Q stringlengths 26 3.6k | A stringlengths 1 9.94k | Result stringclasses 3
values |
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Let \( {S}^{r} \) denote the \( r \) -sphere. Then \( {\pi }_{1}\left( {S}^{1}\right) \cong \mathbb{Z} \), while \( {S}^{r} \) is simply-connected if \( r \geq 2 \) . | We may identify the circle \( {S}^{1} \) with the unit circle in \( \mathbb{C} \) . Then \( x \mapsto {\mathrm{e}}^{2\pi ix} \) is a covering map \( \mathbb{R} \rightarrow {S}^{1} \) . The space \( \mathbb{R} \) is contractible and hence simply-connected, so it is the universal covering space. If we give \( {S}^{1} \s... | Yes |
Show that there is a set \( A \) of reals of cardinality \( \mathfrak{c} \) such that \( A \cap C \) is countable for every closed, nowhere dense set. | Null | No |
Show that the discriminant is well-defined. In other words, show that given \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{n} \) and \( {\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{n} \), two integral bases for \( K \), we get the same discriminant for \( K \). | Null | No |
Lemma 3.2. \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n}\left( x\right) \) exist in \( \mathfrak{A} \) and depends only on \( x \) and \( \Phi \), not on the choice of \( \left\{ {f}_{n}\right\} \) . | Proof of Lemma 3.2. Choose \( \gamma \) as in Exercise 3.2. Then
\[
\begin{Vmatrix}{{f}_{n}\left( x\right) - \frac{1}{2\pi i}{\int }_{\gamma }\frac{\Phi \left( t\right) {dt}}{t - z}}\end{Vmatrix} = \begin{Vmatrix}{\frac{1}{2\pi i}{\int }_{\gamma }\frac{{f}_{n}\left( t\right) - \Phi \left( t\right) }{t - x}{dt}}\end{Vm... | No |
Exercise 4.4.7. Let \( X \) be a compact metric space, and assume that \( {\nu }_{n} \rightarrow \mu \) in the weak*-topology on \( \mathcal{M}\left( X\right) \) . Show that for a Borel set \( B \) with \( \mu \left( {\partial B}\right) = 0 \) , | Show that for a Borel set \( B \) with \( \mu \left( {\partial B}\right) = 0 \) ,\[
\mathop{\lim }\limits_{{n \rightarrow \infty }}{\nu }_{n}\left( B\right) = \mu \left( B\right)
\] | Yes |
Theorem 14 The list-chromatic index of a bipartite graph equals its chromatic index. | Let \( G \) be a bipartite graph with bipartition \( \left( {{V}_{1},{V}_{2}}\right) \), and let \( \lambda : E\left( G\right) \rightarrow \) \( \left\lbrack k\right\rbrack \) be an edge-colouring of \( G \), where \( k \) is the chromatic index of \( G \) . Define preferences on \( G \) as follows: let \( a \in {V}_{1... | Yes |
Theorem 11 (Dual to Theorem 8). In order that a G-module A be cohomo-logically trivial, it is necessary and sufficient that there be an exact sequence \( 0 \rightarrow \mathrm{A} \rightarrow {\mathrm{I}}_{0} \rightarrow {\mathrm{I}}_{1} \rightarrow 0 \), where the \( {\mathrm{I}}_{i} \) are injective \( \mathbf{Z}\left... | As before, there is an exact sequence\n\[
0 \rightarrow \mathrm{A} \rightarrow {\mathrm{I}}_{0} \rightarrow \mathrm{R} \rightarrow 0
\]\nwith \( {\mathrm{I}}_{0}\mathbf{Z}\left\lbrack \mathrm{G}\right\rbrack \) -injective. Since \( \mathrm{A} \) is cohomologically trivial, so is \( \mathrm{R} \) ; on the other hand, \(... | No |
Every non-orientable path connected CW n-manifold has an orientable path connected double cover. | Null | No |
Proposition 8.4.2. Ordinary reduction, supersingular reduction, and multiplicative reduction are well defined on equivalence classes of \( \mathfrak{p} \) -minimal Weierstrass equations. If \( E \) and \( {E}^{\prime } \) are equivalent \( \mathfrak{p} \) -minimal Weierstrass equations with good reduction at \( \mathfr... | Proof. If \( E \) and \( {E}^{\prime } \) are equivalent then by Exercise 8.1.1(b)
\[
{u}^{12}{\Delta }^{\prime } = \Delta ,\;{u}^{4}{c}_{4}^{\prime } = {c}_{4},
\]
where \( u \) comes from the admissible change of variable taking \( E \) to \( {E}^{\prime } \) . Recall the disjoint union mentioned early in this sect... | No |
Let \( D < 0 \) be a fundamental discriminant, and denote by \( h\left( D\right) \) the class number of \( K = \mathbb{Q}\left( \sqrt{D}\right) \) . Denote by \( \mathcal{Q}\left( D\right) \) the set of equivalence classes of quadratic numbers \( \tau = \left( {-b + \sqrt{D}}\right) /\left( {2a}\right) \) of discrimina... | L\left( {{\chi }_{D},1}\right) = \frac{{2\pi h}\left( D\right) }{w\left( D\right) {\left| D\right| }^{1/2}},\;L\left( {{\chi }_{D},0}\right) = \frac{{2h}\left( D\right) }{w\left( D\right) },\]
\[
\frac{{L}^{\prime }\left( {{\chi }_{D},1}\right) }{L\left( {{\chi }_{D},1}\right) } = \gamma - \log \left( 2\right) - \frac{... | Yes |
Theorem 6.3.1 Let \( X, Y \), and \( Z \) be graphs. If \( f : Z \rightarrow X \) and \( g : Z \rightarrow Y \), then there is a unique homomorphism \( \phi \) from \( Z \) to \( X \times Y \) such that \( f = {p}_{X} \circ \phi \) and \( g = {p}_{Y} \circ \phi \) . | Proof. Assume that we are given homomorphisms \( f : Z \rightarrow X \) and \( g : Z \rightarrow Y \) . The map\\ \[\\ \phi : z \mapsto \left( {f\left( z\right), g\left( z\right) }\right)\\ \] is readily seen to be a homomorphism from \( Z \) to \( X \times Y \) . Clearly, \( {p}_{X} \circ \phi = f \) and \( {p}_{Y} \c... | Yes |
If \( A \subseteq \mathbb{N} \) has positive upper density, then \( A \) contains infinitely many arithmetic progressions of length 3. | Null | No |
Corollary 1.141. A multimap \( F : X \rightrightarrows Y \) between two metric spaces is c-subregular at \( \left( {\bar{x},\bar{y}}\right) \in F \) if and only if \( {F}^{-1} \) is \( c \) -calm at \( \left( {\bar{y},\bar{x}}\right) \) . | Null | No |
We have \({B}_{n} = \mathop{\sum }\limits_{{\pi \in \bar{S}\left( n\right) }}W\left( \pi \right)\) | The proof of recurrence (6) for \( \mathop{\sum }\limits_{{\pi \in \bar{S}\left( n\right) }}W\left( \pi \right) \) is analogous, by considering separately the cases \( {\pi }_{n + 1} = n + 1 \) and \( {\pi }_{n + 1} \neq n + 1 \), and is left to the exercises. | No |
We have \[
\mathop{\sum }\limits_{\lambda }{s}_{\lambda }\left( \mathbf{x}\right) {s}_{\lambda }\left( \mathbf{y}\right) = \mathop{\prod }\limits_{{i, j \geq 1}}\frac{1}{1 - {x}_{i}{y}_{j}}.
