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Lemma 6.15. \( \operatorname{CR}\left( {\left. f\right| }_{\mathrm{{CR}}\left( f\right) }\right) = \mathrm{{CR}}\left( f\right) \). | Proof. We prove \( \operatorname{CR}\left( f\right) \subset \operatorname{CR}\left( {\left. f\right| }_{\mathrm{{CR}}\left( f\right) }\right) \). The other direction is obvious.\n\nLet \( x \in \operatorname{CR}\left( f\right) \) be given. For every \( n \geq 1 \), there is a periodic \( \frac{1}{n} \) -chains\n\n\[ {C... | Yes |
Theorem 6.16. The following conditions are equivalent:\n\n(1) \( \operatorname{CR}\left( f\right) \) is hyperbolic.\n\n(2) \( \operatorname{CR}\left( f\right) \) is quasi-hyperbolic.\n\n(3) \( \operatorname{CR}\left( f\right) \) satisfies the linear transversality condition.\n\nNote that, in our use of terminologies, c... | Proof. That \( \left( 1\right) \Rightarrow \left( 2\right) \) is obvious. We prove \( \left( 2\right) \Rightarrow \left( 1\right) \) . Assume \( f \) is quasi-hyperbolic on \( \mathrm{{CR}}\left( f\right) \) . By Theorem 6.11, \( \mathrm{{CR}}\left( {\left. f\right| }_{\mathrm{{CR}}\left( f\right) }\right) \) is hyperb... | Yes |
Theorem 6.17. \( f \) is quasi-Anosov if and only if \( f \) satisfies Axiom A and \( {T}_{x}\left( {{W}^{s}\left( x\right) }\right) \cap {T}_{x}\left( {{W}^{u}\left( x\right) }\right) = \{ 0\} \) for all \( x \in M \) . | Proof. Let \( f \) satisfy Axiom A, and let\n\n\[ \n{T}_{x}\left( {{W}^{s}\left( x\right) }\right) \cap {T}_{x}\left( {{W}^{u}\left( x\right) }\right) = \{ 0\} \n\]\n\nfor all \( x \in M \) . By Theorem 5.5,\n\n\[ \n{B}^{s}\left( x\right) = {T}_{x}\left( {{W}^{s}\left( x\right) }\right) ,{B}^{u}\left( x\right) = {T}_{x... | Yes |
Theorem 6.18. \( f \) satisfies the linear transversality condition if and only if \( f \) satisfies Axiom A and the strong transversality condition. | Proof. Let \( f \) satisfy Axiom A and the strong transversality condition. By Theorem 5.5,\n\n\[ \n{D}^{s}\left( x\right) = {T}_{x}\left( {{W}^{s}\left( x\right) }\right) ,{D}^{u}\left( x\right) = {T}_{x}\left( {{W}^{u}\left( x\right) }\right) \n\]\n\nfor all \( x \in M \) . Thus \( f \) satisfies the linear transvers... | Yes |
For any integer \( N \geq 1 \), defining\n\n\[ \n{\zeta }_{N}\left( s\right) = \mathop{\prod }\limits_{{p \mid N}}{\left( 1 - {p}^{-s}\right) }^{-1},\;\text{ so that }\frac{1}{{\zeta }_{N}\left( s\right) } = \mathop{\sum }\limits_{{1 \leq T \mid {N}_{ \bullet }}}\mu \left( T\right) {T}^{-s}, \n\]\n\nwhere \( \mu \) is ... | Proof. Using the equation \( {T}^{-x\frac{d}{dx}}\delta \left( {x - j}\right) = \delta \left( {\frac{x}{T} - j}\right) = {T\delta }\left( {x - {Tj}}\right) \), one has with \( \mathfrak{D} \) as introduced in (3.4)\n\n\[ \n\frac{1}{2\pi }\mathop{\prod }\limits_{{p \mid N}}\left( {1 - {p}^{-{2i\pi }\mathcal{E}}}\right) ... | Yes |
Setting \( {v}_{Q}\left( x\right) = v\left( {Qx}\right) \), one has the identity\n\n\[ \left( {v \mid \mathcal{B}v}\right) = \left( {{v}_{Q} \mid \mathcal{A}{v}_{Q}}\right) \] | Proof. Starting from\n\n\[ \left( {\mathcal{B}v}\right) \left( {Qx}\right) = \frac{1}{2}{\int }_{{\mathbb{R}}^{2}}\mathfrak{S}\left( {\frac{Q\left( {x + y}\right) }{2},{Q\xi }}\right) {e}^{{i\pi }\left( {x - y}\right) \xi }v\left( {Qy}\right) {dyd\xi } \]\n\n(5.6)\n\none obtains (5.4). From (5.3),\n\n\[ \left( {v \mid ... | Yes |
Lemma 5.2. Set, for \( w \in \mathcal{S}\left( \mathbb{R}\right) \) , \[ \left( {{\theta }_{N}w}\right) \left( n\right) = \mathop{\sum }\limits_{{\ell \in \mathbb{Z}}}w\left( {\frac{n}{N} + 2\ell N}\right) ,\;n{\;\operatorname{mod}\;2}{N}^{2}. \] (5.8) On the other hand, with \( v \in \mathcal{S}\left( \mathbb{R}\right... | Proof. It is immediate. | No |
Lemma 6.2. Let \( Q \) be a squarefree odd number and \( \beta > 0 \) be given. There exist \( R \), with \( N = {RQ} \) odd squarefree divisible by all odd primes \( < {\beta Q} \), and \( A \in \mathbb{Z} \) such that \( {R}^{2} - {2A}{Q}^{2} = 1 \) . Given two integers \( {n}_{R} \) and \( {n}_{Q} \), the number \( ... | Proof. Choose \( {R}_{1} \) odd and squarefree, relatively prime to \( Q \), divisible by all odd primes \( < {\beta Q} \) relatively prime to \( Q \), and \( {\bar{R}}_{1} \) such that \( {\bar{R}}_{1}{R}_{1} \equiv \) \( 1{\;\operatorname{mod}\;2}{Q}^{2} \) and \( {\bar{R}}_{1} \equiv 1{\;\operatorname{mod}\;{R}_{1}}... | Yes |
Lemma 6.3. Assume that \( w \) is supported in \( \left\lbrack {0,\beta }\right\rbrack \) for some \( \beta > 0 \), that \( Q \) is an odd squarefree integer and that \( N = {RQ} \) is squarefree, odd and divisible by all odd primes \( < {\beta Q} \) . Finally, as made possible by Lemma 6.2, assume that \( {R}^{2} = 1 ... | Proof. Set \( S = R - {R}^{-1} \), a large number equivalent to \( R \) . One rewrites the double inequality \( 0 < {R}^{2}x - {2A}{Q}^{2}m + 2{\ell }_{1}{N}^{2} < {\beta N} \) as\n\n\[ 0 < x + {2A}{Q}^{2}\left( {x - m}\right) + 2{\ell }_{1}{N}^{2} < {\beta N} \]\n\n(6.15)\n\nhence\n\n\[ 0 < \frac{x}{{2A}{Q}^{2}} + x -... | Yes |
Lemma 6.4. One has\n\n\[ \left( {w \mid \Psi \left( {{Q}^{{2i\pi }\mathcal{E}}{\mathfrak{T}}_{N}}\right) w}\right) = \mathfrak{A}\mathfrak{B} \] | Proof. Using (5.14),(6.1) and Lemma 6.3, and with \( {m}^{\sharp } = \left( {m, x}\right) \in \mathbb{Z}/{R}^{2}\mathbb{Z} \times \) \( \mathbb{Z}/\left( {2{Q}^{2}}\right) \mathbb{Z},{n}^{\sharp } = \left( {n, y}\right) \), one writes\n\n\[ \left( {w \mid \Psi \left( {{Q}^{{2i\pi }\mathcal{E}}{\mathfrak{T}}_{N}}\right)... | Yes |
Lemma 6.5. One has \( \mathfrak{B} = {2Q} \) . | Proof. Let us specify the divisibility by powers of a prime \( q \) in terms of the \( q \) -adic absolute value (such that \( {\left| q\right| }_{q} = \frac{1}{q} \) ). Let us denote as Dom \( \left( \mathfrak{B}\right) \) the domain of the sum (6.25) and use the partition\n\n\[ \operatorname{Dom}\left( \mathfrak{B}\r... | Yes |
Lemma 7.2. Consider a product \( h\left( \nu \right) f\left( {s - \nu - 1}\right) \), where the function \( f \), defined and meromorphic near the point 1, has a simple pole at that point, and the function \( h \), defined and meromorphic near a point \( {\rho }_{0} \), has at that point a pole of order \( \ell \geq 1 ... | Proof. If \( h\left( \nu \right) = \mathop{\sum }\limits_{{j = 1}}^{\ell }{a}_{j}{\left( \nu - {\rho }_{0}\right) }^{-j} + \mathrm{O}\left( 1\right) \) as \( \nu \rightarrow {\rho }_{0} \), one has for \( s \) near \( 2 + {\rho }_{0}, s \neq 2 + {\rho }_{0} \)\n\n\[ {\operatorname{Res}}_{\nu = {\rho }_{0}}\left( {h\lef... | Yes |
设 \( a \) 与 \( M \) 互素,初值 \( {x}_{0} \) 与 \( M \) 互素,则乘同余法的周期为 \( a \) 对模 \( M \) 的阶数 \( V \) 。 | 证明: 由同余的传递性可知 \( {x}_{V} \equiv {a}^{V}{x}_{0}\left( {\;\operatorname{mod}\;M}\right) \) ,而 \( {a}^{V} \equiv 1\left( {\;\operatorname{mod}\;M}\right) \) 所以\n\n\[ \n{a}^{V}{x}_{0} \equiv {x}_{0}\;\left( {\;\operatorname{mod}\;M}\right) \n\]\n\n于是\n\n\[ \n{x}_{V} \equiv {x}_{0}\;\left( {\;\operatorname{mod}\;M}\right) \... | Yes |
