text stringlengths 10k 10k | response stringlengths 67 6.5k |
|---|---|
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"
} |
Lemma 10.55. If \( \alpha \) and \( \beta \) are distinct variables, \( \alpha \) does not occur bound in \( \varphi \) , and \( \beta \) does not occur in \( \varphi \) at all, then \( \vdash \forall {\alpha \varphi } \leftrightarrow \forall \beta {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi \) .
Proof
\[
\vda... | {
"question": "Lemma 10.55. If \( \alpha \) and \( \beta \) are distinct variables, \( \alpha \) does not occur bound in \( \varphi \) , and \( \beta \) does not occur in \( \varphi \) at all, then \( \vdash \forall {\alpha \varphi } \leftrightarrow \forall \beta {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi \) ... |
Theorem 2.6 (Homomorphism Theorem). If \( \varphi : A \rightarrow B \) is a homomorphism of left \( R \) -modules, then
\[
A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi
\]
in fact, there is an isomorphism \( \theta : A/\operatorname{Ker}f \rightarrow \operatorname{Im}f \) unique such that \( \varphi = ... | {
"question": "Theorem 2.6 (Homomorphism Theorem). If \( \varphi : A \rightarrow B \) is a homomorphism of left \( R \) -modules, then\n\n\[ A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi \]\n\nin fact, there is an isomorphism \( \theta : A/\operatorname{Ker}f \rightarrow \operatorname{Im}f \) unique su... |
Exercise 1.3.2 Let \( p \) be an odd prime. Suppose that \( {2}^{n} \equiv 1\left( {\;\operatorname{mod}\;p}\right) \) and \( {2}^{n} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . Show that \( {2}^{d} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) where \( d \) is the order of 2 \( \left( {\;\operatornam... | {
"question": "Exercise 1.3.2 Let \( p \) be an odd prime. Suppose that \( {2}^{n} \equiv 1\\left( {\\;\\operatorname{mod}\\;p}\\right) \) and \( {2}^{n} ≢ 1\\left( {\\;\\operatorname{mod}\\;{p}^{2}}\\right) \) . Show that \( {2}^{d} ≢ 1\\left( {\\;\\operatorname{mod}\\;{p}^{2}}\\right) \) where \( d \) is the orde... |
Proposition 2.7 Suppose \( U \) is a simply connected domain in \( {\mathbb{R}}^{n} \) and \( \mathbf{F} \) is a smooth, \( {\mathbb{R}}^{n} \) -valued function on \( U \) . Then \( \mathbf{F} \) is conservative if and only if \( \mathbf{F} \) satisfies
\[
\frac{\partial {F}_{j}}{\partial {x}_{k}} - \frac{\partial {... | {
"question": "Proposition 2.7 Suppose \( U \) is a simply connected domain in \( {\mathbb{R}}^{n} \) and \( \mathbf{F} \) is a smooth, \( {\mathbb{R}}^{n} \) -valued function on \( U \) . Then \( \mathbf{F} \) is conservative if and only if \( \mathbf{F} \) satisfies\n\n\[ \frac{\partial {F}_{j}}{\partial {x}_{k}}... |
Theorem 3.2.2. There exist finite constants \( {C}_{n} \) and \( {C}_{n}^{\prime } \) such that the following statements are valid:
(a) Given \( b \in {BMO}\left( {\mathbf{R}}^{n}\right) \), the linear functional \( {L}_{b} \) lies in \( {\left( {H}^{1}\left( {\mathbf{R}}^{n}\right) \right) }^{ * } \) and has norm a... | {
"question": "Theorem 3.2.2. There exist finite constants \( {C}_{n} \) and \( {C}_{n}^{\prime } \) such that the following statements are valid:\n\n(a) Given \( b \in {BMO}\left( {\mathbf{R}}^{n}\right) \), the linear functional \( {L}_{b} \) lies in \( {\left( {H}^{1}\left( {\mathbf{R}}^{n}\right) \right) }^{ * ... |
Theorem 3.3.4. Suppose that \( X \) is a Banach space with an unconditional basis. If \( X \) is not reflexive, then either \( {c}_{0} \) is complemented in \( X \), or \( {\ell }_{1} \) is complemented in \( X \) (or both). In either case, \( {X}^{* * } \) is nonseparable.
Proof. The first statement of the theorem ... | {
"question": "Theorem 3.3.4. Suppose that \( X \) is a Banach space with an unconditional basis. If \( X \) is not reflexive, then either \( {c}_{0} \) is complemented in \( X \), or \( {\ell }_{1} \) is complemented in \( X \) (or both). In either case, \( {X}^{* * } \) is nonseparable.",
"proof": "Proof. The... |
Proposition 19. Let \( R \) be a commutative ring with 1 .
(1) Prime ideals are primary.
(2) The ideal \( Q \) is primary if and only if every zero divisor in \( R/Q \) is nilpotent.
(3) If \( Q \) is primary then rad \( Q \) is a prime ideal, and is the unique smallest prime ideal containing \( Q \) .
(4) If \( ... | {
"question": "Prime ideals are primary.",
"proof": "The first two statements are immediate from the definition of a primary ideal."
} |
Corollary 13.1.1. If \( u \in {W}^{1,2}\left( \Omega \right) \) is a weak solution of \( {\Delta u} = f \) with \( f \in \) \( {C}^{k,\alpha }\left( \Omega \right), k \in \mathbb{N},0 < \alpha < 1 \), then \( u \in {C}^{k + 2,\alpha }\left( \Omega \right) \), and for \( {\Omega }_{0} \subset \subset \Omega \) ,
\[
\... | {
"question": "Corollary 13.1.1. If \( u \in {W}^{1,2}\left( \Omega \right) \) is a weak solution of \( {\Delta u} = f \) with \( f \in \) \( {C}^{k,\alpha }\left( \Omega \right), k \in \mathbb{N},0 < \alpha < 1 \), then \( u \in {C}^{k + 2,\alpha }\left( \Omega \right) \), and for \( {\Omega }_{0} \subset \subset ... |
Theorem 5.9 (Stone [1934]). Every Boolean lattice is isomorphic to the lattice of closed and open subsets of its Stone space.
Readers may prove a converse: every Stone space is homeomorphic to the Stone space of its lattice of closed and open subsets.
Proof. Let \( L \) be a Boolean lattice and let \( X \) be its S... | {
"question": "Theorem 5.9 (Stone [1934]). Every Boolean lattice is isomorphic to the lattice of closed and open subsets of its Stone space.",
"proof": "Proof. Let \( L \) be a Boolean lattice and let \( X \) be its Stone space. For every \( a \in L, V\left( a\right) \) is open in \( X \), and is closed in \( X... |
Lemma 12. Let \( M, B \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) be matrices, with \( M \) irreducible and \( \left| B\right| \leq M \) . Then \( \rho \left( B\right) \leq \rho \left( M\right) \) .
In the case of equality \( \left( {\rho \left( B\right) = \rho \left( M\right) }\right) \), the following hold.
