Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 1.3. If \( X, Y \) are right-invariant vector fields on \( G \), then \( \left\lbrack {X, Y}\right\rbrack \) is also a right-invariant vector field on \( G \) . | Proof. By Theorem 2.3 in Chapter 3,\n\n\[ \left\langle {X \land Y, d{\omega }^{i}}\right\rangle = X\left\langle {Y,{\omega }^{i}}\right\rangle - Y\left\langle {X,{\omega }^{i}}\right\rangle - \left\langle {\left\lbrack {X, Y}\right\rbrack ,{\omega }^{i}}\right\rangle . \]\n\n\( \left( {1.23}\right) \)\n\nFrom the struc... | Yes |
Calculation of the structure constants for the general linear group \( \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \) . | The elements of GL \( \left( {n;\mathbb{R}}\right) \) are \( n \times n \) nondegenerate real matrices. Suppose \( A = \left( {A}_{i}^{j}\right) \in \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \) . Then \( \left( {A}_{i}^{j}\right) ,\left( {1 \leq i, j \leq n}\right) \) is a coordinate system on the manifold \( \operator... | Yes |
Theorem 1.4. Suppose \( f : H \rightarrow G \) is a Lie group homomorphism. Then \( f \) induces a homomorphism \( {f}_{ * } : \mathcal{H} \rightarrow \mathcal{G} \) between the Lie algebras. If \( f \) is a Lie group isomorphism, then \( {f}_{ * } \) is an isomorphism between the Lie algebras. | Proof. Let \( {f}_{ * } \) denote the tangent map of the smooth map \( f \) . First we show that \( {f}_{ * } \) maps the right-invariant vector fields of the Lie group \( H \) to the right-invariant vector fields of the Lie group \( G \) . Choose any \( {X}_{e} \in {H}_{e} \), and let\n\n\[ \n{Y}_{\bar{e}} = {f}_{ * }... | Yes |
Suppose \( G = {\mathbb{T}}^{2} \) . Choose an irrational number \( \alpha \), and let\n\n\[ H = \{ \left( {t,{\alpha t}}\right), t \in \mathbb{R}\} /L \]\n\nwhere \( L = \left\{ {\left( {{n}_{1},{n}_{2}}\right) ,{n}_{i} \in \mathbb{Z}}\right\} \) . Then \( H \) is a Lie subgroup of \( G \), but not a regular submanifo... | By Theorem 1.4, because the imbedding map id \( : H \rightarrow G \) is a homomorphism of Lie groups, it induces a homomorphism of Lie algebras from \( \mathcal{H} \) to \( \mathcal{G} \) . Because \( {H}_{e} \) is a subspace of \( {G}_{e} \), the Lie algebra multiplication (1.26) of \( {G}_{e} \) restricted to \( {H}_... | No |
Theorem 2.1. Suppose \( X \) is a smooth tangent vector field on \( M \) . Then for any point \( p \in M \) there exists a neighborhood \( U \) and a local one-parameter group \( {\varphi }_{t} \) of diffeomorphisms on \( U,\left| t\right| < \epsilon \), such that \( {\left. X\right| }_{U} \) is precisely the tangent v... | Proof. Choose a local coordinate system \( \left( {V;{x}^{i}}\right) \) at \( p \) . Consider the system of ordinary differential equations\n\n\[\n\frac{d{x}^{i}}{dt} = {X}^{i},\;1 \leq i \leq m\n\]\n\n(2.9)\n\nwhere \( {X}^{i} \) are the components of the tangent vector field \( X \) with respect to the natural basis ... | Yes |
Theorem 2.2. Suppose \( {\varphi }_{t} \) is a one-parameter group of diffeomorphisms on a smooth manifold \( M \), and \( X \) is the induced tangent vector field of \( {\varphi }_{t} \) on \( M \) . If \( \psi : M \rightarrow M \) is a diffeomorphism, then \( {\psi }_{ * }X \) is the tangent vector field induced by t... | Proof. Suppose \( f \) is any smooth function on \( M \) . Then by definition we have\n\n\[ \left( {{\psi }_{ * }{X}_{p}}\right) f = {X}_{p}\left( {f \circ \psi }\right) \]\n\n\[ = {\left. \frac{d}{dt}f \circ \psi \left( {\varphi }_{t}\left( p\right) \right) \right| }_{t = 0} \]\n\n\[ = {\left. \frac{d}{dt}f\left( \psi... | Yes |
Theorem 2.3. Suppose \( X, Y \) are any two smooth tangent vector fields on a manifold \( M \) . If the local one-parameter group of diffeomorphisms generated by \( X \) is \( {\varphi }_{t} \), then\n\n\[ \left\lbrack {X, Y}\right\rbrack = \mathop{\lim }\limits_{{t \rightarrow 0}}\frac{Y - {\left( {\varphi }_{t}\right... | Proof. We only need to show the first equality. Suppose \( p \in M \), and \( f \) is a smooth function defined near \( p \) . Let\n\n\[ F\left( t\right) = f\left( {{\varphi }_{t}\left( p\right) }\right) \]\n\nSince\n\n\[ F\left( t\right) - F\left( 0\right) = {\int }_{0}^{1}\frac{{dF}\left( {st}\right) }{ds}{ds} = {\le... | Yes |
Suppose \( \operatorname{Ad} : G \rightarrow {GL}\left( {r;\mathbb{R}}\right) \) is the adjoint representation of the \( r \) -dimensional Lie group \( G \), and\n\n\[ \text{ ad } = {\left( \text{ Ad }\right) }_{ * } : {G}_{e} \rightarrow {gl}\left( {r;\mathbb{R}}\right) \]\n\nis the adjoint representation of the Lie a... | Proof. Suppose the one-parameter subgroup determined by \( X \) is \( {a}_{t} \) . Then the one-parameter group of diffeomorphisms determined by the corresponding right-invariant vector field \( \widetilde{X} \) is \( {\varphi }_{t} = {L}_{{a}_{t}} \) . Suppose the corresponding right-invariant vector field for \( Y \)... | Yes |
Theorem 3.1. Suppose there exist \( r \) differential 1-forms \( {\psi }^{i}\left( {1 \leq i \leq r}\right) \) on \( M \) that satisfy Equations (3.3), where \( {c}_{jk}^{i} \) are the structure constants of a Lie group \( G \) . Then there exists for every point \( p \in M \) a neighborhood \( U \) and a smooth map \(... | Proof. Consider the system of Pfaffian equations of \( m + r \) independent variables on \( M \times G \) :\n\n\[ \n{\theta }^{i} \equiv {\psi }^{i} - {\omega }^{i} = 0,\;1 \leq i \leq r. \n\]\n\n(3.6)\n\nSince the \( {\omega }^{i} \) are linearly independent everywhere, the \( {\theta }^{i} \) are also linearly indepe... | Yes |
Theorem 3.2. Suppose \( {\psi }_{\alpha } \) and \( {\psi }_{\beta \gamma } = - {\psi }_{\gamma \beta }\left( {1 \leq \alpha ,\beta ,\gamma \leq N}\right) \) are differential 1-forms depending on \( n \) variables. Then there exists a family of orthogonal frames depending on \( n \) parameters and having the given diff... | Proof. Let \( M \) denote the space of variables \( x = \left( {{x}^{1},\ldots ,{x}^{n}}\right) \), and \( G = E\left( N\right) \) . By theorem 3.1 there exists a map \( f : M \rightarrow E\left( N\right) \) such that\n\n\[ \n{f}^{ * }{\omega }_{\alpha } = {\psi }_{\alpha },\;{f}^{ * }{\omega }_{\alpha \beta } = {\psi ... | Yes |
Lemma 1. Suppose \( M \) is a 2-dimensional compact surface with constant total curvature \( K \) . Then \( K \) must be greater than zero. | Proof. Consider the function \( r\left( x\right) = x \cdot x \) . Since \( M \) is compact, there must exist a point \( {x}_{0} \in M \) such that \( r\left( x\right) \) attains its maximum value at \( x = {x}_{0} \) . Therefore\n\n\[{\left. dr\right| }_{{x}_{0}} = 0,{\left. \;{d}^{2}r\right| }_{{x}_{0}} \leq 0.\]\n\nS... | Yes |
