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Example 76.6.2 It can be shown that the Stefan problem can be reduced to the consideration of an equation of the form\n\n\[ \n{w}_{t} - \Delta \left( {\alpha \left( w\right) }\right) = f, w\left( 0\right) = {w}_{0} \n\] \n\nwhere \( \alpha : {L}^{2}\left( {0, T,{L}^{2}\left( U\right) }\right) \rightarrow {L}^{2}\left( ... | This example can be included in the above general theory because\n\n\[ \n\left( {\left( {-{\Delta }^{-1}}\right) u - \left( {-{\Delta }^{-1}}\right) v, u - v}\right) \geq \parallel u - v{\parallel }_{{V}^{\prime }}^{2},{V}^{\prime } \equiv {H}^{-1}, V = {H}_{0}^{1}\left( U\right) \n\] \n\nThis is seen as follows. \( V,... | Yes |
Example 76.6.3 Suppose \( U \) is a bounded open set in \( {\mathbb{R}}^{3} \) and \( b\left( \mathbf{x}\right) \geq 0, b \in {L}^{p}\left( U\right) \) , \( p \geq 4 \) for simplicity. Consider the degenerate stochastic initial boundary value problem\n\n\[ b\left( \cdot \right) u\left( {t, \cdot }\right) - b\left( \cdo... | To consider this equation and initial condition, it suffices to let \( W = {H}_{0}^{1}\left( U\right), V = \) \( {W}_{0}^{1, p}\left( U\right) ,\n\n\[ A : V \rightarrow {V}^{\prime },\langle {Au}, v\rangle = {\int }_{U}{\left| \nabla u\right| }^{p - 2}\nabla u \cdot \nabla {vdx}, \]\n\n\[ B : W \rightarrow {W}^{\prime ... | Yes |
Theorem 76.7.2 In the situation of Corollary 76.4.9 where \( V \) is a closed subspace of \( {W}^{\sigma, p}\left( U\right) ,\sigma > 1 \) and \( W \) is as described above for \( U \) a bounded open set, \( {u}_{0} \in \) \( {L}^{2}\left( {\Omega, W}\right) ,{u}_{0}{\mathcal{F}}_{0} \) measurable. Suppose \( {\lambda ... | Note that one can replace\n\n\[ \Phi \in {L}^{\infty }\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ,{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) }\right) \]\n\nwith \( \Phi \in {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ,{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) }\right) \) along wit... | Yes |
Theorem 77.2.1 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, and let \( Y \in \) \( {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \) and\n\n\[ \n{Bu}\left( t\right) = B{u}_{0} + {\int }_{0}^{t}Y\left( s\right) {ds}\text{ in }{V}^{\prime },{u}_{0} \in W,{Bu}\left( t\ri... | Note that the formula 77.2.1 shows that \( B{u}_{0} = {Bu}\left( 0\right) \) . Also it shows that \( t \rightarrow \langle {Bu}, u\rangle \left( t\right) \) is continuous. To emphasize this a little more, \( {Bu} \) is the name of a function. \( {Bu}\left( t\right) = B\left( {u\left( t\right) }\right) \) for a.e. \( t ... | Yes |
In the above corollary, the map \( u \rightarrow {Bu}\left( t\right) \) is continuous as a map from \( X \) to \( {V}^{\prime } \) . Also if \( Y \) denotes those \( f \in {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack ;V}\right) \) for which \( {f}^{\prime } \in \) \( {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack ;... | First, why is \( u \rightarrow {Bu}\left( 0\right) \) continuous? Say \( u, v \in X \) and say \( p \geq 2 \) first.\n\n\[ {Bu}\left( t\right) - {Bv}\left( t\right) = {Bu}\left( 0\right) - {Bv}\left( 0\right) + {\int }_{0}^{t}{\left( Bu\right) }^{\prime }\left( s\right) - {\left( Bv\right) }^{\prime }\left( s\right) {d... | Yes |
Theorem 77.2.4 Let \( V \) be a reflexive separable Banach space with dual \( {V}^{\prime } \), and let \( p,{p}^{\prime } \) be such that \( p > 1 \) and \( \frac{1}{p} + \frac{1}{{p}^{\prime }} = 1 \) . Let the functions \( t \rightarrow {u}_{n}\left( {t,\omega }\right) \), for \( n \in \mathbb{N} \), be in \( {L}^{{... | We prove the theorem in steps given below. Let \( X = \mathop{\prod }\limits_{{k = 1}}^{\infty }C\left( \left\lbrack {0, T}\right\rbrack \right) \) and note that when it is equipped with the product topology, then one can consider \( X \) as a metric space using the metric\n\n\[ \nd\left( {\mathbf{f},\mathbf{g}}\right)... | Yes |
Lemma 77.2.5 Let \( \left\{ {\mathbf{f}}_{n}\right\} \) be a sequence in \( X \) and suppose that each one of the components \( {f}_{nk} \) is bounded by \( C = C\left( k\right) \) in \( {C}^{0,1}\left( \left\lbrack {0, T}\right\rbrack \right) \) . Then, there exists a subsequence \( \left\{ {\mathbf{f}}_{{n}_{j}}\righ... | Proof: By the Ascoli-Arzelà theorem, there exists a subsequence \( \left\{ {\mathbf{f}}_{{n}_{1}}\right\} \) such that the sequence of the first components \( {f}_{{n}_{1}1} \) converges in \( C\left( \left\lbrack {0, T}\right\rbrack \right) \) . Then, taking a subsequence, one can obtain \( \left\{ {n}_{2}\right\} \) ... | Yes |
Lemma 77.2.9 Suppose that \( {u}_{n\left( \omega \right) } \rightarrow u \) weakly in \( {L}^{{p}^{\prime }}\left( {\left\lbrack {0, T}\right\rbrack ;{V}^{\prime }}\right) \), where \( u \) is product measurable, and \( \left\{ {u}_{n\left( \omega \right) }\right\} \) is a subsequence of \( \left\{ {u}_{n}\right\} \), ... | Proof: Assume that \( f, g \in {L}^{{p}^{\prime }}\left( {\left\lbrack {0, T}\right\rbrack ;{V}^{\prime }}\right) \) and let \( \left\{ {\phi }_{k}\right\} \) be a countable dense subset of \( {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack ;V}\right) \) . Then, a bounded set in \( {L}^{{p}^{\prime }}\left( {\left\lbra... | Yes |
Theorem 77.2.10 Let \( V \) be a reflexive separable Banach space with dual \( {V}^{\prime } \), and let \( p,{p}^{\prime } \) be such that \( p > 1 \) and \( \frac{1}{p} + \frac{1}{{p}^{\prime }} = 1 \) . Let the functions \( t \rightarrow {u}_{n}\left( {t,\omega }\right) \), for \( n \in \mathbb{N} \) , be in \( {L}^... | \[ {\begin{Vmatrix}{u}_{n}\left( \cdot ,\omega \right) \end{Vmatrix}}_{\mathcal{V}} \leq C\left( \omega \right) \] for all \( n \) . (Thus, by weak compactness, for each \( \omega \), each subsequence of \( \left\{ {u}_{n}\right\} \) has a further subsequence that converges weakly in \( \mathcal{V} \) to \( v\left( {\c... | Yes |
Lemma 77.2.11 Let \( \\left\\{ {\\mathbf{f}}_{n}\\right\\} \) be a sequence in \( X \) and suppose that each one of the components \( {f}_{nk} \) is bounded by \( C = C\\left( k\\right) \) in \( {C}^{0,\\left( {1/{p}^{\\prime }}\\right) }\\left( \\left\\lbrack {0, T}\\right\\rbrack \\right) \) . Then, there exists a su... | Proof: This follows right away from Tychonoff's theorem and the compactness of the embedding of the Holder space into \( C\\left( \\left\\lbrack {0,1}\\right\\rbrack \\right) \) . ∎ | No |