-... | {
"question": "Lemma 12. Let \( M, B \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) be matrices, with \( M \) irreducible and \( \left| B\right| \leq M \) . Then \( \rho \left( B\right) \leq \rho \left( M\right) \) .",
"proof": "In order to establish the inequality, we proceed as above. If \( \lambda \) is an ... |
Proposition 4.51. If \( S \) is compact, under assumptions (P1),(P2), one has
\[
{\partial }_{F}m\left( \bar{x}\right) = \overline{\operatorname{co}}\left\{ {D{f}_{s}\left( \bar{x}\right) : s \in M\left( \bar{x}\right) }\right\}
\]
\[
{\partial }_{F}p\left( \bar{x}\right) = \mathop{\bigcap }\limits_{{s \in P\left( ... | {
"question": "Proposition 4.51. If \( S \) is compact, under assumptions (P1),(P2), one has\n\n\[ \n{\partial }_{F}m\left( \bar{x}\right) = \overline{\operatorname{co}}\left\{ {D{f}_{s}\left( \bar{x}\right) : s \in M\left( \bar{x}\right) }\right\} \n\]\n\n\[ \n{\partial }_{F}p\left( \bar{x}\right) = \mathop{\bigca... |
Proposition 6.4.14. Assume that neither \( b/a, c/a \), nor \( c/b \) is the cube of \( a \) rational number. If the elliptic curve \( E \) with affine equation \( {Y}^{2} = {X}^{3} + {\left( 4abc\right) }^{2} \) has zero rank then the equation \( a{x}^{3} + b{y}^{3} + c{z}^{3} \) has no nontrivial rational solutions... | {
"question": "Assume that neither \( b/a, c/a \), nor \( c/b \) is the cube of \( a \) rational number. If the elliptic curve \( E \) with affine equation \( {Y}^{2} = {X}^{3} + {\left( 4abc\right) }^{2} \) has zero rank then the equation \( a{x}^{3} + b{y}^{3} + c{z}^{3} \) has no nontrivial rational solutions.",... |
Proposition 9.5 THE FAN LEMMA
Let \( G \) be a \( k \) -connected graph, let \( x \) be a vertex of \( G \), and let \( Y \subseteq V \smallsetminus \{ x\} \) be a set of at least \( k \) vertices of \( G \) . Then there exists a \( k \) -fan in \( G \) from \( x \) to \( Y \) .
We now give the promised application... | {
"question": "Proposition 9.5 THE FAN LEMMA\n\nLet \( G \) be a \( k \) -connected graph, let \( x \) be a vertex of \( G \), and let \( Y \subseteq V \smallsetminus \{ x\} \) be a set of at least \( k \) vertices of \( G \) . Then there exists a \( k \) -fan in \( G \) from \( x \) to \( Y \) .",
"proof": "Nu... |
Theorem 1. \( {\lambda }_{m}\left( x\right) \) is irreducible in the rational field.
Proof. We observe first that \( {\lambda }_{m}\left( x\right) \) has integer coefficients. For, assuming this holds for every \( {\lambda }_{d}\left( x\right), d < m \), and setting \( p\left( x\right) = \mathop{\prod }\limits_{\sub... | {
"question": "Theorem 1. \( {\lambda }_{m}\left( x\right) \) is irreducible in the rational field.",
"proof": "Proof. We observe first that \( {\lambda }_{m}\left( x\right) \) has integer coefficients. For, assuming this holds for every \( {\lambda }_{d}\left( x\right), d < m \), and setting \( p\left( x\right... |
Exercise 9.1.4 Let \( \pi \) be a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . Show that \( {\alpha }^{N\left( \pi \right) - 1} \equiv 1\left( {\;\operatorname{mod}\;\pi }\right) \) for all \( \alpha \in \mathbb{Z}\left\lbrack \rho \right\rbrack \) which are coprime to \( \pi \) .
Exercise 9.1.5 Let \(... | {
"question": "Exercise 9.1.4 Let \( \pi \) be a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . Show that \( {\alpha }^{N\left( \pi \right) - 1} \equiv 1\left( {\;\operatorname{mod}\;\pi }\right) \) for all \( \alpha \in \mathbb{Z}\left\lbrack \rho \right\rbrack \) which are coprime to \( \pi \) .",
... |
Theorem 1. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \) . Then \( {\mathbf{R}}^{2} - J \) has exactly two components.
Proof. Let \( N \) be a "strip neighborhood" of \( J \), formed by small convex polyhedral neighborhoods of the edges and vertices of \( J \) . (More precisely, we mean the edges and vertices o... | {
"question": "Theorem 1. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \) . Then \( {\mathbf{R}}^{2} - J \) has exactly two components.",
"proof": "Proof. Let \( N \) be a \"strip neighborhood\" of \( J \), formed by small convex polyhedral neighborhoods of the edges and vertices of \( J \) . (More precisely... |
Proposition 8.6. \( {f}_{2k} \circ h = {\left( -1\right) }^{k}\mathop{\sum }\limits_{{l = 0}}^{{2k}}{\left( -1\right) }^{l}{f}_{l}^{i}{f}_{{2k} - l}^{i} \) , \( \operatorname{Pf} \circ h = {\left( -1\right) }^{\left\lbrack n/2\right\rbrack }{f}_{n}^{i} \) .
Proof. By Lemma 8.1,
\[
{\left| \det \left( x{I}_{n} - M\r... | {
"question": "Proposition 8.6. \( {f}_{2k} \circ h = {\left( -1\right) }^{k}\mathop{\sum }\limits_{{l = 0}}^{{2k}}{\left( -1\right) }^{l}{f}_{l}^{i}{f}_{{2k} - l}^{i} \) , \( \operatorname{Pf} \circ h = {\left( -1\right) }^{\left\lbrack n/2\right\rbrack }{f}_{n}^{i} \) .",
"proof": "Proof. By Lemma 8.1,\n\n\[ ... |
Theorem 178 I
Idempotent, primitive 698
Image 722
Incidence
isomorphism 82, 169
isomorphism, general 582, 599
number 79, 82
system 84
Inclusion 4, 20, 269, 271
Indeterminacy 722
Injection 55, 269, 271
Integers, twisted 285
Intersection number 509
Inverse, homotopy 23
Inverse limit, derived 271
of fibra... | {
"question": "Excision Theorem",
"proof": "57, 580, 598"
} |
Theorem 2.10 A graph \( G \) is even if and only if \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) .
Proof Suppose that \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . Then, in particular, \( \left| {\partial \left( v\right) }... | {
"question": "Theorem 2.10 A graph \( G \) is even if and only if \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) .",
"proof": "Proof Suppose that \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . Then, in particular, \( \l... |
Lemma 2.12. If \( \mathrm{F} \) is a Galois extension field of \( \mathrm{K} \) and \( \mathrm{E} \) is a stable intermediate field of the extension, then \( \mathrm{E} \) is Galois over \( \mathrm{K} \) .
PROOF. If \( u : E - K \), then there exists \( \sigma \in {\operatorname{Aut}}_{K}F \) such that \( \sigma \le... | {
"question": "Lemma 2.12. If \( \mathrm{F} \) is a Galois extension field of \( \mathrm{K} \) and \( \mathrm{E} \) is a stable intermediate field of the extension, then \( \mathrm{E} \) is Galois over \( \mathrm{K} \) .",
"proof": "PROOF. If \( u : E - K \), then there exists \( \sigma \in {\operatorname{Aut}}... |
Corollary 10.7.7. Set \( b = \log \left( {2\pi }\right) - 1 - \gamma /2 \) . Then for all \( s \in \mathbb{C} \) we have the convergent product
\[
\zeta \left( s\right) = \frac{{e}^{bs}}{s\left( {s - 1}\right) \Gamma \left( {s/2}\right) }\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\right) {e}^{s/\rho }... | {
"question": "Corollary 10.7.7. Set \( b = \log \left( {2\pi }\right) - 1 - \gamma /2 \) . Then for all \( s \in \mathbb{C} \) we have the convergent product\n\n\[ \zeta \left( s\right) = \frac{{e}^{bs}}{s\left( {s - 1}\right) \Gamma \left( {s/2}\right) }\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\r... |
Theorem 2. Let \( \chi \) be a nontrivial Dirichlet character modulo m. Then \( L\left( {1,\chi }\right) \neq 0 \) .
Proof. Having already proved that \( L\left( {1,\chi }\right) \neq 0 \) if \( \chi \) is complex we assume \( \chi \) is real.
