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Exercise 8.1.2 (The maximum modulus principle) If \( f \) is as in the previous exercise, show that \( \left| {f\left( z\right) }\right| < M \) for all interior points \( z \in R \) , unless \( f \) is constant.
Theorem 8.1.3 (Phragmén - Lindelöf) Suppose that \( f\left( s\right) \) is entire in the region
\[
S\lef... | {
"question": "Exercise 8.1.2 (The maximum modulus principle) If \( f \) is as in the previous exercise, show that \( \left| {f\left( z\right) }\right| < M \) for all interior points \( z \in R \) , unless \( f \) is constant.",
"proof": "Null"
} |
Exercise 8.3.7 Let \( K \) be a quadratic field of discriminant \( d \) . Let \( {P}_{0} \) denote the group of principal fractional ideals \( \alpha {\mathcal{O}}_{K} \) with \( \alpha \in K \) satisfying \( {N}_{K}\left( \alpha \right) > 0 \) . The quotient group \( {H}_{0} \) of all nonzero fractional ideals modul... | {
"question": "Exercise 8.3.7 Let \( K \) be a quadratic field of discriminant \( d \) . Let \( {P}_{0} \) denote the group of principal fractional ideals \( \alpha {\mathcal{O}}_{K} \) with \( \alpha \in K \) satisfying \( {N}_{K}\left( \alpha \right) > 0 \) . The quotient group \( {H}_{0} \) of all nonzero fracti... |
Proposition 9.34 Define a domain \( \operatorname{Dom}\left( \Delta \right) \) as follows:
\[
\operatorname{Dom}\left( \Delta \right) = \left\{ {\psi \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \left| {\;{\left| \mathbf{k}\right| }^{2}\widehat{\psi }\left( \mathbf{k}\right) \in {L}^{2}\left( {\mathbb{R}}^{n}\right) }\... | {
"question": "Proposition 9.34 Define a domain \( \operatorname{Dom}\left( \Delta \right) \) as follows:\n\n\[ \operatorname{Dom}\left( \Delta \right) = \left\{ {\psi \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \left| {\;{\left| \mathbf{k}\right| }^{2}\widehat{\psi }\left( \mathbf{k}\right) \in {L}^{2}\left( {\mathb... |
Exercise 9.2.10 Consider the element
\[
\alpha = {\left( x + y\right) }^{\ell - 2}\left( {x + {\zeta y}}\right)
\]
Show that:
(a) the ideal \( \left( \alpha \right) \) is a perfect \( \ell \) th power.
(b) \( \alpha \equiv 1 - {u\lambda }\left( {\;\operatorname{mod}\;{\lambda }^{2}}\right) \) where \( u = {\left(... | {
"question": "Exercise 9.2.10 Consider the element\n\n\\[ \n\\alpha = {\\left( x + y\\right) }^{\\ell - 2}\\left( {x + {\\zeta y}}\\right) \n\\]\n\nShow that:\n\n(a) the ideal \\( \\left( \\alpha \\right) \\) is a perfect \\( \\ell \\) th power.\n\n(b) \\( \\alpha \\equiv 1 - {u\\lambda }\\left( {\\;\\operatorname... |
Proposition 3.2. In a chart \( U \times \mathbf{E} \) for \( {TX} \), let \( f : U \times \mathbf{E} \rightarrow \mathbf{E} \times \mathbf{E} \) represent \( F \), with \( f = \left( {{f}_{1},{f}_{2}}\right) \) . Then \( f \) represents a spray if and only if, for all \( s \in \mathbf{R} \) we have
\[
{f}_{2}\left( ... | {
"question": "In a chart \( U \times \mathbf{E} \) for \( {TX} \), let \( f : U \times \mathbf{E} \rightarrow \mathbf{E} \times \mathbf{E} \) represent \( F \), with \( f = \left( {{f}_{1},{f}_{2}}\right) \) . Then \( f \) represents a spray if and only if, for all \( s \in \mathbf{R} \) we have\n\n\[ \n{f}_{2}\le... |
