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Theorem 2.5. \( \deg \left( {\mathbb{Q}\left( {\zeta }_{n}\right) /\mathbb{Q}}\right) = \phi \left( n\right) \) and \( \operatorname{Gal}\left( {\mathbb{Q}\left( {\zeta }_{n}\right) /\mathbb{Q}}\right) \simeq {\left( \mathbb{Z}/n\mathbb{Z}\right) }^{ \times } \), with \( a{\;\operatorname{mod}\;n} \) corresponding to t... | Proof. Since \( \mathbb{Q}\left( {\zeta }_{m}\right) \) is normal over \( \mathbb{Q} \), Proposition 2.4 implies that if \( \left( {m, n}\right) = 1 \) then \( \deg \left( {\mathbb{Q}\left( {\zeta }_{mn}\right) /\mathbb{Q}}\right) = \deg \left( {\mathbb{Q}\left( {\zeta }_{m}\right) /\mathbb{Q}}\right) \cdot \deg \left(... | No |
There exists a constant \( C = C\left( n\right) < \infty \) such that for all \( j \geq 1 \) and for all \( f \) in \( {L}^{1}\left( {\mathbf{R}}^{n}\right) \) we have | Let \( {K}^{\left( j\right) } = {\left( {\varphi }_{j}\right) }^{ \vee } * {d\sigma } = {\Phi }_{{2}^{-j}} * {d\sigma } \), where \( \Phi \) is a Schwartz function. Setting\n\[
{\left( {K}^{\left( j\right) }\right) }_{t}\left( x\right) = {t}^{-n}{K}^{\left( j\right) }\left( {{t}^{-1}x}\right)\n\]we have that\n\[
{\math... | Yes |
If \( \left( {p - 1}\right) \nmid i \), show that \( {\left| {B}_{i}/i\right| }_{p} \leq 1 \) . | Null | No |
Suppose \( X \) is a paracompact Hausdorff space. If \( \mathcal{U} = {\left( {U}_{\alpha }\right) }_{\alpha \in A} \) is an indexed open cover of \( X \), then \( \mathcal{U} \) admits a locally finite open refinement \( \mathcal{V} = {\left( {V}_{\alpha }\right) }_{\alpha \in A} \) indexed by the same set, such that ... | Proof. By Lemma 4.80, each \( x \in X \) has a neighborhood \( {Y}_{x} \) such that \( {\bar{Y}}_{x} \subseteq {U}_{\alpha } \) for some \( \alpha \in A \) . The open cover \( \left\{ {{Y}_{x} : x \in X}\right\} \) has a locally finite open refinement. Let us index this refinement by some set \( B \), and denote it by ... | Yes |
We have \( {T}^{2}f = m{f}^{ - } \) . | Part (i) is proved as for the finite field case. For (ii), if \( y \) is not prime to \( m \), then \( {T\chi }\left( y\right) = 0 \) by Theorem 1.1. If \( y \) is prime to \( m \) then we can make the usual change of variables to get the right answer. Part (iii) is then proved as in the finite field case. | No |
Let \( {M}^{ \bullet } \) be an exact complex in \( \mathrm{C}\left( \mathrm{A}\right) \) . Then the complex \( \mathcal{F}\left( {M}^{ \bullet }\right) \), obtained by applying \( \mathcal{F} \) to the objects and morphisms of \( {M}^{ \bullet } \), is a zero-object in \( \mathrm{D} \) . | The zero-morphism: \( {M}^{ \bullet } \rightarrow \) \( {M}^{ \bullet } \) is a quasi-isomorphism; hence it is mapped to an invertible morphism by \( \mathcal{F} \) :
\[
\mathcal{F}\left( {M}^{ \bullet }\right) \underset{{\operatorname{id}}_{\mathcal{F}\left( {M}^{ \bullet }\right) }}{\underbrace{\xrightarrow[]{\mathc... | No |
Theorem 10.10. Let \( D \) be a proper subdomain of \( \widehat{\mathbb{C}} \) . Let \( A \) be a subset of \( D \) that has no limit point in \( D \), and let \( v \) be a function mapping \( A \) to \( {\mathbb{Z}}_{ > 0} \) . Then there exists a function \( f \in \mathbf{H}\left( D\right) \) with \( {v}_{z}\left( f\... | Proof. To begin, we make the following observations:
1. \( A \) is either finite or countable.
2. Without loss of generality, we may assume that \( \infty \in D - A \) and that \( A \) is nonempty.
3. If \( A \) is finite, let \( A = \left\{ {{z}_{1},\ldots ,{z}_{n}}\right\} \) . Set \( {v}_{j} = v\left( {z}_{j}\rig... | Yes |
If \( \left| \mathcal{K}\right| \) is a power of a prime for some nonidentity conjugacy class \( \mathcal{K} \) of \( G \) , then \( G \) is not a non-abelian simple group. | Suppose to the contrary that \( G \) is a non-abelian simple group and let \( \left| \mathcal{K}\right| = {p}^{c} \) . Let \( g \in \mathcal{K} \) . If \( c = 0 \) then \( g \in Z\left( G\right) \), contrary to a non-abelian simple group having a trivial center. As above, let \( {\chi }_{1},\ldots ,{\chi }_{r} \) be al... | Yes |
Theorem 10.7.6. Let \( f\left( s\right) \) be an entire function of order at most equal to \( k \in {\mathbb{Z}}_{ \geq 0} \) . For all \( s \in \mathbb{C} \) we have the absolutely convergent product | f\left( s\right) = {s}^{r}{e}^{{P}_{k}\left( s\right) }\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\right) \exp \left( {\mathop{\sum }\limits_{{1 \leq j \leq k}}\frac{{\left( s/\rho \right) }^{j}}{j}}\right) , | No |
Let \( K \) be a pointed convex cone that decomposes into the direct sum (13.7). If \( x \in {K}_{i} \) is a sum \( x = {x}_{1} + \cdots + {x}_{k} \) of elements \( {x}_{j} \in K \) , then each \( {x}_{j} \in {K}_{i} \) . | We have \( 0 = {\Pi }_{{\widehat{E}}_{i}}x = {\Pi }_{{\widehat{E}}_{i}}{x}_{1} + \cdots + {\Pi }_{{\widehat{E}}_{i}}{x}_{k} \) . Each term \( {\widehat{x}}_{j} \mathrel{\text{:=}} {\Pi }_{{\widehat{E}}_{i}}{x}_{j} \) belongs to \( {\widehat{K}}_{i} \subseteq K \), so that \( {\widehat{x}}_{j} \in K \) and \( - {\wideha... | Yes |
Let \( \mathbf{f} \) be a class function on \( {S}_{k} \) . Write \( \mathbf{f} = \mathop{\sum }\limits_{\lambda }{c}_{\lambda }{\mathbf{s}}_{\lambda } \), where the sum is over the partitions of \( k \) . Then | The \( {\mathbf{s}}_{\lambda } \) are orthonormal by Schur orthogonality, so \( {\left| \mathbf{f}\right| }^{2} = \sum {\left| {c}_{\lambda }\right| }^{2} \) . By Theorem 36.2, \( {\mathrm{{Ch}}}^{\left( n\right) }\left( {\mathbf{s}}_{\lambda }\right) \) are distinct irreducible characters when \( \lambda \) runs throu... | Yes |