Example 1.3 (Maxwell equation, Maxwell 1861-1862). Let \( {A}_{\mu } \) be a 1-form on \( {\mathbb{R}}^{1 + 3} \), that is,\n\n\[ A = {A}_{\mu }d{x}^{\mu },\;{x}^{0} = t. \]\n\nDefine \( F = {dA} \) be the exterior derivative of \( A \) which is anti-symmetric 2-form on \( {\mathbb{R}}^{1 + 3} \). Then the Maxwell equa... | It can be checked that the components of the Maxwell fields verify the linear wave equation\n\n\[ ▱{F}_{\mu \nu } = 0. \]\n\nOne can check this by noting that the Maxwell field \( F \) is closed as a 2 -form, which indicates the following Bianchi identity\n\n\[ {\partial }_{\mu }{F}_{\nu \gamma } + {\partial }_{\nu }{F... | No |
Example 1.4 (Yang-Mills equation, Yang-Mills 1954). The Yang-Mills gauge theory is the non-abelian generalization of the above Maxwell equation. Still let \( A \) be a 1-form on \( {\mathbb{R}}^{1 + 3} \) but taking value in a Lie algebra \( \mathfrak{g} \). Then define the covariant derivative\n\n\[ \n{D}_{\mu } = {\p... | We note that the above equation is nonlinear when \( \mathfrak{g} \) is non-abelian. | Yes |
Example 1.5 (Irrotational compressible fluids, Euler,1752). A fluid in \( {\mathbb{R}}^{1 + 3} \) is described by its velocity \( v \) and enthalpy \( h \) . The pressure \( p \) is a positive function of \( h \) such that\n\n\[ \rho \mathrel{\text{:=}} \frac{dp}{dh} > 0,\;{\eta }^{2} \mathrel{\text{:=}} \rho {\left( \... | The third equation means that the fluid is irrotational which in particular implies that \( v = \) \( - \nabla \phi \) for some potential function \( \phi \) (follows from fundamental theorem in calculus). Plugging this into the first equation shows that\n\n\[ \nabla \left( {\frac{\partial \phi }{\partial t} - \frac{1}... | Yes |
Example 1.6 (Einstein vacuum equation, Einstein 1915). The vacuum Einstein equation is to describe the propagation of gravitation waves without matter. The vacuum spacetime is a 4-dimensional Lorentzian manifold with Lorentzian metric \( g \) such that\n\n\[ \operatorname{Ric}\left( g\right) = 0\text{.} \] | In local coordinates, recall the Christoffel symbol\n\n\[ {\Gamma }_{\mu \nu }^{\gamma } = \frac{1}{2}{g}^{\gamma p}\left( {\frac{\partial {g}_{\mu p}}{\partial \nu } + \frac{\partial {g}_{\nu p}}{\partial \mu } - \frac{\partial {g}_{\mu \nu }}{\partial p}}\right) .\n\nThen we can compute that\n\n\[ {Ri}{c}_{\mu \nu } ... | Yes |
Proposition 2.1. A distribution \( u \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) has compact support \( K \) is equivalent to\n\n\[ \left| { < u,\phi > }\right| \leq C\left( {N, U}\right) \mathop{\sup }\limits_{{x \in U}}\mathop{\sum }\limits_{{\left| \alpha \right| \leq N}}\left| {{\partial }^{\alpha ... | Proof. Suppose that \( u \) has compact support \( K \) . We then can choose compact set \( {K}^{\prime } \) such that \( K \subset {K}^{\prime } \) and \( \operatorname{dist}\left( {\partial K,\partial {K}^{\prime }}\right) > 0 \) . Then in particular we can construct smooth function \( \rho \) such that \( {\left. \r... | Yes |
Proposition 2.2. Suppose that the distribution \( u \) is supported at the origin. Then\n\n\[ u = \mathop{\sum }\limits_{{\left| \alpha \right| \leq N}}{a}_{\alpha }{\partial }^{\alpha }\delta \]\n\nfor some integer \( N \) . | Proof. By definition and the previous Proposition, we conclude that there exists nonnegative integer \( N \) such that\n\n\[ \left| {u\left( \phi \right) }\right| \leq C\left( N\right) \mathop{\sum }\limits_{{\left| \alpha \right| \leq N}}\left| {{\partial }^{\alpha }\phi }\right| ,\;\operatorname{supp}\left( \phi \rig... | Yes |
Lemma 2.1. For positive integer \( k \), we have\n\n\[{\chi }_{ + }^{-k} = {\delta }^{\left( k - 1\right) }\left( x\right)\]\n\n\[{\chi }_{ + }^{-k - \frac{1}{2}} = \frac{1}{\sqrt{\pi }}{\left( {x}_{ + }^{-\frac{1}{2}}\right) }^{\left( k\right) }.\] | For the proof, one only needs to note that \( {\chi }_{ + }^{0} = H\left( x\right) \), in particular \( {\chi }_{ + }^{-1} = \delta \) . This implies the integer case. For the half integer case, it suffices to check the case when \( k = 0 \) . By definition\n\n\[{\chi }_{ + }^{-\frac{1}{2}} = \frac{{x}_{ + }^{-\frac{1}... | Yes |
Proposition 2.3. The forward fundamental solution to the wave operator \( ▱ \) is\n\n\[ \n{E}_{ + }\left( {t, x}\right) = - \frac{{\pi }^{\frac{1 - n}{2}}}{2}H\left( t\right) {\chi }_{ + }^{-\frac{n - 1}{2}}\left( {{t}^{2} - {\left| x\right| }^{2}}\right) .\n\]\n\nHere \( n \) is the space dimension, that is, \( x \in ... | Proof. We first check that \( {E}_{ + } \) given in the proposition is indeed a forward fundamental solution. It is easy to see from the definition of \( {\chi }_{ + }^{-k} \) that the support of \( {E}_{ + } \) lies inside of the forward light cone \( \{ 0 \leq \left| x\right| \leq t\} \) . Inside this cone with \( {t... | Yes |
Proposition 2.4. Assume that the initial data \( {\phi }_{0},{\phi }_{1} \) and the inhomogeneous term \( F \) are compactly supported smooth functions. Let \( {E}_{ + }\left( {t, x}\right) \) be the forward fundamental solution to the wave operator we just constructed. Then the Cauchy problem to the linear wave equati... | Proof. In fact we can compute that\n\n\[ \n\phi \left( {t, x}\right) H\left( t\right) = \left( {\phi H}\right) * \delta = \left( {\phi H}\right) * ▱{E}_{ + }\n\]\n\n\[ \n= - \left( {\phi H}\right) * \left( {{\partial }_{tt}{E}_{ + }}\right) + \left( {H\Delta \phi }\right) * {E}_{ + }\n\]\n\n\[ \n= - \left( {H{\partial ... | Yes |
Proposition 2.5 (Duhamel's principle). Consider the linear wave equation\n\n\[ \n▱\phi = f,\;\phi \left( {0, x}\right) = {\phi }_{0}\left( x\right) ,\;{\partial }_{t}\phi \left( {0, x}\right) = {\phi }_{1}\left( x\right) .\n\] | Then\n\n\[ \n\phi \left( {t, x}\right) = {\partial }_{t}F\left( {\phi }_{0}\right) + F\left( {\phi }_{1}\right) - {\int }_{0}^{t}F\left( {f\left( {s, x}\right) }\right) \left( {t - s, x}\right) {ds},\n\]\n\nin which for all \( \psi : {\mathbb{R}}^{n} \rightarrow \mathbb{R}, F\left( \psi \right) \) is the solution to th... | Yes |
Proposition 2.6 (Kirchhoff's representation formula). Consider the following linear wave equation\n\n\[ \n▱\phi = 0,\;\phi \left( {0, x}\right) = 0,\;{\partial }_{t}\phi \left( {0, x}\right) = {\phi }_{1}.\n\]\n\nThen it has the following unique solution | \[ \n\phi \left( {t, x}\right) = \left\{ \begin{array}{l} \frac{1}{2}{\int }_{x - t}^{x + t}{\phi }_{1}\left( y\right) {dy},\;\text{ when }n = 1; \\ \frac{\pi - 2}{4}{\left( \frac{1}{2t}\frac{d}{dt}\right) }^{\frac{n - 3}{2}}\left( {{t}^{n - 2}{\int }_{\left| \omega \right| = 1}{\phi }_{1}\left( {x + {t\omega }}\right)... | Yes |
