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Theorem 4.1.4 (DKW inequality). For \( x \) on the real line anywhere, the Dvoretzky-Kiefer-Wolfowitz (DKW in short) inequality bounds the probability that the random function \( {F}_{n}\left( x\right) \) differs from \( F\left( x\right) \) by more than a given constant \( \varepsilon > 0 \) .:\n\n\[ P\left( {\mathop{\... | By quantifying the rate of convergence, DKW concentration inequality strengthens the Glivenko-Cantelli theorem: \( {\begin{Vmatrix}{F}_{n} - F\end{Vmatrix}}_{\infty } \mathrel{\text{:=}} \mathop{\sup }\limits_{{x \in \mathrm{R}}}\left| {{F}_{n}\left( x\right) - F\left( x\right) }\right| \rightarrow 0 \) a.s. by using a... | Yes |
Show that \( P\left( {{\widehat{\xi }}_{p} > {\xi }_{p} + \varepsilon }\right) \leq P\left( {\mathop{\sup }\limits_{{x \in \mathrm{R}}}\left| {{F}_{n}\left( x\right) - F\left( x\right) }\right| > {\delta }_{\varepsilon }}\right) \) and obtain a similar inequality for \( P\left( {{\widehat{\xi }}_{p} < {\xi }_{p} - \var... | \[ P\left( {{\widehat{\xi }}_{p} > {\xi }_{p} + \varepsilon }\right) = P\left( {{F}_{n}\left( {\widehat{\xi }}_{p}\right) > {F}_{n}\left( {{\xi }_{p} + \varepsilon }\right) \leq P\left( {p > {F}_{n}\left( {{\xi }_{p} + \varepsilon }\right) }\right) }\right. \] \[ = P\left( {F\left( {{\xi }_{p} + \varepsilon }\right) - ... | Yes |
Example 4.1.8 (Kernel density estimation). Let \( {X}_{1},{X}_{2},\cdots ,{X}_{n}\overset{iid}{ \sim }F\left( x\right) \) . Define the Kernel density estimator by \[ {\widehat{f}}_{n, h}\left( x\right) = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}\frac{1}{h}K\left( \frac{x - {X}_{i}}{h}\right) \] where \( K \) is th... | The \( {L}_{1} \) error is \( Z = f\left( {{X}_{1},\ldots ,{X}_{n}}\right) = \int \left| {\phi \left( x\right) - {\phi }_{n}\left( x\right) }\right| {dx} \) The bound difference is \[ \left| {f\left( {{x}_{1},\ldots ,{x}_{n}}\right) - f\left( {{x}_{1},\ldots ,{x}_{i}^{\prime },\ldots ,{x}_{n}}\right) }\right| \leq \fra... | Yes |
Theorem 4.1.11 (Efron-Stein inequality). If \( {x}_{1}^{\prime },\cdots ,{x}_{n}^{\prime } \) are independent copies of \( {x}_{1},\ldots ,{x}_{n} \), let \( {z}_{i}^{\prime } = f\left( {{x}_{1},\cdots ,{x}_{t + 1},{x}_{i}^{\prime },{x}_{i + 1},\ldots ,{x}_{n}}\right) \) then \( \operatorname{var}Z \leq \frac{1}{2}E\le... | Proof: By the above argument, we have \( \operatorname{var}Z \leq E\mathop{\sum }\limits_{{i = 1}}^{n}{\Delta }_{i}^{2} \). Note that, the expectation on single r.v. in \( \mathrm{Z} \) is \( {E}^{\left( i\right) }Z = \int f\left( {{X}_{1},\cdots ,{X}_{i - 1},{x}_{i},{X}_{i + 1},\cdots ,{X}_{n}}\right) d{\mu }_{i}\left... | Yes |
