Q stringlengths 26 3.6k | A stringlengths 1 9.94k | Result stringclasses 3
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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. | Null | No |
Show that the quotient group \( {H}_{0} \) of all nonzero fractional ideals modulo \( {P}_{0} \) is a subgroup of the ideal class group \( H \) of \( K \) and \( \left\lbrack {H : {H}_{0}}\right\rbrack \leq 2 \). | Null | No |
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( {\mathbb{R}}^{n}\right) }\... | The proof of Proposition 9.34 is extremely similar to that of Proposition 9.32 and is omitted. | No |
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. | Null | No |
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 | The proof follows at once from the definitions and the formula giving the chart representation of \( s{\left( {s}_{TX}\right) }_{ * } \) . | No |
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 }_{1} = c{\phi }... | Clearly \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = c{\phi }_{1}{\widetilde{\mathbf{H}}}^{2} \) when \( \left| c\right| = 1 \) . Conversely, suppose that \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) with \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\... | No |
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\n{a}_{n} ≢ 0\;\left... | Proof. Extracting a g.c.d. for the coefficients of \( f \), we may assume without loss of generality that the content of \( f \) is 1 . If there exists a factorization into factors of degree \( \geqq 1 \) in \( K\left\lbrack X\right\rbrack \), then by the corollary of Gauss’ lemma there exists a factorization in \( A\l... | Yes |
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. | Suppose \( p \in D \) is arbitrary. If \( p \) is in the manifold boundary of \( D \), Theorem 4.15 shows that there exist a smooth boundary chart \( \left( {U,\varphi }\right) \) for \( D \) centered at \( p \) and a smooth chart \( \left( {V,\psi }\right) \) for \( M \) centered at \( p \) in which \( F \) has the co... | Yes |
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 \( s \leq 0 \). | By the continuity of \( f \rightarrow \widehat{f} \), we may suppose \( \int \in {\mathfrak{A}}_{N} \). Then for \( s \leq 0, F\left( z\right) = f\left( z\right) {e}^{-{2\pi isz}} \) is also in \( {\mathfrak{A}}_{N} \). The result now follows from Cauchy’s theorem because \({\int }_{0}^{\pi }\left| {F\left( {R{e}^{i\th... | No |
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
\[
\left\langle {f * {\delta }_{{x}_{0}}, h}\right\rangle = \left\langle {{\delta }_{{x}_{0}},\widetilde{f} * h}\right\rangle = \left( {\widetilde{f} * h}\right) \left( {x}_{0}\right) = {\int }_{{\mathbf{R}}^{n}}f\left( {x - {x}_{0}}\right) h\left( x\right) {dx}.
\] | No |
Show that every element of R can be written as a product of irreducible elements. | Suppose b is an element of R . We proceed by induction on the norm of b . If b is irreducible, then we have nothing to prove, so assume that b is an element of R which is not irreducible. Then we can write b = ac where neither a nor c is a unit. By condition (i), n(b) = n(ac) = n(a)n(c) and since a, c are not units, th... | Yes |
Theorem 11.5 For each \( h > 0 \) the difference equations (11.10)-(11.11) have a unique solution. | 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. | Yes |
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 | Since for every point in \( {\mathbf{R}}^{n} \) there is a sequence of dyadic cubes shrinking to it, the Lebesgue differentiation theorem yields that for almost every point \( x \) in \( {\mathbf{R}}^{n} \) the averages of the locally integrable function \( f \) over the dyadic cubes containing \( x \) converge to \( f... | Yes |
The set of lines spanned by the vectors of \( {D}_{n} \) is star-closed. | Lemma 12.2.2 The set of lines spanned by the vectors of \( {D}_{n} \) is star-closed. | No |