Proposition 2.7 (Finite speed of propagation). Consider the linear wave equation\n\n\[ \n▱\phi = F,\;\phi \left( {0, x}\right) = {\phi }_{0},\;{\partial }_{t}\phi \left( {0, x}\right) = {\phi }_{1}.\n\]\n\nSuppose that \( \left( {{\phi }_{0},{\phi }_{1}}\right) = 0 \) on \( \{ \left| x\right| \leq R\} \) and \( F = 0 \... | This is clear from the representation formula. This property even holds for nonlinear equations for which we will return to this later. | No |
Proposition 2.9. Assume that the initial data \( \left( {{\phi }_{0},{\phi }_{1}}\right) \) are smooth and have compact support. Then the solution to the homogeneous linear wave equation \( ▱\phi = 0 \) verifies the following decay estimates\n\n\[ \left| {\phi \left( {t, x}\right) }\right| \leq C{\left( 1 + t\right) }^... | Proof. One can use Fourier method to prove the above decay estimates. Alternatively we can directly use the representation formula. We only need to note that for compactly supported smooth function \( \psi \left( x\right) \) on \( {\mathbb{R}}^{n} \), we have the following estimates\n\n\[ \left| {{\int }_{\left| \omega... | Yes |
Theorem 3.2 (Gagliardo-Nirenberg interpolation).\n\n\[ \n{\begin{Vmatrix}{\partial }^{j}u\end{Vmatrix}}_{{L}^{p}} \leq C{\begin{Vmatrix}{\partial }^{m}u\end{Vmatrix}}_{{L}^{r}}^{\alpha }\parallel u{\parallel }_{{L}^{q}}^{1 - \alpha },\;u \in {C}_{0}^{\infty } \n\]\n\n(10)\n\nfor all\n\n\[ \n\frac{1}{p} = \frac{j}{n} + ... | The proof for this complete version mainly relies on the classical Sobolev embedding together with induction on \( m, j \), see the extra reading materials. | No |
Proposition 3.1. We have\n\n\[ \parallel {fg}{\parallel }_{{H}^{s}} \leq C\left( {\parallel g{\parallel }_{{L}^{\infty }}\parallel f{\parallel }_{{H}^{s}} + \parallel f{\parallel }_{{L}^{\infty }}\parallel g{\parallel }_{{H}^{s}}}\right) . \] | Proof. For \( j \leq k \), using Gagliardo-Nirenberg interpolation theorem we have\n\n\[ {\begin{Vmatrix}{\partial }^{j}f\end{Vmatrix}}_{{L}^{\frac{2k}{j}}} \leq C{\begin{Vmatrix}{\partial }^{k}f\end{Vmatrix}}_{{L}^{2}}^{{\theta }_{j}}\parallel f{\parallel }_{{L}^{\infty }}^{1 - {\theta }_{j}},\;\frac{j}{2k} = \frac{j}... | Yes |
Proposition 4.2 (Classical energy estimates). Assume that \( ▱\phi = F \) . Then\n\n\[ \parallel \partial \phi {\parallel }_{{L}^{2}\left( {\sum }_{t}\right) } \leq \parallel \partial \phi {\parallel }_{{L}^{2}\left( {\sum }_{0}\right) } + {\int }_{0}^{t}\parallel F{\parallel }_{{L}^{2}\left( {\sum }_{s}\right) }{ds}. ... | Proof. Indeed, the above energy identity and computations show that\n\n\[ \frac{d}{dt}\parallel \partial \phi {\parallel }_{{L}^{2}\left( {\sum }_{t}\right) }^{2} = - 2{\int }_{{\sum }_{t}}▱\phi \cdot {\partial }_{t}{\phi dx} \leq 2\parallel F{\parallel }_{{L}^{2}}\parallel \partial \phi {\parallel }_{{L}^{2}}, \]\n\nf... | Yes |
Proposition 4.5 (Decay of linear wave revisit). Assume that \( ▱\phi = 0 \) . Then\n\n\[ \left| {\partial \phi }\right| \left( {t, x}\right) \leq C{\left( 1 + \left| t - \right| x\left| \right| \right) }^{-\frac{1}{2}}{\left( 1 + t + \left| x\right| \right) }^{-\frac{n - 1}{2}}\mathop{\sum }\limits_{{\left| \alpha \rig... | Proof. The first decay estimate for \( \partial \phi \) follows directly from the previous Klainerman-Sobolev embedding together with the energy conservation for \( {\Gamma }^{\alpha }\phi \) while for the second one, we need further treatment due to the fact that we do not have the uniform \( {L}^{2} \) bound for \( {... | Yes |
Lemma 5.1 (Gronwall’s inequality). Let \( f \) and \( g \) be nonnegative functions and \( f \) is continuous, \( g \) is integrable such that\n\n\[ f\left( t\right) \leq A + {\int }_{0}^{t}f\left( s\right) g\left( s\right) {ds},\;\forall t \geq 0. \]\n\nThen we have\n\n\[ f\left( t\right) \leq A\exp \left( {{\int }_{0... | Proof. The proof is simple. | No |
Lemma 5.2 (Energy estimate for solutions of general linear wave equation). We have\n\n\[ \parallel \partial \phi {\parallel }_{{L}^{2}} \leq C{e}^{{\int }_{0}^{t}\parallel \partial g{\parallel }_{{L}^{\infty }}{ds}}\left( {{\begin{Vmatrix}{\phi }_{0}\end{Vmatrix}}_{{H}^{1}} + {\begin{Vmatrix}{\phi }_{1}\end{Vmatrix}}_{... | Proof. In view of the energy identity, we have\n\n\[ {\iint }_{\mathcal{D}}{▱}_{g}\phi \cdot X\left( \phi \right) + T{\left\lbrack \phi \right\rbrack }_{\mu \nu }{\left( {\pi }^{X}\right) }^{\mu \nu } = {\int }_{\partial \mathcal{D}}{i}_{T{\left\lbrack \phi \right\rbrack }_{X}}d\text{vol.} \]\n\nTake the region \( \mat... | Yes |
Corollary 5.1 (Higher order energy estimates). For all positive integer \( k \), we have\n\n\[ \n\parallel \phi \left( {t, x}\right) {\parallel }_{{H}^{k}\left( {\mathbb{R}}^{n}\right) } + {\begin{Vmatrix}{\partial }_{t}\phi \end{Vmatrix}}_{{H}^{k - 1}\left( {\mathbb{R}}^{n}\right) }\n\]\n\n\[ \n\leq C\left( {1 + t}\ri... | Proof. First notice that\n\n\[ \n\parallel \phi \left( {t, x}\right) {\parallel }_{{L}^{2}\left( {\mathbb{R}}^{n}\right) } \leq \parallel \phi \left( {0, x}\right) {\parallel }_{{L}^{2}\left( {\mathbb{R}}^{n}\right) } + {\int }_{0}^{t}{\begin{Vmatrix}{\partial }_{t}\phi \left( s, x\right) \end{Vmatrix}}_{{L}^{2}\left( ... | Yes |
Proposition 5.1. Assume that \( \phi \in {C}_{0}^{\infty }\left( {\left( {-\infty, T}\right) \times {\mathbb{R}}^{n}}\right) \) and the background metric \( g \) is smooth. Then\n\n\[ \parallel \phi \left( {t, \cdot }\right) {\parallel }_{{H}^{s}} + {\begin{Vmatrix}{\partial }_{t}\phi \end{Vmatrix}}_{{H}^{s - 1}} \leq ... | Proof. We only prove the case when \( s \) is integer. The fraction case can follow by using a type of interpolation theorem. The case when \( k \) is positive integer follows from the previous Corollary for higher order energy estimates, for which we view the initial data on the slice \( \{ t = T\} \) and solve the eq... | Yes |
Proposition 5.2 (Existence of solution to general linear wave equation). For given initial data \( \left( {{\phi }_{0},{\phi }_{1}}\right) \in {H}^{k} \times {H}^{k - 1} \) and function \( F \in {L}^{1}\left( {\left\lbrack {0, T}\right\rbrack ,{H}^{k - 1}}\right) \), there exists a unique solution\n\n\[ \left( {\phi ,{... | Proof. First we assume that \( \left( {{\phi }_{0},{\phi }_{1}}\right) = 0 \) . Consider the function space \( \mathcal{M} = {C}_{0}^{\infty }(( - \infty, T\rbrack \times \) \( {\mathbb{R}}^{n} \) ). For all \( \psi \in \mathcal{M} \), consider the map\n\n\[ {▱}_{g}\mathcal{M} \rightarrow \mathbb{R} \]\n\n\[ {▱}_{g}\ps... | Yes |