Assume \( L\left( {1,\chi }\right) = 0 \) and consider the function
\[
... | {
"question": "Theorem 2. Let \( \chi \) be a nontrivial Dirichlet character modulo m. Then \( L\left( {1,\chi }\right) \neq 0 \) .",
"proof": "Proof. Having already proved that \( L\left( {1,\chi }\right) \neq 0 \) if \( \chi \) is complex we assume \( \chi \) is real.\n\nAssume \( L\left( {1,\chi }\right) = 0... |
Theorem 4.2.2 Let \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{n} \) be a set of generators for a finitely generated \( \mathbb{Z} \) -module \( M \), and let \( N \) be a submodule.
(a) \( \exists {\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{m} \) in \( N \) with \( m \leq n \) such that
\[
N = \mathbb{Z}{\bet... | {
"question": "Theorem 4.2.2 Let \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{n} \) be a set of generators for a finitely generated \( \mathbb{Z} \) -module \( M \), and let \( N \) be a submodule.\n\n(a) \( \exists {\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{m} \) in \( N \) with \( m \leq n \) such that\n\n\... |
Lemma 14.1.7. Let \( f : X \rightarrow Y \) be a map between two metric spaces.
(i) Suppose \( \omega : \lbrack 0,\infty ) \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is a function such that \( d\left( {f\left( x\right), f\left( y\right) }\right) \leq \omega \left( {d\left( {x, y}\right) }\right) \) for eve... | {
"question": "Lemma 14.1.7. Let \( f : X \rightarrow Y \) be a map between two metric spaces.\n\n(i) Suppose \( \omega : \lbrack 0,\infty ) \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is a function such that \( d\left( {f\left( x\right), f\left( y\right) }\right) \leq \omega \left( {d\left( {x, y}\right) ... |
Proposition 4.4. For all ideals \( \mathfrak{a},\mathfrak{b} \) of \( R \) and \( \mathfrak{A} \) of \( {S}^{-1}R \) :
(1) \( {\mathfrak{a}}^{E} = {S}^{-1}R \) if and only if \( \mathfrak{a} \cap S \neq \varnothing \) ;
(2) if \( \mathfrak{a} = {\mathfrak{A}}^{C} \), then \( \mathfrak{A} = {\mathfrak{a}}^{E} \) ;
... | {
"question": "Proposition 4.4. For all ideals \( \mathfrak{a},\mathfrak{b} \) of \( R \) and \( \mathfrak{A} \) of \( {S}^{-1}R \) :\n\n(1) \( {\mathfrak{a}}^{E} = {S}^{-1}R \) if and only if \( \mathfrak{a} \cap S \neq \varnothing \) ;\n\n(2) if \( \mathfrak{a} = {\mathfrak{A}}^{C} \), then \( \mathfrak{A} = {\ma... |
Proposition 11.101. \( \mathfrak{B} \) is generated by \( {\mathfrak{B}}_{0} \) and any set of representatives for \( {W}^{\prime } \) in \( N \) .
## Exercises
11.102. Show that there is a short exact sequence \( 1 \rightarrow {\mathfrak{B}}^{\prime } \rightarrow \mathfrak{B} \rightarrow {W}^{\prime } \rightarrow ... | {
"question": "Proposition 11.101. \( \\mathfrak{B} \) is generated by \( {\\mathfrak{B}}_{0} \) and any set of representatives for \( {W}^{\\prime } \) in \( N \) .",
"proof": "Null"
} |
Corollary 1.10. If \( k \) is a finite field, then \( {k}^{ * } \) is cyclic.
An element \( \zeta \) in a field \( k \) such that there exists an integer \( n \geqq 1 \) such that \( {\zeta }^{n} = 1 \) is called a root of unity, or more precisely an \( n \) -th root of unity. Thus the set of \( n \) -th roots of un... | {
"question": "Corollary 1.10. If \( k \) is a finite field, then \( {k}^{ * } \) is cyclic.",
"proof": "Null"
} |
Proposition 1.1. Let \( \alpha : S \rightarrow T \) be a morphism of algebraic sets. If \( A \) is an irreducible subset of \( S \), then \( \alpha \left( A\right) \) is an irreducible subset of \( T \) .
Proof. This follows direcly from the definition of irreducibility; using only the continuity of \( \alpha \) .
... | {
"question": "Proposition 1.1. Let \( \\alpha : S \\rightarrow T \) be a morphism of algebraic sets. If \( A \) is an irreducible subset of \( S \), then \( \\alpha \\left( A\\right) \) is an irreducible subset of \( T \) .",
"proof": "Proof. This follows direcly from the definition of irreducibility; using on... |
Proposition 8.44. 1. If \( \mu \) and \( \lambda \) are dominant, then \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) if and only if \( \lambda \preccurlyeq \mu \) .
2. Let \( \mu \) and \( \lambda \) be elements of \( E \) with \( \mu \) dominant. Then \( \lambda \) belongs to \( \op... | {
"question": "Proposition 8.44. 1. If \( \mu \) and \( \lambda \) are dominant, then \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) if and only if \( \lambda \preccurlyeq \mu \) .",
"proof": "Null"
} |
Lemma 7.2. Let \( S \) be a closed subset of a Banach space \( X \) and let \( {W}_{0} \in \mathcal{S}\left( X\right) \) . Then there exists \( W \in \mathcal{S}\left( X\right) \) containing \( {W}_{0} \) such that \( d\left( {x, S}\right) = d\left( {x, S \cap W}\right) \) for all \( x \in W \) .
Proof. Starting wit... | {
"question": "Lemma 7.2. Let \( S \) be a closed subset of a Banach space \( X \) and let \( {W}_{0} \in \mathcal{S}\left( X\right) \) . Then there exists \( W \in \mathcal{S}\left( X\right) \) containing \( {W}_{0} \) such that \( d\left( {x, S}\right) = d\left( {x, S \cap W}\right) \) for all \( x \in W \) .",
... |
Corollary 3.9.7. Let \( F/K \) be a function field whose full constant field is \( K \) .
(a) Suppose that \( {F}^{\prime } = {F}_{1}{F}_{2} \) is the compositum of two finite separable extensions \( {F}_{1}/F \) and \( {F}_{2}/F \) . Assume that there exists a place \( P \in {\mathbb{P}}_{F} \) of degree one which ... | {
"question": "Corollary 3.9.7. Let \( F/K \) be a function field whose full constant field is \( K \) .\n\n(a) Suppose that \( {F}^{\prime } = {F}_{1}{F}_{2} \) is the compositum of two finite separable extensions \( {F}_{1}/F \) and \( {F}_{2}/F \) . Assume that there exists a place \( P \in {\mathbb{P}}_{F} \) o... |
Corollary 10.3.13. We have
\[
{\mathcal{H}}_{2}\left( \tau \right) = \frac{{5\theta }\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) - {\theta }^{5}\left( \tau \right) }{480},
\]
\[
{\mathcal{H}}_{3}\left( \tau \right) = - \frac{7{\theta }^{3}\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) +... | {
"question": "Corollary 10.3.13. We have\n\n\[ \n{\\mathcal{H}}_{2}\\left( \\tau \\right) = \\frac{{5\\theta }\\left( \\tau \\right) {\\theta }^{4}\\left( {\\tau + 1/2}\\right) - {\\theta }^{5}\\left( \\tau \\right) }{480}, \n\] \n\n\[ \n{\\mathcal{H}}_{3}\\left( \\tau \\right) = - \\frac{7{\\theta }^{3}\\left( \\... |
Theorem 5.4.4 The classification space \( \operatorname{irr}\left( n\right) / \sim \) is standard Borel.