Theorem 2.2.8. If \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| = 1 \), a.e., then \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) if and only if there is a constant \( c \) of modulus 1 such that \( {\phi... | {
"question": "Theorem 2.2.8. If \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| = 1 \), a.e., then \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) if and only if there is a constant \( c \) of modulus 1 s... |
Theorem 3.1. (Eisenstein’s Criterion). Let \( A \) be a factorial ring. Let \( K \) be its quotient field. Let \( f\left( X\right) = {a}_{n}{X}^{n} + \cdots + {a}_{0} \) be a polynomial of degree \( n \geqq 1 \) in \( A\left\lbrack X\right\rbrack \) . Let \( p \) be a prime of \( A \), and assume:
\[
{a}_{n} ≢ 0\;\l... | {
"question": "Theorem 3.1. (Eisenstein’s Criterion). Let \( A \) be a factorial ring. Let \( K \) be its quotient field. Let \( f\left( X\right) = {a}_{n}{X}^{n} + \cdots + {a}_{0} \) be a polynomial of degree \( n \geqq 1 \) in \( A\left\lbrack X\right\rbrack \) . Let \( p \) be a prime of \( A \), and assume:\n\... |
Proposition 5.46. Suppose \( M \) is a smooth manifold without boundary and \( D \subseteq M \) is a regular domain. The topological interior and boundary of \( D \) are equal to its manifold interior and boundary, respectively.
Proof. Suppose \( p \in D \) is arbitrary. If \( p \) is in the manifold boundary of \( ... | {
"question": "Proposition 5.46. Suppose \( M \) is a smooth manifold without boundary and \( D \subseteq M \) is a regular domain. The topological interior and boundary of \( D \) are equal to its manifold interior and boundary, respectively.",
"proof": "Proof. Suppose \( p \in D \) is arbitrary. If \( p \) is... |
Lemma 3.7. Let \( f\left( z\right) \in {H}^{1} \) . Then the Fourier transform
\[
\widehat{f}\left( s\right) = {\int }_{-\infty }^{\infty }f\left( t\right) {e}^{-{2\pi ist}}{dt} = 0
\]
for all \( s \leq 0 \) .
Proof. By the continuity of \( f \rightarrow \widehat{f} \), we may suppose \( \int \in {\mathfrak{A}}_{N... | {
"question": "Lemma 3.7. Let \( f\left( z\right) \in {H}^{1} \) . Then the Fourier transform\n\n\[ \n\widehat{f}\left( s\right) = {\int }_{-\infty }^{\infty }f\left( t\right) {e}^{-{2\pi ist}}{dt} = 0 \n\]\n\nfor all \( s \leq 0 \) .",
"proof": "Proof. By the continuity of \( f \rightarrow \widehat{f} \), we m... |
Example 2.3.14. Let \( u = {\delta }_{{x}_{0}} \) and \( f \in \mathcal{S} \) . Then \( f * {\delta }_{{x}_{0}} \) is the function \( x \mapsto f\left( {x - {x}_{0}}\right) \) , for when \( h \in \mathcal{S} \), we have
\[
\left\langle {f * {\delta }_{{x}_{0}}, h}\right\rangle = \left\langle {{\delta }_{{x}_{0}},\wi... | {
"question": "Example 2.3.14. Let \( u = {\delta }_{{x}_{0}} \) and \( f \in \mathcal{S} \) . Then \( f * {\delta }_{{x}_{0}} \) is the function \( x \mapsto f\left( {x - {x}_{0}}\right) \) , for when \( h \in \mathcal{S} \), we have",
"proof": "\[ \left\langle {f * {\delta }_{{x}_{0}}, h}\right\rangle = \left... |
Exercise 1.4.13 Use Exercises 1.2.7 and 1.2.8 to show that there are infinitely many primes \( \equiv 1\left( {\;\operatorname{mod}\;{2}^{r}}\right) \) for any given \( r \) .