\] | Note that just as the Robinson-Schensted correspondence is gotten by restricting Knuth's generalization to the case where all entries are distinct, we can obtain
\[
n! = \mathop{\sum }\limits_{{\lambda \vdash n}}{\left( {f}^{\lambda }\right) }^{2}
\]
by taking the coefficient of \( {x}_{1}\cdots {x}_{n}{y}_{1}\cdots ... | No |
Proposition 3.4. Let \( B \) be a \( \Lambda \) -module and \( \left\{ {A}_{j}\right\}, j \in J \) be a family of \( \Lambda \) - modules. Then there is an isomorphism\n\[
\eta : {\operatorname{Hom}}_{A}\left( {{\bigoplus }_{j \in J}{A}_{j}, B}\right) \rightarrow \mathop{\prod }\limits_{{j \in J}}{\operatorname{Hom}}_{... | The proof reveals that this theorem is merely a restatement of the universal property of the direct sum. For \( \psi : {\bigoplus }_{j \in J}{A}_{j} \rightarrow B \), define \( \eta \left( \psi \right) = {\left( \psi {\iota }_{j} : {A}_{j} \rightarrow B\right) }_{j \in J} \) . Conversely a family \( \left\{ {{\psi }_{j... | Yes |
Proposition 15.36. Let the notation be as in Proposition 13.32. Then \( {\varepsilon }_{j}{\mathcal{X}}_{\infty } \) has no nonzero finite submodules. | Null | No |
Theorem 10.5. Let \( D \) be a domain in \( \mathbb{C} \) and suppose that \( \left\{ {f}_{n}\right\} \) is a sequence in \( \mathbf{H}\left( D\right) \) with \( {f}_{n} \) not identically 0 for all \( n \) . | For part (a), we only have to verify the formula for the order of \( z \) . We note that the sum in that formula is finite (i.e., all but finitely many summands are zero). Let \( {z}_{0} \in D \) and let \( K \subset D \) be a compact set containing a neighborhood of \( {z}_{0} \) . There is an \( N \) in \( {\mathbb{Z... | No |
Proposition 12.6. For a group extension \( G\overset{\kappa }{ \rightarrow }E\overset{\rho }{ \rightarrow }Q \) the following conditions are equivalent: | Proof. (1) implies (2). If (1) holds, then \( {p}_{a} = \mu \left( a\right) \) is a cross-section of \( E \) , relative to which \( {s}_{a, b} = 1 \) for all \( a, b \), since \( \mu \left( a\right) \mu \left( b\right) = \mu \left( {ab}\right) \) .
(2) implies (3). If \( {s}_{a, b} = 1 \) for all \( a, b \), then \( \... | Yes |
Let \( E \) and \( F \) be affine spaces, and \( A : E \rightarrow F \) a multivalued map whose graph \( \operatorname{gr}\left( A\right) = C \) is a nonempty convex set in \( E \times F \) . Then | The inclusion follows immediately from Lemma 5.13. If \( x \in \operatorname{rai}\operatorname{dom}\left( A\right) \) and \( y \in \operatorname{rai}A\left( x\right) \neq \varnothing \), then Lemma 5.13 implies that \( \left( {x, y}\right) \in \operatorname{raigr}\left( A\right) \), and we have \( x \in \operatorname{d... | Yes |
Let \( p \) be a prime number and \( \alpha \) an algebraic number. The following conditions are equivalent: | Clear and left to the reader (Exercise 7). | No |
Suppose \( X \) is a locally convex space over \( \mathbb{R} \) or \( \mathbb{C} \). Then the topology of \( X \) is given by a directed family of seminorms. This family can be chosen to be countable if \( X \) is first countable. | Proof. \( {\mathcal{B}}_{1} \) exists by Proposition 3.1. By the way, Reed and Simon [29] define a locally convex space this way. What does one do if the family is not directed? There is a standard construction that goes as follows, if \( {\mathcal{F}}_{0} \) is any family of seminorms. 1. If \( {\mathcal{F}}_{0} \) is... | Yes |
Show that \( \mathbb{Z}\left\lbrack \rho \right\rbrack /\left( \lambda \right) \) has order 3. | Null | No |
Proposition 4.3.11 Let \( A \subseteq \left\lbrack {0,1}\right\rbrack \) be a strong measure zero set and \( f \) : \( \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) a continuous map. Then the set \( f\left( A\right) \) has strong measure zero. | Proof. Let \( \left( {a}_{n}\right) \) be any sequence of positive real numbers. We have to show that there exist open intervals \( {J}_{n}, n \in \mathbb{N} \), such that \( \left| {J}_{n}\right| \leq {a}_{n} \) and \( f\left( A\right) \subseteq \mathop{\bigcup }\limits_{n}{J}_{n} \) . Since \( f \) is uniformly conti... | Yes |
Let \( G \) be a compact Abelian group and let \( {L}_{a} \) be the Koopman operator induced by the rotation by \( a \in G \) . Since every character \( \chi \in {G}^{ * } \) is an eigenfunction of \( {L}_{a} \) corresponding to the eigenvalue \( \chi \left( a\right) \in \mathbb{T} \) and since \( \operatorname{lin}{G}... | Direct proof of Theorem 17.11. Let \( T \) be the Koopman operator of the ergodic system \( \left( {\mathrm{X};\varphi }\right) \) with discrete spectrum, and let \( \Gamma \mathrel{\text{:=}} {\sigma }_{\mathrm{p}}\left( T\right) \) be its point spectrum. By Proposition 7.18, each eigenvalue is unimodular and simple, ... | No |
Theorem 5. Let \( X = X\left( \omega \right) \) be a random element with values in the Borel space \( \left( {E,\mathcal{E}}\right) \) . Then there is a regular conditional distribution of \( X \) with respect to \( \mathcal{G} \subseteq \mathcal{F} \) . | Let \( \varphi = \varphi \left( e\right) \) be the function in Definition 9. By (2) in this definition \( \varphi \left( {X\left( \omega \right) }\right) \) is a random variable. Hence, by Theorem 4, we can define the conditional distribution \( Q\left( {\omega ;A}\right) \) of \( \varphi \left( {X\left( \omega \right)... | Yes |
Let \( \left( {U, x}\right) \) be a chart around \( p \) . Then any tangent vector \( v \in {M}_{p} \) can be uniquely written as a linear combination \( v = \mathop{\sum }\limits_{i}{\alpha }_{i}\partial /\partial {x}^{i}\left( p\right) \) . In fact, \( {\alpha }_{i} = v\left( {x}^{i}\right) \) . | We may assume without loss of generality that \( x\left( p\right) = 0 \), and that \( x\left( U\right) \) is star-shaped. By Lemma 3.1, any \( f \in \mathcal{F}M \) satisfies \( f \circ {x}^{-1} = f\left( p\right) + \sum {u}^{i}{\psi }_{i} \), with \( {\psi }_{i}\left( 0\right) = \partial /\partial {x}^{i}\left( p\righ... | Yes |
Theorem 1.29. Every bounded linear functional \( \Lambda \) on a Hilbert space \( \mathcal{H} \) is given by inner product with a (unique) fixed vector \( {h}_{0} \) in \( \mathcal{H} : \Lambda \left( h\right) = \left\langle {h,{h}_{0}}\right\rangle \) . Moreover, the norm of the linear functional \( \Lambda \) is \( \... | Suppose \( \Lambda \) is a bounded linear functional on \( \mathcal{H} \) . If \( \Lambda \) is identically 0, choose \( {h}_{0} = 0 \) . Otherwise, set
\[
M = \ker \Lambda \equiv \{ h \in \mathcal{H} : \Lambda \left( h\right) = 0\} .
