Q stringlengths 26 3.6k | A stringlengths 1 9.94k | Result stringclasses 3
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Let \( {\psi }_{1} \) denote the scaled Airy function in (15.26), let \( {\widetilde{\psi }}_{1} \) denote the same function with the Airy function replaced by the right-hand side of (15.33), and let \( {\psi }_{2} \) denote the oscillatory WKB function in (15.27). If \( x - a \) is positive and of order \( {\hslash }^... | Proof of Lemma 15.9. We consider only the estimates for the derivatives of the functions involved. The analysis of the functions themselves is similar (but easier) and is left as an exercise to the reader (Exercise 11).\n\nWe begin by considering \( {\psi }_{1}^{\prime } - {\widetilde{\psi }}_{1}^{\prime } \) . With a ... | No |
Theorem 8. Let \( 0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0 \) be a short exact sequence of \( R \) -modules. Then there is a long exact sequence of abelian groups | Proof: Take a simultaneous projective resolution of the short exact sequence as in Proposition 7 and take homomorphisms into \( D \) . To obtain the cohomology groups \( {\operatorname{Ext}}_{R}^{n} \) from the resulting diagram, as noted in the discussion preceding Proposition 3 we replace the lowest nonzero row in th... | Yes |
Singular homology satisfies Axiom 4. | We have defined \( {C}_{n}\left( {X, A}\right) = {C}_{n}\left( X\right) /{C}_{n}\left( A\right) \) . Thus for every \( n \), we have a short exact sequence\n\[
0 \rightarrow {C}_{n}\left( A\right) \rightarrow {C}_{n}\left( X\right) \rightarrow {C}_{n}\left( {X, A}\right) \rightarrow 0.
\]\nIn other words, we have a sho... | Yes |
Every affine morphism of the multiplicative group \( {\mathbb{G}}_{m} \) has the form \( \psi \left( z\right) = a{z}^{d} \) for some nonzero \( a \) and some \( d \in \mathbb{Z} \) . More generally, for any commutative group \( G \), any \( a \in G \), and any \( d \in \mathbb{Z} \) there is an affine morphism \( \psi ... | Notice that it is easy to compute the iterates of this map,\n\[
{\psi }^{n}\left( z\right) = {a}^{1 + d + \cdots + {d}^{n - 1}}{z}^{{d}^{n}}.
\] | No |
Theorem 3.4.23 (The isomorphism theorem for measure spaces) If \( \mu \) is a continuous probability on a standard Borel space \( X \), then there is a Borel isomorphism \( h : X \rightarrow I \) such that for every Borel subset \( B \) of \( I,\lambda \left( B\right) = \) \( \mu \left( {{h}^{-1}\left( B\right) }\right... | Proof. By the Borel isomorphism theorem (3.3.13), we can assume that \( X = I \) . Let \( F : I \rightarrow I \) be the distribution function of \( \mu \) . So, \( F \) is a continuous, nondecreasing map with \( F\left( 0\right) = 0 \) and \( F\left( 1\right) = 1 \) . Let
\[
N = \left\{ {y \in I : {F}^{-1}\left( {\{ y... | Yes |
Theorem 2.4.1 For any \( y > 0 \), \(\mathop{\sum }\limits_{{n \leq x}}f\left( n\right) = \mathop{\sum }\limits_{{d \leq y}}g\left( d\right) H\left( \frac{x}{d}\right) + \mathop{\sum }\limits_{{d \leq \frac{x}{y}}}h\left( d\right) G\left( \frac{x}{d}\right) - G\left( y\right) H\left( \frac{x}{y}\right) \). | Proof. We have \(\mathop{\sum }\limits_{{n \leq x}}f\left( n\right) = \mathop{\sum }\limits_{{{de} \leq x}}g\left( d\right) h\left( e\right)\) \(\mathop{\sum }\limits_{{{de} \leq x}}g\left( d\right) h\left( e\right) + \mathop{\sum }\limits_{{{de} \leq x}}g\left( d\right) h\left( e\right)\) \(\mathop{\sum }\limits_{{d \... | Yes |
Proposition 22.8 Take the symplectic potential \( \theta = {p}_{j}d{x}_{j} \) . Then the position, momentum, and holomorphic subspaces may be computed as follows. The position subspace consists of smooth functions \( \psi \) on \( {\mathbb{R}}^{2n} \) of the form | Proof. Since \( \theta \left( {\partial /\partial {p}_{j}}\right) = 0 \), we have \( {\nabla }_{\partial /\partial {p}_{j}} = \partial /\partial {p}_{j} \), so that functions that are covariantly constant in the \( \mathbf{p} \) -directions are actually constant in the \( \mathbf{p} \) -directions. Meanwhile, \( \theta... | No |
Lemma 5.9.1 An independent set \( C \) in a Moore graph of diameter two and valency seven contains at most 15 vertices. If \( \left| C\right| = {15} \), then every vertex not in \( C \) has exactly three neighbours in \( C \) . | Let \( X \) be a Moore graph of diameter two and valency seven. Suppose that \( C \) is an independent set in \( X \) with \( c \) vertices in it. Without loss of generality we may assume that the vertices are labelled so that the vertices \( \{ 1,\ldots ,{50} - c\} \) are the ones not in \( C \) . If \( i \) is a vert... | Yes |
A connected s-arc transitive graph with girth \( {2s} - 2 \) is distance-transitive with diameter \( s - 1 \) . | Let \( X \) satisfy the hypotheses of the lemma and let \( \left( {u,{u}^{\prime }}\right) \) and \( \left( {v,{v}^{\prime }}\right) \) be pairs of vertices at distance \( i \) . Since \( X \) has diameter \( s - 1 \) by Lemma 4.1.4, we see that \( i \leq s - 1 \) . The two pairs of vertices are joined by paths of leng... | No |
Proposition 3.2. In a ring extension \( E \) of \( R \), the elements of \( E \) that are integral over \( R \) constitute a subring of \( E \) . | Null | No |
If \( n \equiv 1\left( {\;\operatorname{mod}\;p}\right) \), prove that \( {n}^{{p}^{m}} \equiv 1\left( {\;\operatorname{mod}\;{p}^{m + 1}}\right) \) . | Null | No |
Proposition 8.5.2 Let \( v \in {\widetilde{S}}_{n}^{B} \) . Then, | Proof. This follows immediately from equations (8.68), (8.69), and (8.71) and the fact that \( v\left( 0\right) = 0 \) and \( v\left( {n + 2}\right) = N - v\left( {n - 1}\right) \) . \( ▱ \) | Yes |