If \( \left( {{g}_{1},{S}_{1}}\right) \sim \left( {{g}_{2},{S}_{2}}\right) \) then for all \( i\left( {{g}_{1},{e}_{i}\left( {S}_{1}\right) }\right) \sim \left( {{g}_{2},{e}_{i}\left( {S}_{2}\right) }\right) \) . | There exist \( e \) and \( T \) such that \( \left( {e\left( {S}_{i}\right), T}\right) \) is a balanced pair representing \( {g}_{i} \) for \( i = 1,2 \) . We write \( \left( {{g}_{1},{S}_{1}}\right) \underset{k}{ \sim }\left( {{g}_{2},{S}_{2}}\right) \) if the length of this \( e \) is \( \leq k \) . The lemma is prov... | No |
Every module can be embedded into an injective module. | By 4.11 there is a monomorphism of \( R/{L}_{n} \) into an injective \( R \) -module \( {J}_{n} \). By the hypothesis, \( J = {\bigoplus }_{n > 0}{J}_{n} \) is injective. Construct a module homomorphism \( \varphi : L \rightarrow J \) as follows. Let \( {\varphi }_{n} \) be the homomorphism \( R \rightarrow R/{L}_{n} \... | Yes |
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\n\[
\vdash \forall {\alpha \varphi } \rightarrow {\operatorname{Subf}}_{\beta }^{\alpha }\varphi
\]\n10.53\n\[
\vdash \forall \beta \forall {\alpha \varphi } \rightarrow \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi
\]\nusing \( {10.23}\left( 2\right) \)\n\[
\vdash \forall {\alpha \varphi } \right... | Yes |
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 Theorem 2.6, there is by Theorem I.5.2 a unique isomorphism \( \theta \) of abelian groups such that \( \varphi = \iota \circ \theta \circ \pi \) . Then \( \theta \left( {a + \operatorname{Ker}\varphi }\right) = \varphi \left( a\right) \) for all \( a \in A \) . Therefore \( \theta \) is a module homomorphism. | Yes |
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( {\;\operatorname{mod}\;p}\right)... | Solution. Suppose \( b \) is an element of \( R \) . We proceed by induction on the norm of \( b \) . If \( b \) is irreducible, then we have nothing to prove, so assume that \( b \) is an element of \( R \) which is not irreducible. Then we can write \( b = {ac} \) where neither \( a \) nor \( c \) is a unit. By condi... | No |
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 {F}_{k}}{\partial {x}... | If \( \mathbf{F} \) is conservative, then \( \frac{\partial {F}_{j}}{\partial {x}_{k}} = - \frac{{\partial }^{2}V}{\partial {x}_{k}\partial {x}_{j}} = - \frac{{\partial }^{2}V}{\partial {x}_{j}\partial {x}_{k}} = \frac{\partial {F}_{k}}{\partial {x}_{j}} \) at every point in \( U \). In the other direction, if \( \math... | Yes |
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 at ... | Proof. We have already proved that for all \( b \in {BMO}\left( {\mathbf{R}}^{n}\right) ,{L}_{b} \) lies in \( {\left( {H}^{1}\left( {\mathbf{R}}^{n}\right) \right) }^{ * } \) and has norm at most \( {C}_{n}\parallel b{\parallel }_{BMO} \) . The embedding \( b \mapsto {L}_{b} \) is injective as a consequence of Exercis... | Yes |
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. | The first statement of the theorem follows immediately from Theorem 3.2.19, Theorem 3.3.1, and Theorem 3.3.2. Now, for the latter statement, if \( {c}_{0} \) were complemented in \( X \), then \( {X}^{* * } \) would contain a (complemented) copy \( {\ell }_{\infty } \) . If \( {\ell }_{1} \) were complemented in \( X \... | Yes |
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 \( Q \) i... | The first two statements are immediate from the definition of a primary ideal. For (3), suppose \( {ab} \in \operatorname{rad}Q \) . Then \( {a}^{m}{b}^{m} = {\left( ab\right) }^{m} \in Q \), and since \( Q \) is primary, either \( {a}^{m} \in Q \), in which case \( a \in \operatorname{rad}Q \), or \( {\left( {b}^{m}\r... | Yes |
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 \( {\Omega }_{0} \subset \subset \Omega \), | Since \( u \in {C}^{2,\alpha }\left( \Omega \right) \) by Theorem 13.1.2, we know that it weakly solves \[ \Delta \frac{\partial }{\partial {x}^{i}}u = \frac{\partial }{\partial {x}^{i}}f \] Theorem 13.1.2 then implies \[ \frac{\partial }{\partial {x}^{i}}u \in {C}^{2,\alpha }\left( \Omega \right) \;\left( {i \in \{ 1,... | No |