Proof. Fix any irreducible \( A \) . Then the \( \sim \) -equivalence class \( \left\lbrack A\right\rbrack \) containing \( A \) equals
\[
{\pi }_{1}\left\{ {\left( {B, U}\right) \in \operatorname{irr}\left( n\r... | {
"question": "Theorem 5.4.4 The classification space \( \operatorname{irr}\left( n\right) / \sim \) is standard Borel.",
"proof": "Proof. Fix any irreducible \( A \) . Then the \( \sim \) -equivalence class \( \left\lbrack A\right\rbrack \) containing \( A \) equals\n\n\[ \n{\pi }_{1}\left\{ {\left( {B, U}\rig... |
Lemma 9.3. Let \( {z}_{0} = {y}_{0}\mathrm{i} \) and \( {z}_{1} = {y}_{1}\mathrm{i} \) with \( 0 < {y}_{0} < {y}_{1} \) . Then
\[
\mathrm{d}\left( {{z}_{0},{z}_{1}}\right) = \log {y}_{1} - \log {y}_{0}
\]
and
\[
\phi \left( t\right) = {y}_{0}{\left( \frac{{y}_{1}}{{y}_{0}}\right) }^{t}\mathrm{i}
\]
for \( t \in \... | {
"question": "Lemma 9.3. Let \( {z}_{0} = {y}_{0}\mathrm{i} \) and \( {z}_{1} = {y}_{1}\mathrm{i} \) with \( 0 < {y}_{0} < {y}_{1} \). Then\n\n\[ \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) = \log {y}_{1} - \log {y}_{0} \]\n\nand\n\n\[ \phi \left( t\right) = {y}_{0}{\left( \frac{{y}_{1}}{{y}_{0}}\right) }^{t}\mathrm... |
Theorem 4.136. Suppose \( X \) is \( F \) -smooth. If \( f \) is locally Lipschitzian and \( q \) is a Lyapunov function for \( S \), then \( S \) is stable.
If \( \left( {p, q, c}\right) \) is a Lyapunov triple for \( S \), then \( S \) is attractive, provided that in the case \( c = 0, f \) is bounded on bounded s... | {
"question": "Theorem 4.136. Suppose \( X \) is \( F \) -smooth. If \( f \) is locally Lipschitzian and \( q \) is a Lyapunov function for \( S \), then \( S \) is stable.",
"proof": "Proof. In order to show the stability of \( S \) in both cases, it suffices to prove that for all \( z \in X \), the function \... |
Proposition 7.23. Let
\[
\begin{matrix} \mathrm{X} = \left( {X,\mathcal{B},\mu, T}\right) \\ \downarrow \\ \mathrm{Y} = \left( {Y,\mathcal{A},\nu, S}\right) \end{matrix}
\]
be a compact extension of invertible measure-preserving systems on Borel probability spaces. If \( \mathrm{Y} \) is \( {SZ} \), then so is \( \... | {
"question": "Proposition 7.23. Let\n\n\[ \n\begin{matrix} \mathrm{X} = \left( {X,\mathcal{B},\mu, T}\right) \\ \downarrow \\ \mathrm{Y} = \left( {Y,\mathcal{A},\nu, S}\right) \end{matrix} \n\]\n\nbe a compact extension of invertible measure-preserving systems on Borel probability spaces. If \( \mathrm{Y} \) is \(... |
Lemma 14.3. If \( F/\mathbb{Q} \) is an extension in which no finite prime ramifies, then \( F = \mathbb{Q} \) .
Proof. A theorem of Minkowski (see Exercise 2.5) states that every ideal class of \( F \) contains an integral ideal of norm less than or equal to
\[
\frac{n!}{{n}^{n}}{\left( \frac{4}{\pi }\right) }^{{r... | {
"question": "Lemma 14.3. If \( F/\mathbb{Q} \) is an extension in which no finite prime ramifies, then \( F = \mathbb{Q} \).",
"proof": "Proof. A theorem of Minkowski (see Exercise 2.5) states that every ideal class of \( F \) contains an integral ideal of norm less than or equal to\n\n\[ \frac{n!}{{n}^{n}}{\... |
Theorem 3.1.1. \( {W}_{G} \) is a finite group and the representation of \( {W}_{G} \) on \( {\mathfrak{h}}^{ * } \) is faithful.
Proof. Let \( s \in {\operatorname{Norm}}_{G}\left( H\right) \) . Suppose \( s \cdot \theta = \theta \) for all \( \theta \in \mathfrak{X}\left( H\right) \) . Then \( {s}^{-1}{hs} = h \) ... | {
"question": "Theorem 3.1.1. \( {W}_{G} \) is a finite group and the representation of \( {W}_{G} \) on \( {\mathfrak{h}}^{ * } \) is faithful.",
"proof": "Proof. Let \( s \in {\operatorname{Norm}}_{G}\left( H\right) \) . Suppose \( s \cdot \theta = \theta \) for all \( \theta \in \mathfrak{X}\left( H\right) \... |
Lemma 5.1.3 With \( \Gamma \) as described in Theorem 5.1.2 satisfying conditions 1 and 2, condition 3 is equivalent to the following requirement:
\( {3}^{\prime } \) . All embeddings \( \sigma \), apart from the identity and \( \mathbf{c} \), complex conjugation, are real and \( {A\Gamma } \) is ramified at all rea... | {
"question": "Lemma 5.1.3 With \( \Gamma \) as described in Theorem 5.1.2 satisfying conditions 1 and 2, condition 3 is equivalent to the following requirement:\n\n\( {3}^{\prime } \) . All embeddings \( \sigma \), apart from the identity and \( \mathbf{c} \), complex conjugation, are real and \( {A\Gamma } \) is ... |
Example 10.5.4. An almost greedy basis that is not greedy.
Aside from being quasi-greedy, the basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in Example 10.2.9 is democratic. Indeed, if \( \left| A\right| = m \), then
\[
{\left( \mathop{\sum }\limits_{{n \in A}}1\right) }^{1/2} = {m}^{1/2}
\]
... | {
"question": "An almost greedy basis that is not greedy.",
"proof": "Aside from being quasi-greedy, the basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in Example 10.2.9 is democratic. Indeed, if \( \left| A\right| = m \), then\n\n\[ \n{\left( \mathop{\sum }\limits_{{n \in A}}1\right) }^{1... |
Theorem 10.71 (Substitutivity of equivalence). Let \( \varphi ,\psi ,\chi \) be formulas and \( \alpha \in {}^{m}\operatorname{Rng}v \) . Suppose that if \( \beta \) occurs free in \( \varphi \) or in \( \psi \) but bound in \( \chi \) then \( \beta \in \left\{ {{\alpha }_{i} : i < m}\right\} \) . Let \( \theta \) be... | {
"question": "Theorem 10.71 (Substitutivity of equivalence). Let \( \varphi ,\psi ,\chi \) be formulas and \( \alpha \in {}^{m}\operatorname{Rng}v \) . Suppose that if \( \beta \) occurs free in \( \varphi \) or in \( \psi \) but bound in \( \chi \) then \( \beta \in \left\{ {{\alpha }_{i} : i < m}\right\} \) . Le... |
Proposition 2.92. Let \( X \) and \( Z \) be Banach spaces, \( Z \) being finite-dimensional, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Z \) be Hadamard differentiable at \( a \in W \) , with \( \operatorname{Dg}\left( a\right) \left( X\right) = Z \) . Then there exist open neighborhoods ... | {
"question": "Proposition 2.92. Let \( X \) and \( Z \) be Banach spaces, \( Z \) being finite-dimensional, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Z \) be Hadamard differentiable at \( a \in W \) , with \( \operatorname{Dg}\left( a\right) \left( X\right) = Z \) . Then there exist op... |
Exercise 2.25. Let \( {x}_{1},\ldots ,{x}_{n} \) be atomic propositions and let \( A \) and \( B \) be two logic statements in CNF. The logic statement \( A \Rightarrow B \) is satisfied if any truth assignment that satisfies \( A \) also satisfies \( B \) . Prove that \( A \Rightarrow B \) is satisfied if and only i... | {
"question": "Exercise 2.25. Let \( {x}_{1},\ldots ,{x}_{n} \) be atomic propositions and let \( A \) and \( B \) be two logic statements in CNF. The logic statement \( A \Rightarrow B \) is satisfied if any truth assignment that satisfies \( A \) also satisfies \( B \) . Prove that \( A \Rightarrow B \) is satisf... |
Theorem 11.1.1. Let \( G \) be a linear algebraic group. For every \( g \in G \) the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is a linear isomorphism from \( \operatorname{Lie}\left( G\right) \) onto \( T{\left( G\right) }_{g} \) . Hence \( G \) is a smooth algebraic set and \( \dim \operatorname{Lie}\left( G... | {
"question": "Theorem 11.1.1. Let \( G \) be a linear algebraic group. For every \( g \in G \) the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is a linear isomorphism from \( \operatorname{Lie}\left( G\right) \) onto \( T{\left( G\right) }_{g} \) . Hence \( G \) is a smooth algebraic set and \( \dim \operator... |
Theorem 5.51. Suppose \( f\left( {e}^{i\theta }\right) = \mathop{\sum }\limits_{{-\infty }}^{\infty }{a}_{n}{e}^{in\theta } \) lies in \( W \) . If \( f \) does not vanish on \( T \) , then \( 1/f \) is also in \( W \), that is, there exist \( \left\{ {b}_{n}\right\} \) with \( \mathop{\sum }\limits_{{-\infty }}^{\in... | {
"question": "Theorem 5.51. Suppose \( f\left( {e}^{i\theta }\right) = \mathop{\sum }\limits_{{-\infty }}^{\infty }{a}_{n}{e}^{in\theta } \) lies in \( W \) . If \( f \) does not vanish on \( T \) , then \( 1/f \) is also in \( W \), that is, there exist \( \left\{ {b}_{n}\right\} \) with \( \mathop{\sum }\limits_... |
Theorem 2.40. If the function \( f \) has continuous first partial derivatives in a neighborhood of \( c \) that satisfy the \( \mathrm{{CR}} \) equations at \( c \), then \( f \) is (complex) differentiable at \( c \) .