Lemma 2. If \( M \) is a connected surface on which every point is an umbilical point, then \( M \) must be a sphere or a plane. | Proof. The condition for \( x \) to be an umbilical point on \( M \) is that the characteristic directions of the Weingarten transformation at \( x \) are not well-defined. Thus\n\n\[ d{e}_{3} + {\kappa dx} = 0 \]\n\n(4.38)\n\nwhere \( {e}_{3} \) is the normal vector to the surface and \( \kappa \) is the principal cur... | Yes |
Define a relation \( \sim \) among the elements in \( {C}_{m + 1} - \{ 0\} \) as follows:\n\n\[ \left( {{z}^{0},{z}^{1},\ldots ,{z}^{m}}\right) \sim \left( {{w}^{0},{w}^{1},\ldots ,{w}^{m}}\right) \]\n\nif and only if there exists a nonzero complex number \( \lambda \) such that\n\n\[ \left( {{z}^{0},{z}^{1},\ldots ,{z... | It is easy to verify that this is an equivalence relation. The \( m \) -dimensional complex projective space \( \mathbb{C}{P}_{m} \) is the quotient space \( \left( {{\mathbb{C}}_{m + 1}-\{ 0\} }\right) / \sim \) . An element in this space is denoted by \( \left\lbrack {{z}^{0},{z}^{1},\ldots ,{z}^{m}}\right\rbrack \) ... | Yes |
The orbit determined by the system of equations\n\n\\[ \n{P}_{l}\left( {{z}^{0},{z}^{1},\ldots ,{z}^{m}}\right) = 0,\;1 \leq l \leq q, \n\\]\n\nwhere each \( {P}_{l} \) is a homogeneous polynomial on \( \mathbb{C}{P}_{m} \), is called an algebraic variety. For instance, the complex manifold given by the equation\n\n\\[... | A theorem of W.-L. Chow (Chow's theorem) states that every compact submanifold imbedded in \( \mathbb{C}{P}_{m} \) is an algebraic variety. | Yes |
Example 4 (Complex torus). \( {\mathbb{C}}_{m} \) can be viewed as a \( {2m} \) -dimensional real vector space \( {\mathbb{R}}^{2m} \) . Choose \( {2m} \) real-linearly independent vectors \( \left\{ {v}_{\alpha }\right\} \) in \( {\mathbb{R}}^{2m} \) . They form the lattice\n\n\[ L = \left\{ {\mathop{\sum }\limits_{{\... | Topologically, the \( m \) -dimensional complex torus and the \( {2m} \) -dimensional real torus are homeomorphic. Yet the former has a complex manifold structure, and thus has richer contents. For instance, when \( m = 1 \), a holomorphic map from a complex torus to itself is conformal (angle preserving). Hence the an... | Yes |
Consider the transformation \( \alpha : {\mathbb{C}}_{m} - \) \( \{ 0\} \rightarrow {\mathbb{C}}_{m} - \{ 0\} \) such that\n\n\[ \alpha \left( {{z}^{1},\ldots ,{z}^{m}}\right) = 2\left( {{z}^{1},\ldots ,{z}^{m}}\right) .\n\]\n\nLet the discrete group generated by \( \alpha \) be denoted by \( \Delta \) . Then the quoti... | Topologically a Hopf manifold is homeomorphic to \( {S}^{{2m} - 1} \times {S}^{1} \) . To see this, we need only consider the annulus between the two concentric spheres \( {S}^{{2m} - 1}\left( 1\right) ,{S}^{{2m} - 1}\left( 2\right) \) with radii 1 and 2, respectively, in \( {\mathbb{C}}_{m} = {\mathbb{R}}^{2m} \) . (s... | No |
Suppose \( M \) is a 2-dimensional oriented surface with the Riemannian metric\n\n\[ d{s}^{2} = {\left( {\omega }_{1}\right) }^{2} + {\left( {\omega }_{2}\right) }^{2}. \]\n\nIf we assume that \( d{s}^{2} \) is analytic, then\n\n\[ d{s}^{2} = \left( {{\omega }_{1} + i{\omega }_{2}}\right) \left( {{\omega }_{1} - i{\ome... | The differential equation \( {\omega }_{1} + i{\omega }_{2} = 0 \) has an integrating factor \( \lambda \) such that\n\n\[ \lambda \left( {{\omega }_{1} + i{\omega }_{2}}\right) = {dz} \]\n\nThus\n\n\[ d{s}^{2} = \frac{1}{{\left| \lambda \right| }^{2}}{dzd}\bar{z} \]\n\nIf we let \( z = x + {iy} \), then\n\n\[ d{s}^{2}... | Yes |
Theorem 2.1. Suppose \( J \) is a complex structure on a real vector space \( V \) . Then the dimension \( m \) of \( V \) must be even, say \( m = {2n} \) . Moreover there exists a basis \( \left\{ {{e}_{j}, J{e}_{j},1 \leq j \leq n}\right\} \) for \( V \), and any two such bases give the same orientation to \( V \) . | Proof. The existence of bases of the form \( \left\{ {{e}_{j}, J{e}_{j},1 \leq j \leq n}\right\} \) was proved in the previous discussion. We only need to show that they determine the same orientation for \( V \) . As described above, there is a dual basis \( \left\{ {{e}^{*j}, - J{e}^{*j},1 \leq j \leq n}\right\} \) o... | Yes |
Theorem 2.2. Suppose \( V \) is a real \( {2n} \) -dimensional vector space. If there is a direct sum decomposition \( {V}_{\mathbb{C}} \oplus {\bar{V}}_{\mathbb{C}} \) of \( {V}^{ * } \otimes \mathbb{C} \) such that there is a one to one correspondence between \( {V}_{\mathbb{C}} \) and \( {\bar{V}}_{\mathbb{C}} \) un... | Proof. Define a linear transformation \( J : {V}^{ * } \otimes \mathbb{C} \rightarrow {V}^{ * } \otimes \mathbb{C} \) as follows:\n\n\[ \left\{ \begin{array}{ll} {Jf} = i \cdot f, & f \in {V}_{\mathbb{C}}, \\ {Jf} = - i \cdot f, & f \in {\bar{V}}_{\mathbb{C}}. \end{array}\right. \]\n\n(2.19)\n\nSince \( {V}^{ * } \otim... | Yes |
Theorem 3.1. An almost complex manifold must be an orientable manifold of even dimension. | Suppose \( M \) is an almost complex manifold of dimension \( m = {2n} \) . Let \( A \) denote the space of smooth complex-valued differential \( \left( {1,0}\right) \) -forms, and \( \bar{A} \) the space of forms which are complex conjugates of elements of \( A \) . Then at every point \( x \in M \) there is a direct ... | Yes |
Theorem 3.3. If there is an integrable almost complex structure on a manifold \( M \), then it must be a canonical almost complex structure induced from a complex manifold structure. | Newlander and Nirenberg gave a proof of the above theorem under the assumption of smoothness of the almost complex structure (Newlander and Nirenberg, 1957). Nijenhuis and Woolf, Kohn, and Hörmander further proved the theorem under weaker differentiability conditions. These theorems are all very difficult, and will not... | No |
Theorem 3.5. Suppose \( J \) is an almost complex structure on a manifold \( M \) . Then a necessary and sufficient condition for \( J \) to be integrable is\n\n\[ d = \partial + \bar{\partial } \] | Proof. (Sufficiency) Suppose \( {\theta }^{k}\left( {1 \leq k \leq n}\right) \) are locally linearly independent differential \( \left( {1,0}\right) \) -forms with respect to \( J \) . Because \( d = \partial + \bar{\partial } \), we have\n\n\[ \mathop{\prod }\limits_{{0,2}}d{\theta }^{k} = 0 \]\n\nthat is, the integra... | Yes |
Theorem 3.6. An almost complex structure \( J \) on a manifold \( M \) is integrable if and only if\n\n\[{\bar{\partial }}^{2} = 0.\] | Proof. (Necessity) Suppose \( J \) is integrable. Then \( d = \partial + \bar{\partial } \) . Hence\n\n\[0 = {d}^{2} = {\partial }^{2} + \left( {\partial \circ \bar{\partial } + \bar{\partial } \circ \partial }\right) + {\bar{\partial }}^{2}.\n\nSuppose \( \omega \in {A}_{p, q} \) . Then\n\n\[{\partial }^{2}\omega \in ... | Yes |
Theorem 3.7. Suppose \( f \) is a smooth, complex-valued function on a complex manifold \( M \) . Then a necessary and sufficient condition for \( f \) to be a holomorphic function is that \( \bar{\partial }f = 0 \) . | Proof. The Cauchy-Riemann condition for \( f \) is\n\n\[\n\frac{\partial g}{\partial {x}^{k}} = \frac{\partial h}{\partial {y}^{k}},\;\frac{\partial g}{\partial {y}^{k}} = - \frac{\partial h}{\partial {x}^{k}},\;1 \leq k \leq n.\n\]\n\nBy (3.32) and (3.36), the above condition is equivalent to \( \bar{\partial }f = 0 \... | Yes |