Theorem 9.6.2 Let \( F \) be a maximal non-elementary Fuchsian subgroup of \( {\Gamma }_{d} \) . Then \( F \) is conjugate in \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) to a Fuchsian group commensurable with \( {F}_{D} \) . | Theorem 9.6.2 completes the proof. | No |
Theorem 5.9. Let \( X \) be Stein, \( {\left( {U}_{\imath },{f}_{\imath }\right) }_{\imath \in I} \) a Cousin II distribution on \( X \) , \( h \in {Z}^{1}\left( {\mathfrak{U},{\mathcal{O}}^{ \star }}\right) \) the corresponding cocycle. Then \( {\left( {U}_{t},{f}_{t}\right) }_{t \in I} \) is solvable if and only if \... | Theorem B and Theorem 4.9. | No |
Corollary 7.4 Let \( f \) be a flow and \( K \) a cut. If \( \operatorname{val}\left( f\right) = \operatorname{cap}\left( K\right) \), then \( f \) is a maximum flow and \( K \) is a minimum cut. | Let \( {f}^{ * } \) be a maximum flow and \( {K}^{ * } \) a minimum cut. By Theorem 7.3,\[
\operatorname{val}\left( f\right) \leq \operatorname{val}\left( {f}^{ * }\right) \leq \operatorname{cap}\left( {K}^{ * }\right) \leq \operatorname{cap}\left( K\right)
\]But, by hypothesis, \( \operatorname{val}\left( f\right) = \... | Yes |
Suppose \( B, C \in {}_{R}\mathbf{M} \) . Then \(\text{ SP-dim }\left( {B \oplus C}\right) = \max \{ \text{ SP-dim }B,\text{ SP-dim }C\} \). | SP-dim \( \left( {B \oplus C}\right) = \max \{ \) SP-dim \( \left( {B \oplus 0}\right) , \) SP-dim \( \left( {B \oplus 0, B \oplus C}\right) \) by Proposition 5.2. But any module between \( B \oplus 0 \) and \( B \oplus C \) corresponds to a submodule of \( C \approx B \oplus C/B \oplus 0 \) by the fundamental isomorph... | Yes |
Corollary 10.2. Under the hypotheses of the Seifert-Van Kampen theorem, the homomorphism \( \Phi \) descends to an isomorphism from the amalgamated free product \( {\pi }_{1}\left( {U, p}\right) { * }_{{\pi }_{1}\left( {U \cap V, p}\right) }{\pi }_{1}\left( {V, p}\right) \) to \( {\pi }_{1}\left( {X, p}\right) \) . | Null | No |
Theorem 16.11 (Stokes’s Theorem). Let \( M \) be an oriented smooth \( n \) -manifold with boundary, and let \( \omega \) be a compactly supported smooth \( \left( {n - 1}\right) \) -form on \( M \) . Then | We begin with a very special case: suppose \( M \) is the upper half-space \( {\mathbb{H}}^{n} \) itself. Then because \( \omega \) has compact support, there is a number \( R > 0 \) such that supp \( \omega \) is contained in the rectangle \( A = \left\lbrack {-R, R}\right\rbrack \times \cdots \times \left\lbrack {-R,... | Yes |
Show that the set \({K}_{f}\left( X\right) = \{ L \in K\left( X\right) : L\text{ is finite }\}\) is an \( {F}_{\sigma } \) set. | Null | No |
Let the rod be the interval \( \left( {0,2}\right) \), and the end points are kept each at a constant temperature, but these are different at the two ends. To be specific, say that \( u\left( {0, t}\right) = 2 \) and \( u\left( {2, t}\right) = 5 \) . Let us take the initial temperature to be given by \( f\left( x\right... | The unique solution is easily found to be \( \varphi \left( x\right) = \frac{3}{2}x + 2 \) . Substituting this into the initial condition of the original problem, we have
\[
1 - {x}^{2} = u\left( {x,0}\right) = v\left( {x,0}\right) + \varphi \left( x\right) = v\left( {x,0}\right) + \frac{3}{2}x + 2.
\]
We collect all... | Yes |
If \( E \) is a finitepotent subspace of \( {\operatorname{End}}_{k}\left( V\right) \), then \( \operatorname{tr} : E \rightarrow k \) is \( k \) -linear. | Take \( y, x \in E \) and for any nonnegative integer \( n \), put
\[
{V}_{n} \mathrel{\text{:=}} \mathop{\sum }\limits_{w}w\left( V\right)
\]
where the sum is taken over all words \( w \) of length \( n \) in \( x \) and \( y \) . If \( {w}_{0} \) is any initial segment of \( w \), then \( w\left( V\right) \subseteq... | No |
Let \( X \) be a standard Borel space, \( Y \) Polish, and \( A \subseteq X \times Y \) analytic with \( {\pi }_{X}\left( A\right) \) uncountable. Suppose that for every \( x \in {\pi }_{X}\left( A\right) \) , the section \( {A}_{x} \) is perfect. Then there is a \( C \subseteq {\pi }_{X}\left( A\right) \) homeomorphic... | Fix a compatible complete metric on \( Y \) and a countable base \( \left( {U}_{n}\right) \) for the topology of \( Y \) . For each \( s \in {2}^{ < \mathbb{N}} \), we define a map \( {n}_{s}\left( x\right) : {\pi }_{X}\left( A\right) \rightarrow \mathbb{N} \) satifying the following conditions.
(i) \( x \rightarrow {... | No |
Let \( A, B \) be sets, and suppose that \( \operatorname{card}\left( A\right) \leqq \operatorname{card}\left( B\right) \), and \( \operatorname{card}\left( B\right) \leqq \operatorname{card}\left( A\right) \) . Then | Then trivially, \( h \) is injective. We must prove that \( h \) is surjective. Let \( b \in B \) . If, when we try to lift back \( b \) by a succession of maps \[ \cdots \circ {f}^{-1} \circ {g}^{-1} \circ {f}^{-1} \circ {g}^{-1} \circ {f}^{-1}\left( b\right) \] we can lift back indefinitely, or if we get stopped in \... | Yes |
Theorem 17.21 (Hurewicz Isomorphism Theorem). Let \( X \) be a simply connected path-connected CW complex. Then the first nontrivial homotopy and homology occur in the same dimension and are equal, i.e., given a positive integer \( n \geq 2 \), if \( {\pi }_{q}\left( X\right) = 0 \) for \( 1 \leq q < n \), then \( {H}_... | Proof. To start the induction, consider the case \( n = 2 \) . The \( {E}^{2} \) term of the homology spectral sequence of the path fibration is
Thus
\[
{H}_{2}\left( X\right) = {H}_{1}\left( {\Omega X}\right) \;... | Yes |
Theorem 8.7. In a field extension, all transcendence bases have the same number of elements. | Theorem 8.7 is similar to the statement that all bases of a vector space have the same number of elements, and is proved in much the same way. First we establish an exchange property.