Lemma 4.1.12 (Hoeffding’s lemma). For any \( \lambda > 0 \), and \( i \in \{ 1,\cdots, n\} \), if \( \left| {\Delta }_{i}\right| \leq {C}_{i} \), \( E\left\lbrack {{e}^{\lambda {\Delta }_{i}} \mid {\mathcal{F}}_{i - 1}}\right\rbrack \leq {e}^{\frac{1}{8}{\lambda }^{2}{c}_{i}^{2}} \) | Proof: By Doob's martingale representation\n\n\[ Z - {EZ} = \mathop{\sum }\limits_{{i = 1}}^{n}{\Delta }_{i}\text{ with }\left| {\Delta }_{i}\right| \leq {C}_{i}\left( {why}\right) ? \]\n\nSetting \( {F}_{i}\left( {{X}_{1},\cdots ,{X}_{i}}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{E}_{i}Z = \int f\left( {{x}_{1},\cd... | No |
Lemma 4.1.14 (Lemma 6.1 in Rigollet (2012)). Let \( {\left\{ {Y}_{i}\right\} }_{i = 1}^{n} \) be a sequence of random variables whose distribution belongs to canonical exponential family with \( f\left( {{y}_{i};{\theta }_{i}}\right) = \) \( c\left( {y}_{i}\right) \exp \left( {{y}_{i}{\theta }_{i} - b\left( {\theta }_{... | Proof: Let \( {Y}_{i} = \dot{\psi }\left( {\theta }_{i}\right) + {Z}_{i} \), where \( {Z}_{1},{Z}_{2},\cdots ,{Z}_{n} \) are centralized and independent exponential random variables. From the MGF \( {M}_{X}\left( t\right) = \mathrm{E}\exp \left( {t{Y}_{i}}\right) = \exp \left\{ {b\left( {{\theta }_{i} + t}\right) - b\l... | Yes |
Lemma 4.1.17. Let \( {X}_{1},\ldots ,{X}_{n} \) be identically distributed but not necessarily independent and assume that \( \mathrm{E}\left( {\left| {X}_{1}\right| }^{p}\right) < \infty \) with \( p \geq 1 \) . Then, we have\n\n\[ \mathrm{E}\left( {\mathop{\max }\limits_{{1 \leq k \leq n}}{X}_{i}}\right) = o\left( {n... | Proof: Put \( {M}_{n} = \mathop{\max }\limits_{{k \leq n}}\left| {X}_{k}\right| \) . For any \( \epsilon > 0 \), we get by truncation\n\n\[ \mathrm{E}\left( {M}_{n}\right) = {\int }_{0}^{\epsilon {n}^{1/p}}\mathrm{P}\left( {{M}_{n} > t}\right) {dt} + {\int }_{\epsilon {n}^{1/p}}^{\infty }\mathrm{P}\left( {{M}_{n} > t}\... | Yes |
Theorem 4.1.18 (Maximal inequality). (a) Using \( \mathrm{E}\left\lbrack {\mathop{\max }\limits_{{1 \leq i \leq N}}{X}_{i}}\right\rbrack = \frac{1}{s}\left\lbrack {\log \exp \left\{ {s\mathop{\max }\limits_{{1 \leq i \leq N}}{X}_{i}}\right\} }\right\rbrack \) and Jensen's inequality to show:\n\n\[ \mathrm{E}\left\lbrac... | \[ \mathrm{E}\mathop{\max }\limits_{{\alpha \in A}}{X}_{\alpha } = \frac{1}{\lambda }\log \left\lbrack {\exp \left( {\lambda \mathrm{E}\mathop{\max }\limits_{{\alpha \in A}}{X}_{\alpha }}\right) \leq \frac{1}{\lambda }\log {\mathrm{{Ee}}}^{\lambda \mathop{\max }\limits_{{\alpha \in A}}{X}_{\alpha }}}\right. \]\n\n\[ \l... | Yes |
Theorem 4.1.20 (Symmetrization theorem). Let \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} \) be a Rademacher sequence, independent of \( {\mathbf{X}}_{1},\ldots ,{\mathbf{X}}_{n} \) and \( f \in \mathcal{F} \) . Then we have\n\n\[ \mathrm{E}\left( {\mathop{\sup }\limits_{{f \in \mathcal{F}}}\left| {\mathop{\sum }\limits_... | Proof: Let \( {E}^{\prime } \) denote the expertation w.r.t \( {X}^{\prime } \), then let \( {\mathcal{F}}_{ \smallsetminus }^{\prime } = \sigma \left( {{\mathcal{X}}_{\infty }^{\prime },\cdots ,{\mathcal{X}}_{ \smallsetminus }^{\prime }}\right) \n\n\[ E\left( {\mathop{\sup }\limits_{{f \in \mathcal{F}}}\mathop{\sum }\... | Yes |