Theorem 5.9 (Stone [1934]). Every Boolean lattice is isomorphic to the lattice of closed and open subsets of its Stone space. | 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 \( X \) since \( X \smallsetminus V\left( a\right) = V\left( {a}^{\prime }\right) \) is open. Conversely, if \( U \in \mathrm{L}\left( X\right) \) is a closed and op... | Yes |
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 order to establish the inequality, we proceed as above. If \( \lambda \) is an eigenvalue of \( B \), of modulus \( \rho \left( B\right) \), and if \( x \) is a normalized eigenvector, then \( \rho \left( B\right) \left| x\right| \leq \left| B\right| \cdot \left| x\right| \leq M\left| x\right| \), so that \( {C}_{\r... | No |
Proposition 4.51. If \( S \) is compact, under assumptions (P1),(P2), one has\n\[
{\partial }_{F}m\left( \bar{x}\right) = \overline{\operatorname{co}}\left\{ {D{f}_{s}\left( \bar{x}\right) : s \in M\left( \bar{x}\right) }\right\}
\]\[
{\partial }_{F}p\left( \bar{x}\right) = \mathop{\bigcap }\limits_{{s \in P\left( \bar... | The proof is left as an exercise that the reader can tackle while reading Sect. 4.7.1 | No |
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. | This is essentially a restatement of Corollary 8.1.15, and also immediately follows from the above proposition after rescaling. Note that Proposition 8.4.3 tells us that the elliptic curve \( {Y}^{2} = {X}^{3} - {432}{\left( abc\right) }^{2} \) is 3 -isogenous with the elliptic curve \( {Y}^{2} = {X}^{3} + {\left( 4abc... | No |
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 of the Fan Lemma. By Theorem 5.1, in a 2-connected graph any two vertices are connected by two internally disjoint paths; equivalently, any two vertices in a 2-connected graph lie on a common cycle. Dirac (1952b) generalized this latter statement to \( k \) -connected graphs.\n\nThe... | Yes |
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 \( p\left( x\right) = \mathop{\prod }\limits_{\substack{{d \mid m} \\ {1 \leq d < m} }}{\lambda }_{d}\left( x\right) \), we obtain by the ... | Yes |
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 \( J \). Below and hereafter, pictures of polyhedra will not necessarily look like polyhedra. Only a sample of \( N \) is indicated in Figure 2.1.
![0d8d6f7c-790d-4773-9cb3-7748c6b57409... | Yes |
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,\[
{\left| \det \left( x{I}_{n} - M\right) \right| }^{2} = \det \left( {h\left( {x{I}_{n} - M}\right) }\right) = \det \left( {x{I}_{2n} - h\left( M\right) }\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{x}^{2\left( {n - k}\right) }{f}_{2k} \circ h\left( M\right) .
\]
On the other hand,\[
{\left| \de... | Yes |
Whitehead product 472, 482 | Whitehead Theorem 181 | No |
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 \( V \) . | Suppose that \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . Then, in particular, \( \left| {\partial \left( v\right) }\right| \) is even for every vertex \( v \) . But, as noted above, \( \partial \left( v\right) \) is just the set of all links incident with \( v \) . Beca... | Yes |
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 \( \mathrm{K} \) . | If \( u : E - K \), then there exists \( \sigma \in {\operatorname{Aut}}_{K}F \) such that \( \sigma \left( u\right) \neq u \) since \( F \) is Galois over \( K \) . But \( \sigma \mid E \) e Aut \( {}_{K}E \) by stability. Therefore, \( E \) is Galois over \( K \) by the Remarks after Definition 2.4. | Yes |
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\[
\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 },
... | Proof. We apply Hadamard's theorem to the function\n\[
f\left( s\right) = s\left( {1 - s}\right) {\pi }^{-s/2}\Gamma \left( {s/2}\right) \zeta \left( s\right) = 2\left( {1 - s}\right) {\pi }^{-s/2}\Gamma \left( {s/2 + 1}\right) \zeta \left( s\right) .