Lemma 77.2.16 \( G \) is product measurable. | Proof: This follows from the formula\n\n\[ E \cap {G}^{C} = { \cap }_{n}{ \cup }_{\psi \in M}\{ \left( {t,\omega }\right) \in E : \left| {\Lambda \left( {t,\omega }\right) \psi }\right| > n\parallel \psi \parallel \} \]\n\nwhich is clearly product measurable because \( \left( {t,\omega }\right) \rightarrow \Lambda \lef... | Yes |
Lemma 77.2.18 Let \( f\left( {\cdot ,\omega }\right) \in {\mathcal{V}}^{\prime } \). Then if \( \omega \rightarrow f\left( {\cdot ,\omega }\right) \) is measurable into \( {\mathcal{V}}^{\prime } \), it follows that for each \( \omega \), there exists a representative \( \widehat{f}\left( {\cdot ,\omega }\right) \in {\... | Proof: If a function \( f \) is measurable into \( {\mathcal{V}}^{\prime } \), then there exist simple functions \( {f}_{n} \)\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{f}_{n}\left( \omega \right) - f\left( \omega \right) \end{Vmatrix}}_{{\mathcal{V}}^{\prime }} = 0,\begin{Vmatrix}{{f}_{n}\l... | Yes |
Theorem 77.3.1 Let \( E \subseteq F \subseteq G \) where the injection map is continuous from \( F \) to \( G \) and compact from \( E \) to \( F \) . Let \( p \geq 1 \), let \( q > 1 \), and define\n\n\[ S \equiv \\left\\{ {u \\in {L}^{p}\\left( {\\left\\lbrack {a, b}\\right\\rbrack, E}\\right) : }\\right. \\text{ for... | We recall the following theorem which is proved in [98] and earlier, Theorem 25.5.2 for what will suffice here. | Yes |
Corollary 77.3.5 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, and \( B \in \) \( \mathcal{L}\left( {W,{W}^{\prime }}\right) \) is nonnegative and self adjoint. Also suppose \( t \rightarrow B\left( {u\left( t\right) }\right) \) has a weak derivative \( {\left( Bu\right) }^{\... | \[ {Bu}\left( t\right) = B{u}_{0} + {\int }_{0}^{t}{\left( Bu\right) }^{\prime }\left( s\right) {ds}\text{ in }{V}^{\prime } \] | Yes |
Lemma 77.4.3 Suppose \( f\left( {\cdot ,\omega }\right) \in {\mathcal{V}}^{\prime } \) for each \( \omega \) and that \( \left( {t,\omega }\right) \rightarrow f\left( {t,\omega }\right) \) is product measurable into \( {V}^{\prime } \) . Also \( {u}_{0} \) is \( \mathcal{F} \) measurable into \( W \) and\n\n\[ B\left( ... | Proof: Let \( {B}_{n}\left( \omega \right) \equiv {k}_{n}\left( \omega \right) B \) where \( \left\{ {{k}_{n}\left( \omega \right) }\right\} \) is an increasing sequence of simple functions converging pointwise to \( k\left( \omega \right) \) . Replace \( B\left( \omega \right) \) with \( {B}_{n}\left( \omega \right) \... | Yes |
Lemma 77.5.2 Suppose \( \omega \rightarrow A\left( {u,\omega }\right) \) has a measurable selection in \( {\mathcal{V}}^{\prime } \) for \( u \) a given element of \( \mathcal{V} \) not dependent on \( \omega \) and for each \( \omega, A\left( {u,\omega }\right) \) is a closed bounded, convex set in \( {\mathcal{V}}^{\... | Proof: Let \( \omega \rightarrow u\left( \omega \right) \) be measurable into \( \mathcal{V} \) and let \( {u}_{n}\left( \omega \right) \rightarrow u\left( \omega \right) \) in \( \mathcal{V} \) where \( {u}_{n} \) is a simple function\n\n\[ \n{u}_{n}\left( \omega \right) = \mathop{\sum }\limits_{{k = 1}}^{{m}_{n}}{c}_... | Yes |
Proposition 77.5.4 Suppose \( A\left( {\cdot ,\omega }\right) : \mathcal{V} \rightarrow \mathcal{P}\left( {\mathcal{V}}^{\prime }\right) \) is upper semicontinuous from the strong topology of \( \mathcal{V} \) to the weak topology of \( {\mathcal{V}}^{\prime } \) and has closed convex values. Then if\n\n\[ \n{u}^{ * }{... | Proof: Let \( {\widehat{T}}_{n} \uparrow T \) such that \( {u}^{ * }{\mathcal{X}}_{\left\lbrack 0,{\widehat{T}}_{n}\right\rbrack } \in A\left( {u{\mathcal{X}}_{\left\lbrack 0,{\widehat{T}}_{n}\right\rbrack },\omega }\right) \) . Then if \( {u}^{ * } \notin A\left( {u,\omega }\right) \) , there exists \( z \in \mathcal{... | Yes |
Lemma 77.5.5 For each \( \varepsilon > 0 \) there exists a solution to\n\n\[ \n{\left( B\left( \omega \right) u\left( \cdot ,\omega \right) \right) }^{\prime } + {\varepsilon Fu}\left( {\cdot ,\omega }\right) + {u}^{ * }\left( {\cdot ,\omega }\right) = f\left( {\cdot ,\omega }\right) \text{ in }{\mathcal{U}}^{\prime },... | Next it is desired to remove the regularizing term \( {\varepsilon Fu} \) . This will involve another use of Theorem 77.2.10. Denote by \( {u}_{\varepsilon } \) the solution to the above lemma. Then act on \( {u}_{\varepsilon } \) on both sides. This yields\n\n\[ \n\frac{1}{2}\left\langle {B{u}_{\varepsilon },{u}_{\var... | Yes |
Theorem 77.5.6 Let the conditions on A hold 77.5.25 - 77.5.28, 77.5.30 - 77.5.34. Also let \( B \) satisfy 77.3.6 and assume, if it depends on \( \omega \), it is of the form\n\n\[ B\left( \omega \right) = k\left( \omega \right) B, k\left( \omega \right) \geq 0, k\text{ measurable } \]\n\nLet \( {u}_{0} \) be \( \mathc... | Proof of Theorem 77.5.6: First consider the claim about replacing the coercivity condition. Returning to 77.5.50, one obtains by integrating up to \( t \) and adding \( \lambda {\int }_{0}^{t}\left\langle {B{u}_{\varepsilon },{u}_{\varepsilon }}\right\rangle {ds} \) to both sides,\n\n\[ \frac{1}{2}\left\langle {B{u}_{\... | Yes |
Theorem 77.5.7 In the context of Theorem 77.5.6, let \( q\left( {t,\omega }\right) \) be a product measurable function into \( V \) such that \( t \rightarrow q\left( {t,\omega }\right) \) is continuous, \( q\left( {0,\omega }\right) = 0 \) . Then, there exists a solution \( u \) of the integral equation \[ {Bu}\left( ... | Proof: Define a stopping time \[ {\tau }_{r} \equiv \inf \{ t : \left| {q\left( {t,\omega }\right) }\right| > r\} \] Then this is the first hitting time of an open set by a continuous random variable and so it is a valid stopping time. Then for each \( r \), let \[ {A}_{r}\left( {\omega, w}\right) \equiv A\left( {\omeg... | Yes |
Theorem 77.6.1 Suppose \( A\left( {\cdot ,\omega }\right) \) is monotone hemicontinuous bounded and single valued and coercive as a map from \( \mathcal{V} \) to \( {\mathcal{V}}^{\prime } \) . Suppose also that for \( \omega \rightarrow u\left( \omega \right) \) measurable into \( \mathcal{V} \), it follows that \( \o... | Proof: This follows from Theorem 77.2.10. This is because there is an estimate of the right sort for the measurable functions \( {u}_{n}\left( {\cdot ,\omega }\right) \) and \( {u}_{n}^{ * }\left( {\cdot ,\omega }\right) \) and the above argument which shows that a subsequence has a convergent subsequence which converg... | No |