Theorem 4.1.21 (Contraction theorem). Let \( {\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{n} \) be the non-random elements of \( \mathcal{X} \) and \( {\varepsilon }_{1},\ldots ,{\varepsilon }_{n} \) be Rademacher sequence. Consider c-Lipschitz functions \( {g}_{i} \), i.e. \( \left| {{g}_{i}\left( s\right) - {g}_{i}\left( t... | \[ \mathrm{E}\left( {\mathop{\sup }\limits_{{f \in \mathcal{F}}}\left| {\mathop{\sum }\limits_{{i = 1}}^{n}{\varepsilon }_{i}\left\{ {{g}_{i}\left( {f\left( {\mathbf{x}}_{i}\right) }\right) - {g}_{i}\left( {h\left( {\mathbf{x}}_{i}\right) }\right) }\right\} }\right| }\right) \leq {2c}\mathrm{E}\left( {\mathop{\sup }\li... | Yes |
Theorem 4.2.6. Let \( A \) and \( B \) be two \( p \times p \) Hermitian matrices. Then\n\n\[ \mathop{\sup }\limits_{x}\left| {{F}_{A}\left( x\right) - {F}_{B}\left( x\right) }\right| \leq \frac{1}{p}\operatorname{rank}\left( {A - B}\right) . | Proof: Use the Interlacing Theorem. | No |
Theorem 4.2.9 (Bai-Yin Law, Bai and Yin (2010)). Assume \( \\left\\{ {X}_{ij}\\right\\} \) is a double array of IID complex-valued random vectors with zero means, variance \( {\\sigma }^{2} = E\\left( {{X}_{ij}{\\bar{X}}_{ij}}\\right) \) and finite fourth moment. Let \( {X}_{n} = {\\left( {X}_{jk}\\right) }_{p \\times ... | \[ \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}{\\lambda }_{\\min }\\left( {S}_{n}\\right) = {\\sigma }^{2}{\\left( 1 - \\sqrt{y}\\right) }^{2} \] and \[ \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}{\\lambda }_{\\max }\\left( {S}_{n}\\right) = {\\sigma }^{2}{\\left( 1 + \\sqrt{y}\\right) }^{2}. \] If \(... | Yes |
Lemma 4.3.2 (Computing the spectral norm on a net). Let B be an \( p \times p \) matrix, and let \( {\mathcal{N}}_{\varepsilon } \) be an \( \varepsilon \) -net of \( {S}^{p - 1} \) for some \( \varepsilon \in \lbrack 0,1) \) . Then\n\n\[ \parallel \mathbf{B}\parallel \mathrel{\text{:=}} \mathop{\max }\limits_{{\parall... | Note that:\n\[ \left\langle {\frac{1}{n}{A}^{\prime }{Ax} - x, x}\right\rangle = \left\langle {\frac{1}{n}{A}^{\prime }{Ax}, x}\right\rangle - 1 = \frac{1}{n}\parallel {Ax}{\parallel }_{2}^{2} - 1 \]\n\nThen setting \( \varepsilon = 1/4 \) in Lemma 4.3.2, we can get:\n\n\[ \begin{Vmatrix}{\frac{1}{n}{A}^{\prime }A - {I... | Yes |
A notable property of the expectation and variance of \( Y \) ’s in (4.4.1) is computed by \( \mathrm{E}\left( Y\right) = \dot{b}\left( \theta \right) \) and \( \operatorname{Var}\left( Y\right) = \ddot{b}\left( \theta \right) \). | To see this, from relation\n\n\[\n\int d{F}_{Y}\left( y\right) = 1 \Leftrightarrow \log \int c\left( y\right) {e}^{y\theta }\nu \left( {dy}\right) = b\left( \theta \right) ,\n\]\n\n(4.4.2)\n\nit gives\n\n\[\n\mathrm{E}\left( Y\right) = \int y\exp \{ {y\theta } - b\left( \theta \right) \} c\left( y\right) \nu \left( {dy... | Yes |