Lemma 8.8. Let \( B \) and \( C \) be transcendence bases of a field extension \( E \) of \( K \) . For every \( \beta \in B \) there e... | Yes |
Theorem 7. Let \( p \) and \( {p}^{\prime } \) be PL paths in \( \mathrm{{CP}}\left( {U,{P}_{0}}\right) \), where \( U \) is open in \( {\mathbf{R}}^{3} \) and \( {P}_{0} \in U \) . If \( p \cong {p}^{\prime } \), then there is a PL mapping \( f : {\left\lbrack 0,1\right\rbrack }^{2} \rightarrow U \), under which \( p ... | Null | No |
If \( {H}_{\phi } \) and \( {H}_{\psi } \) are Hankel operators and \( U \) is the unilateral shift, then \( {H}_{\phi }{H}_{\psi } - {U}^{ * }{H}_{\phi }{H}_{\psi }U = \left( {P\breve{\phi }}\right) \otimes \left( {P\bar{\psi }}\right) \) | Note that \( {H}_{\phi }{H}_{\psi } - {U}^{ * }{H}_{\phi }{H}_{\psi }U = {H}_{\phi }{H}_{\psi } - {H}_{\phi }U{U}^{ * }{H}_{\psi }\;\text{ (by Theorem 4.1.7) } \) \[= {H}_{\phi }\left( {I - U{U}^{ * }}\right) {H}_{\psi } \] Recall that \( I - U{U}^{ * } \) is the projection of \( {\widetilde{\mathbf{H}}}^{2} \) onto th... | Yes |
Corollary 14. (Eisenstein’s Criterion for \( \mathbb{Z}\left\lbrack x\right\rbrack \) ) Let \( p \) be a prime in \( \mathbb{Z} \) and let \( f\left( x\right) = {x}^{n} + {a}_{n - 1}{x}^{n - 1} + \cdots + {a}_{1}x + {a}_{0} \in \mathbb{Z}\left\lbrack x\right\rbrack, n \geq 1 \) . Suppose \( p \) divides \( {a}_{i} \) f... | This is simply a restatement of Proposition 13 in the case of the prime ideal \( \left( p\right) \) in \( \mathbb{Z} \) together with Corollary 6. | Yes |
Let \( \left( {\widetilde{P},\widetilde{H}}\right) \) be a general Hartogs figure in \( {\mathbb{C}}^{n} \) , \( f \) holomorphic in \( \widetilde{H} \) . Then there is exactly one holomorphic function \( F \) on \( \widetilde{P} \) with \( F \mid \widetilde{H} = f \) . | Let \( \left( {\widetilde{P},\widetilde{H}}\right) = \left( {g\left( P\right), g\left( H\right) }\right), g : P \rightarrow {\mathbb{C}}^{n} \) be biholomorphic. Then \( f \circ g \) is holomorphic in \( H \) and by Theorem 5.5 of Chapter I there is exactly one holomorphic function \( {F}^{ \star } \) on \( P \) with \... | Yes |
Corollary 2.5.4. Let \( f, g : \left( {{B}^{n},{S}^{n - 1}}\right) \rightarrow \left( {{e}_{\alpha }^{n},{\overset{ \bullet }{e}}_{\alpha }^{n}}\right) \) be characteristic maps inducing \( {f}^{\prime },{g}^{\prime } : {B}^{n}/{S}^{n - 1} \rightarrow {e}_{\alpha }^{n}/{e}_{\alpha }^{n} \) . For \( t \in \left( {0,1}\r... | Proof. Since \( {B}^{n}/{S}^{n - 1} \) and \( {e}_{\alpha }^{n}/{\mathbf{e}}_{\alpha }^{n} \) are homeomorphic to \( {S}^{n} \), the result follows from 2.5.3. In fact, \( f\left| { \simeq g}\right| \) as maps from \( {S}_{t} \) to \( {\overset{ \circ }{e}}_{\alpha }^{n} - \{ z\} \). | Yes |
Show that \( \mathbb{Q}\left( {\zeta }_{m}\right) \) is normal over \( \mathbb{Q} \) . | Solution. \( {\zeta }_{m} \) is a root of the \( m \) th cyclotomic polynomial, which we have shown to be irreducible. Thus, the conjugate fields are \( \mathbb{Q}\left( {\zeta }_{m}^{j}\right) \) where \( \left( {j, m}\right) = 1 \) and these are identical with \( \mathbb{Q}\left( {\zeta }_{m}\right) \) . | No |
Show that if P has the integer decomposition property, then P is an integral polyhedron. | 1. If P has the integer decomposition property, then P is an integral polyhedron. | No |
Theorem 2. Let \( \mathrm{K} \) be a field that is complete under a discrete valuation and has perfect residue field \( \overline{\mathbf{K}} \) . Let \( \mathfrak{g} \) be the Galois group of the algebraic closure of \( \overline{\mathbf{K}} \) over \( \overline{\mathbf{K}} \), and let \( \mathbf{X}\left( \mathrm{g}\r... | As we have seen, the splitting of this sequence comes from choosing a uniformizer of \( \mathrm{K} \) . | No |
Let \( \operatorname{Conj}\left( G\right) \) be the set of conjugacy classes in \( G \) . For each \( C \in \operatorname{Conj}\left( G\right) \) let \( {\varphi }_{C} \) be the characteristic function of \( C \) . Then the set \( {\left\{ {\varphi }_{C}\right\} }_{C \in \operatorname{Conj}\left( G\right) } \) is a bas... | f = \mathop{\sum }\limits_{{C \in \operatorname{Conj}\left( G\right) }}f\left( C\right) {\varphi }_{C} | Yes |
Let \( {w}_{2},\ldots ,{w}_{n} \) be smooth boundary functions with \( \left| {w}_{j}\right| < \delta \) for all \( j \) and such that \( {w}_{2} \) is schlicht, i.e., its analytic extension is one-one in \( \left| \zeta \right| \leq 1 \) . Put \( A = {A}_{w} \) . Suppose \( {x}^{ * } \in {H}_{1},\left| {x}^{ * }\right... | Since \( A{x}^{ * } = {x}^{ * },{x}^{ * } = - T\left\{ {h\left( {{X}^{ * }, w}\right) }\right\} \), and so \( {x}^{ * } + {ih}\left( {{x}^{ * }, w}\right) \) is a boundary function by (2). Let \( \psi \) be the analytic extension of \( {x}^{ * } + {ih}\left( {{x}^{ * }, w}\right) \) to \( \left| \zeta \right| < 1 \) . ... | Yes |
Theorem 4.2.1 (Duality). Each multiplicity space \( {E}^{\lambda } \) is an irreducible \( {\mathcal{R}}^{G} \) module. Furthermore, if \( \lambda ,\mu \in \operatorname{Spec}\left( \rho \right) \) and \( {E}^{\lambda } \cong {E}^{\mu } \) as an \( {\mathcal{R}}^{G} \) module, then \( \lambda = \mu \) . | We first prove that the action of \( {\mathcal{R}}^{G} \) on \( {\operatorname{Hom}}_{G}\left( {{F}^{\lambda }, L}\right) \) is irreducible. Let \( T \in {\operatorname{Hom}}_{G}\left( {{F}^{\lambda }, L}\right) \) be nonzero. Given another nonzero element \( S \in {\operatorname{Hom}}_{G}\left( {{F}^{\lambda }, L}\rig... | Yes |
Theorem 27.2. Let \( G \) be the complexification of the compact connected Lie group \( K \) . With \( B, N, I \) as above, \( \left( {B, N, I}\right) \) is a Tits’ system in \( G \) . | The proof of this is identical to Theorem 27.1. The analog of Lemma 27.2 is true, and the proof is the same except that we use Lemma 27.3 instead of Lemma 27.1. All other details are the same. | No |
Proposition 24. Fix a monomial ordering on \( R = F\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) and let \( I \) be a nonzero ideal in \( R \). (1) If \( {g}_{1},\ldots ,{g}_{m} \) are any elements of \( I \) such that \( {LT}\left( I\right) = \left( {{LT}\left( {g}_{1}\right) ,\ldots ,{LT}\left( {g}_{m}\right... | Suppose \( {g}_{1},\ldots ,{g}_{m} \in I \) with \( {LT}\left( I\right) = \left( {{LT}\left( {g}_{1}\right) ,\ldots ,{LT}\left( {g}_{m}\right) }\right) \). We need to see that \( {g}_{1},\ldots ,{g}_{m} \) generate the ideal \( I \). If \( f \in I \), use general polynomial division to write \( f = \mathop{\sum }\limit... | Yes |
The function \( f\left( z\right) = \frac{1}{1 - z} \) is analytic on \( \mathbb{D} \) but is not in \( {\mathbf{H}}^{2} \) . | Since \( \frac{1}{1 - z} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{z}^{n} \), the coefficients of \( f \) are not square-summable. | Yes |
Show that conditions (1) and (2) in Definition 1.4 may be replaced by the single condition that the map \( \left( {{g}_{1},{g}_{2}}\right) \rightarrow {g}_{1}{g}_{2}^{-1} \) is smooth. | Null | No |
Theorem 2. Let \( \Omega \) be a bounded open set in \( {\mathbb{R}}^{2} \) whose boundary \( \partial \Omega \) consists of a finite number of simple closed curves. Assume the existence of a number \( r > 0 \) such that at each point of \( \partial \Omega \) there are two circles of radius \( r \) tangent to \( \parti... | Null | No |
If all colength n ideals were radical, then the Hilbert scheme Hn would be easy to describe, as follows. Every unordered list of n distinct points in ℂ2 corresponds to a set of n! points in (ℂ2)n, or alternatively, to a single point in the nth symmetric product Snℂ2, defined as the quotient (ℂ2)n/Sn by the symmetric gr... | Null | No |
Theorem 18.2. (a) The value of \( {\Phi }_{0}\left( {d, p}\right) \) equals | Proof. (a) By the definition (2) we have\n\[
{\Phi }_{0}\left( {d, p}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}\left( \begin{matrix} p - d + i - 1 \\ i \end{matrix}\right) + \mathop{\sum }\limits_{{i = 0}}^{m}\left( \begin{matrix} p - d + i - 1 \\ i \end{matrix}\right) .