Theorem 5.1. Suppose \( M \) is a Hermitian manifold. A necessary and sufficient condition for \( D \) to be a type \( \left( {1,0}\right) \) connection on the tangent bundle of \( M \) is that its torsion matrix is composed of differential \( \left( {2,0}\right) \) -forms. | Proof. Suppose \( \sigma = \left( {{\sigma }^{1},\ldots ,{\sigma }^{m}}\right) \) is the coframe field dual to the holomorphic frame field \( S \) . Then every \( {\sigma }^{i} \) is a holomorphic differential \( \left( {1,0}\right) \) -form, that is, for the complex local coordinate system \( {z}^{i},{\sigma }^{i} \) ... | Yes |
Theorem 5.2. A necessary and sufficient condition for a Hermitian manifold \( M \) to be a Kählerian manifold is that the torsion matrix of the Hermitian connection on \( M \) is zero. | Proof. Obviously, the two conditions mentioned in this theorem are both independent of the choice of frame fields. Hence we only need to prove the theorem with respect to the natural frame field (5.1). Suppose the coframe field dual to \( \left( {5.1}\right) \) is\n\n\[ \sigma = \left( {d{z}^{1},\ldots, d{z}^{m}}\right... | Yes |
Theorem 5.3. Suppose \( N \) is a Kählerian manifold, and \( f : M \rightarrow N \) a holomorphic immersion. Then there is an induced Kählerian structure on \( M \) from \( N \) . | Proof. Suppose \( p \in M,\left( {{z}^{1},\ldots ,{z}^{n}}\right) \) are the complex coordinates of the point \( q = f\left( p\right) \) in \( N \), and \( \left( {{w}^{1},\ldots ,{w}^{m}}\right) \) the complex coordinates of the point \( p \) in \( M \) . Then the map \( f \) can be expressed locally by\n\n\[ \n{z}^{\... | Yes |
Lemma 1. The Hilbert form on PTM given by\n\n\[ \n\\omega = \\frac{\\partial F}{\\partial {X}^{i}}d{u}^{i} \n\]\n\nsatisfies the condition\n\n\[ \n\\omega \\land {\\left( d\\omega \right) }^{m - 1} \\neq 0.\n\] | Proof. Let \( A = \\omega \\land {\\left( d\\omega \right) }^{m - 1} \) . We use (2.19) for \( {d\\omega },\\left( {2.20}\\right) \) for \( {\\omega }_{\\alpha }^{m} \), and the choice \( {\\omega }^{m} = \\omega \) to obtain\n\n\[ \nA = \\pm \\left( {m - 1}\\right) !\\mathop{\\bigwedge }\\limits_{i}{\\omega }^{i}\\mat... | Yes |
Theorem 3.1 (Chern). Let \( M \) be an \( m \) -dimensional Finsler manifold with Finsler function \( F \) . Suppose \( {e}_{i} = {p}_{i}^{j}\frac{\partial }{\partial {u}^{j}}, i = 1,\ldots, m \), is an orthonormal frame field on the bundle \( {p}^{ * }{TM} \rightarrow {PTM} \) and \( {\omega }^{i} = {q}_{j}^{i}d{u}^{j... | We will now fix the \( {\xi }_{i}^{\alpha } \) introduced in (3.7) and (3.8) by requiring that \( {\omega }_{\alpha }^{m} \) and \( {\omega }_{m}^{\alpha } \) are negatives of each other, that is, \[ {\omega }_{\alpha }^{m} + {\omega }_{m}^{\alpha } = 0. \] (3.9) Note that this condition, together with the condition \(... | Yes |
Theorem 3.2 (Chern). The Chern connection forms \( {\omega }_{i}^{j} \) on the Finsler bundle \( {p}^{ * }{TM} \rightarrow {PTM} \) are unique solutions of the structure equations\n\n\[ d{\omega }^{i} = {\omega }^{j} \land {\omega }_{j}^{i},\;\text{(torsion-freeness)} \]\n\nand\n\n\[ {\omega }_{ij} + {\omega }_{ji} = -... | The Finsler metric \( {g}_{ij} \) is Riemannian if and only if the Cartan tensor vanishes. Furthermore, no torsion free connection can be entirely metric-compatible at the same time unless the Finsler structure is Riemannian.\n\nThus our development includes Riemannian geometry as a special case, and the Chern connecti... | No |
Lemma 1. The second Chern curvature tensor \( {P}_{ik\alpha }^{j} \) vanishes if and only if the horizontal part of the covariant derivative of the Cartan tensor, \( {Q}_{ki\alpha s} \) , vanishes. | \[ {P}_{kjl}^{i} = - \frac{\delta {\Gamma }_{kj}^{i}}{\delta {X}^{l}} = - F\frac{\partial {\Gamma }_{kj}^{i}}{\partial {X}^{l}}. \] | No |
Theorem 4.1. This first Chern curvature tensor \( {R}_{ijkl} \) (the \( h \) - \( h \) part) on a Finsler manifold satisfies the following relations:\n\n1) \( {R}_{ijkl} + {R}_{jikl} = - 2{A}_{ija}{R}_{bkl}^{a}\frac{{X}^{b}}{F} \equiv 2{B}_{ijkl} \), \n\n2) \( {R}_{ijkl} + {R}_{kjli} + {R}_{ljik} = 0\; \) (Bianchi iden... | Proof. Property 1) is the counterpart of (4.16) and follows directly from it on recalling (2.14). Properties 2) and 4) are counterparts of (4.7) and (4.4), respectively. To obtain property 3), we cyclically permute the set (ijkl) to obtain three more versions of the Bianchi identity in addition to 2) and then add the f... | Yes |
Lemma 2. Let \( \left( {M, F}\right) \) be a Finsler manifold. The following statements are equivalent.\n\n1) \( M \) is a Berwald space;\n\n2) \( Q \) (the horizontal part of the covariant derivative of the Cartan tensor) vanishes;\n\n3) The Chern connection coefficients \( {\Gamma }_{kj}^{i} \) are independent of \( ... | In view of property 3) and (4.29), we have\n\n\[ \frac{\delta {\Gamma }_{kl}^{i}}{\delta {u}^{j}} = \frac{\partial {\Gamma }_{kl}^{i}}{\partial {u}^{j}} \]\n\n\( \left( {4.40}\right) \)\n\nand\n\n\[ {R}_{kjl}^{i} = \frac{\partial {\Gamma }_{kl}^{i}}{\partial {u}^{j}} - \frac{\partial {\Gamma }_{kj}^{i}}{\partial {u}^{l... | No |
Lemma 1 (The Gauss Lemma). A radial geodesic \( \sigma \left( t\right) \) with velocity \( T\left( t\right) \) intersects the geodesic spheres orthogonally with respect to the scalar product \( {G}_{T} \) . | Proof. For \( X \in {S}_{x}\left( \delta \right) \) and \( \tau \in \left\lbrack {0,1}\right\rbrack \), consider the radial geodesic\n\n\[ \sigma \left( t\right) = {\exp }_{x}\left( {t\tau X}\right) ,\;0 \leq t \leq 1. \]\n\n(5.40)\n\nNote that \( {\tau X} \in {S}_{x}\left( {\tau \delta }\right) \) . Let \( Y\left( u\r... | Yes |
Lemma 2. For \( \left( {u, X}\right) ,\left( {u, V}\right) \in {PTM} \) ,\n\n\[ \n{\left( \frac{\partial F}{\partial {X}^{i}}\right) }_{\left( u, X\right) }{V}^{i} \leq F\left( {u, V}\right) \n\] | Proof. It follows from the positive-definiteness of the Finsler metric that, for \( u \in M \) and any \( X, V, W \in {T}_{u}M \) ,\n\n\[ \n{G}_{\left( u, X\right) }\left( {V, W}\right) \leq \sqrt{{G}_{\left( u, X\right) }\left( {V, V}\right) {G}_{\left( u, X}\right) }\left( {W, W}\right) } \n\]\n\n(5.44)\n\nwhere the ... | Yes |
Lemma 5. Suppose \( \sigma \left( t\right) ,0 \leq t \leq a \), is a geodesic containing no conjugate points. Let \( W \) and \( J \) be smooth fields on \( \sigma \) such that \( W\left( 0\right) = J\left( 0\right), W\left( a\right) = \) \( J\left( a\right) \), and \( J \) is a Jacobi field. Then\n\n\[ I\left( {W, W}\... | Proof. By assumption \( J - W \in {\mathcal{V}}_{0} \) . Lemma 3 then implies\n\n\[ 0 \leq I\left( {J - W, J - W}\right) \]\n\n\[ = I\left( {J, J}\right) - {2I}\left( {J, W}\right) + I\left( {W, W}\right) \]\n\n\[ = {\left. \frac{1}{F}{G}_{T}\left( {D}_{T}J, J\right) \right| }_{0}^{a} - {\left. \frac{2}{F}{G}_{T}\left(... | Yes |