\]\nSince the zeros of \( \zeta \left( s\right) \) for \( s = - {2k}... | Yes |
Theorem 2. Let \( \chi \) be a nontrivial Dirichlet character modulo m. Then \( L\left( {1,\chi }\right) \neq 0 \) . | 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
\[
\psi \left( s\right) = \frac{L\left( {s,\chi }\right) L\left( {s,{\chi }_{0}}\right) }{L\left( {{2s},{\chi }_{0}}\right) }
\]
... | Yes |
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. \( \exists {\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{m} \) in \( N \) with \( m \leq n \) such that \( N = \mathbb{Z}{\beta }_{1} + \mathbb{Z}{\... | Proof. (a) We will proceed by induction on the number of generators of a \( \mathbb{Z} \) -module. This is trivial when \( n = 0 \). We can assume that we have proved the above statement to be true for all \( \mathbb{Z} \) -modules with \( n - 1 \) or fewer generators, and proceed to prove it for \( n \). We define \( ... | Yes |
Let \( f : X \rightarrow Y \) be a map between two metric spaces. 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 every \( x, y \in X \), ... | We do (iii) and leave the other statements as an exercise. We need to show that for \( s > 0 \) there is \( {C}_{s} > 0 \) such that \( d\left( {f\left( x\right), f\left( y\right) }\right) \leq {C}_{s} \) whenever \( d\left( {x, y}\right) \leq s \) . From the definition of uniform continuity there exists \( {\delta }_{... | No |
Proposition 4.4. For all ideals \( \mathfrak{a},\mathfrak{b} \) of \( R \) and \( \mathfrak{A} \) of \( {S}^{-1}R \) : | The proofs make good exercises. | No |
Proposition 11.101. \( \mathfrak{B} \) is generated by \( {\mathfrak{B}}_{0} \) and any set of representatives for \( {W}^{\prime } \) in \( N \) . | Null | No |
If \( k \) is a finite field, then \( {k}^{ * } \) is cyclic. | Corollary 1.10. | No |
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 \( T \) . | This follows direcly from the definition of irreducibility; using only the continuity of \( \alpha \) . | No |
If \( \mu \) and \( \lambda \) are dominant, then \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) if and only if \( \lambda \preccurlyeq \mu \) . | Since \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) is convex and Weyl invariant, we see that if \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \), then every point in \( \operatorname{Conv}\left( {W \cdot \lambda }\right) \) also belongs to \( \operatorname{Conv}\left( {W \cdot \mu... | Yes |
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 \( x \in W \). | Starting with \( {W}_{0} \), we define inductively an increasing sequence \( {\left( {W}_{n}\right) }_{n \geq 1} \) of \( \mathcal{S}\left( X\right) \) such that \( d\left( {\cdot, S}\right) = d\left( {\cdot, S \cap {W}_{n}}\right) \) on \( {W}_{n - 1} \). Assuming that \( {W}_{1},\ldots ,{W}_{n} \) satisfying this pro... | Yes |
Let \( F/K \) be a function field whose full constant field is \( K \). 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 splits completely in \( {F... | We only have to show that \( K \) is the full constant field of \( {F}^{\prime } = {F}_{1}{F}_{2} \); the remaining assertions follow immediately from Proposition 3.9.6. We choose a place \( {P}^{\prime } \) of \( {F}^{\prime } \) lying above \( P \), then \( f\left( {{P}^{\prime } \mid P}\right) = 1 \) and therefore t... | Yes |
Corollary 10.3.13. We have\\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{\\mathcal{H}}_{3}\\left( \\tau \\right) = - \\frac{7{\\theta }^{3}\\left( \\tau \\right) {\\theta }^{4}\\le... | Null | Yes |
Theorem 5.4.4 The classification space \( \operatorname{irr}\left( n\right) / \sim \) is standard Borel. | Fix any irreducible \( A \) . Then the \( \sim \) -equivalence class \( \left\lbrack A\right\rbrack \) containing \( A \) equals
\[
{\pi }_{1}\left\{ {\left( {B, U}\right) \in \operatorname{irr}\left( n\right) \times U\left( n\right) : A = {UB}{U}^{ * }}\right\} ,
\]
where \( {\pi }_{1} : \operatorname{irr}\left( n\r... | Yes |
Let \( {z}_{0} = {y}_{0}\mathrm{i} \) and \( {z}_{1} = {y}_{1}\mathrm{i} \) with \( 0 < {y}_{0} < {y}_{1} \) . Then | It is readily checked that the path \( \phi \) defined in the lemma has constant speed equal to \( \log {y}_{1} - \log {y}_{0} = \mathrm{L}\left( \phi \right) \) as claimed. It follows that
\[
\mathrm{d}\left( {{z}_{0},{z}_{1}}\right) \leq \log {y}_{1} - \log {y}_{0}
\]
Suppose now that \( \eta : \left\lbrack {0,1}\r... | Yes |
Suppose \( X \) is \( F \) -smooth. If \( f \) is locally Lipschitzian and \( q \) is a Lyapunov function for \( S \), then \( S \) is stable. | In order to show the stability of \( S \) in both cases, it suffices to prove that for all \( z \in X \), the function \( t \mapsto {e}^{ct}q\left( {{x}_{z}\left( t\right) }\right) + {\int }_{0}^{t}{e}^{cs}p\left( {{x}_{z}\left( s\right) }\right) \mathrm{d}s \) is nonincreasing on \( {\mathbb{R}}_{ + } \) . Since \( {x... | Yes |