\]\nThe desired expression then follows easil... | No |
If \( \deg f < \deg {gh} \) and \( \gcd \left( {g, h}\right) = 1 \), then there exist unique polynomials \( a, b \) such that \( \deg a < \deg g,\deg b < \deg h \), and \( f/\left( {gh}\right) = \) \( \left( {a/g}\right) + \left( {b/h}\right) \) . If \( \gcd \left( {f,{gh}}\right) = 1 \), then \( \gcd \left( {a, g}\rig... | Proof. Since \( \gcd \left( {g, h}\right) = 1 \), there exist polynomials \( s, t \) such that \( {gs} + {ht} = f \) . Polynomial division yields \( t = {gp} + a, s = {hq} + b \), where \( \deg a < \deg g \) and \( \deg b < \deg h \) . Then \( f = {gh}\left( {p + q}\right) + {ah} + {bg} \), with \( \deg \left( {{ah} + ... | Yes |
The hull complex of a lattice module is locally finite. | We claim that the vertex \( \mathbf{0} \in L \) is incident to only finitely many edges of hull \( \left( {M}_{L}\right) \). This claim implies the theorem because (i) the lattice \( L \) acts transitively on the vertices of hull \( \left( {M}_{L}\right) \), so it suffices to consider the vertex \( \mathbf{0} \), and (... | Yes |
Determine the minimizers and the minimum value of the function f(x1,...,xn) = 1/2∑j=1nxj^2 - ∑1≤i<j≤nln|x_i - x_j|. | The solution to this differential equation is the Hermite polynomial of order n, Hn(x) = n!∑0≤k≤[n/2] ((-1)^k(2x)^(n-2k))/(k!(n-2k)!). Therefore, the solutions xj are the roots of the Hermite polynomial Hn(x). The discriminant of Hn is given by ∏i<j(xi - xj)^2 = 2^(-(n(n-1)/2)) ∏j=1n j^j. The above formula for Hn gives... | Yes |
Theorem 3.1.9 (Bertrand’s postulate) For \( n \) sufficiently large, there is a prime between \( n \) and \( {2n} \) . | Proof: (S. Ramanujan) Observe that if \({a}_{0} \geq {a}_{1} \geq {a}_{2} \geq \cdots\) is a decreasing sequence of real numbers tending to zero, then \({a}_{0} - {a}_{1} \leq \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}{a}_{n} \leq {a}_{0} - {a}_{1} + {a}_{2}\) This is the starting point of Ramanuj... | No |
Let \( L/K \) be a finite extension of algebraic number fields. Suppose that \( {\mathcal{O}}_{L} = {\mathcal{O}}_{K}\left\lbrack \alpha \right\rbrack \) for some \( \alpha \in L \) . If \( f\left( x\right) \) is the minimal polynomial of \( \alpha \) over \( {\mathcal{O}}_{K} \), show that \( {\mathcal{D}}_{L/K} = \le... | This result is identical to Exercises 5.6.6 and 5.6.7. More generally, one can show the following. For each \( \theta \in {\mathcal{O}}_{L} \) which generates \( L \) over \( K \), let \( f\left( x\right) \) be its minimal polynomial over \( {\mathcal{O}}_{K} \) . Define \( {\delta }_{L/K}\left( \theta \right) = {f}^{\... | No |
Corollary 3.111. The closed unit ball of the dual \( {X}^{ * } \) of a WCG space is sequentially compact for the weak* topology in the sense that every sequence of \( {B}_{{X}^{ * }} \) has a weak* convergent subsequence. | Given a bounded sequence \( \left( {x}_{n}^{ * }\right) \) of \( {X}^{ * } \), let \( F\left( n\right) \mathrel{\text{:=}} \left\{ {{x}_{p} : p \geq n}\right\} \) for \( n \in \mathbb{N} \) and let \( {x}^{ * } \) be a weak* cluster point of \( \left( {x}_{n}^{ * }\right) \), i.e., a point in \( {\operatorname{cl}}^{ *... | Yes |
Let \( R \) be a ring with derivation \( D \) and \( S = R\left\lbrack t\right\rbrack ,{tr} = {rt} + D\left( r\right) \) for all \( r \) in \( R \), the corresponding Ore extension. Several authors, among them [4] and [6], have noted the inequality \(\text{l.gl.dim}S \leq \text{l.gl.dim}R + 1\). | In this chapter we show that if \( R \) is left and right noetherian and of finite left global dimension, a necessary condition for equality to hold is the existence of a left \( S \) -module \( M \) that is finitely generated as an \( R \) -module and with \( {\operatorname{ldim}}_{R}M = \operatorname{l.gl.dim}R \) . ... | No |
Theorem 4.3. Let \( K \) be an imaginary abelian extension of \( \mathbf{Q} \) . Then the norm map | We have to use class field theory, which gives the more general statement:\n\nLemma. Let \( K \) be an abelian extension of a number field \( F \) . Let \( H \) be the Hilbert class field of \( F \) (maximal abelian unramified extension of \( F \) ). If \( K \cap H = F \) then the norm map \( {N}_{K/F} : {C}_{K} \right... | Yes |
Show that two metrics \( d \) and \( \rho \) on a set \( X \) are equivalent if and only if for every sequence \( \left( {x}_{n}\right) \) in \( X \) and every \( x \in X \) , \( d\left( {{x}_{n}, x}\right) \rightarrow 0 \Leftrightarrow \rho \left( {{x}_{n}, x}\right) \rightarrow 0 \). | Null | No |
Theorem 19.1. Let \( \gamma \) be an oriented simple closed curve in \( {\mathbb{C}}^{2} \) with a finite number of self-intersections. Then a necessary and sufficient condition that there exists a bounded analytic variety \( \sum \) in \( {\mathbb{C}}^{2} \) with \( {b\sum } = \pm \gamma \) is that \( \gamma \) satisf... | The complete proof of this theorem involves a considerable number of technical details, and we shall refer the reader to the paper of Harvey and Lawson [HarL2] for these. Here we shall present a sketch that we hope conveys the essential aspects of the construction. | No |
Proposition 17.9. Let \( \left( {X,\mathbf{S},\mu }\right) \) be a measure space, let \( \mathcal{E} \) be a normed space, and let \( p \) be a positive real number. Then the collection \( {\mathcal{L}}_{p}\left( {X;\mathcal{E}}\right) = {\mathcal{L}}_{p}\left( {X,\mathbf{S},\mu ;\mathcal{E}}\right) \) of all those mea... | If \( \Phi \) and \( \Psi \) are arbitrary \( \mathcal{E} \) -valued mappings, then \( {N}_{\Phi + \Psi } \leq {N}_{\Phi } + {N}_{\Psi } \) by the triangle inequality in \( \mathcal{E} \), and since \( {\mathcal{L}}_{p}\left( X\right) \) is a linear space, it follows at once that \( {\mathcal{L}}_{p}\left( {X;\mathcal{... | No |