Corollary 77.6.3 Suppose \( A\left( {\cdot ,\omega }\right) \) is monotone hemicontinuous bounded, single valued, and coercive as a map from \( \mathcal{V} \) to \( {\mathcal{V}}^{\prime } \) . Suppose also that for \( \omega \rightarrow u\left( \omega \right) \) measurable into \( \mathcal{V} \), it follows that \( \o... | Proof: The proof is identical to the above. One obtains a measurable solution to 77.6.64 in which \( P \) is replaced with \( P\left( {\cdot ,\omega }\right) \) . Then one proceeds in exactly the same steps as before and finally uses Theorem 77.2.10 to obtain the measurability of a solution to the variational inequalit... | No |
Theorem 77.7.3 Assume the above conditions, 77.7.66, 77.7.67, 77.7.68, 77.7.70, 77.7.71, and the Condition 77.7.2. Let \( {u}_{0} \) be \( {\mathcal{F}}_{0} \) measurable and \( \omega \rightarrow B\left( \omega \right) \) also \( {\mathcal{F}}_{0} \) measurable and \( \left( {t,\omega }\right) \rightarrow {\mathcal{X}... | Proof: Let \( \mathcal{T} \) denote subsets of \( (0, T\rbrack \) which contain \( T \) such that for \( S \in \mathcal{T} \) , there exists a solution \( {u}_{S} \) for each \( \omega \) to the above integral equation on \( \left\lbrack {0, T}\right\rbrack \) such that \( \left( {t,\omega }\right) \rightarrow {\mathca... | Yes |
Assume the above conditions, 77.7.66, 77.7.67, 77.7.68, 77.7.70, 77.7.71, and the Condition 77.7.2. Let \( {u}_{0} \) be \( {\mathcal{F}}_{0} \) measurable and \( \omega \rightarrow B\left( \omega \right) \) also \( {\mathcal{F}}_{0} \) measurable and \( \left( {t,\omega }\right) \rightarrow {\mathcal{X}}_{\left\lbrack... | Proof: By Theorem 77.5.7 there exists a solution to 77.7.72 which is \( \mathcal{B}\left( \left\lbrack {0, T}\right\rbrack \right) \times \) \( {\mathcal{F}}_{T} \) measurable. Now, as in the proof of Theorem 77.5.7 one can define a new operator\n\n\[ {A}_{r}\left( {w,\omega }\right) \equiv A\left( {\omega, w + {q}^{{\... | Yes |
Lemma 77.8.2 Suppose \( \lim \mathop{\sup }\limits_{{n, n \rightarrow \infty }}{a}_{mn} \leq 0 \) . Then \( \lim \mathop{\sup }\limits_{{m \rightarrow \infty }}\left( {\lim \mathop{\sup }\limits_{{n \rightarrow \infty }}{a}_{mn}}\right) \leq \) 0. | Proof: Suppose the first inequality. Then for \( \varepsilon > 0 \), there exists \( N \) such that if \( n, m \) are both as large as \( N \), then \( {a}_{mn} \leq \varepsilon \) . Thus \( \mathop{\sup }\limits_{{n \geq N}}{a}_{mn} \leq \varepsilon \) provided \( m \geq N \) also. Hence for such \( m \) ,\n\n\[ \math... | Yes |
Lemma 77.8.3 Let \( G : D\left( G\right) \subseteq X \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) where \( X \) is a Banach space be maximal monotone and let \( {v}_{n} \in G{u}_{n} \) and\n\n\[ \n{u}_{n} \rightarrow u,{v}_{n} \rightarrow v\text{weakly.} \n\]\n\nAlso suppose that\n\n\[ \n\lim \mathop{\sup }\li... | Proof: By monotonicity,\n\n\[ \n0 \geq \lim \mathop{\sup }\limits_{{m, n \rightarrow \infty }}\left\langle {{v}_{n} - {v}_{m},{u}_{n} - {u}_{m}}\right\rangle \n\]\n\n\[ \n\geq \lim \mathop{\inf }\limits_{{m, n \rightarrow \infty }}\left\langle {{v}_{n} - {v}_{m},{u}_{n} - {u}_{m}}\right\rangle \geq 0 \n\]\n\nand so\n\n... | Yes |
Lemma 77.8.4 Suppose \( A \) is a set valued operator, \( A : X \rightarrow \mathcal{P}\left( X\right) \) and \( {u}_{n}^{ * } \in A{u}_{n} \) .\n\nSuppose also that \( {u}_{n} \rightarrow u \) weakly and \( {u}_{n}^{ * } \rightarrow {u}^{ * } \) weakly. Suppose also that\n\n\[ \n\lim \mathop{\sup }\limits_{{m, n \righ... | Proof: It is assumed that\n\n\[ \n\lim \mathop{\sup }\limits_{{m, n \rightarrow \infty }}\left( {\left\langle {{u}_{n}^{ * },{u}_{n}}\right\rangle + \left\langle {{u}_{m}^{ * },{u}_{m}}\right\rangle - \left( {\left\langle {{u}_{n}^{ * },{u}_{m}}\right\rangle + \left\langle {{u}_{m}^{ * },{u}_{n}}\right\rangle }\right) ... | Yes |
Theorem 77.8.6 The following hold. Here \( V \) is a reflexive Banach space with strictly convex norm. \( G : D\left( G\right) \rightarrow \mathcal{P}\left( {V}^{\prime }\right) \) is maximal monotone. Then 1. \( {J}_{\mu } \) and \( {G}_{\mu } \) are bounded single valued operators defined on \( V \) . Bounded means t... | Then \( A\left( {\cdot ,\omega }\right) + {G}_{\mu } \) will be bounded and have the same limit properties as \( A\left( {\cdot ,\omega }\right) \) . As to measurability, \( G \) and hence \( {G}_{\mu } \) do not depend on \( \omega \) and so the measurability condition will hold. What about the estimates? We need to c... | Yes |
Lemma 77.8.7 Suppose \( 0 \geq {\left| a - b\right| }^{p} - {b}^{p} - {\alpha a} \) for \( a, b \geq 0 \) and \( \alpha > 0 \) . Then there exists a constant \( C \) such that\n\n\[ a \leq {Cb} + C \] | Proof: If \( b \geq a \), then there is nothing to show. Therefore, it suffices to show that the desired inequality holds for \( a > b \) . Thus from now on, \( a > b \).\n\n\[ 0 \geq {\left( a - b\right) }^{p} - {b}^{p} - {\alpha a} \]\n\nSuppose \( a > {nb} + n \) . Let \( x = b/a \) . Then for \( x \in \left\lbrack ... | Yes |
Corollary 77.8.11 Suppose 77.8.73 - 77.8.78 and \( B \) as described above and \( {u}_{0} \) is \( \mathcal{F} \) measurable. Also let \( G : D\left( G\right) \subseteq V \rightarrow \mathcal{P}\left( {V}^{\prime }\right) \) be maximal monotone and quasi-bounded. Let \( \left( {t,\omega }\right) \rightarrow q\left( {t,... | Proof: Define a stopping time\n\n\[ \n{\tau }_{n}\left( \omega \right) \equiv \inf \{ t : q\left( {t,\omega }\right) > n\}\n\]\n\nThen let \( \widetilde{A}\left( {\cdot ,\omega }\right) \equiv A\left( {{q}^{{\tau }_{n}}\left( {\cdot ,\omega }\right) + w,\omega }\right) \) . Then \( \widetilde{A} \) satisfies the same p... | Yes |
Lemma 78.1.5 Let \( f\left( {\cdot ,\omega }\right) \in {\mathcal{V}}^{\prime } \). Then if \( \omega \rightarrow f\left( {\cdot ,\omega }\right) \) is measurable into \( {\mathcal{V}}^{\prime } \), it follows that for each \( \omega \), there exists a representative \( \widehat{f}\left( {\cdot ,\omega }\right) \in {\m... | Proof: If a function \( f \) is measurable into \( {\mathcal{V}}^{\prime } \), then there exist simple functions \( {f}_{n} \)\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{f}_{n}\left( \omega \right) - f\left( \omega \right) \end{Vmatrix}}_{{\mathcal{V}}^{\prime }} = 0,\begin{Vmatrix}{{f}_{n}\l... | Yes |
Corollary 78.1.9 If \( {Bu}\left( 0\right) = 0 \) for \( u \in X \), then \( \langle {Bu}, u\rangle \left( 0\right) = 0 \) . The converse is also true. An analogous result will hold with 0 replaced with \( T \) . | Proof: Let \( {u}_{n} \rightarrow u \) in \( X \) with \( {u}_{n}\left( t\right) = 0 \) for all \( t \) close enough to 0 . For \( t \) off a set of measure zero consisting of the union of sets of measure zero corresponding to \( {u}_{n} \) and \( u \) ,\n\n\[ \left\langle {B{u}_{n},{u}_{n}}\right\rangle \left( t\right... | Yes |