Theorem 4.4.2. Let \( X \) be a random \( {p}_{n} \) -dimensional vector. Suppose that \( E\left\lbrack X\right\rbrack = 0 \) and \( \parallel X\parallel \leq \beta \) almost surely for some \( \beta > 0 \) . Let \( \sum = \operatorname{Cov}\left( X\right) \), and let \( G \sim \mathcal{N}\left( {0,\sum }\right) \) be ... | \[ {\mathcal{W}}_{2}\left( {{S}_{n}, G}\right) \leq \frac{\beta \sqrt{{p}_{n}}\sqrt{{32} + 2{\log }_{2}\left( n\right) }}{\sqrt{n}}. \] | Yes |
Lemma 5.1.1 (a) \( \mathrm{E}\left( \overline{\mathbf{X}}\right) = \mu \) ; (b) \( \operatorname{Var}\left( \overline{\mathbf{X}}\right) = \frac{1}{n}\mathbf{\sum } \) | Remark: Hence, \( \overline{\mathbf{X}} \) is an unbiased and mean square consistent estimator of \( \mu \) . The latter is because \( \operatorname{Var}\left( \overline{\mathbf{X}}\right) \rightarrow 0 \) as \( n \rightarrow \infty \) . Note that \[ \mathrm{E}\parallel \overline{\mathbf{X}} - \mu {\parallel }^{2} = \m... | No |
Theorem 5.1.1. If \( {\mathbf{X}}_{1},\ldots ,{\mathbf{X}}_{n} \) are IID random vectors of fixed dimensions \( p \) with finite mean vector \( \mu \) and finite covariance matrix \( \sum \), then\n\n\[ \sqrt{n}\left( {\overline{\mathbf{X}} - \mu }\right) \overset{d}{ \rightarrow }{N}_{p}\left( {\mathbf{0},\sum }\right... | Proof: Apply the Cramér-Wold device, for all \( \lambda = {\left( {\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{p}\right) }^{T} \)\n\n\[ \sqrt{n}{\lambda }^{\mathbf{T}}\left( {\overline{\widetilde{X}} - \mu }\right) = \sqrt{n}\left( {{\lambda }_{1}\left( {{\overline{\widetilde{X}}}_{1} - {\mu }_{1}}\right) + \cdots... | Yes |
Theorem 5.1.2. Assume that all the fourth order moments of \( {\mathbf{X}}_{i} \) exist. Then, \( \sqrt{n}\left( {{\mathbf{S}}_{n} - \mathbf{\sum }}\right) \overset{d}{ \rightarrow } \) \( {N}_{{p}^{2}}\left( {\mathbf{0},\mathbf{\Omega }}\right) \) . | Proof: Similar to that of Theorem 5.1.1 by applying Cramër-Wold device. | No |
For all \( {\mathbf{A}}_{p \times p} > 0 \) and for a constant \( c > 0 \), let\n\n\[ f\left( \mathbf{A}\right) = c{\left| \mathbf{A}\right| }^{n/2}\exp \left( {-\frac{1}{2}\operatorname{tr}\left( \mathbf{A}\right) }\right) .\n\]\n\nThe maximum of \( f\left( \mathbf{A}\right) \) is achieved at \( \mathbf{A} = n{\mathbf... | Proof: Let \( {\lambda }_{1},\ldots ,{\lambda }_{p} \) be the eigenvalues of \( \mathbf{A} \) . Clearly, \( {\lambda }_{i} > 0 \) .\n\n\[ \operatorname{tr}\left( \mathbf{A}\right) = \mathop{\sum }\limits_{{i = 1}}^{p}{\lambda }_{i}\;\text{ and }\;\left| \mathbf{A}\right| = \mathop{\prod }\limits_{{i = 1}}^{p}{\lambda }... | Yes |