Theorem 2.16. Let \( V \subset {\mathbb{C}}^{n} \) be any nonempty irreducible variety of dimension \( d \), and let \( {V}_{{d}_{1}} \supsetneq \ldots \supsetneq {V}_{{d}_{2}} \) be any strict chain of nonempty irreducible subvarieties of \( V \) . This chain can be extended (or refined) to a maximal chain of irreduci... | Proof of Theorem 2.16. It suffices to show that if \( {W}_{1} \subset {W}_{2} \) are irreducible nonempty subvarieties of \( V \) of dimension \( {d}_{1} \) and \( {d}_{2} \) respectively, then there is a strict chain of irreducible varieties from \( {W}_{2} \) to \( {W}_{1} \) of length \( {d}_{2} - {d}_{1} \) ; or wh... | Yes |
Theorem 2.2.15. If \( D \) is a fundamental discriminant, the Kronecker symbol \( \left( \frac{D}{n}\right) \) defines a real primitive character modulo \( m = \left| D\right| \) . Conversely, if \( \chi \) is a real primitive character modulo \( m \) then \( D = \chi \left( {-1}\right) m \) is a fundamental discrimina... | The definition of the Kronecker symbol and Theorem 2.2.9 show that \( \left( \frac{D}{n}\right) \) is a character modulo \( \left| D\right| \) . To show that it is primitive, it is sufficient to show that for any prime \( p \mid D \) it cannot be defined modulo \( D/p \) . Assume first that \( p \neq 2 \), and let \( a... | Yes |
The edges of \( {K}_{10} \) cannot be partitioned into three copies of the Petersen graph. | Let \( P \) and \( Q \) be two copies of Petersen’s graph on the same vertex set and with no edges in common. Let \( R \) be the subgraph of \( {K}_{10} \) formed by the edges not in \( P \) or \( Q \) . We show that \( R \) is bipartite.
Let \( {U}_{P} \) be the eigenspace of \( A\left( P\right) \) with eigenvalue 1,... | Yes |
Proposition 1. Aut \( \left( K\right) \) is a group under composition and \( \operatorname{Aut}\left( {K/F}\right) \) is a subgroup. | It is clear that \( \operatorname{Aut}\left( K\right) \) is a group. If \( \sigma \) and \( \tau \) are automorphisms of \( K \) which fix \( F \) then also \( {\sigma \tau } \) and \( {\sigma }^{-1} \) are the identity on \( F \), which shows that \( \operatorname{Aut}\left( {K/F}\right) \) is a subgroup. | Yes |
If \( 1 \leq p < \infty \) and if \( f\left( x\right) \in {L}^{p} \), then \( {\begin{Vmatrix}{P}_{y} * f - f\end{Vmatrix}}_{p} \rightarrow 0\;\left( {y \rightarrow 0}\right) \). | Let \( f \in {L}^{p},1 \leq p \leq \infty \) . When \( p = \infty \) we suppose in addition that \( f \) is uniformly continuous. Then
\[
{P}_{y} * f\left( x\right) - f\left( x\right) = \int {P}_{y}\left( t\right) \left( {f\left( {x - t}\right) - f\left( x\right) }\right) {dt}.
\]
Minkowski's inequality gives
\[
{\b... | No |
Proposition 31.1. The group \( \widetilde{T} \) is connected and is a maximal torus of \( \widetilde{G} \) . | Let \( \Pi \subset \widetilde{G} \) be the kernel of \( p \) . The connected component \( {\widetilde{T}}^{ \circ } \) of the identity in \( \widetilde{T} \) is a torus of the same dimension as \( T \), so it is a maximal torus in \( \widetilde{G} \) . Its image in \( G \) is isomorphic to \( {\widetilde{T}}^{ \circ }/... | Yes |
Let \( \mathbf{H} = {L}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \) and let \( A \) be the operator on \( \mathbf{H} \) defined \( {by} \)
\[
\left( {A\psi }\right) \left( x\right) = {x\psi }\left( x\right)
\]
Then this operator is bounded and self-adjoint, and its spectrum is given by
\[
\sigma \left( A\ri... | Proof. It is apparent that \( \parallel {A\psi }\parallel \leq \parallel \psi \parallel \) and that \( \langle \phi ,{A\psi }\rangle = \langle {A\phi },\psi \rangle \) for all \( \phi ,\psi \in \mathbf{H} \), so that \( A \) is bounded and self-adjoint. Given \( \lambda \in \left( {0,1}\right) \), consider the function... | Yes |
Corollary 2.28 A graph \( G \) is edge reconstructible if either \( m > \frac{1}{2}\left( \begin{array}{l} n \\ 2 \end{array}\right) \) or \( {2}^{m - 1} > n! \) | Null | No |
If the \( ON \) system \( \left\{ {\varphi }_{j}\right\} _{j = 1}^{\infty } \) is complete in \( V \), then \( \langle u, v\rangle = \mathop{\sum }\limits_{{j = 1}}^{\infty }\left\langle {u,{\varphi }_{j}}\right\rangle \overline{\left\langle v,{\varphi }_{j}\right\rangle } \) for all \( u, v \in V \) . | Let \( {P}_{n}\left( u\right) \) be the projection of \( u \) on to the subspace spanned by the \( n \) first \( \varphi \) ’s:\n\n\( {P}_{n}\left( u\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\left\langle {u,{\varphi }_{j}}\right\rangle {\varphi }_{j} \)\n\nBy Theorem 5.2 we have\n\n\( \left\langle {{P}_{n}\left( u\r... | Yes |
The coordinate frame \( \left( {\partial /\partial {x}^{i}}\right) \) is a global orthonormal frame for \( {\mathbb{R}}^{n} \) with the Euclidean metric. | Null | No |
Proposition 8.11 Let \( S \) be multigraded by a torsion-free abelian group \( A \) . All associated primes of multigraded S-modules are multigraded. | This is [Eis95, Exercise 3.5]. The proof, based on that of the corresponding \( \mathbb{Z} \) -graded statement in [Eis95, Section 3.5], is essentially presented in the aforementioned exercise from [Eis95]. It works because torsion-free grading groups \( A \cong {\mathbb{Z}}^{d} \) can be totally ordered, for instance ... | No |
Proposition 4.39. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( X \) is infrabarreled. Suppose \( T : X \rightarrow Y \) is a linear transformation for which \( f \circ T \in {X}^{ * } \) whenever \( f \in {Y}^{ * } \) Then \( T \) is continuous. | First of all, if \( f \in {Y}^{ * } \), then \( x \in \{ f{\} }_{ \circ } \Leftrightarrow \left| {f\left( x\right) }\right| \leq 1 \), so
\[
{T}^{-1}\left( {\{ f{\} }_{ \circ }}\right) = {T}^{-1}\left( {{f}^{-1}\left( {\{ z : \left| z\right| \leq 1\} }\right) }\right)