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 Y is SZ, then so is X. | Proof of Proposition 7.23 using van der Waerden. By Lemma 7.24, we may assume that f = χB is AP and that there exists some set A ∈ A of positive measure with μy(A)(B) > 1/2μ(B) for y ∈ A. We will use SZ for A (for arithmetic progressions of quite large length K) to show SZ for B (for arithmetic progressions of length k... | Yes |
If \( F/\mathbb{Q} \) is an extension in which no finite prime ramifies, then \( F = \mathbb{Q} \). | 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
\[
\frac{n!}{{n}^{n}}{\left( \frac{4}{\pi }\right) }^{{r}_{2}}\sqrt{{d}_{F}}
\]
where \( n = \left\lbrack {F : \mathbb{Q}}\right\rbrack ,{d}_{F} \) is the absolute value of the d... | No |
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 \( {s}^{-1}{hs} = h \) for all \( h \in H \), and hence \( s \in H \) by Theorem 2.1.5. This proves that the representation of \( {W}_{G} \) on \( {\mathfra... | Yes |
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 real p... | If condition \( {\mathcal{3}}^{\prime } \) holds and \( \sigma : {k\Gamma } \rightarrow \mathbb{R} \), then there exists \( \tau : {A\Gamma } \rightarrow \) \( \mathcal{H} \), Hamilton’s quaternions, such that \( \sigma \left( {\operatorname{tr}f}\right) = \operatorname{tr}\left( {\tau \left( f\right) }\right) \) for e... | Yes |
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
\[
{\left( \mathop{\sum }\limits_{{n \in A}}1\right) }^{1/2} = {m}^{1/2}
\]
while
\[
\mathop{\sum }\limits_{{n \in A}}\frac{1}{\sqrt{n}} ... | No |
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 o... | Proof. We proceed by induction on \( \chi \) . We may assume that \( \theta \neq \chi \) . If \( \chi \) is atomic, then \( \chi = \varphi \) and \( \psi = \theta \) ; this case is trivial. Suppose \( \chi \) is \( \neg {\chi }^{\prime } \) . Then \( \theta \) is of the form \( \neg {\theta }^{\prime } \), and the indu... | Yes |
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 \(... | Null | No |
one about a system with \( n - 1 \) variables. Namely, we determine necessary and sufficient conditions for which, given a vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \in {\mathbb{R}}^{n - 1} \), there exists \( {\bar{x}}_{n} \in \mathbb{R} \) such that \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{... | o one about a system with \( n - 1 \) variables. Namely, we determine necessary and sufficient conditions for which, given a vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \in {\mathbb{R}}^{n - 1} \), there exists \( {\bar{x}}_{n} \in \mathbb{R} \) such that \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}... | Yes |
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\r... | Proof. We first show that for fixed \( g \in G \), the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is injective from \( \operatorname{Lie}\left( G\right) \) to \( T{\left( G\right) }_{g} \) . Suppose \( {\left( {X}_{A}\right) }_{g} = 0 \) . Then for \( x \in G \) and \( f \in \mathcal{O}\left\lbrack {\mathbf{{GL}}... | Yes |
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 }}^{\infty }\left| {b}_{... | The hypothesis on \( f \) says that \( {\varphi }_{\lambda }\left( f\right) = f\left( \lambda \right) \) does not vanish as \( \lambda \) ranges over \( T \) . Since we have shown that the functionals \( {\varphi }_{\lambda } \) exhaust \( {\mathcal{M}}_{W} \), we may apply Corollary 5.29 to conclude that \( f \) is in... | Yes |
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 \( c \) . | The theorem is an immediate consequence of (2.12), since in this case \( {f}_{\bar{z}}\left( c\right) = 0 \) and hence \( {f}^{\prime }\left( c\right) = {f}_{z}\left( c\right) \) . | Yes |
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 | SKETCH OF PROOF OF 2.7. Once we have shown that multiplication in \( R/I \) is well defined, the proof that \( R/I \) is a ring is routine. (For example, if \( R \) has identity \( {1}_{R} \), then \( {1}_{R} + I \) is the identity in \( R/I \) .) Suppose \( a + I = {a}^{\prime } + I \) and \( b + I = {b}^{\prime } + I... | No |
If \( V \) is an irreducible finite-dimensional representation of a Lie group \( G \), show that \( {V}^{ * } \) is also irreducible. | Null | No |