Proof. The theorem is an immediate consequence of (2.12), since in this case \( {f}_{\bar{z}}\le... | {
"question": "Theorem 2.40. If the function \( f \) has continuous first partial derivatives in a neighborhood of \( c \) that satisfy the \( \mathrm{{CR}} \) equations at \( c \), then \( f \) is (complex) differentiable at \( c \) .",
"proof": "Proof. The theorem is an immediate consequence of (2.12), since ... |
Theorem 2.7. Let \( \mathrm{R} \) be a ring and \( \mathrm{I} \) an ideal of \( \mathrm{R} \) . Then the additive quotient group \( \mathrm{R}/\mathrm{I} \) is a ring with multiplication given by
\[
\left( {a + I}\right) \left( {b + I}\right) = {ab} + I
\]
If \( \mathbf{R} \) is commutative or has an identity, then... | {
"question": "Theorem 2.7. Let \( \mathrm{R} \) be a ring and \( \mathrm{I} \) an ideal of \( \mathrm{R} \). Then the additive quotient group \( \mathrm{R}/\mathrm{I} \) is a ring with multiplication given by\n\n\[ \left( {a + I}\right) \left( {b + I}\right) = {ab} + I \]\n\nIf \( \mathbf{R} \) is commutative or h... |
Exercise 2.16 If \( V \) is an irreducible finite-dimensional representation of a Lie group \( G \), show that \( {V}^{ * } \) is also irreducible.
Exercise 2.17 This exercise considers a natural generalization of \( {V}_{n}\left( {\mathbb{C}}^{2}\right) \) . Let \( W \) be a representation of a Lie group \( G \) . ... | {
"question": "Exercise 2.16 If \( V \) is an irreducible finite-dimensional representation of a Lie group \( G \), show that \( {V}^{ * } \) is also irreducible.",
"proof": "Null"
} |
Proposition 6.45. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a Lattès map that fits into a commutative diagram (6.22). Then
\[
{\operatorname{CritVal}}_{\pi } = {\operatorname{PostCrit}}_{\phi }
\]
In particular, a Lattès map is postcritically finite.
Proof. The key to the proof of this prop... | {
"question": "Proposition 6.45. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a Lattès map that fits into a commutative diagram (6.22). Then \[ {\operatorname{CritVal}}_{\pi } = {\operatorname{PostCrit}}_{\phi } \] In particular, a Lattès map is postcritically finite.",
"proof": "Proof. The... |
Theorem 7.3.12 If \( G \) is an infinite connected solvable group of finite Morley rank with finite center, then \( G \) interprets an algebraically closed field.
Proof We will prove this by induction on the rank of \( G \) . We first argue that we may, without loss of generality, assume that \( G \) is centerless. ... | {
"question": "Theorem 7.3.12 If \( G \) is an infinite connected solvable group of finite Morley rank with finite center, then \( G \) interprets an algebraically closed field.",
"proof": "Proof We will prove this by induction on the rank of \( G \) . We first argue that we may, without loss of generality, ass... |
Lemma 3. If Dedekind's functional equation
(5)
\[
\eta \left( {A\tau }\right) = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \tau \right) ,
\]
is satisfied for some \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in \( \Gamma \) with \( c > 0 \) and \( ... | {
"question": "Lemma 3. If Dedekind's functional equation\n\n(5)\n\n\[ \eta \left( {A\tau }\right) = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \tau \right) ,\]\n\nis satisfied for some \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in \( \Gamma \) wi... |
Corollary 4.1.1. The discrete Dirichlet problem
\[
{\Delta }_{h}{u}^{h} = 0\;\text{ in }{\Omega }_{h}
\]
\[
{u}^{h} = {g}^{h}\;\text{ on }{\Gamma }^{h},
\]
for given \( {g}^{h} \) has at most one solution.