\]
Since \( \Lambda \) is linear, \( M \) is a subspace of \( \mathcal{H} \), and s... | Yes |
Corollary 1.4. Theorem 1.1 follows from Theorem 1.3. | Proof. By the mean value theorem there exists \( \bar{x} \in \left( {a,{s}_{n - 1}}\right) \) such that\n\[
{\int }_{a}^{{s}_{n - 1}}{f}^{\left( n\right) }\left( {s}_{n}\right) d{s}_{n} = {f}^{\left( n\right) }\left( \bar{x}\right) \left( {{s}_{n - 1} - a}\right) = {\int }_{a}^{{s}_{n - 1}}{f}^{\left( n\right) }\left( ... | No |
Let \( V \) be a one-dimensional analytic subvariety of an open subset of \( {\mathbb{C}}^{n} \) . Then \( \operatorname{area}\left( V\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\text{ area-with-multiplicity }\left( {{z}_{j}\left( V\right) }\right) \). | This is a local result and so we can assume that \( V \) can be parameterized by a one-one analytic map \( f : W \rightarrow V \subseteq {\mathbb{C}}^{n} \), where \( W \) is a domain in the complex plane. Let \( \zeta \in W \) be \( \zeta = s + {it} \) and let \( f = \left( {{f}_{1},{f}_{2},\ldots ,{f}_{n}}\right) \),... | Yes |
Corollary 1. Every irreducible representation \( {\mathrm{W}}_{i} \) is contained in the regular representation with multiplicity equal to its degree \( {n}_{i} \). | According to th. 4, this number is equal to \( \left\langle {{r}_{\mathrm{G}},{\chi }_{i}}\right\rangle \), and we have\\[2mm] \left\langle {{r}_{\mathrm{G}},{\chi }_{i}}\right\rangle = \frac{1}{g}\mathop{\sum }\limits_{{s \in \mathrm{G}}}{r}_{\mathrm{G}}\left( {s}^{-1}\right) {\chi }_{i}\left( s\right) = \frac{1}{g}g ... | Yes |
Theorem 1.8 Let \( T \) be a compact operator from \( E \) to \( E \) . | 1. Suppose that 0 is not a spectral value of \( T \) . Then \( I = T{T}^{-1} \) is a compact operator by Proposition 1.2. By the Riesz Theorem (page 49), this implies that \( E \) is finite-dimensional.
2. Take \( \lambda \in {\mathbb{K}}^{ * } \) . Then \( \lambda \) is an eigenvalue of \( T \) if and only if \( I - ... | Yes |
Proposition 9.19. If \( \Gamma \leq {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) is a lattice, the hyperbolic measure defined by the volume form \( \mathrm{d}m = \frac{1}{{y}^{2}}\mathrm{\;d}x\mathrm{\;d}y\mathrm{\;d}\theta \) in Lemma 9.16 induces a \( {fi} \) - nite \( {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\rig... | In fact if \[ \pi : {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \rightarrow X \] is the canonical quotient map \( \pi \left( g\right) = {\Gamma g} \) for \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) and \( F \) is a finite volume fundamental domain, then \[ {m}_{X}\left( B\right) = m\left( {F \cap ... | Yes |
Corollary 14.23 Suppose that \( S \) is a lower semibounded self-adjoint extension of the densely defined lower semibounded symmetric operator \( T \) on \( \mathcal{H} \) . Let \( \lambda \in \mathbb{R} \) , \( \lambda < {m}_{T} \), and \( \lambda \leq {m}_{S} \) . Then \( S \) is equal to the Friedrichs extension \( ... | Since then \( \lambda \in \rho \left( {T}_{F}\right) \), Example 14.6, with \( A = {T}_{F},\mu = \lambda \), yields a boundary triplet \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) for \( {T}^{ * } \) such that \( {T}_{0} = {T}_{F} \) . By Propositions 14.7(v) and 14.21, there is a self-adjoint relation... | Yes |
Using Gram-Schmidt orthogonalization, show multiplication induces a diffeomorphism of \( O\left( n\right) \times A \times N \) onto \( {GL}\left( {n,\mathbb{R}}\right) \) . | Theorem 1.15. Let \( G \) be a connected Lie group and \( U \) a neighborhood of \( e \) . Then \( U \) generates \( G \), i.e., \( G = { \cup }_{n = 1}^{\infty }{U}^{n} \) where \( {U}^{n} \) consists of all \( n \) -fold products of elements of \( U \ ). | No |
Find all integer solutions to the equation \( {x}^{2} + {11} = {y}^{3} \) . | In the ring \( \mathbb{Z}\left\lbrack {\left( {1 + \sqrt{-{11}}}\right) /2}\right\rbrack \), we can factor the equation as
\[
\left( {x - \sqrt{-{11}}}\right) \left( {x + \sqrt{-{11}}}\right) = {y}^{3}.
\]
Now, suppose that \( \delta \left| {\left( {x - \sqrt{-{11}}}\right) \text{and}\delta }\right| \left( {x + \sqrt... | No |
The set \( D = \left\{ {\alpha < \kappa : {\bar{d}}_{\beta } \in {B}_{\alpha }}\right. \) for all \( \left. {\beta < \alpha }\right\} \) is closed unbounded. | It is easy to see that \( D \) is closed. Let \( {\alpha }_{0} < \kappa \) . Build a sequence \( {\alpha }_{0} < {\alpha }_{1} < \ldots \) such that for all \( \beta < {\alpha }_{n},{\bar{d}}_{\beta } \in {B}_{{\alpha }_{n + 1}} \) . If \( \alpha = \sup {\alpha }_{i} \), then \( \alpha \in D \) . | No |
Suppose \( \rho \) is a density matrix on \( \mathbf{H} \). Then the map \( {\Phi }_{\rho } : \mathcal{B}\left( \mathbf{H}\right) \rightarrow \mathbb{C} \) given by \( {\Phi }_{\rho }\left( A\right) = \operatorname{trace}\left( {\rho A}\right) = \operatorname{trace}\left( {A\rho }\right) \) is a family of expectation v... | If we define \( {\Phi }_{\rho }\left( A\right) = \operatorname{trace}\left( {\rho A}\right) \), then \( {\Phi }_{\rho }\left( I\right) = \operatorname{trace}\left( \rho \right) = 1 \). For any \( A \in \mathcal{B}\left( \mathbf{H}\right) \), we have, \[ \operatorname{trace}\left( {\rho {A}^{ * }}\right) = \operatorname... | Yes |
Given a system of linear inequalities \( {Ax} \leq b \), formulate the problem of finding a solution to \( {Ax} \leq b \) as a problem of finding a solution to a system of linear inequalities in \( n - 1 \) variables using Fourier's elimination method. | Given a system of linear inequalities \( {Ax} \leq b \), let \( {A}^{n} \mathrel{\text{:=}} A,{b}^{n} \mathrel{\text{:=}} b \) ;
For \( i = n,\ldots ,1 \), eliminate variable \( {x}_{i} \) from \( {A}^{i}x \leq {b}^{i} \) with the above procedure to obtain system \( {A}^{i - 1}x \leq {b}^{i - 1} \) .