Lemma 6.1 (Banach fixed point theorem). Let \( B \) be complete Banach space. Let \( T : B \rightarrow B \) be the map such that\n\n\[ \parallel T\left( x\right) - T\left( y\right) \parallel \leq c\parallel x - y\parallel ,\;\forall x, y \in B \]\n\nfor some constant \( 0 < c < 1 \) . Then there exists a unique \( x \i... | Proof. The proof is same as the Picard iteration process for proving the existence of solution of ODE. Take \( {x}_{0} \in B \), define \( {x}_{1} = T\left( {x}_{0}\right) \) and \( {x}_{n} = T\left( {x}_{n - 1}\right) \) for \( n \geq 1 \) . Since by the assumption\n\n\[ \begin{Vmatrix}{{x}_{n} - {x}_{n - 1}}\end{Vmat... | Yes |
Proposition 6.2. Let \( {T}^{ * } < \infty \) be the life span of the solution to the general quasilinear wave equation. Then\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow {T}^{ * }}}\mathop{\sum }\limits_{{\left| \alpha \right| \leq 1}}{\begin{Vmatrix}{\partial }^{\alpha }\phi \left( t, x\right) \end{Vmatrix}}_{{L}_{... | Proof. We prove by contradiction. Assume that\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow {T}^{ * }}}\mathop{\sum }\limits_{{\left| \alpha \right| \leq 1}}{\begin{Vmatrix}{\partial }^{\alpha }\phi \left( t, x\right) \end{Vmatrix}}_{{L}_{x}^{\infty }} = M < \infty . \]\n\nFor simplicity, we assume that \( s > \frac{n... | Yes |
Theorem 7.3 (Klainerman, 1985). Consider the nonlinear wave equation in dimension \( 3 + 1 \) verifying the above null condition. Then there exists a small positive constant \( {\epsilon }_{0} > 0 \) such that if\n\n\[ \n{E}_{4,2} < {\epsilon }_{0} \n\]\n\nthen the solution is globally in time. | Proof. First recall the energy estimate\n\n\[ \n{\int }_{{C}_{u, t}}{\left| {\bar{\partial }}_{v}\phi \right| }^{2}{d\sigma } + \int {\left| \partial \phi \left( t, x\right) \right| }^{2}{dx} \leq \parallel \partial \phi \left( {0, x}\right) {\parallel }_{{L}^{2}}^{2} + {\int }_{0}^{t}\left| {▱\phi }\right| \left| {\pa... | Yes |
Theorem 7.4 (Christodoulou [6] 86). Consider the Cauchy problem to the nonlinear wave equation with at least quadratic nonlinearities on \( {\mathbb{R}}^{1 + n} \) with \( n \) odd. Then for sufficiently small and compactly supported initial data, the solution exists globally and obeys pointwise decay estimates\n\n\[ \... | Proof. The assumption implies that for sufficiently small initial data, \( \widetilde{\phi } \) exists on the whole region \( \mathbf{\Phi }\left( \mathbf{D}\right) \) . Thus \( \phi \) exists on the whole region \( \mathbf{D} \) . Moreover since \( \widetilde{\phi } \) is uniformly bounded, we thus conclude that\n\n\[... | Yes |
Consider the equation\n\n\[ \n▱\phi = {\left( 1 - t - \left| x\right| \right) }^{-\frac{1}{2}}{\left( 1 - t + \left| x\right| \right) }^{-\frac{1}{2}}{\left| \phi \right| }^{2},\n\]\n\n\[ \n\phi \left( {0, x}\right) = {\phi }_{0},\;{\partial }_{t}\phi \left( {0, x}\right) = {\phi }_{1},\;t + \left| x\right| \leq 1,\;t ... | By doing conformal reversion, the above statement is true in view of the result of Klainerman. The question is whether one can prove this by using vector field method directly and avoid the use of conformal transformation. | No |
Theorem 8.4 (Christodoulou [8]). Consider the characteristic initial value problem for the Einstein scalar field system with spherical symmetry. Assume that on the initial out going null cone there are two spheres \( {S}_{10},{S}_{20} \) with radius \( {r}_{10} < {r}_{20} \) respectively. Denote\n\n\[ \n{\delta }_{0} =... | Proof. (1) We prove by contradiction. Since on the initial cone \( {C}_{0},{\partial }_{v}r = \frac{1}{2} \) . We in particular have that \( r = \frac{v}{2} \) . Let \( {v}_{10} = 2{r}_{10},{v}_{20} = 2{r}_{20} \) . Assume that the region \( \{ 0 \leq u \leq {u}_{ * },{v}_{10} \leq v \leq \) \( \left. {v}_{20}\right\} ... | Yes |
Theorem 9.1 (Levine [36]). Consider the Cauchy problem to the focusing semilinear wave equation with smooth and compactly supported initial data. If the initial energy \( E\left\lbrack \phi \right\rbrack < 0 \) , then the solution must blow up in finite time. | Proof. We prove by contradiction. Suppose that \( \phi \left( {t, x}\right) \) solves the equation. In view of finite speed of propagation, the solution is also compactly supported at any time \( t \) . Define\n\n\[ m\left( t\right) = {\int }_{{\mathbb{R}}^{d}}{\left| \phi \left( t, x\right) \right| }^{2}{dx}. \]\n\nTh... | Yes |
Theorem 9.2 (Kenig-Merle 08). Consider the Cauchy problem to the energy critical focusing semilinear wave equation in \( {\mathbb{R}}^{1 + d} \) with \( 3 \leq d \leq 5 \) with data \( {\phi }_{0} \in {\dot{H}}^{1},{\phi }_{1} \in {L}^{2} \). Assume that\n\n\[ E\left\lbrack \phi \right\rbrack < E\left\lbrack W\right\rb... | However, we can sketch the proof for the blowing up result. The argument is quite similar to Theorem 9.1 once we have the following fact. Lemma 9.1. Assume that\n\n\[ E\left\lbrack \phi \right\rbrack \leq \left( {1 - {\delta }_{0}}\right) E\left\lbrack W\right\rbrack ,\;{\int }_{{\mathbb{R}}^{d}}{\left| \nabla \phi \ri... | No |
Theorem 9.3 (Local well-posedness for energy subcritical nonlinear wave equations). Consider the Cauchy problem to the above semilinear wave equation on \( {\mathbb{R}}^{1 + d} \) with energy subcritical power, that is \( 1 < p < \frac{d + 2}{d - 2} \) when \( d \geq 3 \) and \( 1 < p < \infty \) for the case \( d = 1,... | The case when \( d = 1 \) is trivial as the energy controls the \( {L}^{\infty } \) norm of the solution. For the higher dimension case, the proof relies on the following spacetime dispersive estimates for wave operator. | No |
Theorem 9.4 (Strichartz estimates). For solutions of linear wave equations, we have the following Strichartz estimates: | For the proof, we refer to a lecture note by Jacek Jendrej on the website of BiCMR under minicourse \ | No |
Theorem 9.6 (Grillakis [23]). The energy critical defocusing semilinear wave equation\n\n\[ \n▱\phi = {\left| \phi \right| }^{4}\phi ,\;\phi \left( {0, x}\right) = {\phi }_{0},\;{\partial }_{t}\phi \left( {0, x}\right) = {\phi }_{1} \n\]\n\nadmits a global in time solution \( \phi \) for all finite energy data \( \left... | We prove by contradiction. Assume that the life span is \( \left\lbrack {0,{T}^{ * }}\right) \) with \( {T}^{ * } < \infty \) and we have a solution \( \phi \) on \( \left\lbrack {0,{T}^{ * }}\right) \times {\mathbb{R}}^{3} \) . For \( t < {T}^{ * } \) and \( x \in {\mathbb{R}}^{3} \), denote the spacetime region\n\n\[... | No |
Proposition 9.2. Blow up in finite time implies the concentration of the mixed spacetime norm, that is, there exists \( x \in {\mathbb{R}}^{3} \) such that\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow {T}^{ * }}}\parallel \phi {\parallel }_{{L}_{t}^{4}{L}_{x}^{12}\left( {D\left( {x, t,{T}^{ * }}\right) }\right) } \ge... | Otherwise, the solution can be extended to time \( {T}^{ * } \) . Indeed, suppose the above claim does not hold. Then for all \( x \in {\mathbb{R}}^{3} \), for \( t \) sufficiently close to \( {T}^{ * } \), we conclude that the spacetime norm \( \parallel \phi {\parallel }_{{L}_{t}^{4}{L}_{x}^{12}\left( {D\left( {x, t,... | Yes |