Proof. This follows in the usual manner by applying the maximum principle to the difference of two solutions... | {
"question": "Corollary 4.1.1. The discrete Dirichlet problem\n\n\\[ \n{\\Delta }_{h}{u}^{h} = 0\\;\\text{ in }{\\Omega }_{h} \n\\]\n\n\\[ \n{u}^{h} = {g}^{h}\\;\\text{ on }{\\Gamma }^{h}, \n\\]\n\nfor given \\( {g}^{h} \\) has at most one solution.",
"proof": "Proof. This follows in the usual manner by applyi... |
Theorem 11.6.9. Let \( g \) and \( h \) be hyperbolic with axes and translation lengths \( {A}_{g},{A}_{h},{T}_{g} \) and \( {T}_{h} \) respectively. Suppose that \( \langle g, h\rangle \) is discrete and nonelementary and that no images of \( {A}_{g} \) and \( {A}_{h} \) cross. Then
\[
\sinh \left( {\frac{1}{2}{T}_... | {
"question": "Theorem 11.6.9. Let \( g \) and \( h \) be hyperbolic with axes and translation lengths \( {A}_{g},{A}_{h},{T}_{g} \) and \( {T}_{h} \) respectively. Suppose that \( \langle g, h\rangle \) is discrete and nonelementary and that no images of \( {A}_{g} \) and \( {A}_{h} \) cross. Then\n\n\[ \sinh \lef... |
Example 1.119. We illustrate the concepts in this section by applying them to the case that \( W \) is the symmetric group on \( n \) letters acting on \( {\mathbb{R}}^{n} \) as in Examples 1.10 and 1.81. Recall that a permutation \( \pi \) acts on \( {\mathbb{R}}^{n} \) by \( \pi {e}_{i} = {e}_{\pi \left( i\right) }... | {
"question": "We illustrate the concepts in this section by applying them to the case that \( W \) is the symmetric group on \( n \) letters acting on \( {\mathbb{R}}^{n} \) as in Examples 1.10 and 1.81. Recall that a permutation \( \pi \) acts on \( {\mathbb{R}}^{n} \) by \( \pi {e}_{i} = {e}_{\pi \left( i\right)... |
Theorem 1 Given an edge \( {ab} \), denote by \( N\left( {s, a, b, t}\right) \) the number of spanning trees of \( G \) in which the (unique) path from \( s \) to \( t \) contains a and \( b \), in this order. Define \( N\left( {s, b, a, t}\right) \) analogously and write \( N \) for the total number of spanning tree... | {
"question": "Theorem 1 Given an edge \( {ab} \), denote by \( N\left( {s, a, b, t}\right) \) the number of spanning trees of \( G \) in which the (unique) path from \( s \) to \( t \) contains a and \( b \), in this order. Define \( N\left( {s, b, a, t}\right) \) analogously and write \( N \) for the total number... |
Proposition 1.2 Assume \( \varphi \in {\mathcal{D}}^{m} \), for some \( m \in \mathbb{N} \) . For every integer \( n \geq 1 \), the convolution \( \varphi * {\chi }_{n} \) belongs to \( \mathcal{D} \) and
\[
\mathop{\lim }\limits_{{n \rightarrow + \infty }}\varphi * {\chi }_{n} = \varphi \;\text{ in }{\mathcal{D}}^{... | {
"question": "Proposition 1.2 Assume \( \varphi \in {\mathcal{D}}^{m} \), for some \( m \in \mathbb{N} \) . For every integer \( n \geq 1 \), the convolution \( \varphi * {\chi }_{n} \) belongs to \( \mathcal{D} \) and \[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}\varphi * {\chi }_{n} = \varphi \;\text{ in ... |
Corollary 10.67. If no free occurrence of \( \alpha \) in \( \varphi \) is within the scope of a quantifier on a variable occurring in \( \sigma \), then \( {\mathrm{{FSubf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi } \) .
Proof
\[
\vdash \forall \alpha \sqsupset \varphi \rightarrow {\operato... | {
"question": "Corollary 10.67. If no free occurrence of \( \alpha \) in \( \varphi \) is within the scope of a quantifier on a variable occurring in \( \sigma \), then \( {\mathrm{{FSubf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi } \) .",
"proof": "\[ \vdash \forall \alpha \sqsupset \var... |
Lemma 4.4.4. Let \( \mathcal{X} \) be a set of 3-connected graphs. Let \( G \) be a graph
\( \left\lbrack {7.3.1}\right\rbrack \)
with \( \kappa \left( G\right) \leq 2 \), and let \( {G}_{1},{G}_{2} \) be proper induced subgraphs of \( G \) such that \( G = {G}_{1} \cup {G}_{2} \) and \( \left| {{G}_{1} \cap {G}_{2... | {
"question": "Lemma 4.4.4. Let \( \mathcal{X} \) be a set of 3-connected graphs. Let \( G \) be a graph with \( \kappa \left( G\right) \leq 2 \), and let \( {G}_{1},{G}_{2} \) be proper induced subgraphs of \( G \) such that \( G = {G}_{1} \cup {G}_{2} \) and \( \left| {{G}_{1} \cap {G}_{2}}\right| = \kappa \left(... |
Proposition 1.1.7. The lightcone \( {\mathcal{L}}_{0} \) is a lightlike submanifold.
By taking \( V \) to be a 3-dimensional Lorentzian vector space and drawing a picture of \( {\mathcal{L}}_{0} \), one can easily convince oneself that indeed \( {\mathcal{L}}_{0} \) is lightlike; for in this case, any tangent plane ... | {
"question": "Proposition 1.1.7. The lightcone \( {\mathcal{L}}_{0} \) is a lightlike submanifold.",
"proof": "Proof. Suppose \( v \in {\mathcal{L}}_{0} \) ; then \( g\left( {v, v}\right) = 0 \) and \( v \neq 0 \) . Let \( \mathcal{U} \) be a neighborhood of \( v \) that does not contain the origin and define ... |
Theorem 14. Let \( f\left( x\right) = {\alpha }_{n}{x}^{n} + {\alpha }_{n - 1}{x}^{n - 1} + \cdots + {\alpha }_{0}, g\left( x\right) = \) \( {\beta }_{m}{x}^{m} + {\beta }_{m - 1}{x}^{m - 1} + \cdots + {\beta }_{0} \) where \( m, n > 0 \) and put
(18)
\[
R\left( {f, g}\right) = \left| \begin{matrix} {\alpha }_{n} &... | {
"question": "Theorem 14. Let \( f\left( x\right) = {\alpha }_{n}{x}^{n} + {\alpha }_{n - 1}{x}^{n - 1} + \cdots + {\alpha }_{0}, g\left( x\right) = \) \( {\beta }_{m}{x}^{m} + {\beta }_{m - 1}{x}^{m - 1} + \cdots + {\beta }_{0} \) where \( m, n > 0 \) and put\n\n(18)\n\n\[ R\left( {f, g}\right) = \left| \begin{ma... |
Exercise 4.4.2 Show that \( \mathfrak{a} \) has an integral basis.
Solution. Let \( \mathfrak{a} \) be an ideal of \( {\mathcal{O}}_{K} \), and let \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{n} \) be an integral basis for \( {\mathcal{O}}_{K} \) . Note that for any \( {\omega }_{i} \) in \( {\mathcal{O}}_{K},... | {
"question": "Exercise 4.4.2 Show that \( \\mathfrak{a} \) has an integral basis.",
"proof": "Solution. Let \( \\mathfrak{a} \) be an ideal of \( {\\mathcal{O}}_{K} \), and let \( {\\omega }_{1},{\\omega }_{2},\\ldots ,{\\omega }_{n} \) be an integral basis for \( {\\mathcal{O}}_{K} \). Note that for any \( {\... |
Exercise 8.4.4. Prove that the group
\[
B = \left\{ {\left. \left( \begin{array}{ll} a & b \\ & 1 \end{array}\right) \right| \;a, b \in \mathbb{R}, a > 0}\right\} ,
\]
which is also called the ’ \( {ax} + b \) ’ group to reflect its natural action on
\[
\left\{ {\left. \left( \begin{array}{l} x \\ 1 \end{array}\ri... | {
"question": "Exercise 8.4.4. Prove that the group\n\n\\[ \nB = \\left\\{ \\left. \\left( \\begin{array}{ll} a & b \\\\ & 1 \\end{array}\\right) \\right| \\;a, b \\in \\mathbb{R}, a > 0\\right\\} ,\n\\]\n\nwhich is also called the ’ \\( {ax} + b \\) ’ group to reflect its natural action on\n\n\\[ \n\\left\\{ \\lef... |
Proposition 18.13. If \( K \) is a compact subset of \( \mathbb{C} \) then the algebra \( \mathcal{P}\left( K\right) \) coincides with the restriction to \( K \) of the subalgebra \( \mathcal{P}\left( \widehat{K}\right) \) of \( \mathcal{C}\left( \widehat{K}\right) \) . Thus the functions in \( \mathcal{P}\left( K\ri... | {
"question": "Proposition 18.13. If \( K \) is a compact subset of \( \mathbb{C} \) then the algebra \( \mathcal{P}\left( K\right) \) coincides with the restriction to \( K \) of the subalgebra \( \mathcal{P}\left( \widehat{K}\right) \) of \( \mathcal{C}\left( \widehat{K}\right) \) . Thus the functions in \( \math... |
Theorem 11.4.11. The action of \( {W}_{G} \) on \( \mathfrak{h} \) coincides with the action of \( W\left( {\mathfrak{g},\mathfrak{h}}\right) \) . Furthermore, every coset in \( {W}_{G} \) has a representative from \( U \) .