System \( {A}^{1... | Yes |
The period 2 elliptic points of \( {\Gamma }_{0}\left( N\right) \) are in bijective correspondence with the ideals \( J \) of \( \mathbb{Z}\left\lbrack i\right\rbrack \) such that \( \mathbb{Z}\left\lbrack i\right\rbrack /J \cong \mathbb{Z}/N\mathbb{Z} \) . | This is an application of beginning algebraic number theory; see for example Chapter 9 of [IR92] for the results to quote. For period 2, the ring \( A = \mathbb{Z}\left\lbrack i\right\rbrack \) is a principal ideal domain and its maximal ideals are - for each prime \( p \equiv 1\left( {\;\operatorname{mod}\;4}\right) \... | No |
Theorem 13.31. \( {\mathcal{X}}_{\infty } \sim {\Lambda }^{{r}_{2}} \oplus \) ( \( \Lambda \) -torsion). | Null | No |
Theorem 2.8. For any compact subset \( M \) of \( {\mathbb{R}}^{d} \), the convex hull conv \( M \) is again compact. | Let \( {\left( {y}_{v}\right) }_{v \in \mathbb{N}} \) be any sequence of points from conv \( M \) . We shall prove that the sequence admits a subsequence which converges to a point in conv \( M \) . Let the dimension of aff \( M \) be denoted by \( n \) . Then Corollary 2.5 shows that each \( {y}_{v} \) in the sequence... | Yes |
Theorem 4 (Lindenstrauss-Pelczynski). Let \( \left( {x}_{n}\right) \) be a normalized unconditional basis of \( {c}_{0} \) . Then \( \left( {x}_{n}\right) \) is equivalent to the unit vector basis. | Null | No |
Let \( \left( {M, F}\right) \) be a Finsler manifold. Suppose that at some \( p \in M \), the exponential map \( {\exp }_{p} : {T}_{p}M \rightarrow M \) is a covering projection. Let \( {\sigma }_{0}\left( t\right) \mathrel{\text{:=}} {\exp }_{p}\left( {t{T}_{0}}\right) \) and \( {\sigma }_{1}\left( t\right) \mathrel{\... | The contrapositive of the first conclusion encompasses the second conclusion. So it suffices to establish the first one. Suppose \( {\sigma }_{0} \) is homotopic to \( {\sigma }_{1} \), through a homotopy \( h\left( {t, u}\right) ,0 \leq t \leq L \) , \( 0 \leq u \leq 1 \) with fixed endpoints \( p \) and \( q \) . Usi... | Yes |
Let \( M \) be a compact connected oriented manifold of dimension \( {2n}, n \) odd. Then for \( G = \mathbb{Z} \) or any field \( \mathbb{F} \) of characteristic not equal to 2, \( \operatorname{rank}\left( {{K}^{n}\left( {M;G}\right) }\right) \) is even. Also, the Euler characteristic \( \chi \left( M\right) \) is ev... | By Theorem 6.4.8, \( \langle \) , \( \rangle {isanonsingularskew} - {symmetricbilinearformon} \) \( {K}^{n}\left( {M;G}\right) \), so by Theorem B.2.1, \( {K}^{n}\left( {M;G}\right) \) must have even rank.
We may use any field to compute Euler characteristic. Choosing \( \mathbb{F} = \mathbb{Q} \), say, and using Poin... | Yes |
Proposition 12.22. Every continuous action of a compact topological group on a Hausdorff space is proper. | Suppose \( G \) is a compact group acting continuously on a Hausdorff space \( E \), and let \( \Theta : G \times E \rightarrow E \times E \) be the map defined by (12.5). Given a compact set \( L \subseteq E \times E \), let \( K = {\pi }_{2}\left( L\right) \), where \( {\pi }_{2} : E \times E \rightarrow E \) is the ... | Yes |
Let \( {a}_{1},\ldots ,{a}_{n} \) for \( n \geq 2 \) be nonzero integers. Suppose there is a prime \( p \) and positive integer \( h \) such that \( {p}^{h} \mid {a}_{i} \) for some \( i \) and \( {p}^{h} \) does not divide \( {a}_{j} \) for all \( j \neq i \) . Then show that \( S = \frac{1}{{a}_{1}} + \cdots + \frac{... | Theorem 1.1.14 Given \( a, n \in \mathbb{Z},{a}^{\phi \left( n\right) } \equiv 1\left( {\;\operatorname{mod}\;n}\right) \) when \( \gcd \left( {a, n}\right) = 1 \). This is a theorem due to Euler.\n\nProof. The case where \( n \) is prime is clearly a special case of Fermat’s little Theorem. The argument is basically t... | No |
Let \( X \) be strongly regular with eigenvalues \( k > \theta > \tau \) . Suppose that \( x \) is an eigenvector of \( {A}_{1} \) with eigenvalue \( {\sigma }_{1} \) such that \( {\mathbf{1}}^{T}x = 0 \). If \( {Bx} = 0 \), then \( {\sigma }_{1} \in \{ \theta ,\tau \} \), and if \( {Bx} \neq 0 \), then \( \tau < {\sig... | Since \( {\mathbf{1}}^{T}x = 0 \), we have
\[
\left( {{A}_{1}^{2} - \left( {a - c}\right) {A}_{1} - \left( {k - c}\right) I}\right) x = - {B}^{T}{Bx}
\]
and since \( X \) is strongly regular with eigenvalues \( k,\theta \), and \( \tau \), we have
\[
\left( {{A}_{1}^{2} - \left( {a - c}\right) {A}_{1} - \left( {k - c... | Yes |
Let \( A \in B\left( {{L}_{2}\left( {\mathbb{R}}^{m}\right) }\right) \) be a real \( {}^{3} \) positivity improving selfadjoint operator. Assume that \( \parallel A\parallel \) is an eigenvalue of \( A \) . Then the multiplicity of the eigenvalue \( \parallel A\parallel \) equals 1 and there is an \( f > 0 \) that span... | Assume that \( f \neq 0 \) and \( {Af} = \parallel A\parallel f \) . Since \( A \) is real, we may assume that \( f \) is real (otherwise we could replace \( f \) by \( \operatorname{Re}f \) or \( \operatorname{Im}f \), because \( A\left( {\operatorname{Re}f}\right) = A\left( {f + {Kf}}\right) /2 = \left( {{Af} + {KAf}... | Yes |
Theorem 8.21 METRIC TSP admits a polynomial-time 2-approximation algorithm. | Applying the Jarnik-Prim Algorithm (6.9), we first find a minimum-weight spanning tree \( T \) of \( G \) . Suppose that \( C \) is a minimum-weight Hamilton cycle of \( G \) . By deleting any edge of \( C \), we obtain a Hamilton path \( P \) of \( G \) . Because \( P \) is a spanning tree of \( G \) and \( T \) is a ... | Yes |
Corollary 11.3.2. There exists a point \( x \in M \) such that \( G \cdot x \) is closed in \( M \). | Let \( y \in M \) and let \( Y \) be the closure of \( G \cdot y \). Then \( G \cdot y \) is open in \( Y\), by the argument in the proof of Theorem 11.3.1, and hence \( Z = Y - G \cdot y \) is closed in \( Y \) Thus \( Z \) is quasiprojective. Furthermore, \( \dim Z < \dim Y \) by Theorem A.1.19, and \( Z \) is a unio... | No |
Theorem 8.4. Let \( A \) be a finitely generated torsion-free abelian group. Then \( A \) is free. | Assume \( A \neq 0 \) . Let \( S \) be a finite set of generators, and let \( {x}_{1},\ldots ,{x}_{n} \) be a maximal subset of \( S \) having the property that whenever \( {v}_{1}{x}_{1} + \cdots + {v}_{n}{x}_{n} = 0 \) then \( {v}_{j} = 0 \) for all \( j \) . (Note that \( n \geqq 1 \) since \( A \neq 0 \) ). Let \( ... | Yes |