Lemma 78.2.1 Let \( f \) be measurable into \( {\mathcal{V}}^{\prime } \) and let \( A \) satisfy the conditions 78.1.1 - 78.1.1. Then for \( K \) and \( M \) defined as above, it follows there exist measurable \( {u}_{\varepsilon } \) and \( {w}_{\varepsilon }^{ * } \) satisfying 78.2.14. | Note this implies that, suppressing dependence on \( \omega \) , \[ \left\langle {B{u}_{\varepsilon }, v}\right\rangle \left( 0\right) = \left\langle {{Bv}\left( 0\right) ,{u}_{0}}\right\rangle \] for all \( v \in X \) . Thus, letting \( v \) be a smooth function with values in \( V \) \[ \left\langle {B{u}_{\varepsilo... | No |
Lemma 78.3.1 Let \( A \) satisfy 78.3-78.3 and let \( f \) be measurable into \( {\mathcal{V}}^{\prime } \) and let \( {u}_{0} \) be measurable into \( W \) . Then for \( \varepsilon > 0 \), there exists a solution to \[ {Lu} + {\varepsilon Fu} + {u}^{ * } = f,{Bu}\left( {0,\omega }\right) = B{u}_{0}\left( \omega \righ... | Proof: Using easy estimates and the definition that \( r > \max \left( {\widehat{p},2}\right) ,\widehat{p} \geq p \) , (Recall that \( \widehat{p} \) determined the polynomial growth of \( {\begin{Vmatrix}{u}^{ * }\end{Vmatrix}}_{{\mathcal{V}}^{\prime }} \) where \( {u}^{ * } \in A\left( {u,\omega }\right) \) ) it is r... | Yes |
Theorem 78.3.2 Suppose the conditions on A,78.3 - 78.3. Also let \( {u}_{0} \) be measurable into \( W \) and \( f \) measurable into \( {\mathcal{V}}^{\prime } \) . Let \( B \in \mathcal{L}\left( {W,{W}^{\prime }}\right) \) be nonnegative and self adjoint as described above. Let \( \sigma > 0 \) be small. Then there e... | \[ {Bu}\left( t\right) - B{u}_{0} + {\int }_{0}^{t}{u}^{ * }\left( s\right) {ds} = {\int }_{0}^{t}f\left( s\right) {ds} \] | No |
Theorem 78.3.4 In the context of Theorem 78.3.2, let \( q\left( {t,\omega }\right) \) be a product measurable function into \( V \) such that \( t \rightarrow q\left( {t,\omega }\right) \) is continuous, \( q\left( {0,\omega }\right) = 0 \) . Then for each small \( \sigma \), there exists a solution \( u \) of the inte... | Proof: Define a stopping time \[ {\tau }_{r}\left( \omega \right) \equiv \inf \{ t : \left| {q\left( {t,\omega }\right) }\right| > r\} \] Then this is the first hitting time of an open set by a continuous random variable and so it is a valid stopping time. Then for each \( r \), let \[ {A}_{r}\left( {\omega, w}\right) ... | Yes |
Theorem 78.4.3 Assume the above conditions,78.3 - 78.3, and 78.4.2. Let \( {u}_{0} \) be \( {\mathcal{F}}_{0} \) measurable and \( \left( {t,\omega }\right) \rightarrow {\mathcal{X}}_{\left\lbrack 0, t\right\rbrack }\left( t\right) f\left( {t,\omega }\right) \) is \( \mathcal{B}\left( \left\lbrack {0, t}\right\rbrack \... | Proof: First note that Theorem 78.3.2 there exists a solution on \( \left\lbrack {0, T - \sigma }\right\rbrack \) for each small \( \sigma > 0 \) . Then by uniqueness, there exists a solution on \( \left( {0, T}\right) \) . Let \( \mathcal{T} \) denote subsets of \( (0, T - \sigma \rbrack \) which contain \( T - \sigma... | Yes |
Theorem 78.4.4 Assume the above conditions,78.3 - 78.3, and 78.4.2. Let \( {u}_{0} \) be \( {\mathcal{F}}_{0} \) measurable and \( \left( {t,\omega }\right) \rightarrow {\mathcal{X}}_{\left\lbrack 0, t\right\rbrack }\left( t\right) f\left( {t,\omega }\right) \) is \( \mathcal{B}\left( \left\lbrack {0, t}\right\rbrack \... | Proof: By Theorem 78.3.4 there exists a solution to 78.4.33 which is \( \mathcal{B}\left( \left\lbrack {0, T - \sigma }\right\rbrack \right) \times \) \( {\mathcal{F}}_{T - \sigma } \) measurable. Since this is true for all \( \sigma > 0 \), there exists a unique \( \mathcal{B}\left( \left\lbrack {0,\widehat{T}}\right\... | Yes |
Corollary 79.1.1 Let \( {\Phi }_{n} \) be as described above. Then\n\n\[ \n{\begin{Vmatrix}{\Phi }_{n}\left( t,\omega \right) \end{Vmatrix}}_{{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, W}\right) } \leq \parallel \Phi \left( {t,\omega }\right) {\parallel }_{{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, W}\right) }\n\]\n\nwhere \( {\be... | Proof: Let, \( \alpha > 2 \) . As explained above, \( \left| {{\int }_{s}^{r}{\Phi }_{n}{dW}}\right| \leq \left| {{\int }_{s}^{r}{\Phi dW}}\right| \) . Thus by the Burkholder Davis Gundy inequality,\n\n\[ \n\mathop{\sup }\limits_{n}\left| {{\int }_{s}^{r}{\Phi }_{n}{dW}}\right| \leq \left| {{\int }_{s}^{r}{\Phi dW}}\ri... | Yes |
Theorem 79.1.6 In Situation 79.1.5, for \( \omega \) off a set of measure zero, for every \( t \in {N}_{\omega }^{C} \), the measure of \( {N}_{\omega } \) equalling 0, | \[ \langle {BX}\left( t\right), X\left( t\right) \rangle = \left\langle {B{X}_{0},{X}_{0}}\right\rangle + {\int }_{0}^{t}2\langle Y\left( s\right), X\left( s\right) \rangle {ds} + \]\n\[ {\int }_{0}^{t}\langle {BZ}, Z{\rangle }_{{\mathcal{L}}_{2}}{ds} + {2M}\left( t\right) \]\n(79.1.10)\n\nwhere \( M\left( t\right) \) ... | Yes |
Corollary 79.1.8 In the situation of Theorem 79.1.7, change the conditions on \( \Phi \) . Instead of letting\n\n\[ \Phi \in {L}^{\alpha }\left( {\Omega ;{L}^{\infty }\left( {\left\lbrack {0, T}\right\rbrack ,{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, W}\right) }\right) }\right) \cap {L}^{2}\left( {\left\lbrack {0, T}\right\... | Proof: Let \( {\tau }_{m} = \inf \left\{ {t : \parallel \Phi \left( {t,\omega }\right) {\parallel }_{{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, W}\right) } > m}\right\} \) . Then \( {\Phi }^{{\tau }_{m}} \) is uniformly bounded above by \( m \) and \( \mathop{\lim }\limits_{{m \rightarrow \infty }}{\tau }_{m} = \infty \) . H... | Yes |
Theorem 79.2.1 In the situation of Theorem 79.1.7, suppose 1 - 79.4.2 and progressive measurability condition 79.1.2 but \( A \) defined pointwise,\n\n\[ A\left( {u,\omega }\right) \left( t\right) = A\left( {u\left( {t,\omega }\right), t,\omega }\right) \]\n\nand suppose \( f \) is progressively measurable and is in \(... | Proof: For given \( w \in {L}^{2}\left( {\Omega, C\left( {\left\lbrack {0, T}\right\rbrack, W}\right) }\right) \) each \( w \) being progressively measurable, define \( u = \psi \left( w\right) \) as the solution to the integral equation\n\n\[ {Bu}\left( {t,\omega }\right) - B{u}_{0}\left( \omega \right) + {\int }_{0}^... | Yes |
Lemma 79.4.3 Let \( A \) be as described above in 1-4. Then for \( u \in \mathcal{V},\widehat{A}\left( u\right) \) is a closed and convex and bounded subset of \( {\mathcal{V}}^{\prime } \) . | Proof: It is clear that \( \widehat{A}\left( u\right) \) is convex. Indeed, if \( z, w \) are in this set, and \( \lambda \in \left\lbrack {0,1}\right\rbrack \), then \( \left( {{\lambda z} + \left( {1 - \lambda }\right) w}\right) \left( {t,\omega }\right) \in A\left( {u\left( {t,\omega }\right), t,\omega }\right) \) f... | Yes |
