crystal polymer phases, from which the fibers are spun. Graphitic fibers formed from discotic phases form another class of strong, light materials.

Lyotropic phases are not without applications either. Everyone is familiar with the usefulness of detergents in everyday life. The correct use of systems—formed from water, surfactants, and oil—may help to recover more of the oil left in the ground after the primary methods of oil recovery have been exhausted.

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Perfect Forecasting, Behavioral Heterogeneities, and Asset Prices

Jan Wenzelburger , in Handbook of Financial Markets: Dynamics and Evolution, 2009

6.2.3. Existence and Uniqueness of Equilibrium

The literature has addressed the existence and uniqueness of asset-market equilibria in the traditional CAPM and its extensions with great generality (e.g., see Nielsen, 1988, 1990a,b; Allingham, 1991; Dana, 1993, 1999; or Hens, Laitenberger, and Löffler, 2002). For the case under consideration, we follow a basic line of reasoning in Dana (1999, Sec. 3), which has been adapted in Böhm (2002).

Consider an asset market with i = 1,…, I investors who are all characterized by Assumption 6.1. Suppose that utility functions and endowments are heterogeneous but that their expectations regarding future gross returns are identical and given by (

, V). Let e⁽ⁱ⁾ denote investor i's endowment and φ⁽ⁱ⁾ be her willingness to take risk derived from some U _(i) satisfying Assumption 6.1. For fixed endowments e _(i), define aggregate willingness to take risk by

(6.9) $\varphi \left(\rho \right):={\sum }_{i=1}^I{\phi }^{\left(i\right)}\left(e^{\left(i\right)},\rho \right),\rho \in [0,\overline{\rho })$

where

$\overline{\rho }:=min\{{\rho }_U^{(i)}:i=1,\mathrm{\dots },I\}$

is the minimum of all limiting slopes ρ_U ⁽ⁱ⁾ of U ⁽ⁱ⁾ Using Theorem 6.1, the asset-market equilibrium condition takes the form

(6.10) $\frac{\varphi \left({〈\pi ,V^{-1}\pi 〉}^{\frac{1}{2}}\right)}{〈\pi ,V^{-1}\pi 〉}{V}^{-1}\pi =x_m$

where π =

− R_fp ≠ 0 is the vector of excess returns. Computing standard deviations of the wealth associated with x_m and with the portfolio on the LHS of Eq. 6.10, π_* is a solution to Eq. 6.10, if

${\rho }_{*}:={〈{\pi }_{*}{V}^{-1}{\pi }_{*}〉}^{\frac{1}{2}}\in (0,\overline{\rho })$

solves

(6.11) $\varphi (\rho )={〈\text{xm},{\text{Vx}}_m〉}^{\frac{1}{2}}$

Vice versa, if some ρ _* ∈ (0,

) solves (6.11), then

${\pi }_{*}:=\frac{{\rho }_{*}}{{〈x_{m,}V{x}_m〉}^{\frac{1}{2}}}V{x}_m$

is a solution to Eq. 6.10. Hence, in equilibrium, aggregate willingness to take risk must be equal to the aggregate risk of the market (x_m , Vx_m) ^½. Define the upperbound bound of risk the investors are willing to accept by

$\sigma \text{max}:=\text{sup}\left\{\phi (\rho ):\rho \in [0,\overline{\rho })\right\}$

. The existence of ρ_* now follows from the intermediate-value theorem, observing that ϕ(0) = 0. Summarizing, the following result is a refinement of Böhm and Chiarella (2005, Lemma 2.5) that includes the case in which the limiting slope

is finite.

Theorem 6.2. Let (

, V) and e⁽¹⁾,…, e^(I) > 0 be given. Assume that aggregate willingness to take risk Φ : [0,

) → ℝ₊ is a continuous map with respect to ρ. Then the following holds :

1.

For each

$0\ne x_m\in {ℝ}_{+}^{\text{K}}$

with

${〈x_m,V{x}_m〉}^{\frac{1}{2}}$

< σ_max, there exists an asset-market equilibrium with market-clearing prices:

(6.12) $p_{*}=\frac{1}{R_f}\left[\overline{q}-\frac{{\rho }_{*}}{{〈x_{m,}V{x}_m〉}^{\frac{1}{2}}}V{x}_m\right]$

where ρ* ∈ (0,

) is a solution to Eq. 6.11.