Lemma 6. \( I\left( {V, W}\right) = 0 \) for all \( W \in {\mathcal{V}}_{0} \) if and only if \( V \) is a Jacobi field. | Proof. If \( V \) is a Jacobi field, \( I\left( {V, W}\right) = 0 \) for all \( W \in {\mathcal{V}}_{0} \) follows directly from (6.19). Conversely, assume \( I\left( {V, W}\right) = 0 \) for all \( W \in {\mathcal{V}}_{0} \) . Let \( f\left( t\right) \) be a smooth function on \( \left\lbrack {0, a}\right\rbrack \) sa... | Yes |
Not every metric space has rays. For instance, there are no rays in a compact metric space. | Indeed, suppose \( p\left( t\right) ,0 \leq t < 1 \), is a ray in a compact metric space \( M \) . Then the limit \( \mathop{\lim }\limits_{{t \rightarrow 1}}p\left( t\right) = {p}_{0} \in M \) exists. Since any ray is a closed subset of \( M,{p}_{0} \) is on the ray, i.e., there is a \( {t}_{0},0 \leq {t}_{0} < 1 \) ,... | Yes |
Lemma 1. Suppose there is a sequence of points \( {a}_{1},\ldots ,{a}_{n} \) in \( M \) which satisfies the condition\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{{n - 1}}\rho \left( {{a}_{i},{a}_{i + 1}}\right) = \rho \left( {{a}_{1},{a}_{n}}\right) \]\n\n(7.2)\n\nThen for any set of integers \( 1 \leq {i}_{1} \leq \cdots \... | Proof. Suppose the lemma is false. Then there exists a set of integers \( 1 \leq {i}_{1} \leq \) \( \cdots \leq {i}_{k} \leq n \) such that\n\n\[ \mathop{\sum }\limits_{{r = 1}}^{{k - 1}}\rho \left( {{a}_{{i}_{r}},{a}_{{i}_{r + 1}}}\right) > \rho \left( {{a}_{{i}_{1}},{a}_{{i}_{k}}}\right) \]\n\nThus\n\n\[ \mathop{\sum... | Yes |
Lemma 2. Suppose \( {a}_{k}{a}_{k + 1}\left( {1 \leq k \leq n - 1}\right) \) are line segments in a metric space \( M \), and\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{{n - 1}}\rho \left( {{a}_{k},{a}_{k + 1}}\right) = \rho \left( {{a}_{1},{a}_{n}}\right) \]\n\n(7.4)\n\nThen \( \gamma = \mathop{\sum }\limits_{{k = 1}}^{{n... | Proof. Suppose \( n = 3 \) . Obviously \( {a}_{1}{a}_{2} + {a}_{2}{a}_{3} \) is an arc. We need to show that it is also a line segment. Suppose \( x, y, z \) are any three points in the arc, and \( y \) lies between \( x \) and \( z \) . If \( x \) and \( z \) both belong to \( {a}_{1}{a}_{2} \) or \( {a}_{2}{a}_{3} \)... | No |
Lemma 4. Suppose \( \gamma \) is a curve in \( M \) connecting points \( p \) and \( q \) with measurable arc length equal to \( \rho \left( {p, q}\right) \) . Then \( \gamma \) is a geodesic curve in the Finsler manifold \( M \), and is also a line segment in the metric space \( M \) . Conversely, if \( \gamma \) is a... | Proof of Lemma 4. Let \( r \) be a point on the curve \( \gamma \) . Choose a normal coordinate neighborhood \( U \) of \( r \) such that it satisfies the requirements of Theorem 2.4 in Chapter Five. Then choose a point \( {r}_{1} \in \gamma \cap U \) such that the part of \( \gamma \) between \( r \) and \( {r}_{1} \)... | Yes |
Theorem 7.2 (Hopf-Rinow). Suppose \( M \) is a connected Finsler manifold. Then the following statements are equivalent\n\n1) \( M \) is complete;\n\n2) any geodesic curve in \( M \) can be infinitely extended;\n\n3) every closed and bounded subset of \( M \) is compact. | Proof. 1) \( \Rightarrow \) 2). Suppose there is a geodesic curve \( \gamma \) in \( M \) of finite length \( L \) that starts from \( p \in M \) and that cannot extended. Then the curve can be expressed as \( p\left( t\right) ,0 \leq t < 1 \), and \( \mathop{\lim }\limits_{{t \rightarrow 1}}p\left( t\right) \) does no... | No |
Corollary 1. In a complete Finsler manifold, any two points can be connected by a minimal geodesic curve. | Proof. Because geodesic curves can be extended infinitely in a complete Finsler manifold, there are no rays with finite length in such a manifold. By Theorem 7.1, any two points can be connected by a line segment, that is, by a minimal geodesic. | Yes |
Corollary 2. A compact connected Finsler manifold is complete. | Proof. Because any infinite subset of a compact metric space has a limit point, a compact connected Finsler manifold is complete by the Hopf-Rinow Theorem. | Yes |
Theorem 7.3. A complete Finsler manifold is non-extendable. | Proof. Let \( M \) be a complete Finsler manifold. If \( M \) is a proper open sub-manifold of another connected Finsler manifold \( {M}^{\prime } \), then we may choose a boundary point \( p \in \left( {{M}^{\prime } - M}\right) \cap \bar{M} \) . By Lemma 3, there exist an \( \epsilon \) -ballshaped neighborhood \( U ... | Yes |
Theorem 8.1 (Bonnet-Myers). Let \( M \) be a complete connected Finsler manifold. If the Ricci curvature along any direction in \( {T}_{p}M \) for all \( p \in M \) has a positive lower bound:\n\n\[ {\operatorname{Ric}}_{p} \geq \frac{1}{{r}^{2}} > 0 \]\n\nthen\n\na) \( M \) is compact with diameter \( \leq {\pi r} \),... | Proof of Theorem 8.1. a) Let \( p, q \) be any two points in \( M \) . By Corollary 1 to the Hopf-Rinow Theorem (Theorem 7.2), there exists a minimal geodesic connecting \( p \) and \( q \) . We shall suppose this to be a unit speed geodesic and call it \( \gamma : \left\lbrack {0, a}\right\rbrack \rightarrow M,\gamma ... | Yes |
Theorem 8.2 (Synge). If a Finsler manifold \( M \) is compact, orientable, even-dimensional, and has positive flag curvatures, then \( M \) is simply connected. | To prove this theorem, we need the following lemma.\n\nLemma 1. In a compact, | No |
Lemma 1. In a compact, connected Finsler manifold \( M \), every free homotopy class of loops has a minimal, closed geodesic. | Proof. Suppose \( {\pi }_{1}\left( M\right) \neq 1 \) . As in the proof of part b) of Theorem 8.1, introduce the universal covering space \( \widetilde{M} \) of \( M \) with covering projection \( \pi : \widetilde{M} \rightarrow M \) and pull-back Finsler structure \( \widetilde{F} = {\pi }^{ * }F \) . Then \( \pi \) i... | Yes |
Lemma 1.4 (Riemann-Lebesgue). If \( f \in {L}^{1}\left( \mathbb{T}\right) \), then \( \mathop{\lim }\limits_{{k \rightarrow \infty }}\widehat{f}\left( k\right) = 0 \) . | Proof. \( \widehat{f}\left( k\right) = {\int }_{0}^{1}f\left( x\right) {e}^{-{2\pi ikx}}{dx} = - {\int }_{0}^{1}f\left( x\right) {e}^{-{2\pi ik}\left( {x + \frac{1}{2k}}\right) }{dx} = \n\n- {\int }_{0}^{1}f\left( {x - \frac{1}{2k}}\right) {e}^{-{2\pi ikx}}{dx},\widehat{f}\left( k\right) = \frac{1}{2}{\int }_{0}^{1}\le... | Yes |
Lemma 1.7. \( {L}_{N} = \frac{4}{{\pi }^{2}}\ln N + O\left( 1\right) \) . | Proof. \( {L}_{N} = 2{\int }_{0}^{1/2}\left| \frac{\sin \left( {\pi \left( {{2N} + 1}\right) t}\right) }{\pi t}\right| {dt} + {O}_{1}\left( 1\right) = 2{\int }_{0}^{N + 1/2}\left| \frac{\sin \left( {\pi t}\right) }{\pi t}\right| {dt} + {O}_{1}\left( 1\right) = \n\n2\mathop{\sum }\limits_{{k = 1}}^{N}{\int }_{k - 1/2}^{... | Yes |