Lemma 79.4.5 Let \( \Omega \) be a set and let \( \mathcal{F} \) be a \( \sigma \) algebra of subsets of \( \Omega \) . Let \( U \) be a separable reflexive Banach space. Suppose there is a sequence \( {\left\{ {u}_{j}\left( \omega \right) \right\} }_{j = 1}^{\infty } \) in \( U \) , where each \( \omega \rightarrow {u... | \[ \mathop{\lim }\limits_{{n\left( \omega \right) \rightarrow \infty }}{u}_{n\left( \omega \right) }\left( \omega \right) = u\left( \omega \right) \] | No |
Lemma 79.4.6 There is a set of measure zero \( N \), an enlargement of the earlier set such that for \( \omega \notin N \), and a suitable subsequence, still denoted with \( n \), such that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{u}_{n}\left( {t,\omega }\right) + {\int }_{0}^{t}{z}_{n}\left( {s,\o... | Proof: The formula 79.4.51 follows from the above discussion. The second claim follows from the first and the equation satisfied by \( u \) 79.4.49. \( \blacksquare \) | No |
Lemma 79.4.7 There exists a set \( M \subseteq \left\lbrack {0, T}\right\rbrack \) having measure zero such that for \( t \notin M \) | \[ \lim \mathop{\inf }\limits_{{n \rightarrow \infty }}{\int }_{\Omega }{\left| {u}_{n}\left( t,\omega \right) \right| }^{2}{dP} \geq {\int }_{\Omega }{\left| u\left( t,\omega \right) \right| }_{H}^{2}{dP} \] | No |
Lemma 79.4.9 Suppose conditions 1 - 79.4.2 hold. Also, suppose \( U \) is a separable Banach space dense in \( V \), a reflexive separable Banach space and \( V \) is dense in a Hilbert space \( H \) identified with its dual space. Thus\n\n\[ U \subseteq V \subseteq H = {H}^{\prime } \subseteq {V}^{\prime } \subseteq {... | Proof: In the following argument, \( N \) will be a set of measure zero containing the one of Lemma 79.4.6 and the sequence will always be a subsequence of the subsequence of that lemma. Recall that from Lemma 79.4.6, that for \( \omega \notin N \) ,\n\n\[ {\begin{Vmatrix} {u}_{n}\left( {t,\omega }\right) - {u}_{0}\lef... | Yes |
Proposition 1.4. For any tape description \( F \) and any \( e \in \mathbb{Z},\langle \left( {F,0, e}\right) \) , \( \left( {F,1, e - 1}\right) \rangle \) is a computation of \( {T}_{\text{right }} \) . | Thus \( {T}_{\text{right }} \) merely moves the tape one square to the right, and then stops. | No |
Proposition 2.2. Suppose \( f \) is \( m \) -ary, \( {g}_{0},\ldots ,{g}_{m - 1} \) are \( n \) -ary, and \( {h}_{0},\ldots ,{h}_{n - 1} \) are p-ary. Then\n\n\[ \n{\mathrm{K}}_{p}^{n}\left( {{\mathrm{\;K}}_{n}^{m}\left( {f;{g}_{0},\ldots ,{g}_{m - 1}}\right) ;{h}_{0},\ldots ,{h}_{n - 1}}\right) = {\mathrm{K}}_{p}^{m}\... | Proof. If \( {x}_{0},\ldots ,{x}_{p - 1} \in \omega \), then, with \( l = \) left hand side and \( r = \) right hand side,\n\n\[ \nl\left( {{x}_{0},\ldots ,{x}_{p - 1}}\right) = \left( {{\mathrm{K}}_{n}^{m}\left( {f;{g}_{0},\ldots ,{g}_{m - 1}}\right) }\right) \left( {{h}_{0}\left( {{x}_{0},\ldots ,{x}_{p - 1}}\right) ... | Yes |
Proposition 2.3. A number-theoretic function \( f \) is elementary iff there is a finite sequence \( \left\langle {{g}_{0},\ldots ,{g}_{k - 1}}\right\rangle \) of number-theoretic functions such that \( {g}_{k - 1} = f \), and for each \( i < k \) one of the following conditions holds:\n\n(i) \( {g}_{i} = + \) ,\n\n(ii... | Proof. Let \( A \) be the set of all \( f \) such that there is a finite sequence of the kind described in the theorem. By considering 1-termed sequences it is easy to see that \( + , \cdot \), subtraction, division, and \( {\mathrm{U}}_{j}^{n} \) are all in \( A \) (for any \( n > 0 \) and \( j < n \) ). Suppose \( f ... | Yes |
Proposition 2.4. Let \( A \) be closed under elementary recursive operations. If \( f \) is \( m \) -ary and \( f \in A \), and \( \pi \) is a permutation of \( \{ 0,\ldots, m - 1\} \), then the \( m \) -ary function \( g \) such that \( g\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) = f\left( {{x}_{▀},\ldots ,{x}_{\pi \... | Proof. \( g = {\mathrm{K}}_{m}^{m}\left( {f;{\mathrm{U}}_{▀}^{m},\ldots ,{\mathrm{U}}_{{nm} - 1)}^{m}}\right) \) . | Yes |
Proposition 2.5 (Identification of variables). Let \( A \) be closed under elementary recursive operations. If \( f \) is \( m \) -ary, \( m > 1 \), and \( f \in A \), then the \( \left( {m - 1}\right) \) -ary function \( g \) such that \( g\left( {{x}_{0},\ldots ,{x}_{m - 2}}\right) = f\left( {{x}_{0},\ldots ,{x}_{m -... | Proof. \( g = {\mathrm{K}}_{m - 1}^{m}\left( {f;{\mathrm{U}}_{0}^{m - 1},\ldots ,{\mathrm{U}}_{m - 2}^{m - 1},{\mathrm{U}}_{0}^{m - 1}}\right) \) . | Yes |
Proposition 2.6 (Adjoining apparent variables). Let \( A \) be closed under elementary recursive operations. If \( f \) is \( m \) -ary and \( f \in A \), then the \( \left( {m + 1}\right) \) -ary function \( g \) such that \( g\left( {{x}_{0},\ldots ,{x}_{m}}\right) = f\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) \) fo... | Proof. \( g = {\mathrm{K}}_{m + 1}^{m}\left( {f;{\mathrm{U}}_{0}^{m + 1},\ldots ,{\mathrm{U}}_{m - 1}^{m + 1}}\right) \) . | Yes |
Proposition 2.8. The following functions are elementary:\n\n(i) \( {\mathrm{C}}_{m}^{n}\left( {\text{for}n \neq 0}\right) \)\n\n(ii) \( s \)\n\n(iii) \( \mathrm{{sg}} \)\n\n(iv) \( \overline{\mathrm{{sg}}} \)\n\n(v) exponentiation\n\n(vi) factorial\n\n(vii) \( {p\not{} } \) | Proof\n\n(1) \( {\mathrm{C}}_{0}^{1} \) is elementary: \( {\mathrm{C}}_{0}^{1}x = \left| {x - x}\right| \) for all \( x \in \omega \) .\n\n(2) \( \overline{\mathrm{{sg}}} \) is elementary: \( \overline{\mathrm{{sg}}}x = \mathop{\prod }\limits_{{y < x}}{\mathrm{C}}_{0}^{1}y \), for all \( x \in \omega \) .\n\n(3) \( \ma... | Yes |
Proposition 2.10. 0 and \( \omega \) are elementary; if \( x \in \omega \) then \( \{ x\} \) is elementary. | Proof. \( {\chi }_{0} = {\mathrm{C}}_{0}^{1} \) and \( {\chi }_{\omega } = {\mathrm{C}}_{1}^{1} \) . If \( x \in \omega \), then for any \( y \in \omega ,{\chi }_{\left( x\right) }y = \) \( \overline{\operatorname{sg}}\left( \left| {x - y}\right| \right) \) ; hence \( {\chi }_{\{ x\} }y = \overline{\operatorname{sg}}\l... | Yes |
Proposition 2.11. Let \( A \) be closed under elementary recursive operations. If \( R \) and \( S \) are \( A \) -relations, then so are \( R \cap S, R \cup S \), and \( {}^{m}\omega \sim R \) . | Proof. For all \( {x}_{0},\ldots ,{x}_{m - 1},{\chi }_{R \cap S}\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) = {\chi }_{R}\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) \cdot {\chi }_{S}\left( {x}_{0}\right. \) , \( \left. {\ldots ,{x}_{m - 1}}\right) ,{\chi }_{T}\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) = \overline{\mat... | Yes |