Blow up in finite time implies the concentration of the potential energy, that is, there exists \( x \in {\mathbb{R}}^{3} \) such that\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow {T}^{ * }}}{\int }_{B\left( {x,{T}^{ * } - t}\right) }{\left| \phi \right| }^{6}{dy} \geq {\epsilon }_{2}\left( {E\left\lbrack \phi \right... | The proof for this claim relies on the result of the previous Proposition. Still assume that the above claim does not hold. Then fix \( t \) close to \( {T}^{ * } \) . We can show that\n\n\[ \parallel \phi {\parallel }_{{L}_{t}^{4}{L}_{x}^{12}\left( {D\left( {x, t,{T}^{ * }}\right) }\right) } \lesssim {\left( {\int }_{... | Yes |
Proposition 9.5. For solution of the energy subcritical semilinear wave equation with \( d \geq 2 \) , the potential energy enjoys the following decay estimates\n\n\[ \n{\int }_{{\mathbb{R}}^{d}}{\left| \phi \right| }^{p + 1}{dx} \leq C{\mathcal{E}}_{2}\left\lbrack \phi \right\rbrack {\left( 1 + t\right) }^{\max \{ - 2... | Proof. The proof is based on the conformal symmetry, obtained by using the conformal Killing vector field \( K = \left( {{t}^{2} + {r}^{2}}\right) {\partial }_{t} + {2tr}{\partial }_{r} \) as multiplier. For solution of the nonlinear wave equation, we can show the following energy identity\n\n\[ \n{Q}_{0}\left( t\right... | Yes |
Proposition 9.8. Let \( \phi \) be solution to the defocusing semilinear wave equation with initial data \( \left( {{\phi }_{0},{\phi }_{1}}\right) \in \left( {{\dot{H}}_{x}^{1} \cap {\dot{H}}_{x}^{{s}_{p}}}\right) \times \left( {{L}_{x}^{2} \cap {\dot{H}}_{x}^{{s}_{p} - 1}}\right) \) . Suppose that the spacetime norm ... | Proof. Since the wave equation is time invertible, it suffices to prove the scattering result in the future direction. Moreover, by interpolation, we only need to prove the solution scatters in the endpoint case when \( s = {s}_{p} \) and \( s = 1 \) . For the case when \( s = {s}_{p} \) and \( 1 < p \leq 1 + \frac{4}{... | Yes |
Theorem 9.7 (Yang [58]). Consider the defocusing semilinear wave equation on \( {\mathbb{R}}^{1 + d}, d \geq 3 \) with initial data \( \left( {{\phi }_{0},{\phi }_{1}}\right) \) bounded in \( {\mathcal{E}}_{{\gamma }_{0}} \) . If \( p\left( d\right) = \frac{1 + \sqrt{{d}^{2} + {4d} - 4}}{d - 1} < p < \frac{d + 2}{d - 2... | Proof. In view of the previous proposition, it suffices to bound the spacetime norm of the solution. For the sub-conformal case \( p\left( d\right) < p \leq \frac{d + 3}{d - 1} \), note that\n\n\[ \n\frac{\left( {d + 1}\right) \left( {p - 1}\right) }{2} \leq p + 1 \n\]\n\nBy using the time decay of the potential energy... | Yes |
Mittag-Leffler theorem: Let \( f\left( z\right) \) be a meromorphic function whose poles are \( {a}_{1},{a}_{2},{a}_{3},\ldots \), and \( 0 < \left| {a}_{1}\right| \leq \left| {a}_{2}\right| \leq \left| {a}_{3}\right| \leq \ldots \) . If there exists a sequence of contours \( \left\{ {C}_{m}\right\} \) such that\n\n(i)... | Proof: Let \( m \) be sufficiently large such that \( z \) lies within \( {C}_{m} \), then, by Cauchy's theorem,\n\n\[ \frac{1}{2\pi i}{\oint }_{{C}_{m}}\frac{f\left( \zeta \right) {d\zeta }}{\zeta - z} = \frac{1}{2\pi i}{\int }_{\left( {z}^{ + }\right) }\frac{f\left( \zeta \right) {d\zeta }}{\zeta - z} + \mathop{\sum ... | Yes |
Example 1. \( y = F\left( {\alpha ,\beta ,\alpha + \beta + \frac{1}{2}, x}\right) \) belongs to the case \( \mu = \frac{1}{2} \), i.e., to III in the table; \( A + B + C = 0, m = \frac{1}{2} \) . Hence, from the second equation of (5), \( m = {2q} + \frac{1}{2} \), we determine \( q = 0;p \) is, however, arbitrary. Let... | \[ x\left( {1 - x}\right) \frac{{d}^{2}P}{d{t}^{2}} + \left\lbrack {\alpha + \beta + \frac{1}{2} - \left( {\alpha + \beta + 1}\right) x}\right\rbrack \frac{dP}{dx} - {\alpha \beta P} = 0. \] Making the transformation \( x = {4t}\left( {1 - t}\right) \) [III in the table], this equation becomes \[ t\left( {1 - t}\right)... | Yes |
Example 2. \( y = F\\left( {\\alpha ,\\beta ,{2\\beta }, x}\\right) \) belongs to the case \( \\mu = \\nu \), i.e., to IX in the table; \( B + C = 0, l + {2m} = 2 \) . By (5), we find \( q = - \\left( {p + \\alpha }\\right) /2 \) . Take \( p = 0 \) , then \( q = - \\alpha /2 \) and \( A = \\frac{\\alpha }{2}\\left( {\\... | \[ {x}^{2}\\left( {1 - {x}^{2}}\\right) \\frac{{d}^{2}P}{d{x}^{2}} + \\left\\lbrack {{2\\beta } - \\left( {\\beta + 1}\\right) x}\\right\\rbrack x\\left( {1 - x}\\right) \\frac{dP}{dx} + \\frac{\\alpha }{2}\\left( {\\beta - \\frac{\\alpha }{2}}\\right) {x}^{2}P = 0. \] Making the transformation \( x = 2\\sqrt{t}/\\left... | Yes |
Let \( R \) be a ring, \( R \) maybe viewed as a left \( R \) -module when the left \( R \) -action (0.3.1.1) is given by left multiplication. | More generally, for \( r \in \mathbb{N} \), let \( M \) denote \( \left\{ {\left( {{m}_{1},\ldots ,{m}_{r}}\right) \mid {m}_{1},\ldots ,{m}_{r} \in R}\right\} \), the set of \( r \) -tuples in \( R \), equipped with an abelian group structure given by termwise addition, namely\n\n\[ \left( {{m}_{1},\ldots ,{m}_{r}}\rig... | Yes |
In the first example, we discuss the five platonic solids:\n\nTetrahedron, Hexahedron, Octahedron, Dodecahedron, and Isocahedron.\n\nThe symmetry of each platonic solid is the set of ways to rotate the space at the center of the solid yet keeping the solid stable. For example, if we label the four vertices of the tetra... | By giving our three dimensional space a Cartesian coordinate system centered at the center of a platonic solid, we may represent each symmetry as a \( 3 \times 3 \) real matrix, and all symmetries form a finite group in \( {\mathrm{{GL}}}_{3}\left( \mathbb{R}\right) \) . In some sense, the classification of platonic so... | Yes |
Example 1.2.4. Fix a positive integer \( n \) . Let \( {\mathbf{Z}}_{n} = \{ \overline{0},\overline{1},\ldots ,\overline{n - 1}\} \) denotes the full set of residual classes modulo \( n \), where \( \bar{i} \) is short for \( i{\;\operatorname{mod}\;n} \) . One can add two residual classes, writing \( { + }_{n} \) for ... | See also Example 0.2.3 . | No |
Example 1.4.2. The group \( {\mathbf{Z}}_{6} = \left\langle {t \mid {t}^{6} = 1}\right\rangle \) . | We may also write \( {\mathbf{Z}}_{6} = \left\langle {r, s \mid {r}^{3} = {s}^{2} = 1,{rs} = {sr}}\right\rangle \) (if \( r \) represents \( \overline{2} \) and \( s \) represents \( \overline{3} \) ). So there might be many ways to represent the same group using generators and relations. | Yes |