Lemma 1.8. 若 \( 1 \leq p < \infty \) 则(1) 成立 \( \Leftrightarrow \exists {C}_{p} > 0 \) (只与 \( p \) 有关) 使得\n\n\[{\begin{Vmatrix}{S}_{N}f\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p},\;\forall f \in {L}^{p}\left( \mathbb{T}\right) . | (3)对 \( 1 < p < \infty \) 成立(see Corollary 3.10).\n\n若 \( p = 1 \) 则 \( {\begin{Vmatrix}{S}_{N}\end{Vmatrix}}_{{L}^{1} \rightarrow {L}^{1}} = {L}_{N} \rightarrow \infty \Rightarrow \left( 1\right) 不成立.\n\n\( \left( {{f}_{n} = n{\chi }_{\left( 0,1/n\right) },{\begin{Vmatrix}{S}_{N}{f}_{n}\end{Vmatrix}}_{1} \rightarrow {... | No |
Theorem 1.9. If \( f \in {L}^{p}\left( \mathbb{T}\right) ,1 \leq p < \infty \) or \( f \in C\left( \mathbb{T}\right), p = \infty \), then \( \mathop{\lim }\limits_{{N \rightarrow \infty }}{\begin{Vmatrix}{\sigma }_{N}f - f\end{Vmatrix}}_{p} = 0. \) | Proof. Key point: \( \mathop{\lim }\limits_{{t \rightarrow 0}}\parallel f\left( {\cdot - t}\right) - f\left( \cdot \right) {\parallel }_{p} = 0.\;{\int }_{-1/2}^{1/2}{F}_{N}\left( t\right) {dt} = 0 \Rightarrow \)\n\n\( {\sigma }_{N}f\left( x\right) - f\left( x\right) = {\int }_{-1/2}^{1/2}\left( {f\left( {x - t}\right)... | Yes |
Corollary 1.10. (i)三角多项式 \( {\mathcal{P}}_{1} \mathrel{\text{:=}} \left\{ {\mathop{\sum }\limits_{{k = - N}}^{N}{c}_{k}{e}^{2\pi ikx} \mid {c}_{k} \in \mathbb{C}, N \in {\mathbb{Z}}_{ + }}\right\} \) 在 \( {L}^{p}\left( \mathbb{T}\right) \) 中稠密 \( \left( {1 \leq p < \infty }\right) \) . (ii) 若 \( f \in {L}^{1}\left( \ma... | Key point: as \( {\sigma }_{N}f\left( x\right) = \mathop{\sum }\limits_{{k = - N}}^{N}\frac{N + 1 - \left| k\right| }{N + 1}\widehat{f}\left( k\right) {e}^{2\pi ikx} \) then (i) \( {\sigma }_{N}f \in {\mathcal{P}}_{1} \) ;\n\n(ii) \( \widehat{f}\left( k\right) = 0,\forall k \in \mathbb{Z} \Rightarrow {\sigma }_{N}f = 0... | Yes |
Theorem 1.11. If \( f \in {L}^{p}\left( \mathbb{T}\right) ,1 \leq p < \infty \) or \( f \in C\left( \mathbb{T}\right), p = \infty \), then \[ \mathop{\lim }\limits_{{r \rightarrow 1 - }}{\begin{Vmatrix}{P}_{r} * f - f\end{Vmatrix}}_{p} = 0. \] | Proof of \( \mathop{\lim }\limits_{{N \rightarrow \infty }}{\sigma }_{N}f\left( x\right) = f\left( x\right) ,\mathop{\lim }\limits_{{r \rightarrow 1 - }}{P}_{r} * f\left( x\right) = f\left( x\right) \), a.e. \( x \), in section 2.4. (在Lebesgue点成立) | No |
Lemma 1.12. 若 \( f\left( x\right) = {e}^{-\pi {\left| x\right| }^{2}} \) 则 \( \widehat{f}\left( \xi \right) = {e}^{-\pi {\left| \xi \right| }^{2}} \) . | Proof. 由Fubini定理,只需证明 \( n = 1 \) 时成立. 由 \( {f}^{\prime } + {2\pi xf} = 0 \) 和(12), (13)得 \( {2\pi i\xi }\widehat{f} + i\frac{\partial \widehat{f}}{\partial \xi } = 0 \) ,解得 \( \widehat{f}\left( \xi \right) = {e}^{-\pi {\left| \xi \right| }^{2}}\widehat{f}\left( 0\right) \) . 而 \( \widehat{f}\left( 0\right) = {\int }_{... | Yes |
Theorem 1.14. 若 \( f \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \) 则 \( \widehat{f} \in {L}^{2}\left( {\mathbb{R}}^{n}\right) ,\parallel \widehat{f}{\parallel }_{2} = \parallel f{\parallel }_{2} \) . | Proof. \( \forall \phi \in \mathcal{S} \) 有 \( \left| {\langle \widehat{f},\phi \rangle }\right| = \left| {\langle f,\widehat{\phi }\rangle }\right| \leq \parallel f{\parallel }_{2}\parallel \widehat{\phi }{\parallel }_{2} = \parallel f{\parallel }_{2}\parallel \phi {\parallel }_{2} \) ,这说明 \( \widehat{f} \in {L}^{2}\l... | Yes |
Corollary 1.16 (Hausdorff-Young不等式). 若 \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right), p \in \left\lbrack {1,2}\right\rbrack \) ,则 \( \widehat{f} \in {L}^{{p}^{\prime }}\left( {\mathbb{R}}^{n}\right) ,\parallel \widehat{f}{\parallel }_{{p}^{\prime }} \leq \parallel f{\parallel }_{p} \) . (最佳常数 \( {\left| \frac{{p}^{1/p... | Proof. 只需证明若 \( f, g \) 是简单可测函数则 \( \left| {{\int }_{{\mathbb{R}}^{n}}\widehat{f}{gdx}}\right| \leq \parallel f{\parallel }_{p}\parallel g{\parallel }_{p} \) . Normalize \( \parallel f{\parallel }_{p} = \parallel g{\parallel }_{p} = 1 \) . \( F\left( z\right) = {\int }_{{\mathbb{R}}^{n}}{\left| \widehat{{\left| f\right... | Yes |
Corollary 1.18 (卷积Young不等式). 若 \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right), g \in {L}^{q}\left( {\mathbb{R}}^{n}\right), p, q, r \in \)\n\n\( \left\lbrack {1,\infty }\right\rbrack ,1 + \frac{1}{r} = \frac{1}{p} + \frac{1}{q} \) 则 \( \parallel f * g{\parallel }_{r} \leq \parallel f{\parallel }_{p}\parallel g{\paralle... | Proof. \( f * g\left( x\right) = {\int }_{{\mathbb{R}}^{n}}f\left( y\right) g\left( {x - y}\right) {dy} \) . Normalize \( \parallel f{\parallel }_{p} = \parallel g{\parallel }_{q} = 1 \) .\n\nCase 1: \( r = \infty \) . Then \( q = {p}^{\prime } \) ,(by Hölder) \( \left| {f * g\left( x\right) }\right| \leq \parallel f{\... | Yes |
Theorem 2.1. 若 \( \phi \in {L}^{1}\left( {\mathbb{R}}^{n}\right) ,\int \phi = A \) ,则 \( \mathop{\lim }\limits_{{t \rightarrow 0 + }}{\begin{Vmatrix}{\phi }_{t} * f - Af\end{Vmatrix}}_{p} = 0,\forall f \in \) \( {L}^{p}\left( {\mathbb{R}}^{n}\right) ,1 \leq p < \infty \) or \( f \in {C}_{0}\left( {\mathbb{R}}^{n}\right... | Proof. \( {\phi }_{t} * f\left( x\right) - {Af}\left( x\right) = {\int }_{{\mathbb{R}}^{n}}\phi \left( y\right) \left( {f\left( {x - {ty}}\right) - f\left( x\right) }\right) {dy} \) . 由Minkowski不等式得 \( {\begin{Vmatrix}{\phi }_{t} * f - Af\end{Vmatrix}}_{p} \leq {\int }_{{\mathbb{R}}^{n}}\left| {\widetilde{\phi }\left( ... | Yes |
Theorem 2.3. 若 \( T \) 是弱 \( \left( {p, q}\right) \) 型次线性算子,则\n\n\( \left\{ {f \in {L}^{p}\left( {X,\mu }\right) \mid {Tf}\left( x\right) = 0\text{a.e.}}\right\} \) 是 \( {L}^{p}\left( {X,\mu }\right) \) 中的闭集. | Proof. 若 \( {f}_{n} \in {L}^{p}\left( {X,\mu }\right), T{f}_{n}\left( x\right) = 0 \) a.e., \( {f}_{n} \rightarrow f \) in \( {L}^{p} \) ,则 \( \left| {{Tf}\left( x\right) }\right| \leq \n\n\( \left| {T\left( {f - {f}_{n}}\right) \left( x\right) }\right| + \left| {T{f}_{n}\left( x\right) }\right| = \left| {T\left( {f - ... | Yes |
Lemma 2.4. 若 \( \mathcal{F} = {\left\{ {B}_{j} = B\left( {x}_{j},{r}_{j}\right) \right\} }_{j = 1}^{N} \) 是度量空间 \( \left( {X, d}\right) \) 中的开球, \( m{B}_{j} \) \( = B\left( {{x}_{j}, m{r}_{j}}\right) ,\left( {B\left( {x, r}\right) = \{ y \in X : d\left( {x, y}\right) < r\} }\right) \) . 则 \( \exists {\left\{ {B}_{i}^{\... | Proof. 不妨设 \( {r}_{1} \geq {r}_{2} \geq \cdots \geq {r}_{N} > 0 \) . 归纳定义 \( {B}_{N + 1} = \varnothing ,{j}_{1} = 1,{j}_{k + 1} = \) \( \min \left\{ {j : {B}_{j} \cap {B}_{{j}_{m}} = \varnothing ,\forall 1 \leq m \leq k}\right\}, l = \sup \left\{ {k : {j}_{k} \leq N}\right\} \) . 则 \( {B}_{m}^{\prime } = {B}_{{j}_{m}} ... | Yes |
Theorem 2.5. 若 \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \) ,则 \( \parallel {Mf}{\parallel }_{1,\infty } \leq {3}^{n}\parallel f{\parallel }_{1} \) . | Proof. \( \forall \lambda > 0 \) 设 \( {E}_{\lambda } = \left\{ {x \in {\mathbb{R}}^{n} : \widetilde{M}f\left( x\right) > \lambda }\right\} \) ,\n\n\( \mathcal{F} = \left\{ {B\left( {x, r}\right) : {\int }_{B\left( {x, r}\right) }\left| f\right| {dy} > \lambda \left| {B\left( {x, r}\right) }\right| }\right\} \) ,则 \( {E... | Yes |