Lemma 11.28. For \( t \in T \), let \( {\operatorname{Ad}}_{{t}^{-1}}^{\prime } \) denote the restriction of \( {\operatorname{Ad}}_{{t}^{-1}} \) to \( \mathfrak{f} \) . | The operator \( {\mathrm{{Ad}}}_{{t}^{-1}}^{\prime } - I \) is invertible provided that the restriction of \( {\mathrm{{Ad}}}_{{t}^{-1}} \) to \( f \) does not have an eigenvalue of 1 . Suppose, then, that \( {\operatorname{Ad}}_{{t}^{-1}}\left( X\right) = X \) for some \( X \in \mathfrak{f} \) . Then for every integer... | Yes |
Let \( A \) and \( B \) be Hopf algebras over a field. Then \({P}_{A \otimes B} = {P}_{A} + {P}_{B}\) | Let \( u = \sum a \otimes b \) be an element of \( {P}_{A \otimes B} \) . Then we have \(\sum \delta \left( a\right) \otimes \delta \left( b\right) = {s}_{23}\left( {\delta \left( u\right) }\right) = \sum a \otimes {1}_{A} \otimes b \otimes {1}_{B} + \sum {1}_{A} \otimes a \otimes {1}_{B} \otimes b\) where \( {s}_{23} ... | Yes |
For all \( n,{P}_{n}^{\text{corr }} \) and \( {P}_{n + 1}^{\text{cut }} \) are linearly isomorphic. | Let \( f : {\mathbb{R}}^{n \times n} \rightarrow {\mathbb{R}}^{{E}_{n + 1}} \) be the linear function that maps each \( x \in {\mathbb{R}}^{n \times n} \) to the element \( y \in {\mathbb{R}}^{{E}_{n + 1}} \) defined by
\[
{y}_{ij} = \left\{ \begin{array}{ll} {x}_{ii} & \text{ if }1 \leq i \leq n, j = n + 1 \\ {x}_{ii... | Yes |
Let \( X \) be a manifold with a spray or covariant derivative \( D \) . There exists a unique vector bundle morphism (over \( \pi \) )
\[
K : {TTX} \rightarrow {TX}
\]
such that for all vector fields \( \xi ,\zeta \) on \( X \), we have
(8)
\[
{D}_{\xi }\zeta = K \circ {T\zeta } \circ \xi ,\;\text{ in other words,... | In a chart \( U \), we let the local representation
\[
{K}_{U,\left( {x, v}\right) } : \mathbf{E} \times \mathbf{E} \rightarrow \mathbf{E}
\]
be given by
\( \left( {8}_{U}\right) \)
\[
{K}_{U,\left( {x, v}\right) }\left( {z, w}\right) = w - {B}_{U}\left( {x;v, z}\right) ,
\]
so \( K = {S}_{2} \) satisfies the requ... | No |
Proposition 7.2. Let \( E \) be free, finite dimensional over \( R \) . Then we have an algebra-isomorphism \( T\left( {L\left( E\right) }\right) = T\left( {{\operatorname{End}}_{R}\left( E\right) }\right) \rightarrow {LT}\left( E\right) = {\bigoplus }_{r = 0}^{\infty }{\operatorname{End}}_{R}\left( {{T}^{r}\left( E\ri... | Proof. By Proposition 2.5, we have a linear isomorphism in each dimension, and it is clear that the map preserves multiplication. | No |
The spheres \( {S}_{r}\left( a\right) \left( {r > 0}\right) \) are both open and closed. | The spheres are closed in all metric spaces, since the distance function \( x \mapsto d\left( {x, a}\right) \) is continuous. A sphere of positive radius is open in an ultrametric space by part \( \left( c\right) \) of the previous lemma. | No |
Theorem 7.11. Let \( S \) be a subgroup of finite index in \( G \) . Let \( F \) be an \( S \) -module, and \( E \) a \( G \) -module (over the commutative ring \( R \) ). Then there is an isomorphism | The \( G \) -module \( {\operatorname{ind}}_{S}^{G}\left( F\right) \) contains \( F \) as a summand, because it is the direct sum \( \bigoplus {\lambda }_{i}F \) with left coset representatives \( {\lambda }_{i} \) as in Theorem 7.3. Hence we have a natural \( S \) -isomorphism\\
\[
f : {\operatorname{res}}_{S}\left( E... | Yes |
Let \( \Omega \) be a bounded domain with \( {C}^{2} \) boundary and \( S \) its Szegő kernel. With \( \mathcal{P}\left( {z,\zeta }\right) \) as defined above, and with \( f \in C\left( \bar{\Omega }\right) \) holomorphic on \( \Omega \) , we have
\[
f\left( z\right) = {\int }_{\partial \Omega }\mathcal{P}\left( {z,\z... | See Proposition 1.2.9. | No |
If \( w \in W \) and \( s \in S \smallsetminus S\left( w\right) \) satisfy \( l\left( {sws}\right) < l\left( w\right) + 2 \), then \( s \) commutes with all elements of \( S\left( w\right) \) . | We have \( l\left( {sw}\right) = l\left( w\right) + 1 = l\left( {ws}\right) \) by Lemma 2.15. Therefore, in view of the folding condition (Section 2.3.1), the hypothesis \( l\left( {sws}\right) < l\left( w\right) + 2 \) is equivalent to the equation \( {sw} = {ws} \) . We now show by induction on \( l \mathrel{\text{:=... | Yes |
Theorem 12.24 (FRIEDRICHS). Let \( S \) be densely defined, symmetric and lower bounded in \( H \) . There exists a selfadjoint extension \( T \) with the same lower bound \( m\left( T\right) = m\left( S\right) \), and with \( D\left( T\right) \) contained in the completion of \( D\left( S\right) \) in the norm \( {\le... | Assume first that \( m\left( S\right) = c > 0 \) . The sesquilinear form
\[
{s}_{0}\left( {u, v}\right) = \left( {{Su}, v}\right)
\]
is then a scalar product on \( D\left( S\right) \) (cf. Theorem 12.12), and we denote the completion of \( D\left( S\right) \) with respect to this scalar product by \( V \) . Hereby \(... | No |
If \( \left( {X,\mu }\right) \) is a measure space, if \( f\left( x\right) \) is measurable, and if \( 0 < p < \infty \), then | We may assume \( f \) vanishes except on a set of \( \sigma \) -finite measure, because otherwise both sides of (4.1) are infinite. Then Fubini's theorem shows that both sides of (4.1) equal the product measure of the ordinate set \( \left\{ {\left( {x,\lambda }\right) : 0 < \lambda < {\left| f\left( x\right) \right| }... | Yes |