Proposition 2.2.3. Two (left) coset \( {g}_{1}H \) and \( {g}_{2}H \) are\n\n- either equal (which is equivalent to \( {g}_{1}^{-1}{g}_{2} \in H \) );\n\n- or disjoint (which is equivalent to \( {g}_{1}^{-1}{g}_{2} \notin H \) ).\n\nIn particular, \( G \) is the disjoint union of left cosets for \( H \) . | Proof. We will prove that\n\n(A) If \( {g}_{1}^{-1}{g}_{2} \in H \), then \( {g}_{1}H = {g}_{1}\left( {{g}_{1}^{-1}{g}_{2} \cdot H}\right) = {g}_{2}H \) (as for any element \( h \in H, h \cdot H = H \) as sets).\n\n(B) If \( {g}_{1}H \cap {g}_{2}H \neq \varnothing \), say \( x = {g}_{1}{h}_{1} = {g}_{2}{h}_{2} \) for \... | Yes |
Theorem 2.3.2 (Lagrange). If \( G \) is a finite group and \( H < G \) is a subgroup, then \( \left| H\right| \) divides \( \left| G\right| \) . | Proof. As each cosets for \( H \) has exactly \( \left| H\right| \) elements, we have \( \left| G\right| = \left\lbrack {G : H}\right\rbrack \cdot \left| H\right| \) . | Yes |
Corollary 2.3.3. (1) If \( G \) is a finite group, then \( \left| x\right| \) divides \( \left| G\right| \) . | Proof. (1) This follows from Lagrange theorem because \( \left| x\right| = \left| {\langle x\rangle }\right| \) and \( \langle x\rangle \) is a subgroup of \( G \) . | Yes |
Corollary 2.3.5. If a group \( G \) has \( p \) elements with \( p \) a prime, then \( G \) is cyclic (and in particular abelian). | Proof. Take an element \( x \in G \) such that \( x \neq e \) . Then \( \left| x\right| \) divides \( \left| G\right| = p \) . Yet \( \left| x\right| \neq 1 \) ; so \( \left| x\right| = p \), i.e. \( \left| {\langle x\rangle }\right| = p \) . So \( G = \langle x\rangle \) is cyclic. | Yes |
Lemma 2.4.2. If \( H \) is a subgroup of \( G \) and \( g \in G \), then \( {gH}{g}^{-1} \mathrel{\text{:=}} \left\{ {{gh}{g}^{-1} \mid h \in H}\right\} \) is a subgroup, called the conjugate of \( H \) by \( g \) . | Proof. Given \( {ga}{g}^{-1},{gb}{g}^{-1} \in {gH}{g}^{-1} \) , \[ \left( {{ga}{g}^{-1}}\right) {\left( gb{g}^{-1}\right) }^{-1} = {ga}{g}^{-1} \cdot g{b}^{-1}{g}^{-1} = {ga}{b}^{-1}{g}^{-1} \in {gH}{g}^{-1}. \] So \( {gH}{g}^{-1} \) is a subgroup of \( G \) . | Yes |
Every subgroup of an abelian group is normal, because \( {gH}{g}^{-1} = H \) automatically holds. | because \( {gH}{g}^{-1} = H \) automatically holds. | Yes |
Proposition 2.5.1. Let \( H \) and \( K \) be subgroups of a group \( G \) . Define \( {HK} : = \{ {hk} \mid h \in \) \( H, k \in K\} \) . When \( G \) is finite, we have\n\n\[ \left| {HK}\right| = \frac{\left| H\right| \cdot \left| K\right| }{\left| H \cap K\right| } \] | Proof. By Proposition 2.2.3, \( {HK} \) is a disjoint union of left cosets of \( K \), namely\n\n\[ {HK} = {h}_{1}K \sqcup {h}_{2}K \sqcup \cdots \sqcup {h}_{m}K. \]\n\nWe claim that for the same \( {h}_{1},\ldots ,{h}_{m} \),\n\n\[ H = {h}_{1}\left( {H \cap K}\right) \sqcup \cdots \sqcup {h}_{m}\left( {H \cap K}\right... | Yes |
Lemma 2.5.3. Let \( H \) and \( K \) be subgroups of \( G \). If \( {HK} = {KH} \) as sets (meaning every product \( {kh} \) with \( k \in K \) and \( h \in H \) can be rewritten as \( {h}^{\prime }{k}^{\prime } \) with \( {k}^{\prime } \in {K}^{\prime } \) and \( {h}^{\prime } \in {H}^{\prime } \) ), then \( {HK} \) i... | Proof. For \( {h}_{1},{h}_{2} \in H \) and \( {k}_{1},{k}_{2} \in K \), we need to check that\n\n\[ \n{h}_{1}{k}_{1} \cdot {\left( {h}_{2}{k}_{2}\right) }^{-1} = {h}_{1}{k}_{1}{k}_{2}^{-1}{h}_{2}^{-1} \n\]\n\nbelongs to \( {HK} \). Yet the condition says that \( {k}_{1}{k}_{2}^{-1} \cdot {h}_{2}^{-1} = {h}^{\prime }{k}... | Yes |
Lemma 2.5.4. If \( H \) and \( K \) are both normal subgroups of \( G \), then \( {HK} \) is also a normal subgroup of \( G \) . | Proof. We have checked that \( {HK} \) is a subgroup. For any \( g \in G \), we check\n\n\[ \n{gHK} = {HgK} = {HKg}.\n\]\n\nSo \( {HK} \) is a normal subgroup of \( G \) . | No |
Lemma 2.6.4. If \( \phi : \left( {G, * }\right) \rightarrow \left( {H, \star }\right) \) and \( \psi : \left( {H, \star }\right) \rightarrow \left( {K, \bullet }\right) \) are two homomorphisms, then the composition \( \psi \circ \psi : \left( {G, * }\right) \rightarrow \left( {K, \bullet }\right) \) is a homomorphism. | Proof. We simply check that for \( x, y \in G \), we have\n\n\[ \psi \circ \phi \left( {x * y}\right) = \psi (\phi \left( x\right) \star \phi \left( y\right) = \psi \left( {\phi \left( x\right) }\right) \bullet \psi \left( {\phi \left( y\right) }\right) . \] | Yes |
Lemma 2.7.2. Let \( \phi : G \rightarrow H \) be a group homomorphism.\n\n(1) The image \( \phi \left( G\right) \) is a subgroup of \( H \) . | Proof. (1) It follows from that \( \phi \left( {g}_{1}\right) \phi {\left( {g}_{2}\right) }^{-1} = \phi \left( {g}_{1}\right) \phi \left( {g}_{2}^{-1}\right) = \phi \left( {{g}_{1}{g}_{2}^{-1}}\right) \in \phi \left( G\right) \) . | Yes |
Lemma 2.7.3. A homomorphism \( \phi : G \rightarrow H \) of groups is injective if and only if \( \ker \phi = \) \( \left\{ {e}_{G}\right\} \) . | Proof. The injectivity \( \Rightarrow \ker \phi = \left\{ {e}_{G}\right\} \) is clear as \( \phi \left( {e}_{G}\right) = {e}_{H} \) .\n\nConversely, suppose that \( \ker \phi = \left\{ {e}_{G}\right\} \), we need to show that \( \phi \) is injective.\n\nSuppose that \( \phi \left( {g}_{1}\right) = \phi \left( {g}_{2}\r... | Yes |
Theorem 3.1.1 (The first isomorphism theorem). If \( \phi : G \rightarrow H \) is a homomorphism of groups, then \( \ker \phi \trianglelefteq G \) and\n\n\[ G/\ker \phi \cong \phi \left( G\right) \] | Proof. It may help to visualize the situation of the theorem as follows:\n\n\n\nWe define the needed map\n\n\[ \psi : G/\ker \phi \rightarrow \phi \left( G\right) \]\n\n\[ g\ker \phi \mapsto \phi \left( g\right) . \]\n... | Yes |
Theorem 3.1.2 (The second isomorphism theorem). Let \( G \) be a group, and let \( A \leq G \) be a subgroup and \( B \trianglelefteq G \) a normal subgroup. Then \( {AB} \) is a subgroup of \( G \) , \( B \trianglelefteq {AB} \) , \( A \cap B \trianglelefteq A \) , and\n\n\[{AB}/B \cong A/\left( {A \cap B}\right) \tex... | Proof. We have shown in Lemma 2.5.3 that \( {AB} \leq G \) .\n\nWe first prove that \( B \trianglelefteq {AB} \) : given \( a \in A \) and \( b \in B \), we have\n\n\[{abB}{\left( ab\right) }^{-1} = {abB}{b}^{-1}{a}^{-1} = {aB}{a}^{-1} = B.\]\n\nThus, the quotient \( {AB}/B \) makes sense.\n\nNow define a homomorphism\... | Yes |
Theorem 3.1.6 (The third isomorphism theorem). Let \( G \) be a group and \( H \) and \( K \) be normal subgroups with \( H \leq K \). Then \( K/H \trianglelefteq G/H \), and\n\n\[ \left( {G/H}\right) /\left( {K/H}\right) \cong G/K \] | Proof. Consider the map\n\n\[ \phi : G/H \rightarrow G/K \]\n\n\[ {gH} \mapsto {gK}\text{.} \]\n\n- \( \phi \) is well-defined. This is because if \( {g}_{1}H = {g}_{2}H \), then \( {g}_{1} = {g}_{2}h \) for some \( h \in H \). Thus\n\n\[ {g}_{1}K = {g}_{2}{hK} = {g}_{2}K \]\n\n- \( \phi \) is a homomorphism. This is b... | Yes |