若 \( \phi \in {\mathcal{V}}_{0}\left( {\mathbb{R}}^{n}\right), f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) 则 \( \mathop{\sup }\limits_{{t > 0}}\left| {{\phi }_{t} * f\left( x\right) }\right| \leq \parallel \phi {\parallel }_{1}{Mf}\left( x\right) \) . | Proof. \( \left\lbrack {{\phi }_{t}\left( x\right) = {t}^{-n}\phi \left( {x/t}\right) ,\phi \in {\mathcal{V}}_{0}}\right\rbrack \Rightarrow \left\lbrack {{\phi }_{t} \in {\mathcal{V}}_{0},{\begin{Vmatrix}{\phi }_{t}\end{Vmatrix}}_{1} = \parallel \phi {\parallel }_{1};\left| {{\phi }_{t} * f\left( x\right) }\right| \leq... | Yes |
若 \( \phi \in {\mathcal{V}}_{1}\left( {\mathbb{R}}^{n}\right) \) 则 \( f \mapsto \mathop{\sup }\limits_{{t > 0}}\left| {{\phi }_{t} * f\left( x\right) }\right| \) 弱 \( \left( {1,1}\right) \) (且强 \( \left( {p, p}\right) ,\forall 1 < \) \( p \leq \infty ).{\mathcal{V}}_{1} = {\mathcal{V}}_{1}\left( {\mathbb{R}}^{n}\right)... | Keypoint: \( \left| {{\phi }_{t} * f\left( x\right) }\right| \leq {\psi }_{t} * \left| f\right| \left( x\right) \leq \parallel \psi {\parallel }_{1}{Mf}\left( x\right) ;{Mf} \) 弱 \( \left( {1,1}\right) \) ,强 \( \left( {p, p}\right) \) | Yes |
Corollary 2.8. 若 \( 1 \leq p \leq \infty, f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) ,\phi \in {\mathcal{V}}_{1}\left( {\mathbb{R}}^{n}\right) \) 则 \( \mathop{\lim }\limits_{{t \rightarrow 0 + }}{\phi }_{t} * f\left( x\right) = \) \( \left( {\int \phi }\right) f\left( x\right) \) a.e. | Proof. 设 \( \int \phi = A \) . (i) \( p = 1 \) 设 \( {\Omega f}\left( x\right) = \mathop{\limsup }\limits_{{t \rightarrow 0 + }}\left| {{\phi }_{t} * f\left( x\right) - {Af}\left( x\right) }\right| \) ,则 \( \Omega \) 次线性, \( \left| {{\Omega f}\left( x\right) }\right| \leq \mathop{\sup }\limits_{{t > 0}}\left| {{\phi }_{... | No |
Proposition 3.1. \( \mathop{\lim }\limits_{{t \rightarrow 0 + }}{Q}_{t} = \frac{1}{\pi } \) p.v. \( \frac{1}{x}\left( {\text{in}{\mathcal{S}}^{\prime }\left( \mathbb{R}\right) }\right) \) . | Proof. \( \forall \epsilon > 0,{\psi }_{\epsilon }\left( x\right) = {x}^{-1}{\chi }_{\left| x\right| > \epsilon } \in {L}^{\infty }\left( \mathbb{R}\right) \Rightarrow {\psi }_{\epsilon } \in \mathcal{S}\left( \mathbb{R}\right) \) . \( \left\langle {\text{p.v. }\frac{1}{x},\phi }\right\rangle = \) \( \mathop{\lim }\lim... | Yes |
Proposition 3.2. \( \forall 1 < p < \infty ,\exists {C}_{p} > 0 \), s.t. \( \forall - \infty \leq a < b \leq + \infty \) 有 \( {\begin{Vmatrix}{S}_{a, b}f\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p},\forall f \in {L}^{p}\left( \mathbb{R}\right) . | 设 \( {S}_{R} = {S}_{-R, R} \) ,则 \( {S}_{R}f = {D}_{R} * f,{\begin{Vmatrix}{S}_{R}f\end{Vmatrix}}_{p} \leq {C}_{p}\parallel f{\parallel }_{p},\forall 1 < p < \infty \) | No |
Lemma 3.6. 若 \( 1 < p < \infty, f, g \in \mathcal{S}\left( \mathbb{R}\right), a \in \mathbb{R} \) ,则 \( \left| {{\int }_{a}^{\infty }\widehat{f}\widehat{g}}\right| \leq {C}_{p}\parallel f{\parallel }_{p}\parallel g{\parallel }_{{p}^{\prime }} \) . | Proof. \( \left| {{\int }_{a}^{\infty }\widehat{f}\widehat{g}}\right| = \left| {{\int }_{\mathbb{R}}{S}_{a,\infty }f \cdot {\sigma g}}\right| \leq {\begin{Vmatrix}{S}_{a,\infty }f\end{Vmatrix}}_{p}\parallel {\sigma g}{\parallel }_{{p}^{\prime }} \leq {C}_{p}\parallel f{\parallel }_{p}\parallel g{\parallel }_{{p}^{\prim... | Yes |
Lemma 3.9. \( \parallel \widetilde{S}\left\lbrack f\right\rbrack {\parallel }_{p} \leq {C}_{p}\parallel f{\parallel }_{p},\forall f \in {L}^{p} \cap {L}^{2}\left( \mathbb{T}\right) ,1 < p < \infty \left( {\exists {C}_{p} > 0}\right) \) . | 设 \( {\widetilde{S}}_{ \pm }\left\lbrack f\right\rbrack \left( x\right) = \widetilde{S}\left\lbrack f\right\rbrack \left( x\right) \pm i\widehat{f}\left( 0\right) \) ,则 \( {\begin{Vmatrix}{\widetilde{S}}_{ \pm }\left\lbrack f\right\rbrack \end{Vmatrix}}_{p} \leq \left( {{C}_{p} + 1}\right) \parallel f{\parallel }_{p},{... | Yes |
Corollary 3.10. \( {\begin{Vmatrix}{S}_{N}f\end{Vmatrix}}_{p} \leq \left( {{C}_{p} + 1}\right) \parallel f{\parallel }_{p},\forall f \in {L}^{p}\left( \mathbb{T}\right) ,1 < p < \infty \) . | 设 \( u\left( {r, t}\right) = {P}_{r} * f\left( t\right), v\left( {r, t}\right) = {Q}_{r} * f\left( t\right) = {P}_{r} * \widetilde{S}\left\lbrack f\right\rbrack \left( t\right), F\left( {r{e}^{2\pi it}}\right) = \) \( \left( {u + {iv}}\right) \left( {r, t}\right) \) ,则 \( F\left( z\right) = \widehat{f}\left( 0\right) +... | Yes |
Lemma 3.11. 若 \( 1 < p \leq 2, u > 0, v \in \mathbb{R},{a}_{p} = {\left( \sin \frac{\pi }{2p}\right) }^{p - 1}/\cos \frac{\pi }{2p},{C}_{p} = \) \( \tan \frac{\pi }{2p} \) ,则 \( {a}_{p}\operatorname{Re}{\left( u + iv\right) }^{p} \leq {C}_{p}^{p}{u}^{p} - {\left| v\right| }^{p} \) . | Proof of Lemma 3.9 for \( 1 < p \leq 2 \) . 不妨设 \( f \geq 0 \) ,则 \( F\left( 0\right) = \widehat{f}\left( 0\right) = {\int }_{\mathbb{T}}f > \) \( 0 \) (否则 \( \parallel \widetilde{S}\left\lbrack f\right\rbrack {\parallel }_{p} = \parallel f{\parallel }_{p} = 0), u\left( {r, t}\right) > 0,\forall r \in \lbrack 0,1) \) ,... | Yes |
Proposition 4.1. 若 \( \forall f \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) ,极限对 a.e. \( x \) 存在,则 \( {\int }_{{S}^{n - 1}}{\Omega d\sigma } = 0 \) . | Proof. 取 \( f \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) s.t. \( f\left( x\right) = 0,\forall \left| x\right| < 2 \) . 则 \( \forall \left| x\right| < 1 \) 有 \( {Tf}\left( x\right) = \) \( {\int }_{\{ \left| y\right| > 1\} }\frac{\Omega \left( {y}^{\prime }\right) }{{\left| y\right| }^{n}}f\left( {x - y}\right) {d... | Yes |
Proposition 4.3. 若 \( T \in {\mathcal{S}}^{\prime }, T : a \) 次齐次,则 \( \widehat{T} : - n - a \) 次齐次. | Proof. \( \forall \phi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) ,\lambda > 0 \) 有 \( \left\langle {\widehat{T},{\phi }_{\lambda }}\right\rangle = \left\langle {T,\widehat{{\phi }_{\lambda }}}\right\rangle = \langle T,\widehat{\phi }\left( {\lambda \cdot }\right) \rangle = {\lambda }^{-n}\left\langle {T,{\widehat{\... | Yes |
Theorem 4.4. 若 \( \Omega \in {L}^{1}\left( {S}^{n - 1}\right) ,{\int }_{{S}^{n - 1}}{\Omega d\sigma } = 0, m\left( \xi \right) = {\int }_{{S}^{n - 1}}\Omega \left( u\right) \left\lbrack {\ln \frac{1}{\left| u \cdot {\xi }^{\prime }\right| } - }\right. \) \( \left. {i\frac{\pi }{2}\operatorname{sgn}\left( {u \cdot {\xi ... | Proof. 设 \( {m}_{\epsilon }\left( \xi \right) = {\int }_{\{ \epsilon < \left| y\right| < 1/\epsilon \} }\frac{\Omega \left( {y}^{\prime }\right) }{{\left| y\right| }^{n}}{e}^{-{2\pi iy} \cdot \xi }{dy} \) ,则 \( {\int }_{\{ \epsilon < \left| x\right| < 1/\epsilon \} }\frac{\Omega \left( {x}^{\prime }\right) }{{\left| x\... | Yes |