2.

If, in addition, Φ is strictly increasing with respect to all ρ for which Φ(ρ) > 0, then the asset-market equilibrium ( Eq. 6.12 ) is uniquely determined.

The pricing rule (Eq. 6.12) reveals three features of the CAPM. First, the market portfolio x_m is efficient and in equilibrium any investor will hold a proportion of x_m , that is., a portfolio with the same mix of risky assets as x_m . Second, the equilibrium price of risk ρ_* responds to changes in second-moment beliefs ν but not to changes in first-moment beliefs

. As a consequence, any change in first-moment beliefs

changes the corresponding market-clearing asset prices in a linear fashion, irrespective of any nonlinearities in investors' utility functions. Third, the equilibrium price of risk is bounded by the lowest limiting slope

. This confirms that Nielsen's earlier requirement that all limiting slopes be infinite is not necessary.

Theorem 6.2 reduces the existence and uniqueness of an equilibrium in K markets to an invertibility condition of a one-dimensional demand function Eq. 6.9. It is illustrated in Figure 6.2 with two panels displaying "aggregate offer curves" for risk. Panel (a) depicts a backward-bending offer curve implying the existence of multiple equilibria. Panel (b) depicts a situation in which aggregate willingness to take risk is increasing but not surjective. Then an asset-market equilibrium does not exist if the aggregate risk of the market (x_m, Vx_m )^½ is above the upper bound of risk σ_max that investors are prepared to accept.

Eq. 6.12 is a vector version of what is known as the certainty equivalent pricing formula of the CAPM (see Luenberger, 1998). To see this, denote by

$r_m=\frac{〈q,x_m〉}{〈p_{*,}{x}_m〉}-1$

the return of the market portfolio x_m and by

${\sigma }_m=\frac{{〈x_{\mathrm{m,}}V{x}_m〉}^{\frac{1}{2}}}{〈p_{*,}{x}_m〉}$

its standard deviation. Using the pricing formula (Eq. 6.12), it is readily seen that the expected rate of return μ_m of x_m satisfies μ_m = r_f + ρ_* σm, so that the risk-return characteristics of x_m lie on the capital market line. Thus the equilibrium price of the k ^th asset (k = 1,…, K) takes the form

$p_{*}^{(k)}=\frac{1}{R_f}\left[{\overline{q}}^{(k)}-(\frac{{\mu }_m-r_f}{{\sigma }_m^2})ℂov[q^{(k)},r_m]\right]$

The term in the bracket is called the certainty equivalent of the k ^th asset because this value may be treated as the certain amount of the asset's proceeds before discounting it to obtain p^(k) _*.

The next corollary is immediate from Proposition 6.1 and Theorem 6.2 and a refinement of Böhm (2002, Thm. 3.2). Closely related results are Dana (1999, Prop. 3.4) and Hens, Laitenberger, and Löffler (2002, Thm. 1). The key observation is that aggregate willingness to take risk ϕ is invertible with respect to all ρ ∈ (0,

), if all individual demand functions φ⁽ⁱ⁾ are nondecreasing in ρ with at least one demand function being increasing for positive ρ and surjective on R₊.

Corollary 6.1. Under the hypotheses of Theorem 6.2, suppose that the willingness to take risk of all investors is nondecreasing in p and that the preferences of at least one investor satisfy the conditions of Proposition 6.1 stated in 2 and 3. Then, for any market portfolio

$0\ne x_m\in {ℝ}_{+}^{\text{K}}$

, there exists a unique asset-market equilibrium.

Although Corollary 6.1 is quite elementary, it reveals that two tasks have to be tackled when addressing existence and uniqueness issues in more general setups with heterogeneous beliefs, asset endowments, or both. First, invertibility and surjectivity of the aggregate asset demand function, given beliefs about future gross returns. Second, the specification of preferences that guarantees these two properties. Although the first task can be addressed using the Global Inverse Function Theorem (e.g., see Deimling, 1980, Thm. 15.4, p. 153, or Gale and Nikaido, 1965), the second one is significantly harder, mainly because the dimensionality of the first problem cannot be reduced to one as in Theorem 6.2, if investors have either initial endowments of assets or heterogeneous beliefs.

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