Proposition 7.1. (1) \( {H}^{0}\left( {G, A}\right) \cong \{ a \in A \mid {xa} = a \) for all \( x \in G\} \) . | Null | No |
Let \( k \geq 2 \) be an even integer. We have \( {B}_{k} \equiv k + 1/2\left( {\;\operatorname{mod}\;{2}^{2 + {v}_{2}\left( k\right) }}\right) \), so in particular \( {B}_{k} \equiv k + 1/2 \) \( \left( {\;\operatorname{mod}\;4}\right) \) and \( {B}_{k} \equiv 1/2\left( {\;\operatorname{mod}\;2}\right) \) . | Null | No |
A (non-empty) closed, convex, locally compact, and line-free set \( A \) in \( X \) has an extreme point. | We may assume that \( A \) is not compact. Then \( {C}_{A} \) is a non-trivial closed cone in \( X \) (closure follows from equation (8.5)). Further \( {C}_{A} \) is itself locally compact since a translate of it lies in \( A \) . Let \( \phi \in {X}^{ * } \) be a strictly positive linear functional and let \( K \) be ... | Yes |
Theorem 10.33. Let \( S \) be defined as above with \( {b}_{j} = 0\left( {j = 1,2,\ldots, m}\right) \) , \( q \in {M}_{\rho ,\text{ loc }}\left( {\mathbb{R}}^{m}\right) \) and \( {q}_{ - } \in {M}_{\rho }\left( {\mathbb{R}}^{m}\right) \) for some \( \rho < 4 \) . Then \( S \) is bounded from below. If the lowest point ... | By Theorem 10.29(a) the operators \( S \) and \( {S}_{0} - {q}_{ - } \) are bounded from below. The lower bound of \( {S}_{0} - {q}_{ - } \) is, at the same time, a lower bound of the operators \( {S}_{n} \) and \( S - {Q}_{n} \) used in steps 2 and 3 . These operators therefore have a common lower bound, so that Theor... | Yes |
Theorem 4.7.3 \( \left| \sum \right| < \infty \) . | Suppose that \( \left| \sum \right| = \infty \) . Since there are only finitely many elements in the root poset of any given depth, we conclude that there are small roots of arbitrarily large depth. For each small root \( \alpha \), we have (by the definition of \( \sum \) ) a saturated chain in the root poset, entirel... | Yes |
Theorem 15.2. Let \( {G}_{i}, i = 1,2 \) be two groups, and let \( G = {G}_{1} \times {G}_{2} \) be their direct product. Then the following sequence is exact:\[
{\bigoplus }_{p + q = n}{H}_{p}\left( {G}_{1}\right) \otimes {H}_{q}\left( {G}_{2}\right) \rightarrow {H}_{n}\left( G\right) \rightarrow {\bigoplus }_{p + q =... | Moreover the sequence splits by an unnatural splitting. | No |
Suppose \( M \) is a smooth \( n \) -manifold and \( D \subseteq {TM} \) is a distribution of rank \( k \). Then \( D \) is smooth if and only if each point \( p \in M \) has a neighborhood \( U \) on which there are smooth 1 -forms \( {\omega }^{1},\ldots ,{\omega }^{n - k} \) such that for each \( q \in U \), | First suppose that there exist such forms \( {\omega }^{1},\ldots ,{\omega }^{n - k} \) in a neighborhood of each point. The assumption (19.1) together with the fact that \( D \) has rank \( k \) implies that the forms \( {\omega }^{1},\ldots ,{\omega }^{n - k} \) are independent on \( U \) for dimensional reasons. By ... | Yes |
Theorem 18.6.2 For any nonzero \( a \in {}^{ * }R\left\lbrack x\right\rbrack \), the following are equivalent. | In general, \( \left| a\right| = {2}^{-o\left( a\right) } \) and \( o\left( a\right) \) is a nonnegative hyperinteger, so \( \left| a\right| \) will be appreciable iff \( o\left( a\right) \) is limited, or equivalently, \( \left| a\right| \) will be infinitesimal iff \( o\left( a\right) \) is unlimited. Thus (1) and (2... | Yes |
Suppose \( p \nmid m \) is prime. Show that \( p \mid {\phi }_{m}\left( a\right) \) for some \( a \in \mathbb{Z} \) if and only if \( p \equiv 1\left( {\;\operatorname{mod}\;m}\right) \). Deduce from Exercise 1.2.5 that there are infinitely many primes congruent to 1 (mod \( m \) ). | Solution. If \( p \mid {\phi }_{m}\left( a\right) \), by the previous exercise the order of \( a\left( {\;\operatorname{mod}\;p}\right) \) is \( m \) so that \( m \mid p - 1 \).
Conversely, if \( p \equiv 1\left( {\;\operatorname{mod}\;m}\right) \), there is an element \( a \) of order \( m\left( {\;\operatorname{mod}... | Yes |
If \( p \) is a covering projection, so is \( {p}^{\prime } \) . | Let \( y \in Y \) and let \( U \) be a neighborhood of \( p\left( y\right) \) in \( B \) which is evenly covered by \( p \) . Then \( {p}^{-1}\left( U\right) = \bigcup \left\{ {{U}_{\alpha } \mid \alpha \in \mathcal{A}}\right\} \) where \( \mathcal{A} \) is an indexing set and \( \left\{ {{U}_{\alpha } \mid \alpha \in ... | No |
Prove that the map \( f\left( {x, y}\right) \mapsto f\left( {t,\tau }\right) \) defines an embedding \( k\left( {x, y}\right) \rightarrow k\left( \left( t\right) \right) \) . Thus, there is a discrete valuation on \( k\left( {x, y}\right) \) with residue field \( k \) . Show that this valuation is not obtained by the c... | Null | No |
Theorem 7.2.10. Let \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \) be a tower over \( {\mathbb{F}}_{q} \). | Proof. (a) Above each place \( P \in \operatorname{Split}\left( {\mathcal{F}/{F}_{0}}\right) \) there are exactly \( \left\lbrack {{F}_{n} : {F}_{0}}\right\rbrack \) places of \( {F}_{n} \), and they are all rational. Hence \( N\left( {F}_{n}\right) \geq \left\lbrack {{F}_{n} : {F}_{0}}\right\rbrack \cdot \left| {\oper... | Yes |
Let \( {\left( {c}_{k}\right) }_{k \geq 0} \) be a sequence of elements of \( \mathbb{Z} \), set \( {a}_{k} = \mathop{\sum }\limits_{{0 \leq m \leq k}}{\left( -1\right) }^{k - m}\left( \begin{matrix} k \\ m \end{matrix}\right) {c}_{m} \), and assume that as \( k \rightarrow \infty \) we have \( {v}_{p}\left( {a}_{k}\ri... | Clear from the above results. | No |