\]
\[
= {\left( f \circ T\right) }^{-1}\left( {\{... | Yes |
Proposition 10.4. There are canonical isomorphisms \( {\Lambda }_{k}\left( {V}^{ * }\right) \cong {\Lambda }_{k}{\left( V\right) }^{ * } \cong {A}_{k}\left( V\right) \) . | The second isomorphism is the one induced from Exercise 26. For the first one, there is a unique bilinear map \( b : {\Lambda }_{k}\left( {V}^{ * }\right) \times {\Lambda }_{k}\left( V\right) \rightarrow \mathbb{R} \) which is given on decomposable elements by\\
\[
b\left( {{v}_{1}^{ * } \land \cdots \land {v}_{k}^{ * ... | No |
If \( f \) is in \( {BV}\left( {\mathbf{T}}^{1}\right) \), then \(\left| {\widehat{f}\left( m\right) }\right| \leq \frac{\operatorname{Var}\left( f\right) }{{2\pi }\left| m\right| }\) whenever \( m \neq 0 \) . | Integration by parts gives \(\widehat{f}\left( m\right) = {\int }_{{\mathbf{T}}^{1}}f\left( x\right) {e}^{-{2\pi imx}}{dx} = {\int }_{{\mathbf{T}}^{1}}\frac{{e}^{-{2\pi imx}}}{-{2\pi im}}{df},\) where the boundary terms vanish because of periodicity. The conclusion follows from the fact that the norm of the measure \( ... | Yes |
Show that the theory \( \mathcal{D} \) (Exercise 3.21) of dense linear order admits \( \Pi \) -reduction of quantifiers with \( \Pi = \left\{ {\left( {{x}_{i} = {x}_{j}}\right) ,\left( {{x}_{i} < {x}_{j}}\right) }\right. \) \( i, j \in \mathbf{N}\} \) . Hence show that \( \mathcal{D} \) is decidable and complete. | Null | No |
Suppose that the \( k \) -algebra \( \mathcal{O} \) is a complete discrete \( k \) -valuation ring with residue class map \( \eta : \mathcal{O} \rightarrow F \) . Assume further that \( F \) is a finite separable extension of \( k \) . Given any local parameter \( t \), there is a unique isometric isomorphism \( \wideh... | Let \( \eta : \mathcal{O} \rightarrow F \) be the residue class map, and let \( \mu : F \rightarrow \mathcal{O} \) be the unique splitting given by (1.2.12). Define \( \widehat{\mu } : F\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \rightarrow \mathcal{O} \) via
\[
\mu \left( {\mathop{\sum }\limits_{i}{a}_{i}... | Yes |
Theorem 5.17. 1. If \( G \) is a regular open subset of \( \mathbf{F} \) then \( {G}^{\Delta } \) is regular open. | Proof. 1. Let \( {G}_{1} = {\left( {G}^{\Delta }\right) }^{-0} \) . Then
\[
G = {G}^{\Delta * } \subseteq {\left( {G}^{\Delta }\right) }^{-0 * } = {G}_{1}^{ * }.
\]
If \( {G}_{2} \) is regular open and \( G \cap {G}_{2} = 0 \) then
\[
{\left( {G}^{\Delta } \cap {G}_{2}^{\Delta }\right) }^{ * } \subseteq {G}^{\Delta ... | Yes |
Let \( f\left( x\right) \in k\left\lbrack x\right\rbrack, k \) a field. Suppose that \( \deg f\left( x\right) = n \) . Then \( f \) has at most \( n \) distinct roots. | The proof goes by induction on \( n \) . For \( n = 1 \) the assertion is trivial. Assume that the lemma is true for polynomials of degree \( n - 1 \) . If \( f\left( x\right) \) has no roots in \( k \), we are done. If \( \alpha \) is a root, \( f\left( x\right) = q\left( x\right) \left( {x - \alpha }\right) + r \), w... | Yes |
Proposition 8.1.2. Let t be an element of F such that v_P(i)(t) = 1 for i = 1,…, n. Then the following hold: | Proof. (a) Since t is a prime element of P := P_i, the P-adic power series of η = dt/t with respect to t is
\[
\eta = \frac{1}{t}{dt}
\]
Hence v_P(η) = - 1 and res_P(η) = 1.
(b) Follows immediately from (a) and Proposition 2.2.10. | No |
For each \( r > 0 \), the image under \( q \) of the rectangle \( \left\lbrack {0,1}\right\rbrack \times \left\lbrack {-r, r}\right\rbrack \) is a Möbius band \( {M}_{r} \). Because \( q \) restricts to a smooth covering map from \( \mathbb{R} \times \left\lbrack {-r, r}\right\rbrack \) to \( {M}_{r} \), the same argum... | Null | No |
Proposition 21.7 (Orbits of Proper Actions). Suppose \( \theta \) is a proper smooth action of a Lie group \( G \) on a smooth manifold \( M \) . For any point \( p \in M \), the orbit map \( {\theta }^{\left( p\right) } : G \rightarrow M \) is a proper map, and thus the orbit \( G \cdot p = {\theta }^{\left( p\right) ... | If \( K \subseteq M \) is compact, then \( {\left( {\theta }^{\left( p\right) }\right) }^{-1}\left( K\right) \) is closed in \( G \) by continuity, and since it is contained in \( {G}_{K\cup \{ p\} } \), it is compact by Proposition 21.5. Therefore, \( {\theta }^{\left( p\right) } \) is a proper map, which implies that... | Yes |
Corollary 25.5. Let \( \mathbf{K} \) be a class of \( {\mathcal{L}}^{\prime } \) -structures and let \( \mathcal{L} \) be a reduct of \( {\mathcal{L}}^{\prime } \) . If \( \mathbf{K} \) can be characterized by first-order sentences, then \( \mathbf{S}\left( {\mathbf{K} \upharpoonright \mathcal{L}}\right) \) is a univer... | Proof. By 25.3 and 18.29, \( \mathbf{S}\left( {\mathbf{K} \upharpoonright \mathcal{L}}\right) \) is closed under \( \mathbf{S} \) and \( \mathbf{{Up}} \) . Hence 25.5 is immediate from 25.2. | Yes |
Let \( f \) be a function in \( {C}^{\infty }\left( \Omega \right) \). The multiplication by \( f \) defines a continuous operator \( {M}_{f} : \varphi \mapsto {f\varphi } \) on \( {C}_{0}^{\infty }\left( \Omega \right) \). Since | \[
{\int }_{\Omega }\left( {f\varphi }\right) {\psi dx} = {\int }_{\Omega }\varphi \left( {f\psi }\right) {dx},\varphi ,\psi \in {C}_{0}^{\infty }\left( \Omega \right) ,
\]
we define \( {M}_{f}u = {fu} \) by
\[
\left( {fu}\right) \left( \varphi \right) = u\left( {f\varphi }\right), u \in {\mathcal{D}}^{\prime }\left( ... | No |
If \( \pi \) is a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \), show that \( N\left( \pi \right) \) is a rational prime or the square of a rational prime. | Let \( N\left( \pi \right) = n > 1 \) . Then \( {\pi \pi } = n \) . Now \( n \) is a product of rational prime divisors. Since \( \pi \) is prime, \( \pi \mid p \) for some rational prime \( p \) . Write \( p = {\pi \gamma } \) . Then \( N\left( p\right) = N\left( \pi \right) N\left( \gamma \right) = {p}^{2} \) . Thus,... | Yes |