Proposition 2.13. The binary relations \( \leq , < , \geq , = \) , \( \neq \) are elementary. | Proof. For any \( x, y \in \omega \) , \[ {\chi }_{ < }\left( {x, y}\right) = \overline{\operatorname{sg}}\left\lbrack {{sx}/{sy}}\right\rbrack = \overline{\operatorname{sg}}\left\lbrack {s{\mathrm{\;U}}_{0}^{2}\left( {x, y}\right) /s{\mathrm{\;U}}_{1}^{2}\left( {x, y}\right) }\right\rbrack . \] Thus \( < \) is element... | Yes |
Proposition 2.14 (Bounded existential quantifier). Let \( A \) be closed under elementary recursive operations. Suppose \( R \) is an m-ary A-relation. Let \( S = \left\{ {\left\langle {{x}_{0},\ldots ,{x}_{m - 1}}\right\rangle : }\right. \) there is a \( \left. {y < {x}_{m - 1}\text{such that}\left\langle {{x}_{0},\ld... | Proof. \( {\chi }_{S}\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) = \operatorname{sg}\sum \left\{ {{\chi }_{R}\left( {{x}_{0},\ldots ,{x}_{m - 2}, y}\right) : y < {x}_{m - 1}}\right\} \) | Yes |
Proposition 2.15 (Bounded universal quantifier). Let \( A \) be closed under elementary recursive operations. Suppose \( R \) is an m-ary A-relation. Let \( T = \left\{ {\left\langle {{x}_{0},\ldots ,{x}_{m - 1}}\right\rangle : \text{ for every }y < {x}_{m - 1}\text{ we have }\left\langle {{x}_{0},\ldots ,{x}_{m - 2}, ... | Proof. Let \( S \) be as in 2.14, with \( R \) replaced by \( {}^{m}\omega \sim R \) . Then \( T = \) \( {}^{m}\omega \sim S \) . | Yes |
Proposition 2.17. Let \( A \) be closed under elementary recursive operations. If \( R \) is an m-ary A-relation, then the function \( f \) of 2.16 is a member of \( A \) . | Proof. Note that\n\n(1) \( \overline{\mathrm{{sg}}}\mathop{\sum }\limits_{{y < i}}{\chi }_{R}\left( {{x}_{0},\ldots ,{x}_{m - 2}, y}\right) = \left\{ \begin{array}{ll} 1 & \text{ if }\left\langle {{x}_{0},\ldots ,{x}_{m - 1}, y}\right\rangle \notin R\text{ for all }y < i, \\ 0 & \text{ otherwise. } \end{array}\right. \... | Yes |
Proposition 2.18 (Definition by cases). Let \( A \) be closed under elementary recursive operations. Suppose \( {g}_{0},\ldots ,{g}_{m - 1} \) are n-ary members of \( A \) , \( {R}_{0},\ldots ,{R}_{m - 1} \) are pairwise disjoint n-ary A-relations with \( \mathop{\bigcup }\limits_{{i < m}}{R}_{i} = {}^{n}\omega \) , an... | Proof. For any \( {x}_{0},\ldots ,{x}_{n - 1} \in \omega \) , \[ f\left( {{x}_{0},\ldots ,{x}_{n - 1}}\right) = {\chi }_{R0}\left( {{x}_{0},\ldots ,{x}_{n - 1}}\right) \cdot {g}_{0}\left( {{x}_{0},\ldots ,{x}_{n - 1}}\right) + \cdots \] \[ + {\chi }_{R\left( {m - 1}\right) }\left( {{x}_{0},\ldots ,{x}_{n - 1}}\right) \... | Yes |
Proposition 2.20. All of the functions and relations of 2.19 are elementary. | Proof. Obvious, as concerns \( \left( i\right) - \left( {iv}\right) \) . For \( \left( v\right) \) ,\n\n\[ \operatorname{rm}\left( {x, y}\right) = \left\{ \begin{array}{ll} x - \left( {y \cdot \left\lbrack {x/y}\right\rbrack }\right) & \text{ if }y \neq 0, \\ 0 & \text{ if }y = 0. \end{array}\right. \]\n\nFor \( \left(... | No |
Proposition 2.22 (Number-theoretic). For every \( k,{\mathrm{p}}_{k} \leq \exp \left( {2,{2}^{k}}\right) \) . | Proof. By induction on \( k \) . Trivial for \( k = 0,1 \) . Induction step, \( k > 0 \) :\n\n(Euclid)\n\n\[ \leq \exp \left( {2,{2}^{0}}\right) \cdot \ldots \cdot \exp \left( {2,{2}^{k}}\right) - 1\;\text{(induction hypothesis)} \]\n\n\[ = {2}^{\sum \{ \exp \left( {2, i}\right) : i \leq k\} } - 1 \]\n\n\[ = \exp \left... | No |
Proposition 2.23. \( \mathrm{p} \) is elementary. | Proof. Let \( N = \{ \left( {x, y}\right) : x, y \in \mathrm{{PM}}, x < y \), and \( y \) is the next prime after \( x\} \) . Thus \( N = \{ \left( {x, y}\right) : x, y \in \mathrm{{PM}} \) and \( x < y \) and for all \( z < y \), either \( z \leq x \) or not \( z \notin \mathrm{{PM}}\} \), so \( N \) is elementary. Le... | Yes |
Proposition 2.25. ( ) is elementary. | Proof. \( \;{\left( a\right) }_{i} = {\mu x} < a\left( {{\mathrm{p}}_{i}^{x}\left| {a\text{ and not }{\mathrm{p}}_{i}^{x + 1}}\right| a}\right) \) . | No |
Proposition 2.27. 1 is elementary. | Proof. \( \;\operatorname{la} = {\mu i} < a\left\lbrack {{\mathrm{p}}_{i} \mid x\text{and}\forall j \leq a\left( {i < j \Rightarrow {\mathrm{p}}_{j} \nmid a}\right) }\right\rbrack \) . | No |
Theorem 2.30. If \( A \) is closed under primitive recursive operations, then \( A \) is closed under elementary recursive operations. In particular, every elementary function is primitive recursive. | ## Proof\n\n(1) \( p \) is primitive recursive. For,\n\n\[ \n{10} = 0 \n\] \n\n\[ \n{hsy} = {\mathrm{U}}_{0}^{2}\left( {y,{hy}}\right) \text{.} \n\] \n\n(2) \( \div \) is primitive recursive. For, \n\n\[ \nx - 0 = {\mathrm{U}}_{0}^{1}x, \n\] \n\n\[ \nx \div {sy} = p{\mathrm{\;U}}_{2}^{3}\left( {x, y, x \div y}\right) .... | Yes |
Proposition 2.32. Let \( A \) be closed under primitive recursive operations. Then \( f \in A \) iff \( \widetilde{f} \in A \) . | Proof. Assume first that \( f \in A \) . Then\n\n\[ \widetilde{f}\left( {{x}_{0},\ldots ,{x}_{m - 2},0}\right) = 1, \]\n\n\[ \widetilde{f}\left( {{x}_{0},\ldots ,{x}_{m - 2},{sy}}\right) = \mathop{\prod }\limits_{{i < {sy}}}{\mathrm{p}}_{i}^{f\left( {{x0},\ldots, x\left( {m - 2}\right), i}\right) + 1} \]\n\n\[ = \widet... | Yes |
Proposition 2.33 (Course-of-values recursion). Let \( A \) be closed under primitive recursive operations. Suppose \( f \) is an m-ary function and \( h \) is an \( \left( {m + 1}\right) \) -ary member of \( A \) such that, for all \( {x}_{0},\ldots ,{x}_{m - 1} \in \omega \) ,\n\n\[ \nf\left( {{x}_{0},\ldots ,{x}_{m -... | Proof\n\n\[ \n\widetilde{f}\left( {{x}_{0},\ldots ,{x}_{m - 2},0}\right) = 1, \n\]\n\n\[ \n\widetilde{f}\left( {{x}_{0},\ldots ,{x}_{m - 2},{sy}}\right) = \mathop{\prod }\limits_{{i < {sy}}}{\mathrm{p}}_{i}^{f\left( {{x0},\ldots, x\left( {m - 2}\right), i}\right) + 1} \n\]\n\n\[ \n= \widetilde{f}\left( {{x}_{0},\ldots ... | Yes |
Lemma 2.37. \( m \leq a\left( {m, n}\right) \) for all \( m, n \) . | Proof. We may assume that \( m \neq 0 \) . Now we prove 2.37 by induction on \( n \) : \( a\left( {m,0}\right) = m \) . Assuming \( m \leq a\left( {m, n}\right) \) ,\n\n\[ a\left( {m, n + 1}\right) = {m}^{a\left( {m, n}\right) } \geq {m}^{m} \geq m. \] | Yes |
Lemma 2.38. \( a\left( {m, n}\right) < a\left( {m, n + 1}\right) \) for all \( m > 1 \) and all \( n \in \omega \) . | Proof. \( a\left( {m, n + 1}\right) = {m}^{a\left( {m, n}\right) } > a\left( {m, n}\right) \) . | Yes |