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 | The key to the proof of this proposition is the fact that the map \( \psi : E \rightarrow E \) is unramified, i.e., it has no critical points, see Remark 6.20. (In the language of modern algebraic geometry, the map \( \psi \) is étale.) More precisely, the map \( \psi \) is the composition of an endomorphism of \( E \)... | Yes |
If \( G \) is an infinite connected solvable group of finite Morley rank with finite center, then \( G \) interprets an algebraically closed field. | 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. Suppose that \( Z\left( G\right) \) is finite. We claim that \( G/Z\left( G\right) \) is centerless. Let \( a \in G \) such that \( a/Z\left( G\right) \in Z\left( {G/Z\left... | No |
If Dedekind's functional equation is satisfied for some A = (a b c d) in Γ with c > 0 and ε(A) given by (2), then it is also satisfied for ATm and for AS. | Replace τ by Tmτ in (5) to obtain η(ATmτ) = ε(A){-i(cTmτ + d)}1/2η(Tmτ) = ε(A){-i(cτ + mc + d)}1/2e^πim/12η(τ). Using Lemma 2 we obtain (6). Now replace τ by Sτ in (5) to get η(ASτ) = ε(A){-i(cSτ + d)}1/2η(Sτ). Using Theorem 3.1 we can write this as η(ASτ) = ε(A){-i(cSτ + d)}1/2{-iτ}1/2η(τ). If d > 0, we write cSτ + d ... | Yes |
Corollary 4.1.1. The discrete Dirichlet problem | This follows in the usual manner by applying the maximum principle to the difference of two solutions. | No |
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 | Proof of Theorem 11.6.9. Consider Figure 11.6.4. As \( g \) (or \( {g}^{-1} \) ) is \( {\sigma }_{3}{\sigma }_{1} \) and \( h\left( {\text{or}{h}^{-1}}\right) \) is \( {\sigma }_{2}{\sigma }_{3} \) we see that \( {\sigma }_{2}{\sigma }_{1} \) is in \( G \) . If \( G \) has no elliptic elements, then \( {L}_{1} \) and \... | Yes |
Show that every positive root is a nonnegative linear combination of simple roots. | Null | No |
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 trees.... | Proof. To simplify the situation, multiply all currents by \( N \) . Also, for every spanning tree \( T \) and edge \( {ab} \in E\left( G\right) \), let \( {w}^{\left( T\right) } \) be the current of size 1 along the unique \( s - t \) path in \( T \) : \[
{w}_{ab}^{\left( T\right) } = \left\{ \begin{array}{ll} 1 & \te... | Yes |
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}}^{m} ... | Since the functions \( \varphi \) and \( {\chi }_{n} \) have compact support, so does \( \varphi * {\chi }_{n} \) . More precisely,
\[
\operatorname{Supp}\left( {\varphi * {\chi }_{n}}\right) \subset \operatorname{Supp}\varphi + \operatorname{Supp}{\chi }_{n} \subset \operatorname{Supp}\varphi + \bar{B}\left( {0,1/n}\... | Yes |
If no free occurrence of \( \alpha \) in \( \varphi \) is within the scope of a quantifier on a variable occurring in \( \sigma \), then \( {\mathrm{{FSubf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi } \) . | \[
\vdash \forall \alpha \sqsupset \varphi \rightarrow {\operatorname{Subf}}_{\sigma }^{\alpha } \sqsupset \varphi \;\text{ universal specification }
\]
\[
{ \vdash }^{ \Vdash }{\mathrm{{Subf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi }
\]
suitable tautology | Yes |
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 \( \left| {{G}_{1} \cap {G}_{2}}\right| = \kappa \left( G\right) \) . If \( G \) is edg... | Note first that every vertex \( v \in S \mathrel{\text{:=}} V\left( {{G}_{1} \cap {G}_{2}}\right) \) has a neighbour in every component of \( {G}_{i} - S, i = 1,2 \) : otherwise \( S \smallsetminus \{ v\} \) would separate \( G \), contradicting \( \left| S\right| = \kappa \left( G\right) \) . By the maximality of \( G... | Yes |
The lightcone \( {\mathcal{L}}_{0} \) is a lightlike submanifold. | 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 \( \widetilde{g} : \mathcal{U} \rightarrow \mathbb{R} \) by \( \widetilde{g}w = g\left( {w, w}\right) \forall w \in \mathcal{U}... | Yes |
Show that \( \mathfrak{a} \) has an integral basis. | 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},{a}_{0}{\omega }_{i} = - \left( {{\alpha }^{r} + \cdots + {a}_{1}\alpha }\right)... | Yes |
Prove that the group B = {(a, b) | a, b ∈ ℝ, a > 0} is amenable but not unimodular. | Null | No |
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 \( \mathcal{C}\left( \widehat{K}\right) \) . | A sequence \( \left\{ {p}_{n}\right\} \) of polynomials converging uniformly on \( K \) automatically converges uniformly on the outer boundary of \( K \) . But then, by the maximum modulus principle (Ex. 5M), \( \left\{ {p}_{n}\right\} \) converges uniformly on \( \widetilde{K} \), and the limit is therefore different... | No |
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
\[
{u}_{\alpha } = \frac{1}{2}\left( {{X}_{\alpha } - {X}_{-\alpha }}\right) \;\text{ and }\;{v}_{\alpha } = \frac{1}{2\mathrm{i}}\left( {{X}_{\alpha } + {X}_{-\alpha }}\right) .