Theorem 2.9. Schönflies Theorem. Let \( e : {S}^{2} \rightarrow {S}^{3} \) be any piecewise linear embedding. Then \( {S}^{3} - e{S}^{2} \) has two components, the closure of each of which is a piecewise linear ball. | No proof will be given here for this fundamental, non-trivial result (for a proof see [81]). The piecewise linear condition has to be inserted, as there exist the famous "wild horned spheres" that are are examples of topological embeddings \( e : {S}^{2} \rightarrow {S}^{3} \) for which the complementary components are... | No |
Suppose that \( X \) and \( Y \) are graphs with minimum valency four. Then \( X \cong Y \) if and only if \( L\left( X\right) \cong L\left( Y\right) \) . | Let \( C \) be a clique in \( L\left( X\right) \) containing exactly \( c \) vertices. If \( c > 3 \) , then the vertices of \( C \) correspond to a set of \( c \) edges in \( X \), meeting at a common vertex. Consequently, there is a bijection between the vertices of \( X \) and the maximal cliques of \( L\left( X\rig... | No |
An intersection of closed cells is a closed cell. | Corollary 1.26. An intersection of closed cells is a closed cell. | No |
The LAPLACE transform of a function \( f \) is defined to be another function \( \widetilde{f} \), given by \( \widetilde{f}\left( s\right) = {\int }_{0}^{\infty }f\left( t\right) {e}^{-{st}}{dt} \) for all \( s \) such that the integral is convergent. | With this definition one finds that \( \widetilde{\delta }\left( s\right) = 1 \) for all \( s \) . Similarly, \( {\widetilde{\delta }}_{a}\left( s\right) = {e}^{-{as}} \) , if \( a > 0 \) . | No |
Let \( \varphi : A \rightarrow B \) be a morphism in an additive category. Then \( \varphi \) is a monomorphism if and only if \( 0 \rightarrow A \) is its kernel, and \( \varphi \) is an epimorphism if and only if \( B \rightarrow 0 \) is its cokernel. | First assume \( \varphi : A \rightarrow B \) is a monomorphism. If \( \zeta : Z \rightarrow A \) is any morphism such that the composition \( Z \rightarrow A \rightarrow B \) is 0, then \( \zeta \) is 0 by Lemma 1.3, and in particular \( \zeta \) factors (uniquely) through \( 0 \rightarrow A \) . This proves that \( 0 ... | Yes |
Theorem 6.94 (Sum rule for limiting directional subdifferentials). Let \( X \) be a WCG space, and let \( f = {f}_{1} + \cdots + {f}_{k} \), where \( {f}_{1},\ldots ,{f}_{k} \in \mathcal{L}\left( X\right) \) . Then\n\n{\partial }_{\ell }f\left( \bar{x}\right) \subset {\partial }_{\ell }{f}_{1}\left( \bar{x}\right) + \c... | Proof. We know that \( X \) is H-smooth. Let \( {\bar{x}}^{ * } \in {\partial }_{\ell }f\left( \bar{x}\right) \), and let \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) with \( {x}_{n}^{ * } \in {\partial }_{H}f\left( {x}_{n}\right) \) for all \( ... | Yes |
The closed convex hull of a totally bounded set in a complete locally convex linear topological space is compact. | Let \( K \) be such a set in such a space. By the preceding lemma, \( \operatorname{co}\left( K\right) \) is totally bounded. Hence \( \overline{\mathrm{{co}}}\left( K\right) \) is closed and totally bounded. Since the ambient space is complete, \( \overline{\mathrm{{co}}}\left( K\right) \) is complete and totally boun... | Yes |
For \( k \geq 1 \) we have \[
\mathop{\sum }\limits_{{m = 0}}^{N}\frac{1}{m + x} = - \psi \left( x\right) + \log \left( {N + x}\right) - \mathop{\sum }\limits_{{j = 1}}^{k}\frac{{B}_{j}}{j{\left( N + x\right) }^{j}} + {R}_{k}\left( {-1, x, N}\right) ,
\] | where
\[
{R}_{k}\left( {-1, x, N}\right) = {\int }_{N}^{\infty }\frac{{B}_{k}\left( {\{ t\} }\right) }{{\left( t + x\right) }^{k + 1}}{dt}
\]
and \( \left| {{R}_{k}\left( {-1, x, N}\right) }\right| \leq \left| {{B}_{k + 2}/\left( {\left( {k + 2}\right) {\left( N + x\right) }^{k + 2}}\right) }\right| \) when \( k \) is... | Yes |
If \( M \) is a compact surface with non-empty boundary then each path component of \( \partial M \) is a circle, and if \( C \) is one of those circles then the space obtained from \( M \) by attaching a 2-cell using as attaching map an embedding \( {S}^{1} \rightarrow M \) whose image is \( C \) is again a surface. | Thus, by attaching finitely many 2-cells in this way \( M \) becomes a closed surface, i.e., becomes a \( {T}_{g} \) or a \( {U}_{h} \) . | No |
A universal algebra \( A \) of type \( T \) is isomorphic to a subdirect product of \( {\left( {A}_{i}\right) }_{i \in I} \) if and only if there exist surjective homomorphisms \( {\varphi }_{i} : A \rightarrow {A}_{i} \) such that \( \mathop{\bigcap }\limits_{{i \in I}}\ker {\varphi }_{i} \) is the equality on \( A \)... | Let \( P \) be a subdirect product of \( {\left( {A}_{i}\right) }_{i \in I} \) . The inclusion homomorphism \( \iota : P \rightarrow \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) and projections \( {\pi }_{j} : \mathop{\prod }\limits_{{i \in I}}{A}_{i} \rightarrow {A}_{j} \) yield surjective homomorphisms \( {\rho }_{i}... | Yes |
If \( w \in {\Lambda }^{k}\left( E\right) \) and \( z \in {\Lambda }^{\ell }\left( E\right) \), then \( z \land w = {\left( -1\right) }^{k\ell }w \land z \). | Proposition 4.2 If \( w \in {\Lambda }^{k}\left( E\right) \) and \( z \in {\Lambda }^{\ell }\left( E\right) \), then \( z \land w = {\left( -1\right) }^{k\ell }w \land z.\) | Yes |
Theorem 2.7 Gaussian elimination for the simultaneous solution of an \( n \times n \) system for \( r \) different right-hand sides requires a total of | The computational cost, counting only the multiplications, in Gaussian elimination is \( {n}^{3}/3 + O\left( {n}^{2}\right) \) . It is left to the reader to show that the number of additions is also \( {n}^{3}/3 + O\left( {n}^{2}\right) \) (see Problem 2.7). | No |
Lemma 12.2. \( T : X \rightarrow Y \) is closed if and only if the following holds: When \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence in \( D\left( T\right) \) with \( {x}_{n} \rightarrow x \) in \( X \) and \( T{x}_{n} \rightarrow y \) in \( Y \), then \( x \in D\left( T\right) \) with \( y = {Tx} \... | The closed graph theorem (recalled in Appendix B, Theorem B.16) implies that if \( T : X \rightarrow Y \) is closed and has \( D\left( T\right) = X \), then \( T \) is bounded. Thus for closed, densely defined operators, \( D\left( T\right) \neq X \) is equivalent with unboundedness. | No |
Theorem 3.14 (Gluing distributions together). Let \( {\left( {\omega }_{\lambda }\right) }_{\lambda \in \Lambda } \) be an arbitrary system of open sets in \( {\mathbb{R}}^{n} \) and let \( \Omega = \mathop{\bigcup }\limits_{{\lambda \in \Lambda }}{\omega }_{\lambda } \) . Assume that there is given a system of distrib... | Observe to begin with that there is at most one solution \( u \) . Namely, if \( u \) and \( v \) are solutions, then \( {\left. \left( u - v\right) \right| }_{{\omega }_{\lambda }} = 0 \) for all \( \lambda \) . This implies that \( u - v = 0 \), by Lemma 3.11.