Lemma 2.39. \( a\left( {m, n}\right) < a\left( {m + 1, n}\right) \) for all \( m \neq 0 \) and all \( n \in \omega \) . | Proof. We proceed by induction on \( n : a\left( {m,0}\right) = m < m + 1 = a\left( {m + 1,0}\right) \) . Assuming our result for \( n \) , \[ a\left( {m, n + 1}\right) = {m}^{a\left( {m, n}\right) } \leq {\left( m + 1\right) }^{a\left( {m, n}\right) } \] \[ < {\left( m + 1\right) }^{a\left( {m + 1, n}\right) } = a\lef... | Yes |
Lemma 2.40. \( a\left( {m, n}\right) + a\left( {m, p}\right) \leq a\left( {m,\max \left( {n, p}\right) + 1}\right) \) for all \( m > 1 \) and all \( n, p \in \omega \) . | Proof. \( a\left( {m, n}\right) + a\left( {m, p}\right) \leq {2a}\left( {m,\max \left( {n, p}\right) }\right) \) by 2.38\n\n\[ \leq {2}^{a\left( {m,\max \left( {n, p}\right) }\right) } \leq {m}^{a\left( {m,\max \left( {n, p}\right) }\right) } \]\n\n\[ = a\left( {m,\max \left( {n, p}\right) + 1}\right) \text{.} \] | Yes |
Lemma 2.41. \( a\left( {m, n}\right) \cdot a\left( {m, p}\right) \leq a\left( {m,\max \left( {n, p}\right) + 1}\right) \) for all \( m > 1 \) and all \( n, p \in \omega \) . | Proof. If \( n = p = 0 \) then the inequality is obvious. Hence assume that \( n \neq 0 \) or \( p \neq 0 \) . Then\n\n\[ a\left( {m, n}\right) \cdot a\left( {m, p}\right) \leq a{\left( m,\max \left( n, p\right) \right) }^{2}\text{ by }{2.38} \]\n\n\[ = {\left( {m}^{a\left( {m,\max \left( {n, p}\right) - 1}\right) }\ri... | Yes |
Lemma 2.42. \( a{\left( m, n\right) }^{a\left( {m, p}\right) } \leq a\left( {m,\max \left( {p + 2, n + 1}\right) }\right) \) for all \( m > 1 \) and all \( n, p \in \omega \) . | Proof. For \( n = 0 \) we have\n\n\[ a{\left( m, n\right) }^{a\left( {m, p}\right) } = {m}^{a\left( {m, p}\right) } = a\left( {m, p + 1}\right) \leq a\left( {m,\max \left( {p + 2, n + 1}\right) }\right) \]\n\n(using 2.38). If \( n \neq 0 \) we have\n\n\[ a{\left( m, n\right) }^{\alpha \left( {m, p}\right) } = {m}^{\alp... | Yes |
Lemma 2.43. \( a\left( {a\left( {m, n}\right), p}\right) \leq a\left( {m, n + {2p}}\right) \) for all \( m > 1 \) and all \( n, p \in \omega \) . | Proof. We proceed by induction on \( p \) :\n\n\[ a\left( {a\left( {m, n}\right) ,0}\right) = a\left( {m, n}\right) = a\left( {m, n + 2 \cdot 0}\right) .\n\]\n\nAssuming our result for \( p \), we then have\n\n\[ a\left( {a\left( {m, n}\right), p + 1}\right) = a{\left( m, n\right) }^{a\left( {a\left( {m, n}\right), p}\... | Yes |
Theorem 2.45. There are primitive recursive functions which are not elementary in fact, \( a \) is such a function. | Proof. By 2.36, \( a \) is primitive recursive. Suppose \( a \) is elementary. Let \( {fm} = a\left( {m, m}\right) \) for all \( m \in \omega \) . Thus \( f \) is elementary. By 2.44 choose \( m \in \omega \) such that \( x > 1 \) implies that \( {fx} < a\left( {x, m}\right) \) . Then\n\n\[ a\left( {m + 2, m + 2}\right... | Yes |
Theorem 3.4. Let \( A \) be a set of number-theoretic functions closed under elementary recursive operations. If \( f \) is universal for unary members of \( A \) , then \( f \notin A \) . | Proof. Assume that \( f \in A \) . Let \( {gm} = f\left( {m, m}\right) + 1 \) for all \( m \in \omega \) . Thus \( g \in A \) . Since \( f \) is universal for unary members of \( A \), choose \( m \in \omega \) such that \( f\left( {m, n}\right) = \) \( {gn} \) for all \( n \in \omega \) . Then \( {gm} = f\left( {m, m}... | Yes |
Theorem 3.7. There are exactly \( {\aleph }_{0} \) recursive functions. | Proof. Let \( {A}_{0} \) consist of all of the functions \( s,{\mathrm{U}}_{i}^{n} \) with \( i < n \) . Thus \( \left| {A}_{0}\right| = \) \( {\aleph }_{0} \) . Having defined \( {A}_{n} \), let \( {A}_{n + 1} \) consist of all members of \( {A}_{n} \) together with all functions obtainable from members of \( {A}_{n} ... | Yes |
Theorem 3.8. There is a number-theoretic function which is not recursive. | Although Theorem 3.8 follows from 3.7 purely on grounds of cardinality, we can also explicitly exhibit a nonrecursive function. Let \( {f}_{0},{f}_{1},\ldots \) be an enumeration of all unary recursive functions (by 3.7). Define \( {gm} = {f}_{m}m + 1 \) for all \( m \in \omega \) . Then \( g \) is obviously not in our... | Yes |
Lemma 3.11. \( {\mathrm{U}}_{i}^{n} \) is Turing computable. | Proof. The machine is \( {T}_{\left( {n - i}\right) \text{copy }} \) . | No |
Lemma 3.12. The class of Turing computable functions is closed under composition. | Proof. Suppose \( {fm} \) -ary, \( {g}_{0},\ldots ,{g}_{m - 1}n \) -ary. Suppose \( f,{g}_{0},\ldots ,{g}_{m - 1} \) are computed by \( M,{N}_{0},\ldots ,{N}_{m - 1} \) respectively. Then the following machine computes \( {\mathrm{K}}_{n}^{m}\left( {f;{g}_{0},\ldots ,{g}_{m - 1}}\right) \) :\n\n\[ \n{T}_{\text{left}} \... | Yes |
Lemma 3.13. The class of Turing computable functions is closed under primitive recursion without a parameter. | Proof. Suppose that \( f \) is a binary operation on \( \omega \), computed by a machine\n\n\( M \), and \( a \in \omega \) . Let \( {g0} = a, g\left( {n + 1}\right) = f\left( {n,{gn}}\right) \) for all \( n \in \omega \) . Then the following machine computes \( g \) :\n\n\[ \n{T}_{\text{left }} \rightarrow {T}_{1} \ri... | Yes |
Lemma 3.14. The class of Turing computable functions is closed under primitive recursion with parameters. | Proof. Suppose that \( f \) is \( m \) -ary, \( m > 0, g \) is \( \left( {m + 2}\right) \) -ary and that they are computed by \( M \) and \( N \) respectively. Let \( h\left( {{x}_{0},\ldots ,{x}_{m - 1},0}\right) = f\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) \), \( h\left( {{x}_{0},\ldots ,{x}_{m - 1}, y + 1}\right) ... | Yes |
Lemma 3.15. The class of Turing computable functions is closed under minimalization (applied to special functions). | Proof. Let \( f \) be an \( m \) -ary special function, \( m > 1 \), and suppose that \( f \) is computed by a machine \( M \) . Let \( g\left( {{x}_{0},\ldots ,{x}_{m - 2}}\right) = {\mu y}\left\lbrack {f\left( {{x}_{0},\ldots ,{x}_{m - 2}, y}\right) = }\right. \) 0] for all \( {x}_{0},\ldots ,{x}_{m - 2} \in \omega \... | Yes |
Lemma 3.18. \( {g}^{ * }T \) is elementary. | Proof. For any \( x \in \omega, x \in {\mathcal{J}}^{ * }T \) if \( \ln x \) is odd, \( x > 1 \), for every \( i \leq \ln x \) we have \( {\left( {\left( x\right) }_{i}\right) }_{2} < 5 \), for every \( i \leq {1x} \) there is a \( j \leq {1x} \) such that \( {\left( {\left( x\right) }_{i}\right) }_{3} = {\left( {\left... | Yes |
Lemma 3.21. \( {g}^{ * }\mathbb{C} \) is elementary. | Proof. For any \( x \in \omega, x \in {\mathcal{J}}^{ * }\mathbb{C} \) iff \( \forall i \leq 1{\left( x\right) }_{1}\left( {{\left( {\left( x\right) }_{1}\right) }_{i} < 2}\right) ,{\left( x\right) }_{1} \neq 0,{\left( x\right) }_{0} \in \) \( {\mathcal{J}}^{ * }T \), and there is an \( i \leq \mathrm{l}{\left( x\right... | No |