Corollary 4.7. 若 \( {\int }_{{S}^{n - 1}}\Omega = 0,{\Omega }_{o} \in {L}^{1}\left( {S}^{n - 1}\right) ,{\Omega }_{e} \in {L}^{q}\left( {S}^{n - 1}\right), q > 1 \) , 则 \( \mathcal{F}\left( {p.v.\frac{{\Omega }_{o}\left( {x}^{\prime }\right) }{{\left| x\right| }^{n}}}\right) \in {L}^{\infty }\left( {\mathbb{R}}^{n}\rig... | Keypoint: (i) \( \left| {\operatorname{sgn}\left( {u \cdot {\xi }^{\prime }}\right) }\right| = 1 \) ; (ii) \( {\int }_{{S}^{n - 1}}{\left| \ln \left| u \cdot \xi \right| \right| }^{{q}^{\prime }}{d\sigma }\left( u\right) = {C}_{q} < \infty \) . | No |
若 \( \Omega \in {L}^{1}\left( {S}^{n - 1}\right) \) ,则 \( {M}_{\Omega }^{\prime \prime },{M}_{\Omega }^{\prime },{M}_{\Omega } \) 在 \( {L}^{p}\left( {\mathbb{R}}^{n}\right) \) 有界 \( \left( {\forall p > 1}\right) \) . | 取 \( \Omega = 1 \) 则 \( {M}_{\Omega }^{\prime \prime }f\left( x\right) = {Mf}\left( x\right) ,\left| {S}^{n - 1}\right| = \parallel \Omega {\parallel }_{{L}^{1}\left( {S}^{n - 1}\right) } \), \[ \parallel {Mf}{\parallel }_{p} = \parallel {M}_{\Omega }^{\prime \prime }f{\parallel }_{p} \leq \frac{1}{\left| B\left( 0,1\r... | Yes |
Theorem 4.12. 若 \( {\int }_{{S}^{n - 1}}\Omega = 0,{\Omega }_{o} \in {L}^{1}\left( {S}^{n - 1}\right) ,{\Omega }_{e} \in {L}^{q}\left( {S}^{n - 1}\right), q > 1 \) , 则 \( T : {L}^{p} \rightarrow {L}^{p} \) 有界 \( \left( {1 < p < \infty }\right) \) . | 极大奇异积分算子 \( {T}^{ * }f\left( x\right) = \mathop{\sup }\limits_{{\epsilon > 0}}\left| {{\int }_{\{ \left| y\right| > \epsilon \} }\frac{\Omega \left( {y}^{\prime }\right) }{{\left| y\right| }^{n}}f\left( {x - y}\right) {dy}}\right| \) . 首先证明\n\n(41)\n\n\[ \left| {{T}^{ * }f\left( x\right) - \frac{1}{\ln 2}{T}_{1}^{ * }f... | Yes |
Theorem 4.13. 若 \( m \in {C}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) 是 0 次齐次函数, \( \widehat{{T}_{m}f} = m\widehat{f} \) ,则 \( \exists \Omega \in \)\n\n\( {C}^{\infty }\left( {S}^{n - 1}\right) ,{\int }_{{S}^{n - 1}}\Omega = 0, a \in \mathbb{C} \), s.t. \( {T}_{m}f = {af} + \) p.v. \( \frac{\O... | \( \exists a \in \mathbb{C} \) s.t. \( {\int }_{{S}^{n - 1}}\left( {m\left( {\xi }^{\prime }\right) - a}\right) {d\sigma }\left( {\xi }^{\prime }\right) = 0 \) . 不妨设 \( {\int }_{{S}^{n - 1}}m\left( u\right) {d\sigma }\left( u\right) = 0 \)\n\n(否则考虑 \( m - a \) ). (结合 \( {\mathcal{F}}^{2} = \sigma \) ) 只需证 | No |
Theorem 4.15. \( \mathcal{A} = \left\{ {{T}_{m} \mid m \in {C}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) }\right. \) 是 0 次齐次函数 \( \} \) 是交换代数, \( {T}_{m} \) 是 \( \mathcal{A} \) 的可逆元 \( \Leftrightarrow m \neq 0 \) on \( {S}^{n - 1} \) . | 若 \( \overset{⏜}{Tf}\left( \xi \right) = {i}^{m}\frac{P\left( \xi \right) }{{\left| \xi \right| }^{m}}\widehat{f}\left( \xi \right) \) 则 \( T \) 可逆 \( \Leftrightarrow P\left( \xi \right) \neq 0 \) on \( {S}^{n - 1} \) (此时若 \( P \) 是实值函数则 \( m \) 是偶数, \( \left. {{\Lambda }^{m} = {\left( -\Delta \right) }^{m/2}}\right) \... | Yes |
Theorem 4.16. 若 \( \left( i\right) \Omega \left( {x,{\lambda z}}\right) = \operatorname{sgn}\left( \lambda \right) \Omega \left( {x, z}\right) ,\forall z \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} ,\lambda \in \mathbb{R} \smallsetminus \{ 0\} \) ; (ii) \( {\Omega }^{ * }\left( u\right) = \mathop{\sup }\limits_{x}\left|... | Proof. \( {Tf}\left( x\right) = \frac{\pi }{2}{\int }_{{S}^{n - 1}}\Omega \left( {x, u}\right) {H}_{u}f\left( x\right) {d\sigma }\left( u\right) ,\forall f \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) .\left| {{Tf}\left( x\right) }\right| \leq \frac{\pi }{2}{\int }_{{S}^{n - 1}}{\Omega }_{ * }\left( u\right) \left| {{... | Yes |
Theorem 4.17. 若Therrem 4.16的条件 \( \left( {ii}\right) \) 换成 \( {\sup }_{x}{\left( {\int }_{{S}^{n - 1}}{\left| \Omega \left( x, u\right) \right| }^{q}d\sigma \left( u\right) \right) }^{\frac{1}{q}} \) \( = {B}_{q} < \infty ,1 < q < \infty \) . 则 \( T : {L}^{p} \rightarrow {L}^{p} \) 有界, \( \forall {q}^{\prime } \leq p <... | Proof. 由 \( {q}^{\prime } \leq p \) 得 \( {p}^{\prime } \leq q \) . 由Hölder不等式得 \( \mathop{\sup }\limits_{x}{\left( {\int }_{{S}^{n - 1}}{\left| \Omega \left( x, u\right) \right| }^{{p}^{\prime }}d\sigma \left( u\right) \right) }^{\frac{1}{{p}^{\prime }}} \leq \)\n\n\( {B}_{{p}^{\prime }} = {\left| {S}^{n - 1}\right| }^... | Yes |
Theorem 5.1 (Benedek-Calderon-Panzone原理). 设 \( T \) 是次线性算子. (i) \( T \) 是弱 \( \left( {p, p}\right) \) 型, \( 1 < p < \infty : \parallel {Tf}{\parallel }_{p,\infty } \leq {C}_{1}\parallel f{\parallel }_{p},\forall f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) . (ii) 存在常数 \( {C}_{2} > 1,{C}_{3} > 0 \) s.t. 若 \( \operator... | Proof. \( \forall \lambda > 0 \) ,对 \( f \) 作Calderon-Zygmund分解, \( \exists \) 不交方体 \( \left\{ {Q}_{k}\right\} \) s.t. \( \mathop{\sum }\limits_{k}\left| {Q}_{k}\right| \leq \frac{1}{\lambda }\parallel f{\parallel }_{1},\lambda < \frac{1}{\left| {Q}_{k}\right| }{\int }_{{Q}_{k}}\left| f\right| \leq {2}^{n}\lambda ,\lef... | Yes |
Theorem 5.2 (Calderon-Zygmund). 设 \( K \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right), K \in {L}_{loc}^{1}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) . 满 足(i) \( \parallel \widehat{K}{\parallel }_{\infty } \leq A \) . (ii) Hörmander条件: \( {\int }_{\{ \left| x\right| > 2\left| y\right| \} }\left... | Proof. 设 \( {Tf} = K * f, f \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) . 则 \( \widehat{Tf}\left( \xi \right) = \widehat{K}\left( \xi \right) \widehat{f}\left( \xi \right) \) ,由(i)得 \( \parallel {Tf}{\parallel }_{2} \leq \) \( A\parallel f{\parallel }_{2},\forall f \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) ... | Yes |
Lemma 5.6. 若 \( K \in {L}_{loc}^{1}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) ,{\left\lbrack K\right\rbrack }_{ * } < \infty, f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) ,1 \leq p < \infty \) . 则 \( {\int }_{{\mathbb{R}}^{n}}\left| {{K}_{\epsilon }\left( {x - y}\right) f\left( y\right) }\right| {dy} < \inft... | Key point: \( {K}_{\epsilon }, K \in {L}^{1,\infty }\left( {\mathbb{R}}^{n}\right) ,{K}_{\epsilon } * {\chi }_{B\left( {0,\epsilon }\right) } \in {L}^{q}\left( {\mathbb{R}}^{n}\right) ,\forall 1 < q \leq \infty \) . | No |
若 \( T \) 满足Theorem 5.9的条件, \( a \in \mathcal{A} \) 则 \( \parallel {Ta}{\parallel }_{1} \leq C \) . | Proof. \( \exists \) 方体 \( Q \) s.t. \( a \in {\mathcal{A}}_{Q} \) ,则 \( \parallel a{\parallel }_{2} \leq {\left| Q\right| }^{-1/2},\parallel a{\parallel }_{1} \leq 1 \) . 设 \( Q = Q\left( {c, r}\right) \) , \( {Q}^{ * } = B\left( {c,2\sqrt{n}r}\right) \) ,则 \( \left| {Q}^{ * }\right| = {C}_{n}\left| Q\right| .{\int }_... | Yes |
Corollary 6.2. 若 \( T \) 满足Theorem 5.9的条件, \( f \in {\mathcal{H}}_{at}^{1} \) ,则 \( \parallel {Tf}{\parallel }_{1} \leq C\parallel f{\parallel }_{{\mathcal{H}}_{at}^{1}} \) . | Key point: \( \exists {f}_{k} \in \operatorname{span}\mathcal{A} \) s.t. \( {f}_{k} \rightarrow f \) in \( {L}^{1},{\begin{Vmatrix}T{f}_{k}\end{Vmatrix}}_{1} \leq C\parallel f{\parallel }_{{\mathcal{H}}_{at}^{1}} \) . | No |