We construct \( u \) as follows: Let \( {\left( {K}_{l}\... | Yes |
Let \( {S}_{1} \) and \( {S}_{2} \) be any two projective subspaces of \( {\mathbb{P}}^{n}\left( k\right) \) . Then | Any subspace \( {k}^{r + 1} \) has codimension \( n - r \) in \( {k}^{n + 1} \) ; therefore the associated subspace \( {\mathbb{P}}^{r}\left( k\right) \) has the same codimension \( n - r \) in \( {\mathbb{P}}^{n}\left( k\right) \) . Then apply the corresponding vector space theorem. | No |
Theorem 2.1.4 (Greene-Krantz [GRK2]). Let \( B \subseteq {\mathbb{C}}^{n} \) be the unit ball. Let \( {\rho }_{0}\left( z\right) = {\left| z\right| }^{2} - 1 \) be the usual defining function for \( B \) . If \( \epsilon > 0 \) is sufficiently small, \( k = k\left( n\right) \) is sufficiently large, and \( \Omega \in {... | either \[ \Omega \sim B \] (2.1.4.1) or \( \Omega \) is not biholomorphic to the ball and (2.1.4.2) (a) Aut \( \left( \Omega \right) \) is compact. (b) Aut \( \left( \Omega \right) \) has a fixed point. Moreover, If \( K \subset \subset B,\epsilon > 0 \) is sufficiently small (depending on \( K \) ), and \( \Omega \in ... | Yes |
Theorem 5.3. Let \( \mathrm{S} \) be an extension ring of \( \mathrm{R} \) and \( \mathrm{s} \in \mathrm{S} \) . Then the following conditions are equivalent. | SKETCH OF PROOF. (i) \( \Rightarrow \) (ii) Suppose \( s \) is a root of the monic polynomial \( {f\varepsilon R}\left\lbrack x\right\rbrack \) of degree \( n \) . We claim that \( {1}_{R} = {s}^{0}, s,{s}^{2},\ldots ,{s}^{n - 1} \) generate \( R\left\lbrack s\right\rbrack \) as an \( R \) -module. As observed above, e... | No |
For all \( k,{\dim }_{K}{\widetilde{H}}_{k}\left( {\Delta ;K}\right) \leq {\dim }_{K}{\widetilde{H}}_{k}\left( {\Gamma ;K}\right) \) . | By considering an extension field of \( K \) if necessary, we may assume that \( K \) is infinite. Let \( {\Delta }^{e} \) denote the exterior algebraic shifted complex of \( \Delta \) . By Proposition 11.4.7 we have \( {\widetilde{H}}_{k}\left( {\Delta ;K}\right) \cong {\widetilde{H}}_{k}\left( {{\Delta }^{e};K}\right... | Yes |
Theorem 7. For spaces, connectivity is preserved by surjective mappings. That is, if \( \left\lbrack {X,\mathcal{O}}\right\rbrack \) is connected, and \( f : X \rightarrow Y \) is a mapping, then \( \left\lbrack {Y,{\mathcal{O}}^{\prime }}\right\rbrack \) is connected. | Suppose not. Then \( Y = U \cup V \), where \( U \) and \( V \) are disjoint, open, and nonempty. Therefore \( X = {f}^{-1}\left( U\right) \cup {f}^{-1}\left( V\right) \), and the latter sets are disjoint, open, and nonempty, which is impossible. | Yes |
A vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfies the system (3.2) if and only if there exists \( {\bar{x}}_{n} \) such that \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1},{\bar{x}}_{n}}\right) \) satisfies \( {Ax} \leq b \) . | We already remarked the "if" statement. For the converse, assume there is a vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfying (3.2). Note that the first set of inequalities in (3.2) can be rewritten as \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{kj}^{\prime }{x}_{j} - {b}_{k}^{\prime ... | Yes |
Corollary 4.34. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( Y \) is barreled. Then any linear map \( T \) from \( X \) onto \( Y \) is nearly open. | Use \( {\mathcal{B}}_{0} = \) all convex, balanced neighborhoods of 0 in \( X \) . If \( B \in {\mathcal{B}}_{0} \), and \( x \in X \), then \( x \in {cB} \Rightarrow T\left( x\right) \in T\left( {cB}\right) = {cT}\left( B\right) \), so \( T\left( B\right) \) is convex, balanced, and absorbent (since \( T \) is onto). ... | Yes |
If \( \left( {K,\mu ;\varphi }\right) \) is a faithful topological measure-preserving system, then \( \varphi \left( K\right) = K \), i.e., \( \left( {K;\varphi }\right) \) is a surjective topological system. | Theorem 10.2 of Krylov and Bogoljubov tells that every topological system \( \left( {K;\varphi }\right) \) has at least one invariant probability measure, and hence gives rise to at least one topological measure-preserving system. (By the lemma above, this topological measure-preserving system cannot be faithful if \( ... | No |
Proposition 16.45 (The Riemannian Density). Let \( \left( {M, g}\right) \) be a Riemannian manifold with or without boundary. There is a unique smooth positive density \( {\mu }_{g} \) on \( M \) , called the Riemannian density, with the property that | Uniqueness is immediate, because any two densities that agree on a basis must be equal. Given any point \( p \in M \), let \( U \) be a connected smooth coordinate neighborhood of \( p \). Since \( U \) is diffeomorphic to an open subset of Euclidean space, it is orientable. Any choice of orientation of \( U \) uniquel... | Yes |
Assuming the \( {ABC} \) Conjecture, show that there are infinitely many primes \( p \) such that \( {2}^{p - 1} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . | Null | No |
Proposition 9.32 For each \( j = 1,2,\ldots, n \), define a domain \( \operatorname{Dom}\left( {P}_{j}\right) \subset \) \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) as follows: | Proof of Proposition 9.32. By Proposition 9.30, the operator of multiplication by \( {k}_{j} \) is an unbounded self-adjoint operator on \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \), with domain equal to the set of \( \phi \) for which \( {k}_{j}\phi \left( \mathbf{k}\right) \) belongs to \( {L}^{2}\left( {\mathbb{R}}^{... | Yes |
Proposition 7.4 Suppose \( \mathbf{A} \) and \( {\mathbf{A}}^{\prime } \) are two additive categories, and suppose \( \mathbf{A} \) contains a biproduct of any two objects. Suppose \( F : \mathbf{A} \rightarrow {\mathbf{A}}^{\prime } \) is a covariant functor. Then the following are equivalent. | Proof: (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) \( \Rightarrow \) (i) works the same way here as it did in Proposition 6.1. The technical point-that \( F\left( {\pi }_{1}\right) \) and \( F\left( {\pi }_{2}\right) \) are the \( {\pi }_{1}^{\prime } \) and \( {\pi }_{2}^{\prime } \) for which \( \left( {F\left... | Yes |
Theorem 12.6.2 Let \( \mathcal{Q} \) be the incidence structure whose points are the vectors of \( {C}^{ * } \), and whose lines are triples of mutually orthogonal vectors. Then either \( \mathcal{Q} \) has no lines, or \( \mathcal{Q} \) is a generalized quadrangle, possibly degenerate, with lines of size three. | Proof. A generalized quadrangle has the property that given any line \( \ell \) and a point \( P \) off that line, there is a unique point on \( \ell \) collinear with \( P \) . We show that \( \mathcal{Q} \) satisfies this axiom.