Theorem 3.1 (Divergence Theorem). \[
{\int }_{X}{\mathcal{L}}_{\xi }\Omega = {\int }_{\partial X}\Omega \circ \xi
\] | Suppose that \( \left( {X, g}\right) \) is a Riemannian manifold, assumed oriented for simplicity. We let \( \Omega \) or \( {\operatorname{vol}}_{g} \) be the volume form defined in Chapter XV, §1. Let \( \omega \) be the canonical Riemannian volume form on \( \partial X \) for the metric induced by \( g \) on the bou... | Yes |
Let \( \Psi ,{\Delta }_{j}^{\Psi } \) be as above and \( \gamma > 0 \) . Then there is a constant \( C = \) \( C\left( {n,\gamma ,\Psi }\right) \) such that for all \( f \) in \( {\dot{\Lambda }}_{\gamma } \) we have the estimate | We begin with the proof of (1.4.7). We first consider the case \( 0 < \gamma < 1 \) , which is very simple. Since each \( {\Delta }_{j}^{\Psi } \) is given by convolution with a function with mean value zero, for a function \( f \in {\dot{\Lambda }}_{\gamma } \) and every \( x \in {\mathbf{R}}^{n} \) we write | No |
Corollary 18. If \( A \) is an abelian group then \( A \) is torsion free if and only if \( {\mathrm{{Tor}}}_{1}\left( {A, B}\right) = 0 \) for every abelian group \( B \) (in which case \( A \) is flat as a \( \mathbb{Z} \) -module). | By the proposition, if \( A \) has no elements of finite order then we have \( {\operatorname{Tor}}_{1}\left( {A, B}\right) = {\operatorname{Tor}}_{1}\left( {t\left( A\right), B}\right) = {\operatorname{Tor}}_{1}\left( {0, B}\right) = 0 \) for every abelian group \( B \) . Conversely, if \( {\operatorname{Tor}}_{1}\lef... | Yes |
The algebra \( B\left( {K, S}\right) \) contains the rational functions with poles in \( S \), and is closed under uniform limits in \( K \). | Null | No |
Corollary 25. Let \( R \) be a subring of the commutative ring \( S \) with \( 1 \in R \) . Then the integral closure of \( R \) in \( S \) is integrally closed in \( S \) . | Null | No |
Theorem 3.2. If \( N = p \) is prime \( \geqq 3 \), then for every admissible pair \( \left( {r, s}\right) \) the curve \( F\left( {r, s}\right) \) has genus \( \left( {p - 1}\right) /2 \), and \( K\left( {r, s}\right) = K\left( {1,{s}^{ * }}\right) \) for a uniquely determined integer \( {s}^{ * } \) such that the pai... | The genus can either be computed directly as we did for the Fermat curve, or one can use Theorem 3.1. The number of \( m \) such that \( \left( {\langle {mr}\rangle ,\langle {ms}\rangle }\right) \) is admissible is trivially computed to be \( \left( {p - 1}\right) /2 \), using the remark preceding the theorem. The stat... | Yes |
Corollary 15.2 Let \( A \) be an infinite set of positive integers with \( \gcd \left( A\right) = 1 \). Then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{\log {p}_{A}\left( n\right) }{\log n} = \infty \) | For every sufficiently large integer \( k \) there exists a subset \( {F}_{k} \) of \( A \) of cardinality \( k \) such that \( \gcd \left( {F}_{k}\right) = 1 \) . By Theorem 15.2,\[
{p}_{A}\left( n\right) \geq {p}_{{F}_{k}}\left( n\right) = \frac{{n}^{k - 1}}{\left( {k - 1}\right) !\mathop{\prod }\limits_{{a \in {F}_{... | Yes |
Theorem 9.5. Let the notation be as in Proposition 8.18 and Corollary 8.19. If there exists a prime \( l \equiv 1{\;\operatorname{mod}\;p} \) with \( l < {p}^{2} - p \) such that \({Q}_{i}^{k} ≢ 1{\;\operatorname{mod}\;l}\;\text{ for all }i \in \left\{ {{i}_{1},\ldots ,{i}_{s}}\right\} \), then the second case of Ferma... | Proof. By Corollary \( {8.19}, p \nmid {h}^{ + }\left( {\mathbb{Q}\left( {\zeta }_{p}\right) }\right) \), so Assumption I is satisfied. Suppose that \({x}^{p} + {y}^{p} = {z}^{p},\;p \nmid {xy}, p \mid z, z \neq 0,\) where \( x, y, z \in \mathbb{Z} \) are relatively prime. Let \( l \) be as in the statement of the theo... | No |
Corollary 23. The sequence \( 0 \rightarrow A\overset{\psi }{ \rightarrow }B\overset{\varphi }{ \rightarrow }C \rightarrow 0 \) is exact if and only if \( \psi \) is injective, \( \varphi \) is surjective, and image \( \psi = \ker \varphi \), i.e., \( B \) is an extension of \( C \) by \( A \) . | Null | No |
Theorem 3.22 (Abel’s Limit Theorem). Assume that the power series \( \sum {a}_{n}{z}^{n} \) has finite radius of convergence \( \rho > 0 \) . If \( \sum {a}_{n}{z}_{0}^{n} \) converges for some \( {z}_{0} \) with \( \left| {z}_{0}\right| = \) \( \rho \), then \( f\left( z\right) = \sum {a}_{n}{z}^{n} \) is defined for ... | Proof. By the change of variable \( w = \frac{z}{{z}_{0}} \) we may assume that \( \rho = 1 = {z}_{0} \) (replace \( \left. {{a}_{n}\text{by}{a}_{n}{z}_{0}^{n}}\right) \) . Thus \( \sum {a}_{n} \) converges to \( f\left( 1\right) \) . By changing \( {a}_{0} \) to \( {a}_{0} - f\left( 1\right) \), we may assume that \( ... | Yes |
Let \( K \) be a function field over a perfect ground field \( k \). Then \( K \) is geometric. | Let \( {k}^{\prime } \) be a finite extension of \( k \). Then \( {k}^{\prime }/k \) is separable, so \( {k}^{\prime } = k\left( u\right) \) for some \( u \in {k}^{\prime } \) by (A.0.17). Moreover, \( u \) satisfies an irreducible separable polynomial \( f\left( X\right) \in k\left\lbrack X\right\rbrack \) of degree \... | Yes |
Let \( f \) and \( g \) be two Krasner analytic functions on the set \( \mathcal{D} \) defined above. If \( f \) and \( g \) coincide on some nonempty open subset of \( \mathcal{D} \) then \( f = g \) . | Consider the map from \( \mathcal{D} \) to \( {\mathbb{C}}_{p} \) sending \( x \) to \( t = 1/\left( {x - 1}\right) \) . Since \( \left| {x - 1}\right| \geq 1 \) for \( x \in \mathcal{D} \), this map is a well-defined map from \( \mathcal{D} \) to \( {\mathcal{Z}}_{p} \), and since \( x = 1 + 1/t \) for \( t \neq 0 \),... | Yes |