Proposition 8.2.1. For any \( P = \left( {x, y}\right) \in E \) set \(\phi \left( P\right) = \left( {\widehat{x},\widehat{y}}\right) = \left( {\frac{{y}^{2}}{{x}^{2}},\frac{y\left( {{x}^{2} - b}\right) }{{x}^{2}}}\right)\) for \( P \) not equal to \( T \) or \( \mathcal{O} \), and set \( \phi \left( T\right) = \phi \le... | The proof consists in a series of explicit verifications, where in each case we must separate the points \( \mathcal{O} \) and \( T \) from the other points. It is done with utmost detail in [Sil-Tat], to which we refer. We will simply show that \( \phi \) maps \( E \) to \( \widehat{E} \), and that it maps three colli... | No |
Let \( f \) and \( g \) be convex functions on a normed space \( X \). If \( f \) and \( g \) are finite at \( \bar{x} \) and if \( f \) is continuous at some point of \( \operatorname{dom}f \cap \operatorname{dom}g \), then | The inclusion \( \partial f\left( \bar{x}\right) + \partial g\left( \bar{x}\right) \subset \partial \left( {f + g}\right) \left( \bar{x}\right) \) is an immediate consequence of the definition of the subdifferential. Let us prove the reverse inclusion under the assumptions of the theorem. Let \( {\bar{x}}^{ * } \in \pa... | Yes |
Proposition 1.6. For any tape description \( F \) and any \( e \in \mathbb{Z},\langle \left( {F,0, e}\right) \) , \( \left( {F,1, e + 1}\right) \rangle \) is a computation of \( {T}_{\text{left }} \) . | Thus \( {T}_{\text{left }} \) moves the tape one square to the left and then stops. | No |
For \( 1 \leq i \leq p - 2 \) we have \( \begin{Vmatrix}{f}_{i}\end{Vmatrix} = \left( {p - 1}\right) /2 \) . | For \( 1 \leq t \leq p - 1 \) and \( 1 \leq i \leq p - 1 \) we note that\\ \[ \lfloor {ti}/p\rfloor + \lfloor \left( {p - t}\right) i/p\rfloor = \lfloor {ti}/p\rfloor + i - \lceil {ti}/p\rceil = i - 1 \] \\ since \( p \nmid {ti} \) . It follows that\\ \[ \mathop{\sum }\limits_{{1 \leq t \leq p - 1}}\lfloor {ti}/p\rfloo... | Yes |
Corollary 1. \( A,\mathfrak{m} \) and \( E \) being as in Theorem 7, suppose that \( {G}_{\mathfrak{m}}\left( E\right) \) is a finite \( {G}_{\mathfrak{m}}\left( A\right) \) -module. Then \( E \) is a finite \( A \) -module. | We apply Theorem 7 to the case \( F = E \). | No |
Let \( \left( {d, p}\right) = 1,{q}_{n} = {qd}{p}^{n} \), and \( {h}_{n}^{ - } = {h}^{ - }\left( {\mathbb{Q}\left( {\zeta }_{{q}_{n}}\right) }\right) \) . We assume \( d ≢ 2\left( {\;\operatorname{mod}\;4}\right) \) . Then | Let \( {q}_{n}^{\prime } = \operatorname{lcm}\left( {{q}_{n},2}\right) \) . Theorem 4.17 implies that
\[
{h}_{0}^{ - } = {q}_{0}^{\prime }Q\mathop{\prod }\limits_{\substack{{\theta \neq 1} \\ {{f}_{\theta } \mid {q}_{0}} \\ {\theta \text{ even }} }}\left( {-\frac{1}{2}{B}_{1,\theta {\omega }^{-1}}}\right)
\]
and
\[
{... | Yes |
Theorem 16.8.3 Let \( {Y}_{1} \) and \( {Y}_{2} \) be signed graphs that are related by a Whitney flip. Then their rank polynomials are equal. | The graphs \( {Y}_{1} \) and \( {Y}_{2} \) have the same edge set, and it is clear that a set \( S \subseteq E\left( {Y}_{1}\right) \) is independent in \( M\left( {Y}_{1}\right) \) if and only if it is independent in \( M\left( {Y}_{2}\right) \) . Therefore, the two graphs have the same cycle matroid. | Yes |
Let \( \Omega \) be a smoothly bounded, finite-type domain in \( {\mathbb{C}}^{2} \) . Equip Aut \( \left( \Omega \right) \) with the \( {C}^{k} \) topology, some integer \( k \geq 0 \) . Assume that \( \Omega \) has compact automorphism group in the \( {C}^{k} \) topology. Then there is an \( \epsilon > 0 \) so that i... | The proof is just the same as that for the last theorem. The main point is to have a uniform bound for derivatives of automorphisms (Proposition 5.2.5), so that the smooth-to-the-boundary invariant metric can be constructed. | No |
The complete graph \( {K}_{n} \) is not the edge-disjoint union of \( n - 2 \) complete bipartite graphs. | Suppose that, contrary to the assertion, \( {K}_{n} \) is the edge-disjoint union of complete bipartite graphs \( {G}_{1},\ldots ,{G}_{n - 2} \) . For each \( i \), let \( {H}_{i} \) be obtained from \( {G}_{i} \) by adding to it isolated vertices so that \( V\left( {H}_{i}\right) = V\left( {K}_{n}\right) \) . Note tha... | Yes |
For \( n \geq 2 \) even we have \(\mathop{\sum }\limits_{{k \geq 1}}\frac{\cos \left( {2\pi kx}\right) }{{k}^{n}} = \frac{{\left( -1\right) }^{n/2 + 1}}{2}\frac{{\left( 2\pi \right) }^{n}{B}_{n}\left( {\{ x\} }\right) }{n!}\) and for \( n \geq 1 \) odd we have \(\mathop{\sum }\limits_{{k \geq 1}}\frac{\sin \left( {2\pi... | Proof. (1) and (2). Since \( {B}_{n}\left( 1\right) = {B}_{n}\left( 0\right) \) for \( n \neq 1 \), the function \( {B}_{n}\left( {\{ x\} }\right) \) is piecewise \( {C}^{\infty } \) and continuous for \( n \geq 2 \), with simple discontinuities at the integers if \( n = 1 \) . If \( n \geq 2 \) we thus have \({B}_{n}\... | Yes |
Theorem 5.15 (Dynkin’s Formula). Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \) . For \( X, Y \in \mathfrak{g} \) in a sufficiently small neighborhood of 0,\n\n{e}^{X}{e}^{Y} = {e}^{Z}\n\nwhere \( Z \) is given by the formula\n\nZ = \sum \frac{{\left( -1\right) }^{n + 1}}{n}\frac{1}{\left( {{i}... | Proof. The approach of this proof follows [34]. Using Theorem 4.6, choose a neighborhood \( {U}_{0} \) of 0 in \( \mathfrak{g} \) on which exp is a local diffeomorphism and where \( \ln \) is well defined on \( \exp U \) . Let \( U \subseteq {U}_{0} \) be an open ball about of 0 in \( \mathfrak{g} \), so that \( {\left... | Yes |