Suppose that \( x, y \), and \( a - b - x - y \) are the three points of a line of \( \m... | Yes |
An operator \( T \) from a separable Hilbert space \( H \) into \( {L}_{2}\left( M\right) \) is a Carleman operator if and only if \( T{f}_{n}\left( x\right) \rightarrow 0 \) almost everywhere in \( M \) for every null-sequence \( \left( {f}_{n}\right) \) from \( D\left( T\right) \) . | It is evident from the definition that every Carleman operator has this property. It remains to prove the reverse direction. By Theorem 6.15 it is sufficient to show that the series \( {\sum }_{n}{\left| T{e}_{n}\left( x\right) \right| }^{2} \) is almost everywhere convergent for every ONS \( \left\{ {{e}_{1},{e}_{2},\... | Yes |
Corollary 3.3.6. Let \( G \) be a finite group. Then \( G \) is reductive. | Null | No |
Let \( \Gamma \) be a gallery of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{d}}\right) \) . If \( \Gamma \) is not minimal, then there is a gallery \( {\Gamma }^{\prime } \) with the same extremities as \( \Gamma \) such that \( {\Gamma }^{\prime } \) has type \( {\mathbf{s}}^{\prime } = \left( {{s}_{1},\ldots ,... | Proof. Since \( \Gamma \) is not minimal, Lemma 3.69 implies that the number of walls separating \( {C}_{0} \) from \( {C}_{d} \) is less than \( d \) . Hence the walls crossed by \( \Gamma \) cannot all be distinct; for if a wall is crossed exactly once by \( \Gamma \), then it certainly separates \( {C}_{0} \) from \... | Yes |
Each sequence from a subset \( U \subset C\left\lbrack {a, b}\right\rbrack \) contains a uniformly convergent subsequence; i.e., \( U \) is relatively sequentially compact, if and only if it is bounded and equicontinuous | Null | No |
Theorem 13.5. We have \[{\pi }_{1}\left( {\mathrm{{GL}}\left( {n,\mathbb{C}}\right) }\right) \cong {\pi }_{1}\left( {\mathrm{U}\left( n\right) }\right) ,\;{\pi }_{1}\left( {\mathrm{{SL}}\left( {n,\mathbb{C}}\right) }\right) \cong {\pi }_{1}\left( {\mathrm{{SU}}\left( n\right) }\right) ,\] and \[{\pi }_{1}\left( {\mathr... | Proof. First, let \( G = \mathrm{{GL}}\left( {n,\mathbb{C}}\right), K = \mathrm{U}\left( n\right) \), and \( P \) be the space of positive definite Hermitian matrices. By the Cartan decomposition, multiplication \( K \times P \rightarrow G \) is a bijection, and in fact, a homeomorphism, so it will follow that \( {\pi ... | Yes |
Let \( X = {\mathbb{R}}^{\mathbb{N}}, x = \left( {{x}_{0},{x}_{1},\ldots }\right) \) and \( y = \left( {{y}_{0},{y}_{1},\ldots }\right) \) . Define \( d\left( {x, y}\right) = \mathop{\sum }\limits_{n}\frac{1}{{2}^{n + 1}}\min \left\{ {\left| {{x}_{n} - {y}_{n}}\right| ,1}\right\} \). Then \( d \) is a metric on \( {\ma... | To see (iii), take two open balls \( B\left( {x, r}\right) \) and \( B\left( {y, s}\right) \) in \( X \). Let \( z \in \) \( B\left( {x, r}\right) \cap B\left( {y, s}\right) \). Take any \( t \) such that \( 0 < t < \min \{ r - d\left( {x, z}\right), s - d\left( {y, z}\right) \} \). By the triangle inequality we see th... | No |
Corollary 3.3.4. A point \( x \in X \) belongs to the Shilov boundary of \( A \) if and only if given any open neighbourhood \( U \) of \( x \), there exists \( f \in A \) such that \(\parallel f{\left| {}_{X \smallsetminus U}{\parallel }_{\infty } < \parallel f\right| }_{U}{\parallel }_{\infty }\) | Proof. First, let \( x \in X \smallsetminus \partial \left( A\right) \) . Then \( U = X \smallsetminus \partial \left( A\right) \) is an open neighbourhood of \( x \) and because \( \partial \left( A\right) \) is a boundary, we have for all \( f \in A \), \(\parallel f{\left| {}_{U}{\parallel }_{\infty } \leq \parallel... | Yes |
Theorem 2.2.1 (The division algorithm). Let \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) denote the polynomial ring in \( n \) variables over a field \( K \) and fix a monomial order \( < \) on \( S \) . Let \( {g}_{1},{g}_{2},\ldots ,{g}_{s} \) be nonzero polynomials of \( S \) . Then, given a polyno... | Proof (of Theorem 2.2.1). Let \( I = \left( {{\operatorname{in}}_{ < }\left( {g}_{1}\right) ,\ldots ,{\operatorname{in}}_{ < }\left( {g}_{s}\right) }\right) \) . If none of the monomials \( u \in \operatorname{supp}\left( f\right) \) belongs to \( I \), then the desired expression can be obtained by setting \( {f}^{\pr... | Yes |
Corollary 3. For some constant \( c = c\left( f\right) \), we have \({\operatorname{ord}}_{p}\mathop{\prod }\limits_{\substack{{\text{ cond }\psi = {p}^{t}} \\ {{n}_{0} \leq t \leq n} }}B\left( {\psi ,\mu }\right) = m{p}^{n} + {\lambda n} + c\left( f\right)\) | Since \(\mathop{\prod }\limits_{\substack{{\zeta {p}^{n} = 1} \\ {\zeta \neq 1} }}\left( {\zeta - 1}\right) = {p}^{n}\), the formula is immediate, since the product taken for \( {n}_{0} \leq t \leq n \) differs by only a finite number of factors (depending on \( {n}_{0} \) ) from the product taken over all \( t \), and... | Yes |
Proposition 5.42. Suppose \( X \) is a Hausdorff locally convex space, \( T : X \rightarrow X \) is a continuous linear transformation, and \( U \) is a barrel neighborhood of 0 subject to:\n(α) \( T\left( U\right) \) does not contain a nontrivial subspace of \( X \), and\n(β) \( T\left( U\right) \) is covered by \( N ... | Proof of Proposition 5.42: This is done using a series of steps.\nStep 1: \( \dim {K}_{1} \leq N \) . Suppose \( {v}_{1},\ldots ,{v}_{n} \) is a finite, linearly independent subset of \( {K}_{1} \) . Set \( {M}_{k} = \operatorname{span}\left\{ {{v}_{1},\ldots ,{v}_{k}}\right\} \) . Since \( \left( {I - T}\right) {M}_{k... | Yes |
Example 6.4.4. Example 6.4.3(3) allows us to define a family of orientations on complex projective spaces. | We refer to the orientations obtained in this way as the standard orientations, and the ones with the opposite sign for \( \left\lbrack {\mathbb{C}{P}^{n}}\right\rbrack \) as the nonstandard orientations.