Lemma 3.23. \( {f}_{0} \) and \( {f}_{1} \) are elementary. For any \( e \in \mathbb{Z} \) we have \( {f}_{0}{ge} = g\left( {e + 1}\right) \) and \( {f}_{1}{ge} = g\left( {e - 1}\right) \) . | \[ {f}_{0}{ge} = \left\{ \begin{array}{ll} {f}_{0}{2e} & e \geq 0 \\ {f}_{0}\left( {-{2e} - 1}\right) & e < 0 \end{array}\right\} = \left\{ \begin{array}{ll} 2\left( {e + 1}\right) & e \geq 0 \\ 0 & e = - 1 \\ - {2e} - 3 & e < - 1 \end{array}\right\} = g\left( {e + 1}\right) ; \] \[ {f}_{1}{ge} = \left\{ \begin{array}{... | Yes |
Lemma 3.24. Let \( {R}_{0} = \{ \left( {x, n,\varepsilon, y}\right) : x = {gF} \) for some tape description \( F \) , \( n = {ge} \) for some \( e \in \mathbb{Z},\varepsilon = 0 \) or \( \varepsilon = 1 \), and \( \left. {y = g\left( {F}_{\varepsilon }^{e}\right) }\right\} \) . Then \( {R}_{0} \) is elementary. | Proof. \( \left( {x, n,\varepsilon, y}\right) \in {R}_{0} \) iff \( \forall i \leq \operatorname{lx}\left( {{\left( x\right) }_{i} < 2}\right), x \neq 0,\varepsilon < 2 \), and \( y = \) \( \left\lbrack {x/{\mathrm{p}}_{n}^{\left( x\right) n}}\right\rbrack \cdot {\mathrm{p}}_{n}^{\varepsilon }. \) | No |
Lemma 3.25. Let \( {R}_{1} = \{ \left( {x, y}\right) : x \) is the Gödel number of a complete configuration \( \left( {M, F, d, e}\right), y \) is the Gödel number of a complete configuration \( \left( {M,{F}^{\prime },{d}^{\prime },{e}^{\prime }}\right) \) (same \( M \) ), and \( \left( {\left( {F, d, e}\right) ,\left... | Proof. For any \( x, y,\left( {x, y}\right) \in {R}_{1} \) iff \( x \in {\mathcal{g}}^{ * }\mathbb{C}, y \in {\mathcal{g}}^{ * }\mathbb{C},{\left( x\right) }_{0} = {\left( y\right) }_{0} \), and there is an \( i \leq \mathrm{l}\left( {\left( x\right) }_{0}\right) \) such that \( {\left( x\right) }_{2} = {\left( {\left(... | Yes |
Lemma 3.27. \( {R}_{2} \) is elementary. |  \( {\left( {\left( x\right) }_{0}\right) }_{2} \), and for every \( i < \mathrm{l}x,\left( {{\left( x\right) }_{i},{\left( x\right) }_{i + 1}}\right) \in {R}_{1} \), and there is an \( i \leq \mathrm{l}{\left( {\left(... | No |
Lemma 3.34. Let \( {R}_{3} = \{ \left( {x, y, m, n}\right) : x \) is the Gödel number of a tape description \( F, y \) is the Gödel number of a finite sequence \( h \) of 0 ’s and 1’s, \( m = {ge} \) and \( n = g{e}^{\prime } \) for certain \( e,{e}^{\prime } \in \mathbb{Z} \), and \( h \) lies on \( F \) beginning at ... | Proof. For any \( x, y, m, n\left( {x, y, m, n}\right) \in {R}_{3} \) iff \( y \neq 0,\forall i \leq \operatorname{lx}\left( {{\left( x\right) }_{i} < 2}\right) \) , \( x \neq 0 \), either \( y = 1 \) and \( m = n \), or else \( y > 1 \), for every \( i \leq {1y}\left\lbrack {{\left( y\right) }_{i} = 1}\right. \) or \(... | Yes |
Lemma 3.36. \( {\mathrm{T}}_{m} \) is elementary. | Proof. For any \( e,{x}_{0},\ldots ,{x}_{m - 1}, u,\left( {e,{x}_{0},\ldots ,{x}_{m - 1}, u}\right) \in {\mathrm{T}}_{m} \) iff \( e \in {\mathcal{J}}^{ * }\mathbb{T} \) , \( u \in {R}_{2},{\left( {\left( u\right) }_{0}\right) }_{0} = e,\left( {{\left( {\left( u\right) }_{0}\right) }_{1},{f}_{3}^{m}\left( {{x}_{0},\ldo... | Yes |
Lemma 3.38. Every Turing computable function is recursive. | Proof. Let \( M \) be a Turing machine which computes \( f \) as described in Definition 3.9(ii), and let \( e = {gM} \) . Then for any \( {x}_{0},\ldots ,{x}_{m - 1} \in \omega \) ,\n\n\[ f\left( {{x}_{0},\ldots ,{x}_{m - 1}}\right) = \mathrm{V}{\mu u}\left\lbrack {\left( {e,{x}_{0},\ldots ,{x}_{m - 1}, u}\right) \in ... | Yes |
Lemma 3.41. \( \left\lbrack \sqrt{}\right\rbrack \) and Exc are elementary. | \[ \left\lbrack \sqrt{0}\right\rbrack = 0 \] \[ \left\lbrack {\sqrt{}\left( {n + 1}\right) }\right\rbrack = \left\{ \begin{array}{l} \left\lbrack \sqrt{n}\right\rbrack \;\text{ if }n + 1 \neq {\left( \left\lbrack \sqrt{n}\right\rbrack + 1\right) }^{2}, \\ \left\lbrack \sqrt{n}\right\rbrack + 1\;\text{ otherwise } \end{... | No |
Theorem 3.44 (Number-theoretic: The Chinese remainder theorem). Let \( {m}_{0},\ldots ,{m}_{r - 1} \) be natural numbers \( > 1 \), with \( r > 1 \), the \( {m}_{i} \) ’s pairwise relatively prime. Let \( {a}_{0},\ldots ,{a}_{r - 1} \) be any \( r \) natural numbers. Then there is an \( x \in \omega \) such that \( x \... | Proof. By induction on \( r \) ; we first take the case \( r = 2 \) . Since \( {m}_{0} \) and \( {m}_{1} \) are relatively prime, there exist integers (positive, negative, or zero) \( s \) and \( t \) such that \( 1 = {m}_{0}s + {m}_{1}t \) . Then \( {a}_{0} - {a}_{1} = {m}_{0}s\left( {{a}_{0} - {a}_{1}}\right) + {m}_{... | Yes |
Theorem 3.46 (Number-theoretic: Gödel’s \( \\beta \) -function lemma). For any finite sequence \( {y}_{0},\\ldots ,{y}_{n - 1} \) of natural numbers there is an \( x \\in \\omega \) such that \( \\beta \\left( {x, i}\\right) = {y}_{i} \) for each \( i < n \) . | Proof. Let \( s \) be the maximum of \( {y}_{0},\\ldots ,{y}_{n - 1}, n \) . For each \( i < n \) let \( {m}_{i} = \) \( 1 + \\left( {i + 1}\\right) \\cdot s! \) Then for \( i < j < n \) the integers \( {m}_{i} \) and \( {m}_{j} \) are relatively prime. For, if a prime \( p \) divides both \( {m}_{i} \) and \( {m}_{j} ... | Yes |
Theorem 3.50. The class of 1-place recursive functions is the intersection of all sets \( A \) of 1-place functions such that \( s,\mathrm{{Exc}} \in A \) and \( A \) is closed under \( {\mathrm{K}}_{1}^{1},\mathrm{P} \), and inversion (applied to functions with range \( \omega \) ). | Proof. Clearly the intersection indicated is included in the class of 1-place recursive functions. Now suppose that \( A \) satisfies the conditions of the theorem. Note that \( {\mathrm{U}}_{0}^{1} \in A \), since \( {\mathrm{U}}_{0}^{1} = {\mathrm{K}}_{1}^{1}\left( {\mathrm{{Exc}},{\mathrm{{Exc}}}^{\left( -1\right) }... | Yes |
Lemma 4.4. The set of Gödel numbers of words is elementary. | Proof. \( m \) is the Gödel number of a word iff \( m = 1 \) or \( m > 1 \) and \( \forall i \leq \operatorname{lm} \) \( \left\lbrack {{\left( m\right) }_{i} \leq 3\text{ and }1 \leq {\left( m\right) }_{i}}\right\rbrack \) | Yes |
Lemma 4.6. The set of Gödel numbers of Markov algorithms is elementary. | Proof. \( \;n \) is the Gödel number of a Markov algorithm iff \( n \geq 2 \) and \( \forall i \leq {1n} \) \( \left\lbrack {{\left( {\left( n\right) }_{i}\right) }_{0}\text{ and }{\left( {\left( n\right) }_{i}\right) }_{1}}\right. \) are Gödel numbers of words, \( {\left( {\left( n\right) }_{i}\right) }_{2} \leq 1 \),... | No |
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