Proposition 4.11. The firm and the directional subdifferentials are homotone in the sense that for \( f \geq g \) with \( f\left( \bar{x}\right) = g\left( \bar{x}\right) \) finite one has | Proposition 4.11. If \( f \geq g \) with \( f\left( \bar{x}\right) = g\left( \bar{x}\right) \) finite one has \({\partial }_{F}g\left( \bar{x}\right) \subset {\partial }_{F}f\left( \bar{x}\right) ,\;{\partial }_{D}g\left( \bar{x}\right) \subset {\partial }_{D}f\left( \bar{x}\right) .\) | Yes |
Proposition 8.18. Let \( T \) be a Cesàro bounded operator on some Banach space \( E \) such that \( \frac{1}{n}{T}^{n}h \rightarrow 0 \) for each \( h \in E \) . Then for \( f, g \in E \) the following statements are equivalent: | The implication \( \left( \mathrm{v}\right) \Rightarrow \left( \mathrm{i}\right) \) follows from Theorem 8.5 while the implications (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) are trivial. If (iii) holds, then \( g \in \operatorname{fix}\left( T\right) \) by Lemma 8.17. Moreover,
\[
g \in {\operatorname{cl}}_{\... | Yes |
Theorem 4.4.8 (Miljutin's Theorem). Suppose \( K \) is an uncountable compact metric space. Then \( \mathcal{C}\left( K\right) \) is isomorphic to \( \mathcal{C}\left\lbrack {0,1}\right\rbrack \) . | The first step is to show that \( \mathcal{C}\left( {\left\lbrack 0,1\right\rbrack }^{\mathbb{N}}\right) \) is isomorphic to a complemented subspace of \( \mathcal{C}\left( \Delta \right) \) . By Lemma 4.4.7 there is a continuous surjection \( \psi : \Delta \rightarrow \left\lbrack {0,1}\right\rbrack \) , so that we ca... | Yes |
Consider a sequence of fields \( {F}_{0} \subseteq {F}_{1} \subseteq {F}_{2} \subseteq \ldots \) where \( {F}_{0} \) is a function field with the exact constant field \( {\mathbb{F}}_{q} \) and \( \left\lbrack {{F}_{n + 1} : {F}_{n}}\right\rbrack \) \( < \infty \) for all \( n \geq 0 \) . Suppose that for all \( n \) t... | By the Fundamental Equality we have \( \left\lbrack {{F}_{n + 1} : {F}_{n}}\right\rbrack \geq e\left( {{Q}_{n} \mid {P}_{n}}\right) \) and therefore \( {F}_{n} \subsetneqq {F}_{n + 1} \) . If we assume the equality \( e\left( {{Q}_{n} \mid {P}_{n}}\right) = \left\lbrack {{F}_{n + 1} : {F}_{n}}\right\rbrack \), then \( ... | Yes |
Theorem 1. Let \( \rho : \mathrm{G} \rightarrow \mathrm{{GL}}\left( \mathrm{V}\right) \) be a linear representation of \( \mathrm{G} \) in \( \mathrm{V} \) and let \( \mathrm{W} \) be a vector subspace of \( \mathrm{V} \) stable under \( \mathrm{G} \) . Then there exists a complement \( {\mathrm{W}}^{0} \) of \( \mathr... | Let \( {\mathrm{W}}^{\prime } \) be an arbitrary complement of \( \mathrm{W} \) in \( \mathrm{V} \), and let \( p \) be the corresponding projection of \( \mathrm{V} \) onto \( \mathrm{W} \) . Form the average \( {p}^{0} \) of the conjugates of \( p \) by the elements of \( \mathrm{G} \):
\[
{p}^{0} = \frac{1}{g}\math... | Yes |
a) Show that \( X \subseteq {M}^{n} \) is definable in \( \mathcal{M} \) if and only if it is definable in \( {\mathcal{M}}^{ * } \) if and only if it is definable in \( {\mathcal{M}}_{0} \) . | Null | No |
Theorem 6.3 Let \( A \) and \( B \) be \( n \times n \) Hermitian matrices. Let \( 1 \leq i, j, k \leq n \) be indices. If \( i + j = k + 1 \), we have \({\lambda }_{k}\left( {A + B}\right) \geq {\lambda }_{i}\left( A\right) + {\lambda }_{j}\left( B\right)\). If \( i + j = k + n \), we have \({\lambda }_{k}\left( {A + ... | Proof. Once again, any inequality can be deduced from the other ones by means of \( \left( {A, B}\right) \leftrightarrow \left( {-A, - B}\right) \) . Thus it is sufficient to treat the case where \( i + j = k + n \) . From (6.4), we know that there exists an \( \left( {n - k + 1}\right) \) -dimensional subspace \( H \)... | Yes |
Theorem 3.7. The continued fraction map \( T\left( x\right) = \left\{ \frac{1}{x}\right\} \) on \( \left( {0,1}\right) \) is ergodic with respect to the Gauss measure \( \mu \) . | Proof of Theorem 3.7. The description of the continued fraction map as a shift on the space \( {\mathbb{N}}^{\mathbb{N}} \) described above suggests the method of proof: the measure \( \mu \) corresponds to a rather complicated measure on the shift space, but if we can control the measure of cylinder sets (and their in... | No |
Let \( \psi : {\Omega }_{1} \rightarrow {\Omega }_{2} \) be a \( {C}^{j} \) diffeomorphism that satisfies (2.1.11.1), (2.1.11.2), and (2.1.11.3). Then there is a number \( J = J\left( j\right) \) such that whenever \( g \in {W}_{0}^{j + J}\left( {\Omega }_{2}\right) \), then \( g \circ \psi \in {W}_{0}^{j}\left( {\Omeg... | The proof is given in the text. It involves showing that the composition of a function in \( {W}_{0}^{j + J}\left( {\Omega }_{2}\right) \) with a \( {C}^{j} \) diffeomorphism is in \( {W}_{0}^{j}\left( {\Omega }_{1}\right) \) by using the chain rule and Leibniz's rule, and then applying the Sobolev embedding theorem. | Yes |
Corollary 1.18. Let \( k \subseteq E \subseteq F \) be field extensions. Then \( k \subseteq F \) is algebraic if and only if both \( k \subseteq E \) and \( E \subseteq F \) are algebraic. | If \( k \subseteq F \) is algebraic, then every element of \( F \) is algebraic over \( k \), hence over \( E \), and every element of \( E \) is algebraic over \( k \) ; thus \( E \subseteq F \) and \( k \subseteq E \) are algebraic.