Theorem 3.1.7 (The fourth isomorphism theorem / Lattice isomorphism theorem). Let \( G \) be a group and \( N \trianglelefteq G \) a normal subgroup. Then there is a bijection\n\n\[ \n\{ \text{ subgroups of }G\text{ containing }N\} \leftrightarrow \{ \text{ subgroups of }G/N\}\n\]\n\n\[ \nA \vdash \text{___} \rightarro... | This bijection preserves\n\n- inclusions of groups,\n\n- index of subgroups,\n\n- intersections, and\n\n- normality of subgroups.\n\nVisually, we have\n\nLattice of subgroups of \( G \) containning \( N \leftrightarrow \) Lattice of subgroups of \( G/N \) . | Yes |
Lemma 3.2.1. Such map \( \Phi \) is well-defined if and only if \( N \subseteq \ker \phi \) . In this case, \( \Phi \) is a homomorphism. | Proof. This is because if \( {g}_{1}N = {g}_{2}N \), then \( {g}_{1} = {g}_{2}n \) for some \( n \in N \) . Thus we need to see whether\n\n\[ \phi \left( {g}_{1}\right) = \phi \left( {{g}_{2}n}\right) = \phi \left( {g}_{2}\right) \phi \left( n\right) \neq \phi \left( {g}_{2}\right) . \]\n\nThis happens if and only if \... | Yes |
Consider a homomorphism \( \phi : \mathbb{Z} \rightarrow {\mathbb{C}}^{ \times } \) (such a homomorphism is in fact determined by the value \( \lambda \mathrel{\text{:=}} \phi \left( 1\right) \in {\mathbb{C}}^{ \times } \) . Then we ask the question: which \( \phi \) induces a well-defined homomorphism \( \mathbb{Z}/\l... | For this, we need \( \phi \left( {\langle n\rangle }\right) = 1 \) . This is equivalent to requiring \( \phi \left( n\right) = 1 \), or in other words, \( {\lambda }^{n} = 1 \) . | Yes |
Example 3.4.2. For the dihedral group \( {D}_{8} = \left\langle {r, s \mid {r}^{4} = {s}^{2} = 1,{srs} = {r}^{-1}}\right\rangle \), the following are two composition series (and there are more): | - \( \{ 1\} \vartriangleleft \langle s\rangle \vartriangleleft \left\langle {s,{r}^{2}}\right\rangle \vartriangleleft {D}_{8} \) ,\n- \( \{ 1\} \vartriangleleft \left\langle {r}^{2}\right\rangle \vartriangleleft \langle r\rangle \vartriangleleft {D}_{8} \) . | Yes |
Theorem 3.4.3 (Jordan-Hölder). Let \( G \) be a nontrivial finite group. Then\n\n(1) \( G \) has a composition series, and\n\n(2) the composition factors are unique up to permutation, i.e. if we have two composition\n\nseries\n\n\[ \n\{ 1\} = {A}_{0} \vartriangleleft {A}_{1} \vartriangleleft \cdots \vartriangleleft {A}... | Proof of (1). This is because if \( G \) is simple, then \( \{ 1\} \vartriangleleft G \) itself forms a composition series. If \( G \) has a nontrivial normal subgroup \( N \), then we may immediately reduce to \( N \) and \( G/N \) as follows: writing \( \pi : G \rightarrow G/N \) and giving composition series\n\n\[ \... | No |
Example 3.4.6. The group of upper triangular invertible matrices is solvable. | \[ G = \left\{ {\left. {\left( \begin{matrix} * & * & * \\ 0 & * & * \\ 0 & 0 & * \end{matrix}\right) \in {\mathrm{{GL}}}_{3}\left( \mathbb{C}\right) }\right\} \supseteq N = \left\{ \left. {\left( \begin{matrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{matrix}\right) \in {\mathrm{{GL}}}_{3}\left( \mathbb{C}\right) }\ri... | Yes |
Theorem 4.1.1 (Jordan-Hölder). Assume that a group \( G \) has the following two composition series\n\n\[ \n\{ 1\} = {A}_{0} \vartriangleleft {A}_{1} \vartriangleleft \cdots \vartriangleleft {A}_{m} = G\;\text{ and }\;\{ 1\} = {B}_{0} \vartriangleleft {B}_{1} \vartriangleleft \cdots \vartriangleleft {B}_{n} = G, \n\]\n... | 4.1.3. Proof of Theorem 4.1.1. We prove a slightly stronger version: let \( G \) be a group. Suppose that we are given two chains of subgroups\n\n\[ \n\{ 1\} = {A}_{0} \trianglelefteq {A}_{1} \trianglelefteq \cdots \trianglelefteq {A}_{m} = G,\;\{ 1\} = {B}_{0} \trianglelefteq {B}_{1} \trianglelefteq \cdots \trianglele... | Yes |
Proposition 4.2.5. The map \( \operatorname{sgn} : {S}_{n} \rightarrow \{ \pm 1\} \) is a homomorphism. | Proof. By definition, for \( \sigma ,\tau \in {S}_{n} \), we have\n\n\[ \operatorname{sgn}\left( {\sigma \tau }\right) = \frac{\mathop{\prod }\limits_{{1 \leq i < j \leq n}}\left( {{x}_{{\sigma \tau }\left( i\right) } - {x}_{{\sigma \tau }\left( j\right) }}\right) }{\mathop{\prod }\limits_{{1 \leq i < j \leq n}}\left( ... | Yes |
Lemma 4.4.2. If \( m, n \in {\mathbb{N}}_{ \geq 2} \) satisfying \( \gcd \left( {m, n}\right) = 1 \), then\n\n\[{\mathbf{Z}}_{mn} \cong {\mathbf{Z}}_{m} \times {\mathbf{Z}}_{n}\] | Proof. Consider the group homomorphism\n\n\[{\mathbf{Z}}_{mn}\xrightarrow[]{\phi }{\mathbf{Z}}_{m} \times {\mathbf{Z}}_{n}\]\n\n\[a \mapsto \left( {a{\;\operatorname{mod}\;m}, a{\;\operatorname{mod}\;n}}\right) .\n\nWe compute the kernel of \( \phi \) :\n\n\[\ker \phi = \left\{ {a{\;\operatorname{mod}\;m}n\left| {\;\be... | Yes |
Determine whether \( {\mathbf{Z}}_{30} \times {\mathbf{Z}}_{100} \) is isomorphic to \( {\mathbf{Z}}_{60} \times {\mathbf{Z}}_{50} \) . | We write\n\n\[ \n{\mathbf{Z}}_{30} \times {\mathbf{Z}}_{100} \simeq {\mathbf{Z}}_{2} \times {\mathbf{Z}}_{3} \times {\mathbf{Z}}_{5} \times {\mathbf{Z}}_{4} \times {\mathbf{Z}}_{25} \]\n\n\[ \n{\mathbf{Z}}_{60} \times {\mathbf{Z}}_{50} \simeq {\mathbf{Z}}_{3} \times {\mathbf{Z}}_{4} \times {\mathbf{Z}}_{5} \times {\mat... | No |
List all abelian groups of order \( {72} = 8 \times 9 \) . | We list them using the following table:\n\n<table><thead><tr><th></th><th>\( {\mathbf{Z}}_{2}^{3} \)</th><th>\( {\mathbf{Z}}_{2} \times {\mathbf{Z}}_{4} \)</th><th>\( {\mathbf{Z}}_{8} \)</th></tr></thead><tr><td>\( {\mathbf{Z}}_{3}^{2} \)</td><td>\( {\mathbf{Z}}_{3}^{2} \times {\mathbf{Z}}_{2}^{3} \)</td><td>\( {\mathb... | Yes |
Theorem 5.1.1 (Criterion of direct product group). Suppose that \( G \) is a group with subgroups \( H \) and \( K \) such that\n\n(1) \( H \) and \( K \) are normal subgroups of \( G \), and\n\n(2) \( H \cap K = \{ 1\} \) .\n\nThen \( {HK} \cong H \times K \) (as groups). | Proof. Since both \( H \) and \( K \) are normal subgroups of \( G,{HK} \) is a normal subgroup of \( G \) . Consider the natural map\n\n\[ \phi : H \times K \rightarrow {HK} \]\n\n\[ \left( {h, k}\right) \mapsto {hk} \]\n\n- \( \phi \) is a homomorphism. For this, we need to check that, for \( {h}_{1},{h}_{2} \in H \)... | Yes |
Proposition 5.2.5. Let \( G \) be a group acting on a set \( X \) . Then we have a natural homomorphism from \( G \) to the permutation group of \( X \) :\n\n\[ \Phi : G \rightarrow {S}_{X} \]\n\n\[ g \mapsto \left( {{\phi }_{g} : x \mapsto g \cdot x}\right) . \]\n\nIn fact, given a group action of \( G \) on \( X \) i... | Proof. We need to check that \( {\phi }_{g} \circ {\phi }_{h} = {\phi }_{gh} \) for every \( g, h \in G \) . Indeed, for \( x \in X \) ,\n\n\[ {\phi }_{g} \circ {\phi }_{h}\left( x\right) = {\phi }_{g}\left( {h \cdot x}\right) = g \cdot \left( {h \cdot x}\right) = \left( {gh}\right) \cdot x = {\phi }_{gh}\left( x\right... | Yes |