Proof. For \( \alpha \in \Phi \) and \( {X}_{\alpha } \) as in (7.42), set
\[
{u}_{\alpha }... | {
"question": "Theorem 11.4.11. The action of \( {W}_{G} \) on \( \mathfrak{h} \) coincides with the action of \( W\left( {\mathfrak{g},\mathfrak{h}}\right) \) . Furthermore, every coset in \( {W}_{G} \) has a representative from \( U \) .",
"proof": "Proof. For \( \alpha \in \Phi \) and \( {X}_{\alpha } \) as ... |
Example 1.83. (Type \( {\mathrm{D}}_{n} \) ) Let \( W \) be the subgroup of the signed permutation group consisting of elements that change an even number of signs (Example 1.13). Then \( \mathcal{H} \) consists of the hyperplanes \( {x}_{i} - {x}_{j} = 0 \) and \( {x}_{i} + {x}_{j} = 0\left( {i \neq j}\right) \) . T... | {
"question": "Example 1.83. (Type \( {\mathrm{D}}_{n} \) ) Let \( W \) be the subgroup of the signed permutation group consisting of elements that change an even number of signs (Example 1.13). Then \( \mathcal{H} \) consists of the hyperplanes \( {x}_{i} - {x}_{j} = 0 \) and \( {x}_{i} + {x}_{j} = 0\left( {i \neq... |
Theorem 12.5.1 An internal set \( A \) is hyperfinite with internal cardinality \( N \) if and only if there is an internal bijection \( f : \{ 1,\ldots, N\} \rightarrow A \) .
Proof. Let \( A = \left\lbrack {A}_{n}\right\rbrack \) . If \( A \) is hyperfinite with internal cardinality \( N = \) \( \left\lbrack {N}_{... | {
"question": "Theorem 12.5.1 An internal set \( A \) is hyperfinite with internal cardinality \( N \) if and only if there is an internal bijection \( f : \{ 1,\ldots, N\} \rightarrow A \) .",
"proof": "Proof. Let \( A = \left\lbrack {A}_{n}\right\rbrack \) . If \( A \) is hyperfinite with internal cardinality... |
Example 8.12. Consider the Lagrangian relaxation of the traveling salesman problem proposed in (8.7). As discussed in Sect. 8.1.1, the bound provided by (8.7) is equal to
\[
\min \;\mathop{\sum }\limits_{{e \in E}}{c}_{e}{x}_{e}
\]
\[
\mathop{\sum }\limits_{{e \in \delta \left( i\right) }}{x}_{e} = 2\;i \in V \smal... | {
"question": "Consider the Lagrangian relaxation of the traveling salesman problem proposed in (8.7). As discussed in Sect. 8.1.1, the bound provided by (8.7) is equal to\n\n\\[ \n\\min \\;\\mathop{\\sum }\\limits_{{e \\in E}}{c}_{e}{x}_{e} \n\\]\n\n\\[ \n\\mathop{\\sum }\\limits_{{e \\in \\delta \\left( i\\right)... |
Proposition 23.41 Let \( {\delta }_{P} \) be a fixed square root of \( {\mathcal{K}}_{P} \) . For any vector field \( X \) lying in \( P \), there is a unique linear operator \( {\nabla }_{X} \) mapping sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P} \), such that
\[
{\nabla }_{X}\left( {f{s}_{1}}\ri... | {
"question": "Proposition 23.41 Let \( {\delta }_{P} \) be a fixed square root of \( {\mathcal{K}}_{P} \). For any vector field \( X \) lying in \( P \), there is a unique linear operator \( {\nabla }_{X} \) mapping sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P} \), such that\n\n\[{\nabla }_{X}\le... |
Theorem 7. Let \( M \) and \( {M}^{\prime } \) be totally disconnected compact sets in \( {\mathbf{R}}^{2} \), and let \( f \) be a homeomorphism \( M \leftrightarrow {M}^{\prime } \) . Then \( f \) has an extension \( F : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \) . Proof. (1) Let \( A \) and \( {A}^{\prim... | {
"question": "Theorem 7. Let \( M \) and \( {M}^{\prime } \) be totally disconnected compact sets in \( {\mathbf{R}}^{2} \), and let \( f \) be a homeomorphism \( M \leftrightarrow {M}^{\prime } \) . Then \( f \) has an extension \( F : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \) .",
"proof": "Proof. ... |
Lemma 7.4. If \( {h}_{\mathfrak{p}} \) is the natural homomorphism
\[
{h}_{\mathfrak{p}} : \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \rightarrow \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p},
\]
then \( {h}_{\mathfrak{p}}{}^{-1} \) induces a natural lattice-embeddin... | {
"question": "Lemma 7.4. If \( {h}_{\mathfrak{p}} \) is the natural homomorphism\n\n\[ \n{h}_{\mathfrak{p}} : \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \rightarrow \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p},\n\]\n\nthen \( {h}_{\mathfrak{p}}{}^{-1} \) induces a ... |
Theorem 5.1.15. Singular homology satisfies Axiom 3.
Proof. Immediate from the definition of the boundary map on singular cubes and from the definition of the induced map on singular cubes as composition.
Theorem 5.1.16. Singular homology satisfies Axiom 4.
Proof. We have defined \( {C}_{n}\left( {X, A}\right) = {... | {
"question": "Theorem 5.1.15. Singular homology satisfies Axiom 3.",
"proof": "Proof. Immediate from the definition of the boundary map on singular cubes and from the definition of the induced map on singular cubes as composition."
} |
Exercise 9.1.9 Prove that
\[
\pi \left( {x, z}\right) = {xV}\left( z\right) + O\left( {x{\left( \log z\right) }^{2}\exp \left( {-\frac{\log x}{\log z}}\right) }\right) ,
\]
where
\[
V\left( z\right) = \mathop{\prod }\limits_{{p \leq z}}\left( {1 - \frac{1}{p}}\right)
\]
and \( z = z\left( x\right) \rightarrow \in... | {
"question": "Exercise 9.1.9 Prove that\n\n\[ \pi \left( {x, z}\right) = {xV}\left( z\right) + O\left( {x{\left( \log z\right) }^{2}\exp \left( {-\frac{\log x}{\log z}}\right) }\right) ,\] \n\nwhere\n\n\[ V\left( z\right) = \mathop{\prod }\limits_{{p \leq z}}\left( {1 - \frac{1}{p}}\right) \] \n\nand \( z = z\left... |
Corollary 3.10. Every convex function that is continuous on an open convex subset \( U \) of a normed space is locally Lipschitzian on \( U \) .
Now let us turn to some links between convex functions and continuous affine functions. Hereinafter we say that a convex function is closed if it is lower semicontinuous an... | {
"question": "Corollary 3.10. Every convex function that is continuous on an open convex subset \( U \) of a normed space is locally Lipschitzian on \( U \) .",
"proof": "Null"
} |
Proposition 3.4 The image under \( T \) of the closed unit ball of \( C\left( Y\right) \) is a relatively compact subset of \( C\left( X\right) \) .
We say that \( T \) is a compact operator from \( C\left( Y\right) \) to \( C\left( X\right) \) (see Chapter 6).