We begin with \( n = 1 \) . We have the standard generator \( {\sigma }_{1} \in {H}_{1}\left( {S}^{1}\right) \) of... | Yes |
Prove that the integer program (8.1) can be written equivalently as \[
{z}_{I} = \max \;{cx}
\]
\[
x - y = 0
\]
\[
{A}_{1}x \leq {b}^{1}
\]
\[
{A}_{2}y \leq {b}^{2}
\]
\[
{x}_{j},{y}_{j} \in \mathbb{Z}\text{ for }j = 1,\ldots, p
\]
\[
x, y \geq 0
\] | Let \( \bar{z} \) be the optimal solution of the Lagrangian dual obtained by dualizing the constraints \( x - y = 0 \) . Prove that
\[
\bar{z} = \max \left\{ {{cx} : x \in \operatorname{conv}\left( {Q}_{1}\right) \cap \operatorname{conv}\left( {Q}_{2}\right) }\right\}
\]
where \( {Q}_{i} \mathrel{\text{:=}} \left\{ {... | No |
If \( \left( {T, S}\right) \) satisfies condition \( \left( \mathrm{C}\right) \), then \( T * S = S * T \) . | The second part of Proposition 2.6 allows us, by passing to the limit, to reduce the problem to the case of distributions with compact support, for which these properties were stated in Proposition 2.2. The reasoning is straightforward for the proof of parts 1 and 3 . We spell it out for part 2 . | No |
Proposition 5.5.6. Let \( \mathcal{B} \) be a Banach space and \( \left( {X,\mu }\right) \) a \( \sigma \) -finite measure space. (a) The set \( \left\{ {\mathop{\sum }\limits_{{j = 1}}^{m}{\chi }_{{E}_{j}}{u}_{j} : {u}_{j} \in \mathcal{B},{E}_{j} \subseteq X}\right. \) are pairwise disjoint and \( \left. {\mu \left( {... | If \( F \in {L}^{p}\left( {X,\mathcal{B}}\right) \) for \( 0 < p \leq \infty \), then \( F \) is \( \mathcal{B} \) -measurable; thus there exists \( {X}_{0} \subseteq X \) satisfying \( \mu \left( {X \smallsetminus {X}_{0}}\right) = 0 \) and \( F\left\lbrack {X}_{0}\right\rbrack \subseteq {\mathcal{B}}_{0} \), where \(... | No |
The vibrations of a string are modeled by the so-called wave equation \n\[
\frac{{\partial }^{2}w}{\partial {x}^{2}} = \frac{1}{{c}^{2}}\frac{{\partial }^{2}w}{\partial {t}^{2}}
\]\nwhere \( w = w\left( {x, t}\right) \) denotes the vertical elongation and \( c \) is the speed of sound in the string. | Null | No |
Show that if \( q \) is prime, then \(\frac{\varphi \left( {q - 1}\right) }{q - 1}\mathop{\sum }\limits_{{d \mid q - 1}}\frac{\mu \left( d\right) }{\varphi \left( d\right) }\mathop{\sum }\limits_{{o\left( \chi \right) = d}}\chi \left( a\right) = \left\{ \begin{array}{ll} 1 & \text{ if }a\text{ has order }q - 1 \\ 0 & \... | Null | No |
identify \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \) and \( {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) . | As a result we may identify \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \) and \( {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) . | No |
Corollary 9.25. For all \( s \in \mathbb{C}, s \notin \mathbb{N} \) , \(\Gamma \left( s\right) \Gamma \left( {1 - s}\right) = \frac{\pi }{\sin \left( {\pi s}\right) }.\) | Proof. By Theorem 9.20,\[
\Gamma \left( s\right) \Gamma \left( {-s}\right) = - \frac{1}{{s}^{2}}\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 + \frac{s}{n}\right) }^{-1}{e}^{s/n}\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - \frac{s}{n}\right) }^{-1}{e}^{-s/n}
\]
\[
= - \frac{1}{{s}^{2}}\mathop{\prod }\li... | Yes |
Proposition 11.2.4. Let \( \mathcal{C} \) be \( R \) -modules or Groups. Let \( Z \) be an object of \( \mathcal{C} \) and let \( {g}_{\alpha } : {X}_{\alpha } \rightarrow Z \) be a homomorphism for each \( \alpha \in \mathcal{A} \), such that \( {g}_{\alpha } = {g}_{\beta } \circ {f}_{\beta }^{\alpha } \) whenever \( ... | Null | No |
Proposition 9.2.1. Let \( \mathcal{A} \) (respectively, \( {\mathcal{A}}^{\prime } \) ) be the subalgebra of \( \operatorname{End}\left( Y\right) \) generated by \( \rho \left( K\right) \) (respectively, \( \rho \left( {K}^{\prime }\right) \) ). The following are equivalent: | The implication (2) \( \Rightarrow \) (1) follows directly from Theorem 4.2.1. Now assume that (1) holds and suppose \( {m}_{\pi ,{\pi }^{\prime }} = 1 \) for some pair \( \left( {\pi ,{\pi }^{\prime }}\right) \) . The \( {\pi }^{\prime } \) -isotypic subspace of \( Y \) (viewed as a \( {K}^{\prime } \) -module) is \( ... | Yes |
Theorem 2.2. A field extension \( K \) of a field \( F \) is separable if and only if, for every field \( L \) containing \( F \), the tensor product \( K{ \otimes }_{F}L \) has no non-zero nilpotent element. | Proof. First, suppose that \( K \) is separable over \( F \), and let \( u \) be a nilpotent element of \( K{ \otimes }_{F}L \) . We shall prove that \( u = 0 \) . Clearly, there is a field \( {K}_{1} \) between \( F \) and \( K \) that is finitely field-generated over \( F \) and such that \( u \) belongs to the canon... | Yes |
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