Theorem 5.2.7 (Cayley). Every group is isomorphic to a subgroup of some symmetry group. If \( \left| G\right| = n \), then \( G \) is isomorphic to a subgroup of \( {S}_{n} \) . | Proof. Consider the left translation action; by Proposition 5.2.5, it induces a homomorphism \( G \hookrightarrow {S}_{G} \) . This action is clearly faithful, and thus identify \( \overline{G\text{ as }} \) a subgroup of \( {S}_{G} \) . | Yes |
Proposition 5.4.5 (Recognizing semidirect products). Let \( G \) be a group, and let \( N \trianglelefteq G \) a normal subgroup and \( H \leq G \) a subgroup. Suppose that \( N \cap H = \{ 1\} \) . Then \( {NH} \) is a subgroup of \( G \) and \( {NH} \cong N \rtimes H \) is a semidirect product. | Proof. We have proved that \( {NH} \) is a subgroup of \( G \) . As \( N \) is a normal subgroup of \( G \) , the conjugation action for each \( h \in H \) defines an automorphism \( {\operatorname{Ad}}_{h} : N \rightarrow N \) given by \( {\operatorname{Ad}}_{h}\left( n\right) = {hn}{h}^{-1} \) . Collectively, this de... | Yes |
A typical example of semi-direct product comes from the following.\n\nRecall that \( {\mathbf{Z}}_{n} \) is the group of modulo \( n \) residual classes. Then \( \operatorname{Aut}\left( {{\mathbf{Z}}_{n}, + }\right) \cong {\mathbf{Z}}_{n}^{ \times } = \) \( \{ a{\;\operatorname{mod}\;n} \mid \gcd \left( {a, n}\right) ... | We can visualize the group \( {\mathbf{Z}}_{n} \rtimes {\mathbf{Z}}_{n}^{ \times } \) as\n\n\[ \n{\mathbf{Z}}_{n} \rtimes {\mathbf{Z}}_{n}^{ \times } \cong \left\{ {\left. {\left( \begin{array}{ll} a & b \\ 0 & 1 \end{array}\right) \in {\mathrm{M}}_{2 \times 2}\left( {\mathbf{Z}}_{n}\right) }\right| \;a \in {\mathbf{Z}... | Yes |
Let \( p \) and \( q \) be distinct primes such that \( p \mid \left( {q - 1}\right) \). We may use Example 5.4.7 construct nonabelian groups of order \( {pq} \) which are semidirect products. | It is known (will be proved later) that \( {\mathbf{Z}}_{q}^{ \times } \) is a cyclic group of order \( q - 1 \). So it must contain a unique subgroup of order \( p \). This in particular gives a homomorphism \( {\mathbf{Z}}_{p} \hookrightarrow {\mathbf{Z}}_{q}^{ \times } = \operatorname{Aut}\left( {\mathbf{Z}}_{q}\rig... | No |
Proposition 6.2.4. If a group \( G \) acts transitively on a set \( X \), for every element \( x \in X \), put \( H \mathrel{\text{:=}} {\operatorname{Stab}}_{G}\left( x\right) \) . Then there is a \( G \) -equivariant bijection\n\n\[ \phi : G/H\xrightarrow[]{ \cong }X \]\n\n\[ {gH} \mapsto {gx}\text{.} \] | Proof. First, \( \phi \) is surjective because the \( G \) -action on \( X \) is transitive.\n\nSecond, \( \phi \) is well-defined because if \( {g}_{1}H = {g}_{2}H \), then \( {g}_{1} = {g}_{2}h \) for some \( h \in H \) . Thus\n\n\[ {g}_{1}x = {g}_{2}{hx} = {g}_{2}x. \]\n\nThird, \( \phi \) is injective because if \(... | Yes |
Corollary 6.2.5. Let \( G \) be a group acting on a set \( X \) . Then \( G = \mathop{\coprod }\limits_{{\text{orbits }\mathcal{O}}}\mathcal{O} \) . | For each \( x \in X, G \) acts transitively on \( {\operatorname{Orb}}_{G}\left( x\right) \), we have\n\n\[ \n{\operatorname{Orb}}_{G}\left( x\right) = G/{\operatorname{Stab}}_{G}\left( x\right) \n\] \n\nSumming up this, we have \n\n\[ \nX \simeq \mathop{\coprod }\limits_{{G\text{-orbits }G \cdot x}}G/{\operatorname{St... | Yes |
Theorem 6.3.1. Let \( G \) be a finite group (acting on itself by conjugation).\n\n(1) For each \( g \in G \), the number of elements in its conjugacy class is\n\n\[ \left| {{\operatorname{Ad}}_{G}\left( g\right) }\right| = \left| G\right| /\left| {{C}_{G}\left( g\right) }\right| = \left\lbrack {G : {C}_{G}\left( g\rig... | Proof. (1) is clear from Proposition 6.2.4: \( {\operatorname{Ad}}_{G}\left( g\right) = G/{C}_{G}\left( g\right) \) .\n\n(2) Consider the conjugation action of \( G \) on itself. By Corollary 6.2.5, we have\n\n\[ \left| G\right| = \mathop{\sum }\limits_{\substack{\text{ conjugacy } \\ \text{ classes }{\operatorname{Ad}... | Yes |
Let \( G = {S}_{5} \). Then \( Z\left( G\right) = \{ 1\} \). | <table><thead><tr><th>Partition type</th><th>representative</th><th>stabilizer</th><th>size of conjugacy class</th></tr></thead><tr><td>\( 1 + 1 + 1 + 1 + 1 \)</td><td>(1)</td><td>\( {S}_{5} \)</td><td>\( \frac{120}{120} = 1 \)</td></tr><tr><td>\( 1 + 1 + 1 + 2 \)</td><td>(12)</td><td>\( {S}_{2} \times {S}_{3} \)</td><... | Yes |
Proposition 6.3.4. For a nontrivial p-group \( G, Z\left( G\right) \) is nontrivial. | Proof. We use class formation for the \( p \) -group \( G \) : \n\nFrom this, we see that \( p \) divides \( \left| {Z\left( G\right) }\right| \) . | Yes |
For a group \( G,\operatorname{Inn}\left( G\right) \vartriangleleft \operatorname{Aut}\left( G\right) \) . | Proof. We need to show that, if \( \sigma : G\overset{ \simeq }{ \Rightarrow }G \) is an automorphism, then \( \sigma \operatorname{Inn}\left( G\right) {\sigma }^{-1} = \operatorname{Inn}\left( G\right) \) . (In fact, it suffices to prove \ | No |
Consider \( G = {\mathrm{{GL}}}_{n}\left( \mathbb{Q}\right) \), the conjugation action gives \( \mathrm{{Ad}} : {\mathrm{{GL}}}_{n}\left( \mathbb{Q}\right) \rightarrow \) \( \operatorname{Aut}\left( G\right) \), then | \[ \ker \left( \mathrm{{Ad}}\right) = Z\left( {{\mathrm{{GL}}}_{n}\left( \mathbb{Q}\right) }\right) = \left\{ {A \in {\mathrm{{GL}}}_{n}\left( \mathbb{Q}\right) \mid {AB} = {BA},\text{ for all }B \in {\mathrm{{GL}}}_{n}\left( \mathbb{Q}\right) }\right\} \] \[ = \left\{ {a \cdot {I}_{n} \mid a \in {\mathbb{Q}}^{ \times ... | Yes |
Theorem 7.1.2 (Sylow’s theorem). Let \( G \) be a finite group with \( \left| G\right| = {p}^{r}m \) with \( r, m \in \mathbb{N} \) and \( p \nmid m \) .\n\n- (First Sylow Theorem) Sylow p-subgroups exist. | 7.2.1. Proof of First Sylow Theorem. We use an induction on \( \left| G\right| \) . When \( \left| G\right| = 1 \), there is nothing to prove.\n\nSuppose that the First Sylow Theorem is proved for finite groups of order \( < n \) . Let \( G \) be a finite group of order \( n = {p}^{r}m \) with \( r, m \in \mathbb{N} \)... | Yes |
Corollary 7.2.4. There is only one Sylow p-subgroup if and only if one Sylow p-subgroup \( P \leq G \) is normal. | Proof. \ | No |
Corollary 7.2.6. If \( P \) is a Sylow \( p \) -subgroup, then \( {N}_{G}\left( {{N}_{G}\left( P\right) }\right) = {N}_{G}\left( P\right) \), and \( {N}_{G}\left( P\right) \) contains a unique Sylow p-subgroup, which is \( P \) . | Proof. Note that \( P \trianglelefteq {N}_{G}\left( P\right) \) tautologically holds; so \( P \) is a normal Sylow \( p \) -subgroup of \( {N}_{G}\left( P\right) \) . By the above Corollary, \( P \) is the unique Sylow \( p \) -subgroup of \( {N}_{G}\left( P\right) \) . (This proves the second statement.)\n\nIt is clea... | Yes |
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