Corollary 5.2.2. Let \( {\mathbf{X}}_{1},\ldots ,{\mathbf{X}}_{n}\overset{i.i.d.}{ \sim }F \) with mean \( \mu \) and covariance \( \sum \) . Then, \( \overline{\mathbf{X}} \) has mean \( \mu \) and covariance \( {n}^{-1}\sum \) and | \[ {\mathbf{S}}_{n} = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{{n - 1}}{\mathbf{Z}}_{i}{\mathbf{Z}}_{i}^{\mathbf{T}},\;\overline{\mathbf{X}} = {n}^{-1/2}{\mathbf{Z}}_{n} \] where \( {\mathbf{Z}}_{1},\ldots ,{\mathbf{Z}}_{n} \) are uncorrelated and \( {\mathbf{Z}}_{i} \) has mean \( \mathbf{0} \) and covariance \( \s... | No |
Example 5.4.2 (Poisson regressions). For modelling count data, Poisson regression is a model that the response variables are nonnegative integers (count value) and follow the Poisson distribution. The responses \( {\left\{ {Y}_{i}\right\} }_{i = 1}^{n} \) obeys the Poisson distribution\n\n\[ P\left( {{Y}_{i} = {y}_{i} ... | It worth noting that the Poisson response is mean-variance dependent GLMs by\n\n\[ \mathrm{E}\left( {{Y}_{i} \mid {\mathbf{X}}_{i}}\right) = \operatorname{Var}\left( {Y \mid {\mathbf{X}}_{i}}\right) = \exp \left( {{\mathbf{X}}_{i}^{T}\mathbf{\beta }}\right) \]\n\nwhich is unbound when \( {\mathbf{X}}_{i} \) are unbound... | Yes |
Example 5.4.3 (Probit Models). Let \( \Phi \left( t\right) \) be the cumulative distribution function of \( N\left( {0,1}\right) \) . Similar to logistic regression, suppose the \( n \) independent observed data sets \( {\left\{ {Y}_{i},{\mathbf{X}}_{i}\right\} }_{i = 1}^{n} \) satisfy\n\n\[ P\left( {{Y}_{i} = 1 \mid {... | Unlike the logistic regression, the \( u\left( \cdot \right) \) and \( b\left( \cdot \right) \) in the probit model follow parametric equations :\n\n\[ u\left( {{\mathbf{X}}_{i}^{T}\beta }\right) = \log \left\lbrack {\Phi \left( {{\mathbf{X}}_{i}^{T}\beta }\right) /\left( {1 - \Phi \left( {{\mathbf{X}}_{i}^{T}\beta }\r... | No |
Example 5.4.4 (Negative Binomial Regressions). It is so common that negative-binomial GLMs are more plausible than Poisson regression for modelling overdispersion count data. For example in RNA-Seq gene expression data, the negative binomial (NB) distribution provides a good choice for modeling a count variable and rel... | The logarithm of the maximum likelihood function for \( n \) independent observed data sets \( {\left\{ {Y}_{i},{\mathbf{X}}_{i}\right\} }_{i = 1}^{n} \) for NBR is\n\n\[ n{l}_{n}\left( {\mathbf{\beta }, X, Y}\right) = \log \left\{ {\mathop{\prod }\limits_{{i = 1}}^{n}f\left( {{Y}_{i} \mid \theta ,{\mu }_{i}}\right) }\... | Yes |