Theorem 6.3 (*). \( {\mathcal{H}}^{1}\left( {\mathbb{R}}^{n}\right) = {\mathcal{H}}_{at}^{1}\left( {\mathbb{R}}^{n}\right) \) ,且范数等价. | 注: 由Corollary 6.2得 \( {\mathcal{H}}_{at}^{1}\left( {\mathbb{R}}^{n}\right) \subseteq {\mathcal{H}}^{1}\left( {\mathbb{R}}^{n}\right) \) . 另一方面需要证明 \( f \in {\mathcal{H}}^{1}\left( {\mathbb{R}}^{n}\right) \) \( \Rightarrow {P}^{ * }f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \Rightarrow f \in {\mathcal{H}}_{at}^{1}\left... | No |
Proposition 6.4. (i) \( \frac{1}{2}\parallel f{\parallel }_{ * } \leq \parallel f{\parallel }_{ * }^{\prime } \leq \parallel f{\parallel }_{ * } \) ,(ii) \( {M}^{\# }\left| f\right| \left( x\right) \leq 2{M}^{\# }f\left( x\right) \) . | Proof. \( \forall \) 方体 \( Q \) 和 \( a \in \mathbb{C} \) 有 \( {\int }_{Q}\left| {f - {f}_{Q}}\right| \leq {\int }_{Q}\left| {f - a}\right| + {\int }_{Q}\left| {a - {f}_{Q}}\right| \leq 2{\int }_{Q}\left| {f - a}\right| \) , 其中用到 \( {\int }_{Q}\left| {a - {f}_{Q}}\right| = \left| Q\right| \left| {a - {f}_{Q}}\right| = \... | Yes |
Theorem 6.5. 若 \( T \) 满足 Theorem 5.9 的条件, \( f \in {L}_{c}^{\infty } \) ,则 \( \parallel {Tf}{\parallel }_{ * } \leq C\parallel f{\parallel }_{\infty } \) . | Proof. 给定方体 \( Q \) ,设 \( Q = Q\left( {c, r}\right) ,{Q}^{ * } = B\left( {c,2\sqrt{n}r}\right), f = {f}_{1} + {f}_{2} \), \( {f}_{1} = f{\chi }_{{Q}^{ * }},{f}_{2} = f{\chi }_{{\mathbb{R}}^{n} \smallsetminus {Q}^{ * }}.T{f}_{2}\left( x\right) = {\int }_{{\mathbb{R}}^{n} \smallsetminus {Q}^{ * }}K\left( {x, y}\right) f\... | Yes |
若 \( 1 < p < \infty \) ,则 \( \parallel f{\parallel }_{*, p} = \mathop{\sup }\limits_{Q}{\left( \frac{1}{\left| Q\right| }{\int }_{Q}{\left| f - {f}_{Q}\right| }^{p}\right) }^{\frac{1}{p}} \) 是BMO的范数且与 \( \parallel \cdot {\parallel }_{ * } \) 等价. | 由Hölder不等式得 \( \parallel f{\parallel }_{ * } \leq \parallel f{\parallel }_{*, p} \) ,只需再证 \( \parallel f{\parallel }_{*, p} \leq {C}_{p}\parallel f{\parallel }_{ * } \) . 由 Theorem 6.9得 \( {\int }_{Q}{\left| f - {f}_{Q}\right| }^{p} = {\int }_{0}^{\infty }p{\lambda }^{p - 1}\left| \left\{ {x \in Q : \left| {f\left( x\r... | Yes |
Corollary 6.11. 若 \( f \in {BMO} \) ,则 \( \exists \lambda > 0 \) s.t. \( \forall \) 方体 \( Q \) 有 \( {\int }_{Q}{e}^{\lambda \left| {f - {f}_{Q}}\right| } < \infty \) . | Proof. 若 \( 0 < \lambda < {C}_{2}/\parallel f{\parallel }_{ * } \) ,则由Theorem 6.9得\n\n\( {\int }_{Q}{e}^{\lambda \left| {f - {f}_{Q}}\right| } = \left| Q\right| + {\int }_{0}^{\infty }\lambda {e}^{\lambda s}\left| \left\{ {x \in Q : \left| {f\left( x\right) - {f}_{Q}}\right| > s}\right\} \right| {ds} \leq \)\n\n\( \lef... | Yes |
Corollary 6.12. 若 \( f \in {L}_{loc}^{1},\exists {C}_{1},{C}_{2}, K > 0 \) s.t. \( \forall \) 方体 \( Q,\lambda > 0 \) 有\n\n\( \left| \left\{ {x \in Q : \left| {f\left( x\right) - {f}_{Q}}\right| > \lambda }\right\} \right| \leq {C}_{1}{e}^{-{C}_{2}\lambda /K}\left| Q\right| \) ,则 \( f \in {BMO} \) . | Proof. \( {\int }_{Q}\left| {f - {f}_{Q}}\right| = {\int }_{0}^{\infty }\left| \left\{ {x \in Q : \left| {f\left( x\right) - {f}_{Q}}\right| > \lambda }\right\} \right| {d\lambda } \leq {\int }_{0}^{\infty }{C}_{1}{e}^{-{C}_{2}\lambda /K}\left| Q\right| {d\lambda }\)\n\n\( = {C}_{1}\left| Q\right| K/{C}_{2} \), i.e. \(... | Yes |
\[ \int \rho {\Phi }^{\prime }{dV} + \int \sigma {\Phi }^{\prime }{dA} = \int {\rho }^{\prime }{\Phi dV} + \int {\sigma }^{\prime }{\Phi dA} \] | Green gave us a handy relationship which is useful here. Namely,\n\n\[ {\int }_{V}\left( {\phi {\nabla }^{2}\psi - \psi {\nabla }^{2}\phi }\right) d{V}^{\prime } = {\oint }_{S}\left\lbrack {\phi \frac{\partial \psi }{\partial n} - \psi \frac{\partial \phi }{\partial n}}\right\rbrack {dA} \]\n\nLet \( \phi = \Phi \) and... | Yes |
The Kramer-Krönig relation states:\n\n\[ \Re \left( \frac{\epsilon \left( \omega \right) }{{\epsilon }_{0}}\right) = 1 + \frac{2}{\pi }P{\int }_{0}^{\infty }\frac{{\omega }^{\prime }}{{\omega }^{\prime 2} - {\omega }^{2}}\Im \left( \frac{\epsilon \left( {\omega }^{\prime }\right) }{{\epsilon }_{0}}\right) d{\omega }^{\... | a. \( \Im \left( \frac{\epsilon \left( \omega \right) }{{\epsilon }_{0}}\right) = \lambda \left\lbrack {\Theta \left( {\omega - {\omega }_{1}}\right) - \Theta \left( {\omega - {\omega }_{2}}\right) }\right\rbrack \) .\n\nPlug this into the Kramer-Kronig relationship.\n\n\[ \Re \left( \frac{\epsilon \left( \omega \right... | Yes |
\[ {\wp }_{1} + {\wp }_{2} \rightarrow {\wp }_{3} + {\wp }_{4} \] | I am going to take advantage of the fact that the product of two Lorentz four vectors is invariant or the same in all Lorentz frames. A prime denotes that a quantity is measured in the center of momentum frame. no prime means that the quantity is measured in the lab frame. Sometimes, I get a bit carried away and use bo... | Yes |
Problem 12.14\n\n\[ \mathcal{L} = - \frac{1}{8\pi }{\partial }_{\alpha }{A}_{\beta }{\partial }^{\alpha }{A}^{\beta } - \frac{1}{c}{J}_{\alpha }{A}^{\alpha } \] | a.\n\nThe Euler-Lagrange theorem says\n\n\[ \frac{\partial \mathcal{L}}{\partial {\phi }^{\alpha }} = {\partial }^{\beta }\frac{\partial \mathcal{L}}{\partial \left( {{\partial }^{\beta }{\phi }^{\alpha }}\right) } \]\n\nSo we have \( \frac{\partial \mathcal{L}}{\partial {A}^{\alpha }} = - \frac{1}{c}{J}_{\alpha } \) a... | Yes |
Proposition 1.1 (Euclid’s algorithm). Suppose that \( a, b \) are integers. Then there are integers \( m, n \) such that \( {am} + {bn} = \left( {a, b}\right) \) . | Proof. Replacing \( a \) by \( - a \) and \( b \) by \( - b \) and switching the role of \( a, b \) if necessary, we may assume that \( a \geq b \geq 0 \) . Now perform Euclid’s algorithm:\n\n\[ a = {q}_{1}b + {r}_{1} \]\n\n\[ b = {q}_{2}{r}_{1} + {r}_{2} \]\n\n\[ {r}_{1} = {q}_{3}{r}_{2} + {r}_{3} \]\n\n\[ \vdots \]\n... | No |
Proposition 1.2. An integer \( p \) is irreducible if and only if it is prime. | Proof. Suppose first that \( p \) is prime. We claim that \( p \) is irreducible. Suppose not; then \( p = {ab} \) with neither \( a \) nor \( b \) a unit. Since \( p \) is prime, either \( p \mid a \) or \( p \mid b \) . Suppose without loss of generality that \( p \mid a \) . Thus \( a = {pc} \) for some \( c \in \ma... | Yes |
Proposition 1.3. Every integer other than zero and the units may be factored into primes in an essentially unique way. | Proof. By \ | No |
Proposition 1.4. Suppose that \( a, b, c \) are integers with \( a, b \neq 0 \) . Then there is a solution to the equation \( {am} + {bn} = c \) in integers \( m, n \) if and only if \( \left( {a, b}\right) \mid c \) . Two integers \( {m}^{\prime },{n}^{\prime } \) give another solution if and only if \( {m}^{\prime } ... | Proof. It is obvious that if the equation is soluble then \( \left( {a, b}\right) \) divides \( c \) . Conversely, by Euclid’s algorithm we see that if \( \left( {a, b}\right) \mid c \) then the equation does have a solution: take a solution to \( {au} + {bv} = \left( {a, b}\right) \) and set \( m = \frac{c}{\left( a, ... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.