Conversely, assume \( k \subseteq E \) and \( E \subseteq F \) are both algebraic, a... | Yes |
A closed subgroup of a Lie group is a Lie group in its own right with respect to the relative topology. | It is well known (see [8]) that if a smooth map \( \varphi \) has constant rank, then \( {\varphi }^{-1}\{ e\} \) is a closed regular submanifold of \( G \) of dimension \( \dim G - \operatorname{rk}\varphi \) . Since \( \ker \varphi \) is a subgroup, it suffices to show that \( \varphi \) has constant rank. Write \( {... | Yes |
Theorem 6.3.3. The space \( {L}_{1} \) cannot be embedded in a Banach space with unconditional basis. | Let \( X \) be a Banach space with \( K \) -unconditional basis \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) and suppose that \( T : {L}_{1} \rightarrow X \) is an embedding. We can assume that for some constant \( M \geq 1 \) ,
\[
\parallel f{\parallel }_{1} \leq \parallel {Tf}\parallel \leq M\parallel f{\parall... | No |
Proposition 11.25. Let \( F : M \rightarrow N \) be a smooth map between smooth manifolds with or without boundary. Suppose \( u \) is a continuous real-valued function on \( N \) , and \( \omega \) is a covector field on \( N \) . Then | To prove (11.14) we compute
\[
{\left( {F}^{ * }\left( u\omega \right) \right) }_{p} = d{F}_{p}^{ * }\left( {\left( u\omega \right) }_{F\left( p\right) }\right) \;\left( {\text{by }\left( {11.13}\right) }\right)
\]
\[
= d{F}_{p}^{ * }\left( {u\left( {F\left( p\right) }\right) {\omega }_{F\left( p\right) }}\right) \;\... | Yes |
Show that \( {\Delta }_{p, q} \) is a \( G \) -map from \( {V}_{p, q}\left( {\mathbb{C}}^{n}\right) \) onto \( {V}_{p - 1, q - 1}\left( {\mathbb{C}}^{n}\right) \). | Null | No |
Let \( G \) be the additive group of residue classes mod \( k \) . Show that a sequence of natural numbers \( {\left\{ {x}_{n}\right\} }_{n = 1}^{\infty } \) is equidistributed in \( G \) if and only if \( \mathop{\sum }\limits_{{n = 1}}^{N}{e}^{{2\pi ia}{x}_{n}/k} = o\left( N\right) \) for \( a = 1,2,\ldots, k - 1 \) ... | Let \( n = {p}_{1}^{{\alpha }_{1}}\cdots {p}_{k}^{{\alpha }_{k}} \) be the unique factorization of \( n \) as a product of powers of primes. Let \( N = {p}_{1}\cdots {p}_{k} \) . Then \( \mathop{\sum }\limits_{{d \mid n}}\mu \left( d\right) = \mathop{\sum }\limits_{{d \mid N}}\mu \left( d\right) \) since the Möbius fun... | No |
If \( \mathfrak{a} \) is an ideal of a commutative ring \( R \) [with identity], then \( R/\mathfrak{a} \) is a domain if and only if \( \mathfrak{a} \) is a prime ideal. | Null | No |
Show that \( \mathbb{Z}\left\lbrack \sqrt{-2}\right\rbrack \) is Euclidean. | Null | No |
Show that the sequence of non-zero rational numbers in \( \left\lbrack {0,1}\right\rbrack \) is u.d. mod 1. | Null | No |
Let \( X,\mathcal{B}, N, W \) be as in Chapter 2, and \( \sigma : X \rightarrow X \) an endomorphism such that \( \# {\sigma }^{-1}\left( {\{ x\} }\right) = N, x \in X \), and assume in addition that \( X \) is a compact Hausdorff space. Suppose branches of \( {\sigma }^{-1} \) may be chosen such that, for some measure... | Proof of Theorem 6.1.1. Let \( k \in {\mathbb{N}}_{0} \), and consider the \( N \) -adic representation \( k = \) \( {i}_{1} + {i}_{2}N + \cdots + {i}_{n}{N}^{n - 1} \) . Note that \(\omega \left( k\right) = \left( {{i}_{1},\ldots ,{i}_{n},\underset{\infty \text{ string of zeroes }}{\underbrace{0,0,0,\ldots }}}\right) ... | Yes |
Proposition 16. Let \( V \) and \( W \) be finite dimensional vector spaces over the field \( F \) with bases \( {v}_{1},\ldots ,{v}_{n} \) and \( {w}_{1},\ldots ,{w}_{m} \) respectively. Then \( V{ \otimes }_{F}W \) is a vector space over \( F \) of dimension \( {nm} \) with basis \( {v}_{i} \otimes {w}_{j},1 \leq i \... | Remark: If \( v \) and \( w \) are nonzero elements of \( V \) and \( W \), respectively, then it follows from the proposition that \( v \otimes w \) is a nonzero element of \( V{ \otimes }_{F}W \), because we may always build bases of \( V \) and \( W \) whose first basis vectors are \( v, w \), respectively. In a ten... | No |
A compact Lie group \( G \) possesses a faithful representation, i.e., there exists a (finite-dimensional representation) \( \left( {\pi, V}\right) \) of \( G \) for which \( \pi \) is injective. | By the proof of the Peter-Weyl Theorem, for \( {g}_{1} \in {G}^{0},{g}_{1} \neq e \), there exists a finite-dimensional representation \( \left( {{\pi }_{1},{V}_{1}}\right) \) of \( G \), so that \( {\pi }_{1}\left( {g}_{1}\right) \) is not the identity operator. Thus \( \ker {\pi }_{1} \) is a closed proper Lie subgro... | Yes |
Proposition 36. Let \( F \) be a field of characteristic not dividing \( n \) which contains the \( {n}^{\text{th }} \) roots of unity. Then the extension \( F\left( \sqrt[n]{a}\right) \) for \( a \in F \) is cyclic over \( F \) of degree dividing \( n \). | The extension \( K = F\left( \sqrt[n]{a}\right) \) is Galois over \( F \) if \( F \) contains the \( {n}^{\text{th }} \) roots of unity since it is the splitting field for \( {x}^{n} - a \) . For any \( \sigma \in \operatorname{Gal}\left( {K/F}\right) ,\sigma \left( \sqrt[n]{a}\right) \) is another root of this polynom... | Yes |