When \( P \) is properly supported in \( \Omega \), there is a unique symbol \( p\left( {x,\xi }\right) \in {S}^{\infty }\left( \Omega \right) \) such that \( P = \operatorname{Op}\left( {p\left( {x,\xi }\right) }\right) \), namely, the one determined by (7.29). | Null | No |
Theorem 3.6.6. For all \( r \in \mathbb{Z} \) we have \[ \frac{\tau \left( {\omega }^{-r}\right) }{{\left( \zeta - 1\right) }^{s\left( r\right) }} \equiv - \frac{1}{t\left( r\right) }\left( {\;\operatorname{mod}\;\mathfrak{P}}\right) . \] | Proof. By periodicity we may assume that \( 0 \leq r < q - 1 \) . We prove the theorem by induction on \( s\left( r\right) = {s}_{p}\left( r\right) \) . If \( s\left( r\right) = 0 \) we have \( r = 0 \), hence \( t\left( 0\right) = 1 \) and \( \tau \left( {\omega }^{0}\right) = \tau \left( \varepsilon \right) = - 1 \) ... | Yes |
Let \( K \) be a geometric function field with \( \omega \in {\Omega }_{K} \) and \( P \in {\mathbb{P}}_{K} \) . Then, | Corollary 2.5.8. Let \( K \) be a geometric function field with \( \omega \in {\Omega }_{K} \) and \( P \in {\mathbb{P}}_{K} \) . Then,\n\n\[
{\nu }_{P}\left( {x\omega }\right) = {\nu }_{P}\left( x\right) + {\nu }_{P}\left( \omega \right)
\]\n\n\[
{\nu }_{P}\left( {\omega + {\omega }^{\prime }}\right) \geq \min \{ {\nu... | Yes |
If \( 0 \leftarrow M \leftarrow {N}_{0} \leftarrow {N}_{1} \leftarrow \cdots \leftarrow {N}_{r} \leftarrow 0 \) is an exact sequence of finitely generated positively multigraded modules, then the Hilbert series of \( M \) equals the alternating sum of those for \( {N}_{0},\ldots ,{N}_{r} \) : | For each \( \mathbf{a} \in A \), the degree \( \mathbf{a} \) piece of the given exact sequence of modules is an exact sequence of finite-dimensional vector spaces over \( \mathbb{k} \) . The rank-nullity theorem from linear algebra says that the alternating sum of the dimensions of these vector spaces equals zero. | Yes |
Theorem 8.1.5. ( \( \mathbf{{Sp}}\left( n\right) \rightarrow \mathbf{{Sp}}\left( {n - 1}\right) \) Branching Law) The multiplicity \( m\left( {\lambda ,\mu }\right) \) is nonzero if and only if\n\[
{\lambda }_{j} \geq {\mu }_{j} \geq {\lambda }_{j + 2}\;\text{ for }j = 1,\ldots, n - 1\n\]
(8.4)\n(here \( {\lambda }_{n ... | Null | No |
Let \( X \) be a path-connected space. The inclusion of basepoint preserving maps into the set of all maps induces a bijection \({\pi }_{q}\left( {X, x}\right) /{\pi }_{1}\left( {X, x}\right) \overset{ \sim }{ \rightarrow }\left\lbrack {{S}^{q}, X}\right\rbrack \) | Let \( h : {\pi }_{q}\left( {X, x}\right) \rightarrow \left\lbrack {{S}^{q}, X}\right\rbrack \) be induced by the inclusion of base point preserving maps into the set of all maps. If \( \left\lbrack \alpha \right\rbrack \in {\pi }_{q}\left( {X, x}\right) \) and \( \left\lbrack \gamma \right\rbrack \in {\pi }_{1}\left( ... | Yes |
Proposition 8. Let \( \mathrm{A} \) be a ring that is Hausdorff and complete for the topology defined by a decreasing sequence \( {\mathfrak{a}}_{1} \supset {\mathfrak{a}}_{2} \supset \cdots \) of ideals such that \( {\mathfrak{a}}_{n} \cdot {\mathfrak{a}}_{m} \subset \) \( {\mathfrak{a}}_{n + m} \) . Assume that the r... | Let \( \lambda \in \overline{\mathrm{K}} \) ; for all \( n \geq 0 \), denote by \( {\mathrm{L}}_{n} \) the inverse image of \( {\lambda }^{{p}^{-n}} \) in \( \mathrm{A} \), and by \( {\mathrm{U}}_{n} \) the set of all \( {x}^{{p}^{n}}, x \in {\mathrm{L}}_{n} \) ; the \( {\mathrm{U}}_{n} \) are contained in the residue ... | Yes |
Theorem 11.5.9. The map \( \Phi : U \times \mathfrak{u} \rightarrow G \) defined by \( \Phi \left( {u, X}\right) = u\exp \left( {\mathrm{i}X}\right) \), for \( u \in U \) and \( X \in \mathfrak{u} \), is a diffeomorphism onto \( G \) . In particular, \( U \) is connected. | Proof. By Lemma 11.5.8 we may assume that \( G \subset \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) and \( \tau \left( g\right) = {\left( {g}^{ * }\right) }^{-1} \) . If \( g \in G \) then \( {g}^{ * }g \) is positive definite. Since \( {\left( {g}^{ * }g\right) }^{m} \in G \) for all \( m \in \mathbb{Z} \), Lemma 11.5... | Yes |
Theorem 7. If \( M \) is orientable, then \(\chi \left( M\right) = 2 - {p}^{1}\left( M\right)\). If \( M \) is not orientable, then \(\chi \left( M\right) = 1 - {p}^{1}\left( M\right)\). | Proof. Let \( h \) be the number of handles in \( M \), and let \( m \) be the number of cross-caps, with \( 0 \leq m \leq 2 \). For \( m = 0 \), we have \(\chi \left( M\right) = 2 - {2h} = 2 - {p}^{1}\left( M\right)\). For \( m = 1,\chi \left( M\right) = 2 - \left( {{2h} + 1}\right),{p}^{1}\left( M\right) = {2h}\), an... | Yes |
Let \( Y \) be a regular CW complex structure on an n-manifold. Every cell of \( Y \) is a face of an \( n \) -cell of \( Y \) . Every \( \left( {n - 1}\right) \) -cell of \( Y \) is a face of at most two \( n \) -cells of \( Y \) . An \( \left( {n - 1}\right) \) -cell, \( e \), of \( Y \) is a face of exactly one \( n... | We saw in Sect. 5.1 that every cell of \( Y \) has dimension \( \leq n \) . If some cell were not a face of an \( n \) -cell, there would be \( k < n \) and a \( k \) -cell \( \widetilde{e} \) of \( Y \) which is not a face of any higher-dimensional cell of \( Y \), implying \( \overset{ \circ }{e} \) open in \( Y \), ... | No |
Corollary 4.4.7. The product of two Hankel operators is 0 if and only if one of them is 0 . | If the product of two Hankel operators is the Toeplitz operator 0 , the previous corollary implies that at least one of the Hankel operators is zero. | No |
Let \( {a}^{ * } \in A \) and \( \widehat{A} \mathrel{\text{:=}} A \smallsetminus \left\{ {a}^{ * }\right\} \), and assume that \( G \) has no proper \( A \rightarrow B \) wave. Then \( {a}^{ * } \) is linkable for \( \left( {G,\widehat{A}, B}\right) \) . | Nu | No |
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