\]
(11.13)
Then \( {X}_{\alpha } = {u}_{\alpha } + \mathrm{i}{v... | Yes |
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 \( {x}_{i} + {x}_{j} = 0\left( {i \neq j}\right) \) . To figure out what the chambers look like, consi... | It follows that there are \( {2}^{n - 1}n \) ! chambers, each defined by inequalities of the form
\[
{\epsilon }_{1}{x}_{\pi \left( 1\right) } > {\epsilon }_{2}{x}_{\pi \left( 2\right) } > \cdots > {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > \left| {x}_{\pi \left( n\right) }\right|
\]
(1.21)
with \( {\epsi... | No |
Theorem 12.5.1 An internal set \( A \) is hyperfinite with internal cardinality \( N \) if and only if there is an internal bijection \( f : \{ 1,\ldots, N\} \rightarrow A \) . | Let \( A = \left\lbrack {A}_{n}\right\rbrack \) . If \( A \) is hyperfinite with internal cardinality \( N = \) \( \left\lbrack {N}_{n}\right\rbrack \), then we may suppose that for each \( n \in \mathbb{N},{A}_{n} \) is a finite set of cardinality \( {N}_{n} \) . Thus there is a bijection \( {f}_{n} : \left\{ {1,\ldot... | No |
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 \smallsetminus \{ 1\}
... | ∎ | No |
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 | Proof. If \( V \) is a one-dimensional vector space, then the map \( \otimes : V \times V \rightarrow \) \( V \otimes V \) is commutative: \( u \otimes v = v \otimes u \) for all \( u, v \in V \) . Furthermore, if \( {u}_{0} \) is a nonzero element of \( V \), then the map \( u \mapsto u \otimes {u}_{0} \) is an invert... | Yes |
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 \( F : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \). | 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}^{\prime } ... | Yes |
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-embedding \( \mathfrak{... | Let \( {\mathfrak{a}}_{1} \neq {\mathfrak{a}}_{2} \) be distinct ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \); say \( p \in {\mathfrak{a}}_{1} \) but \( p \notin {\mathfrak{a}}_{2} \). Then for any \( q \in \left\{ {{h}_{\mathfrak{p}}{}^{-1}\left( p\right) }\right\} \), we... | Yes |
Singular homology satisfies Axiom 3. | Immediate from the definition of the boundary map on singular cubes and from the definition of the induced map on singular cubes as composition. | No |
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 \\[\\begin{aligned} V\\left( z\\right) &= \\mathop{\\prod }\\limits_{{p \\leq z}}\\left( {1 - \\frac{1}{p}}\\right) \\end{aligned} \\]and \\( z = z\\left( x\\right) \\rightarrow \\infty \\) as \\( x \\rightarrow \\infty \\) . | No |
Every convex function that is continuous on an open convex subset \( U \) of a normed space is locally Lipschitzian on \( U \). | Null | No |
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. It is clear that \( T\left( {\bar{B}\left( {C\left( Y\right) }\right) }\right) \) is bounded by\n\[
M = \mu \left( Y\right) \mathop{\max }\limits_{{\left( {x, y}\right) \in X \times Y}}\left| {K\left( {x, y}\right) }\right| .
\]\nOn the other hand, \( K \) is uniformly continuous on \( X \times Y \) ; in particu... | Yes |
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 \( {\mathbb{F}}_{p} \) , r... | Since \( E \) is degenerate it has a singular point, and changing coordinates we may assume that it is at the origin, so that we can choose the equation of our curve to be \( {y}^{2} = {x}^{2}\left( {x + a}\right) \) for some \( a \in {\mathbb{F}}_{p} \) . One computes that \( {c}_{6}\left( E\right) = - {64}{a}^{3} \) ... | Yes |
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 linearly ... | Proof of Proposition 11.4. In light of Lemma 11.5, we may reformulate the proposition as follows: The numbers \( 1,{\theta }_{1},\ldots ,{\theta }_{k} \) are linearly dependent over \( \mathbb{Q} \) if and only if there exists a nonconstant homomorphism \( \Phi : T \rightarrow {S}^{1} \) with \( \left( {{e}^{{2\pi i}{\... | No |
Proposition 9.6.22 (Stirling’s formula). As \( n \rightarrow \infty \) we have\n\[
n! \sim {n}^{n}{e}^{-n}\sqrt{2\pi n}
\]\nor equivalently,\n\[
\log \left( {n!}\right) = \left( {n + \frac{1}{2}}\right) \log \left( n\right) - n + \frac{1}{2}\log \left( {2\pi }\right) + o\left( 1\right) .