Let \( A \) be a unital commutative Banach algebra and let \( \varphi \in \) \( \partial \left( A\right) \) . Then \( \ker \varphi \) consists of joint topological zero divisors. | It suffices to show that given \( {a}_{1},\ldots ,{a}_{q} \in A \) such that \( d\left( {{a}_{1},\ldots ,{a}_{q}}\right) > 0 \) , there is no maximal ideal of \( A \) containing all of \( {a}_{1},\ldots ,{a}_{q} \) and corresponding to some point in \( \partial \left( A\right) \) . Of course, we can assume \( d\left( {... | Yes |
Corollary 7.56 Suppose \( \mathbf{A} \) is a balanced pre-Abelian category and \( {\mathbf{A}}^{\prime } \) is Abelian. Suppose \( F : \mathbf{A} \rightarrow {\mathbf{A}}^{\prime } \) is a functor. | a) If \( F \) is contravariant and left exact, and \( \mathbf{A} \) has enough projectives, then \( {\mathcal{L}}^{0}F \approx F. \) b) If \( F \) is covariant and right exact, and \( \mathbf{A} \) has enough projectives, then \( {\mathcal{L}}_{0}F \approx F \) . c) If \( F \) is contravariant and right exact, and \( \... | Yes |
Proposition 2.7.14. Let \( F : X \times I \rightarrow Y \) be a cellular homotopy from \( f \) to g. Define \( {D}_{n} : {C}_{n}\left( X\right) \rightarrow {C}_{n + 1}\left( Y\right) \) by \( {D}_{n}\left( {e}_{\alpha }^{n}\right) = {\left( -1\right) }^{n + 1}{F}_{\# }\left( {{e}_{\alpha }^{n} \times I}\right) \) . The... | Proof. We use the proof of 2.7.10 and 2.7.9 to get:\n\[
\partial {D}_{n}\left( {e}_{\alpha }^{n}\right) = {\left( -1\right) }^{n + 1}\partial {F}_{\# }\left( {{e}_{\alpha }^{n} \times I}\right)
\]\n\[
= {\left( -1\right) }^{n + 1}{F}_{\# }\partial \left( {{e}_{\alpha }^{n} \times I}\right)
\]\n\[
= {F}_{\# }\left( {{e}... | Yes |
If \( M \) is an \( R \) -S-bimodule and \( A \) is an \( R \) -T-bimodule, then \( {\operatorname{Hom}}_{R}\left( {M, A}\right) \) is an \( S \) - \( T \) -bimodule, in which | In the above, \( {s\alpha } \) and \( {\alpha t} \) are homomorphisms of left \( R \) -modules, since \( M \) and \( A \) are bimodules. Moreover, \( s\left( {\alpha + \beta }\right) = {s\alpha } + {s\beta } \), and\\
\[
s\left( {{s}^{\prime }\alpha }\right) \left( x\right) = \left( {{s}^{\prime }\alpha }\right) \left(... | Yes |
Corollary 10.36 (The Normal Bundle to a Submanifold of \( {\mathbb{R}}^{n} \) ). If \( M \subseteq {\mathbb{R}}^{n} \) is an immersed m-dimensional submanifold with or without boundary, its normal bundle \( {NM} \) is a smooth rank- \( \left( {n - m}\right) \) subbundle of \( {\left. T{\mathbb{R}}^{n}\right| }_{M} \) .... | Apply Lemma 10.35 to the smooth subbundle \( {\left. TM \subseteq T{\mathbb{R}}^{n}\right| }_{M} \) . | Yes |
Proposition 13.19. Suppose \( \mathcal{U} \) is any open cover of \( X \) . Then the inclusion map \( {C}_{ * }^{\mathcal{U}}\left( X\right) \rightarrow {C}_{ * }\left( X\right) \) induces a homology isomorphism \( {H}_{p}^{\mathcal{U}}\left( X\right) \cong {H}_{p}\left( X\right) \) for all \( p \) . | The idea of the proof is simple, although the technical details are somewhat involved. If \( \sigma : {\Delta }_{p} \rightarrow X \) is any singular \( p \) -simplex, the plan is to show that there is a homologous \( p \) -chain obtained by "subdividing" \( \sigma \) into \( p \) -simplices with smaller images. If we s... | No |
Theorem 9.21 Every chordal graph which is not complete has two nonadjacent simplicial vertices. | Let \( \left( {{V}_{1},{V}_{2},\ldots ,{V}_{k}}\right) \) be a simplicial decomposition of a chordal graph, and let \( x \in {V}_{k} \smallsetminus \left( {{ \cup }_{i = 1}^{k - 1}{V}_{i}}\right) \) . Then \( x \) is a simplicial vertex. Now consider a simplicial decomposition \( \left( {{V}_{\pi \left( 1\right) },{V}_... | Yes |
Theorem 2.9. Let \( A \) be an entire ring, integrally closed in its quotient field \( K \) . Let \( f\left( X\right) \in A\left\lbrack X\right\rbrack \) have leading coefficient 1 and be irreducible over \( K \) (or \( A \), it’s the same thing). Let \( \mathfrak{p} \) be a maximal ideal of \( A \) and let \( \bar{f} ... | Proof. Let \( \left( {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right) \) be the roots of \( f \) in \( B \) and let \( \left( {{\bar{\alpha }}_{1},\ldots ,{\bar{\alpha }}_{n}}\right) \) be their reductions mod \( \mathfrak{P} \) . Since
\[
f\left( X\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\left( {X - {\alpha }_{i}}\r... | Yes |
If \( M \) is an orientable hyperbolic 3-manifold, then \( M \) is isometric to \( {\mathbf{H}}^{3}/\Gamma \), where \( \Gamma \) is a torsion-free Kleinian group. | Now let us suppose that the manifold \( M = {\mathbf{H}}^{3}/\Gamma \) has finite volume. This means that the fundamental domain for \( \Gamma \) has finite volume and so \( \Gamma \) has finite covolume. Thus \( \Gamma \) is finitely generated. Furthermore, if \( M \) is not compact, then the ends of \( M \) can be de... | No |
n we carry over the max-flow min-cut theorem to this case? Yes, very easily, if we notice that a flow can be interpreted to flow in a vertex as well, namely from the part where all the currents enter it to the part where all the currents leave it. More precisely, we can turn each vertex of \( \overrightarrow{G} \) into... | Can we carry over the max-flow min-cut theorem to this case? Yes, very easily, if we notice that a flow can be interpreted to flow in a vertex as well, namely from the part where all the currents enter it to the part where all the currents leave it. More precisely, we can turn each vertex of \( \overrightarrow{G} \) in... | Yes |
If \( u \in {H}_{0}^{k, p}\left( \Omega \right) \) for some \( p \) and all \( k \in \mathbb{N} \), then \( u \in {C}^{\infty }\left( \Omega \right) \) . | Null | No |