Exercise 1.4.14 Suppose \( p \) is an odd prime such that \( {2p} + 1 = q \) is also prime. Show that the equation
\[
{x}^{p} + 2{y}^{p} + 5... | {
"question": "Exercise 1.4.13 Use Exercises 1.2.7 and 1.2.8 to show that there are infinitely many primes \( \equiv 1\\left( {\\operatorname{mod}\;{2}^{r}}\\right) \) for any given \( r \) .",
"proof": null
} |
Theorem 11.5 For each \( h > 0 \) the difference equations (11.10)-(11.11) have a unique solution.
Proof. The tridiagonal matrix \( A \) is irreducible and weakly row-diagonally dominant. Hence, by Theorem 4.7, the matrix \( A \) is invertible, and the Jacobi iterations converge.
Recall that for speeding up the con... | {
"question": "Theorem 11.5 For each \( h > 0 \) the difference equations (11.10)-(11.11) have a unique solution.",
"proof": "Proof. The tridiagonal matrix \( A \) is irreducible and weakly row-diagonally dominant. Hence, by Theorem 4.7, the matrix \( A \) is invertible, and the Jacobi iterations converge."
} |
Corollary 3.4.6. Let \( 0 < {p}_{0} < \infty \) . Then for any \( p \) with \( {p}_{0} \leq p < \infty \) and for all locally integrable functions \( f \) on \( {\mathbf{R}}^{n} \) with \( {M}_{d}\left( f\right) \in {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) we have
\[
\parallel f{\parallel }_{{L}^{p}\left( {\ma... | {
"question": "Corollary 3.4.6. Let \( 0 < {p}_{0} < \infty \) . Then for any \( p \) with \( {p}_{0} \leq p < \infty \) and for all locally integrable functions \( f \) on \( {\mathbf{R}}^{n} \) with \( {M}_{d}\left( f\right) \in {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) we have\n\n\[ \parallel f{\parallel }_... |
Lemma 12.2.2 The set of lines spanned by the vectors of \( {D}_{n} \) is star-closed.
## 12.3 Reflections
We can characterize star-closed sets of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) in terms of their symmetries. If \( h \) is a vector in \( {\mathbb{R}}^{n} \), then there is a unique hyperplane t... | {
"question": "Lemma 12.2.2 The set of lines spanned by the vectors of \( {D}_{n} \) is star-closed.",
"proof": "Null"
} |
Lemma 9.3.7. If \( \left( {{g}_{1},{S}_{1}}\right) \sim \left( {{g}_{2},{S}_{2}}\right) \) then for all \( i\left( {{g}_{1},{e}_{i}\left( {S}_{1}\right) }\right) \sim \left( {{g}_{2},{e}_{i}\left( {S}_{2}\right) }\right) \) .
Proof. There exist \( e \) and \( T \) such that \( \left( {e\left( {S}_{i}\right), T}\righ... | {
"question": "Lemma 9.3.7. If \( \left( {{g}_{1},{S}_{1}}\right) \sim \left( {{g}_{2},{S}_{2}}\right) \) then for all \( i\left( {{g}_{1},{e}_{i}\left( {S}_{1}\right) }\right) \sim \left( {{g}_{2},{e}_{i}\left( {S}_{2}\right) }\right) \) .",
"proof": "Proof. There exist \( e \) and \( T \) such that \( \left( ... |
Theorem 4.11. Every module can be embedded into an injective module.
Noetherian rings. Our last result is due to Bass (cf. Chase [1961]).
Theorem 4.12. A ring \( R \) is left Noetherian if and only if every direct sum of injective left \( R \) -modules is injective.
Proof. Assume that every direct sum of injective... | {
"question": "Theorem 4.11. Every module can be embedded into an injective module.",
"proof": "Null"
} |
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