\] | Proof. Once again there are several classical proofs. Certainly the most classical is as follows: if we set \( {u}_{n} = \log \left( {n!/\left( {{n}^{n}{e}^{-n}\sqrt{n}}\right) }\right) \) then\n\[
{u}_{n + 1} - {u}_{n} = 1 - \left( {n + \frac{1}{2}}\right) \log \left( {1 + \frac{1}{n}}\right) \sim - \frac{1}{{12}{n}^{... | Yes |
Theorem 5.23 (Jordan Curve Theorem \( {}^{2} \) ). If \( \gamma \) is a simple closed path in \( \mathbb{C} \), then | We shall not prove the above theorem. It is a deep result. In all of our applications, it will be obvious that our Jordan curves have the above properties. | No |
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. Let \( v \in V, v \notin B \) . Then \( B \cup \{ v\} \) is not linearly independent, by the maximality of \( B \) ; therefore, there exist \( {c}_{0},\ldots ,{c}_{t} \in k \) and (distinct) \( {b}_{1},\ldots ,{b}_{t} \in B \) such that
\[
{c}_{0}v + {c}_{1}{b}_{1} + \cdots + {c}_{t}{b}_{t} = 0,
\]
\( {}^{3} \... | Yes |
A scalar product satisfies the Cauchy-Schwarz inequality | The polynomial t ↦ q(tx + y) = q(x)t^2 + 2b(x, y)t + q(y) takes nonnegative values for t ∈ ℝ . Hence its discriminant 4(b(x, y)^2 - q(x)q(y)) is nonpositive. When the latter vanishes, the polynomial has a real root t0, which implies that t0x + y = 0 . The Cauchy-Schwarz inequality implies immediately q(x + y) ≤ (√q(x) ... | Yes |
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 \( {T}_{e}\left( {G/H}\right) \) . On \( G/H \), the left translation \( {L}_{h} \) is induced by conjugation \( {\mathbf{c}}_{h} \) on \( G \) . By Proposition 7.4 and the hypothesis, we have\\
\[
\text{d... | Yes |
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 \( \mathcal{M} \) . | Consider the language \( {\mathcal{L}}^{ * } \) where we have constant symbols for all elements of \( M \) and a new constant symbol \( c \) . Let \( T = {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) \cup \{ c > m \) : \( m \in M\} \), and for \( a \in M \smallsetminus \mathbb{N} \) let \( {p}_{a} \) b... | Yes |
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 \( K \) . | This is Lemma 5.1.5 and Corollary 5.1.6. The fact that \( {q}^{\prime } \) is nondegenerate follows from Proposition 5.1.2. | No |
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 \( \chi \neq {\chi }_{0} \) . | Null | No |
Let \( \left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) be a probability space, and let \( {\left( {X}_{n}\right) }_{n \in \mathbb{N}} \subseteq {\mathrm{L}}^{1}\left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) be a sequence of independent and identically distributed real random variables. Then \( \mathop{\lim }\limit... | Since the \( {X}_{j} \) are identically distributed, \( v \mathrel{\text{:=}} {\mathrm{P}}_{{X}_{j}} \) (the distribution of \( {X}_{j} \) ) is a Borel probability measure on \( \mathbb{R} \), independent of \( j \), and \( \mathrm{E}\left( {X}_{1}\right) = {\int }_{\mathbb{R}}t\mathrm{\;d}v\left( t\right) \) is the co... | Yes |
As an illustration of Theorem 9.1.6 we consider a Goppa code \\[\\left. \\Gamma \\left( L, g\\left( z\\right) \\right) = {C}_{\\Omega }\\left( {D}_{L},{G}_{0} - {P}_{\\infty }\\right) \\right| \\limits_{{\\mathbb{F}}_{q}} \\] (notation as in Definition 2.3.10 and Proposition 2.3.11). Let \\( {g}_{1}\\left( z\\right) \\... | Null | No |
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