Let \( \left( {G, S}\right) \) be a Coxeter system and let \( d\left( { \geq 1}\right) \) be the largest number such that there is a d-element subset \( T \) of \( S \) with \( \langle T\rangle \) finite. Then every torsion free subgroup of finite index in \( G \) has geometric dimension \( \leq d \) and has type \( F ... | The dimension of \( \left| K\right| \) is \( d - 1 \), so the dimension of \( \left| D\right| \) is \( d \) . The torsion free subgroup \( H \) acts freely on \( D \), and \( G \smallsetminus \left| D\right| \) is finite. | No |
Corollary 12.5.2. The Kuratowski set for any minor-closed graph property is finite. | Null | No |
Show that \( L\left( {s,\chi }\right) \) converges absolutely for \( \Re \left( s\right) > 1 \) and that \( L\left( {s,\chi }\right) = \mathop{\prod }\limits_{\wp }{\left( 1 - \frac{\chi \left( \wp \right) }{N{\left( \wp \right) }^{s}}\right) }^{-1} \), in this region. Deduce that \( L\left( {s,\chi }\right) \neq 0 \) ... | We have by multiplicativity of \( \chi \) , \( L\left( {s,\chi }\right) = \mathop{\prod }\limits_{\wp }{\left( 1 - \frac{\chi \left( \wp \right) }{N{\left( \wp \right) }^{s}}\right) }^{-1} \) and the product converges absolutely for \( \Re \left( s\right) > 1 \) if and only if \( \mathop{\sum }\limits_{\wp }\frac{1}{N{... | Yes |
Theorem 4.3.1 Consider play sequences starting from some positive position \( p \in {\mathbb{R}}_{ + }^{S} \) . | The rule for changing the "position" \( p \) to \( {p}^{s} \) by firing node \( s \) coincides with the mapping \( {\sigma }_{s}^{ * } : p \mapsto s\left( p\right) \) considered in Section 4.1 (cf. equation (4.5)). Hence, the point denoted \( {p}^{w} \) here is the same as the point denoted by \( {w}^{-1}\left( p\right... | 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 |
Let \( N, H \) be normal subgroups of a group \( G \) . Then \(\left\lbrack {N, H}\right\rbrack \subseteq N \cap H\) | Proof. It suffices to verify this on generators; that is, it suffices to check that \(\left\lbrack {n, h}\right\rbrack = n\left( {h{n}^{-1}{h}^{-1}}\right) = \left( {{nh}{n}^{-1}}\right) {h}^{-1} \in N \cap H\) for all \( n \in N, h \in H \) . But the first expression and the normality of \( N \) show that \( \left\lbr... | Yes |
Theorem 11.3. Let \( \left( {A, X,\Omega, p}\right) \) be a maximum modulus algebra over \( \Omega \) . Fix \( F \in A \) . Then \( \lambda \mapsto \log {Z}_{F}\left( \lambda \right) \) is subharmonic on \( \Omega \). | Proof. In view of Exercise 11.3, it suffices to show that \( \log {Z}_{F} \) satisfies the inequality (17). We fix a disk \( \Delta = \left\{ {\left| {\lambda - {\lambda }_{0}}\right| \leq r}\right\} \) contained in \( \Omega \) and apply Theorem 11.2 to the function \( F \), a point \( {x}^{0} \in {p}^{-1}\left( {\lam... | "No" |
We have \[ \Delta = \mathop{\sum }\limits_{{w \in W}}{\left( -1\right) }^{l\left( w\right) }{\mathrm{e}}^{w\left( \rho \right) }.\] | The irreducible representation \( \chi \left( 0\right) \) with highest weight vector 0 is obviously the trivial representation. Therefore, \( \chi \left( 0\right) = {\mathrm{e}}^{0} = 1 \) . The formula now follows from (22.4). | No |
Let \( I \subset S \) be a graded ideal. Then \({\alpha }_{ij}\left( {S/I}\right) = {\alpha }_{ij}\left( {S/{\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) }\right) \). | Let \( i < n \) . According to the definition of the generic annihilator numbers we have \( {\alpha }_{ij}\left( {S/I}\right) = {\dim }_{K}{A}_{i}\left( {{x}_{n},{x}_{n - 1},\ldots ,{x}_{1};S/{\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) }\right) \), and \({\alpha }_{ij}\left( {S/{\operatorname{gin}}_{{ < }... | Yes |
whether \( T\left( {{\psi \phi } - \phi }\right) = 0 \) for \( \phi \in \mathfrak{D} \) . We use the definition of the support of \( T \) to answer this. We must verify only that the support of \( \left( {1 - \psi }\right) \phi \) is contained in \( {\mathbb{R}}^{n} \smallsetminus K \) . This is true because \( 1 - \ps... | s whether \( T\left( {{\psi \phi } - \phi }\right) = 0 \) for \( \phi \in \mathfrak{D} \) . We use the definition of the support of \( T \) to answer this. We must verify only that the support of \( \left( {1 - \psi }\right) \phi \) is contained in \( {\mathbb{R}}^{n} \smallsetminus K \) . This is true because \( 1 - \... | Yes |
Let \( {C}_{0} \) and \( {C}_{1} \) be disjoint coanalytic subsets of \( I = \left\lbrack {0,1}\right\rbrack \) that are not Borel separated; i.e., there is no Borel set containing \( {C}_{0} \) and disjoint from \( {C}_{1} \) . Let \( {A}_{0} = \left( {I\times \{ 0\} }\right) \bigcup \left( {{C}_{0} \times \left\lbrac... | Null | No |
Let \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) be a basis for a Banach space \( X \) with biorthogonal functionals \( {\left( {e}_{n}^{ * }\right) }_{n = 1}^{\infty } \) . Then for every \( {x}^{ * } \in {X}^{ * } \) there is a unique sequence of scalars \( {\left( {a}_{n}\right) }_{n = 1}^{\infty } \) such that ... | For every \( x \in X \) , \[ \left| {\left( {{x}^{ * } - {S}_{N}^{ * }\left( {x}^{ * }\right) }\right) \left( x\right) }\right| = \left| \left( {{x}^{ * }\left( {x - {S}_{N}\left( x\right) }\right) \mid \leq \begin{Vmatrix}{x}^{ * }\end{Vmatrix}\begin{Vmatrix}{x - {S}_{N}\left( x\right) }\end{Vmatrix}\overset{N \righta... | Yes |
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