Example 5.4.6 (Sparse approximation for non-parametric regression). The connection between \( \log \mathrm{E}\left( {{Y}_{i} \mid {\mathbf{X}}_{i}}\right) \) and \( {\mathbf{X}}_{i} \) is sometimes unknown, it is unlikely to be linear. Thus, it is too rough and too simple to use a linear function to approximate \( \log... | A crucial point for orthogonal base functions \( \mathbb{D} \) is that many \( f \), which have no sparse representations in the non-orthogonal base but they should be represented as sparse linear combinations of \( \mathbb{D} \) in the orthogonal scenario, see Section 10 of Hastie et al. (2015) for details. In practic... | No |
For each individual \( i \), consider the response \( {Y}_{i} \) conditional on the process \( \left\{ {{X}_{i}\left( s\right) : s \in \left\lbrack {0,1}\right\rbrack }\right\} \), follows the exponential family \( {f}_{{Y}_{i}}\left( {y}_{i}\right) = \exp \left\{ {{\theta }_{i}{y}_{i} - \psi \left( {\theta }_{i}\right... | \[ {\theta }_{i} = {\int }_{\left\lbrack 0,1\right\rbrack }{X}_{i}\left( t\right) \beta \left( t\right) {dt} = : \left\langle {\beta ,{X}_{i}}\right\rangle = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\beta }_{k}\left\langle {{X}_{i},{\phi }_{k}}\right\rangle \] by the expansion \( \beta \left( s\right) = \mathop{\sum }... | Yes |
Example 5.4.8 (Robust Poisson regression with weighted scores functions). | In the era of massive data, one of the popular methods to reduce the computational burden is doing subsampling to downsize the data volume. Introduced by Hajek (1964), Horvitz and Thompson (1952), Poisson sampling is an unequal probability sampling method that overcomes a lot of the difficulties associated with fixed-s... | No |
Example 5.4.9 (Poisson subsampling for quasi-GLMs with Massive Data). Let \( {\pi }_{i} > 0 \) be the inclusion probability for the \( i \) -th data point and \( \mathbf{\pi } = \left( {{\pi }_{1},\ldots ,{\pi }_{n}}\right) \) be the sampling distribution. First, we generate \( n \) independent Bernoulli variables \( {... | \[ {S}_{n}^{sub}\left( \mathbf{\beta }\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{i : {R}_{i} \neq 0}}\frac{-1}{n{\pi }_{i}}{\mathbf{X}}_{i}\dot{u}\left( {{\mathbf{X}}_{i}^{T}\mathbf{\beta }}\right) \left\lbrack {{Y}_{i} - \dot{b}\left( {u\left( {{\mathbf{X}}_{i}^{T}\mathbf{\beta }}\right) }\right) }\right\rbra... | Yes |
Theorem 5.5.1. (Anderson,1963). For Gaussian data, \( {\mathrm{X}}_{i}\overset{iid}{ \sim }\mathrm{N}\left( {\mu ,\sum }\right) \) for \( i = 1,\cdots, n \) and \( n > p \) . Suppose that \( \sum \) has \( k \) distinct eigenvalues \( {\lambda }_{1} > \cdots > {\lambda }_{k} \) with each has multiplicity \( {f}_{i} \),... | Proof: Assume WLOG that \( {\lambda }_{1} > {\lambda }_{2} > \cdots > {\lambda }_{k - 1} > {\lambda }_{k} = {\lambda }_{k + 1} = \cdots = {\lambda }_{p} \) . The likelihood of \( \left( {\mu ,\sum }\right) \) is\n\n\[ \nL\left( {\mu ,\sum }\right) = {\left| \sum \right| }^{-n/2}\exp \left\{ {\operatorname{tr}\left( {-\... | Yes |