Proof. It is clear that \( T\left( {\bar{B}\left( {C\l... | {
"question": "Proposition 3.4 The image under \( T \) of the closed unit ball of \( C\left( Y\right) \) is a relatively compact subset of \( C\left( X\right) \) .",
"proof": "Proof. It is clear that \( T\left( {\bar{B}\left( {C\left( Y\right) }\right) }\right) \) is bounded by\n\n\[ M = \mu \left( Y\right) \ma... |
Lemma 7.1.3. Let \( p \geq 3 \), let \( K = {\mathbb{F}}_{p} \), let \( E \) be a degenerate curve over \( {\mathbb{F}}_{p} \) as above, and let \( {c}_{6} = {c}_{6}\left( E\right) \) be the invariant defined in Section 7.1.2. Then \( E \) has a cusp (respectively a double point with tangents defined over \( {\mathbb... | {
"question": "Lemma 7.1.3. Let \( p \geq 3 \), let \( K = {\mathbb{F}}_{p} \), let \( E \) be a degenerate curve over \( {\mathbb{F}}_{p} \) as above, and let \( {c}_{6} = {c}_{6}\left( E\right) \) be the invariant defined in Section 7.1.2. Then \( E \) has a cusp (respectively a double point with tangents defined... |
Proposition 11.4. Let \( T = {\left( {S}^{1}\right) }^{k} \) and let \( t = \left( {{e}^{{2\pi }{\theta }_{1}},\ldots ,{e}^{{2\pi }{\theta }_{k}}}\right) \) be an element of \( T \) . Then \( t \) generates a dense subgroup of \( T \) if and only if the numbers
\[
1,{\theta }_{1},\ldots ,{\theta }_{k}
\]
are linear... | {
"question": "Proposition 11.4. Let \( T = {\left( {S}^{1}\right) }^{k} \) and let \( t = \left( {{e}^{{2\pi }{\theta }_{1}},\ldots ,{e}^{{2\pi }{\theta }_{k}}}\right) \) be an element of \( T \) . Then \( t \) generates a dense subgroup of \( T \) if and only if the numbers\n\n\[1,{\theta }_{1},\ldots ,{\theta }_... |
Proposition 9.6.22 (Stirling’s formula). As \( n \rightarrow \infty \) we have
\[
n! \sim {n}^{n}{e}^{-n}\sqrt{2\pi n}
\]
or equivalently,
\[
\log \left( {n!}\right) = \left( {n + \frac{1}{2}}\right) \log \left( n\right) - n + \frac{1}{2}\log \left( {2\pi }\right) + o\left( 1\right) .
\]
Proof. Once again there a... | {
"question": "Proposition 9.6.22 (Stirling’s formula). As \( n \rightarrow \infty \) we have\n\n\[ n! \sim {n}^{n}{e}^{-n}\sqrt{2\pi n} \]\n\nor equivalently,\n\n\[ \log \left( {n!}\right) = \left( {n + \frac{1}{2}}\right) \log \left( n\right) - n + \frac{1}{2}\log \left( {2\pi }\right) + o\left( 1\right) . \]",
... |
Theorem 5.23 (Jordan Curve Theorem \( {}^{2} \) ). If \( \gamma \) is a simple closed path in \( \mathbb{C} \), then
(a) \( \mathbb{C} \) - range \( \gamma \) has exactly two connected components, one of which is bounded.
(b) Range \( \gamma \) is the boundary of each of these components, and
(c) \( I\left( {\gamm... | {
"question": "Theorem 5.23 (Jordan Curve Theorem \( {}^{2} \) ). If \( \gamma \) is a simple closed path in \( \mathbb{C} \), then\n\n(a) \( \mathbb{C} \) - range \( \gamma \) has exactly two connected components, one of which is bounded.\n\n(b) Range \( \gamma \) is the boundary of each of these components, and\n... |
Lemma 1.6. Let \( R = k \) be a field, and let \( V \) be a \( k \) -vector space. Let \( B \) be a maximal linearly independent subset of \( V \) ; then \( B \) is a basis of \( V \) .
Again, this should be contrasted with the situation over rings: \( \{ 2\} \) is a maximal linearly independent subset of \( \mathbb... | {
"question": "Lemma 1.6. Let \( R = k \) be a field, and let \( V \) be a \( k \) -vector space. Let \( B \) be a maximal linearly independent subset of \( V \) ; then \( B \) is a basis of \( V \) .",
"proof": "Proof. Let \( v \in V, v \notin B \) . Then \( B \cup \{ v\} \) is not linearly independent, by the... |
Proposition 1.5 A scalar product satisfies the Cauchy-Schwarz \( {}^{1} \) inequality
\[
b{\left( x, y\right) }^{2} \leq q\left( x\right) q\left( y\right) ,\;\forall x, y \in E.
\]
The equality holds true if and only if \( x \) and \( y \) are colinear.
Proof. The polynomial
\[
t \mapsto q\left( {{tx} + y}\right)... | {
"question": "Proposition 1.5 A scalar product satisfies the Cauchy-Schwarz inequality\n\n\\[ b{\\left( x, y\\right) }^{2} \\leq q\\left( x\\right) q\\left( y\\right) ,\\;\\forall x, y \\in E. \\]\n\nThe equality holds true if and only if \\( x \\) and \\( y \\) are colinear.",
"proof": "Proof. The polynomial\... |
Proposition 7.5. Let \( X \) be a homogeneous space for \( G \) . If \( X \) is strictly unimodular, then there exists a left G-invariant volume form on \( X \), unique up to a constant multiple.
Proof. We want to define the invariant form on \( G/H \) by translating a given volume form \( {\omega }_{e} \) on \( {T}... | {
"question": "Proposition 7.5. Let \( X \) be a homogeneous space for \( G \) . If \( X \) is strictly unimodular, then there exists a left G-invariant volume form on \( X \), unique up to a constant multiple.",
"proof": "Proof. We want to define the invariant form on \( G/H \) by translating a given volume fo... |
Theorem 4.2.5 If \( \mathcal{M} \) is a countable model of \( {PA} \), then there is \( \mathcal{M} \prec \mathcal{N} \) such that \( \mathcal{N} \) is a proper end extension of \( \mathcal{M} \) .
Proof Consider the language \( {\mathcal{L}}^{ * } \) where we have constant symbols for all elements of \( M \) and a ... | {
"question": "Theorem 4.2.5 If \( \mathcal{M} \) is a countable model of \( {PA} \), then there is \( \mathcal{M} \prec \mathcal{N} \) such that \( \mathcal{N} \) is a proper end extension of \( \mathcal{M} \) .",
"proof": "Proof Consider the language \( {\mathcal{L}}^{ * } \) where we have constant symbols fo... |
Proposition 5.1.16. If \( q \) is nondegenerate and represents 0, then there exists a hyperbolic form \( h \) and a nondegenerate form \( {q}^{\prime } \) such that \( q \sim h \oplus {q}^{\prime } \) . Furthermore, \( q \) represents all elements of \( K \) .
Proof. This is Lemma 5.1.5 and Corollary 5.1.6. The fact... | {
"question": "Proposition 5.1.16. If \( q \) is nondegenerate and represents 0, then there exists a hyperbolic form \( h \) and a nondegenerate form \( {q}^{\prime } \) such that \( q \sim h \oplus {q}^{\prime } \) . Furthermore, \( q \) represents all elements of \( K \) .",
"proof": "Proof. This is Lemma 5.1... |
Exercise 2.4.4 Using Dirichlet's hyperbola method, show that
\[
\mathop{\sum }\limits_{{n \leq x}}\frac{f\left( n\right) }{\sqrt{n}} = {2L}\left( {1,\chi }\right) \sqrt{x} + O\left( 1\right)
\]
where \( f\left( n\right) = \mathop{\sum }\limits_{{d \mid n}}\chi \left( d\right) \) and \( \chi \neq {\chi }_{0} \) .
E... | {
"question": "Exercise 2.4.4 Using Dirichlet's hyperbola method, show that\n\n\\[ \n\\mathop{\\sum }\\limits_{{n \\leq x}}\\frac{f\\left( n\\right) }{\\sqrt{n}} = {2L}\\left( {1,\\chi }\\right) \\sqrt{x} + O\\left( 1\\right) \n\\]\n\nwhere \\( f\\left( n\\right) = \\mathop{\\sum }\\limits_{{d \\mid n}}\\chi \\left... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.