Theorem 5.5.2. For \( {\mathrm{X}}_{i}\overset{\text{ iid }}{ \sim }\mathrm{N}\left( {\mu ,\sum }\right) \) for \( i = 1,\cdots, n \) and \( n > p \) . Suppose that \( \sum \) has \( k \) distinct eigenvalues \( {\lambda }_{1} > \cdots > {\lambda }_{k - 1} > {\lambda }_{k} = \cdots = {\lambda }_{p} \) . Let \( {x}_{i} ... | \[ \mathop{\prod }\limits_{{i = 1}}^{{k - 1}}\phi \left( {x}_{i}\right) \frac{{\pi }^{\left( {p - k + 1}\right) \left( {p - k}\right) /4}}{{2}^{\left( {p - k + 1}\right) /2}{\Gamma }_{p - k + 1}\left( \frac{p - k + 1}{2}\right) }\exp \left( {-\frac{1}{2}\mathop{\sum }\limits_{{j = k}}^{p}{x}_{j}^{2}}\right) \mathop{\pr... | Yes |
Theorem 6.1.2. Suppose that \( \mathbf{A} \sim {W}_{p}\left( {n,\sum }\right) \) with \( \sum > 0, n > p \), and\n\n\[ \mathbf{A} = \left( \begin{array}{ll} {\mathbf{A}}_{11} & {\mathbf{A}}_{12} \\ {\mathbf{A}}_{21} & {\mathbf{A}}_{22} \end{array}\right) \;\text{ and }\;\mathbf{\sum } = \left( \begin{array}{ll} {\mathb... | Proof: Let \( {\mathbf{X}}_{i} = \left( \begin{array}{l} {\mathbf{X}}_{i1} \\ {\mathbf{X}}_{i2} \end{array}\right) \), where \( {\mathbf{X}}_{i1} \) is \( q \times 1 \) and \( {\mathbf{X}}_{i2} \) is \( \left( {p - q}\right) \times 1 \) . Let \( {\mathbf{X}}_{1}^{\mathbf{T}} = \) \( \left( {{\mathbf{X}}_{11},{\mathbf{X... | Yes |
Theorem 6.1.3. If \( \mathbf{A} \sim {W}_{p}\left( {n,\mathbf{\sum }}\right) ,\mathbf{X} \sim {N}_{p}\left( {\mu, c\mathbf{\sum }}\right) \) for a constant \( c > 0,\mathbf{A} \) and \( \mathbf{X} \) are independent. Let \( {T}^{2}\left( {p, n,\mu }\right) = n{\mathbf{X}}^{T}{\left( c\mathbf{A}\right) }^{-1}\mathbf{X} ... | Proof: Note that\n\n\[ \frac{n - p + 1}{np}{T}^{2}\left( {p, n,\mu }\right) \sim {F}_{p, n - p + 1}\left( \lambda \right) \]\n\nwhere \( \lambda = {\mu }^{T}{\left( c\mathbf{\sum }\right) }^{-1}\mu \) .\n\n\[ {T}^{2}\left( {p, n,\mu }\right) = n \cdot \frac{{\mathbf{X}}^{T}{\left( c\mathbf{A}\right) }^{-1}\mathbf{X}}{{... | Yes |
Theorem 6.2.1. Suppose \( {X}_{i1},\cdots ,{X}_{i{n}_{i}}\overset{i.i.d.}{ \sim }N\left( {{\mu }_{i},\sum }\right) \) for \( i = 1 \) and 2. Among all tests for \( {H}_{0} : {\mu }_{1} - {\mu }_{2} = 0 \) based on \( {\bar{X}}_{1} - {\bar{X}}_{2} \) and \( \mathbf{A} = \mathop{\sum }\limits_{{i = 1}}^{2}\mathop{\sum }\... | Proof: It is readily shown that the \( {T}^{2} \) statistics is invariant with respect to the scale transformations. For any scale invariant statistic \( f\left( {{\bar{X}}_{1} - {\bar{X}}_{2},\mathbf{A}}\right) \), there exists an invertible \( C \) such that\n\n\[ C\left( {{\bar{X}}_{1} - {\bar{X}}_{2}}\right) = {\le... | Yes |
Corollary 6.3.1. Under the same assumption as Lemma 6.3.1, but assume \( \delta = 0 \), then the upper \( \alpha \) -quantile of \( {F}_{p, n - p + 1} \) -distribution admits an expansion | \[ {F}_{p, n - p + 1,\alpha } = 1 + {z}_{\alpha }\sqrt{\frac{2}{y\left( {1 - y}\right) n}} + o\left( \frac{1}{\sqrt{n}}\right) ,\] where \( {z}_{\alpha } \) is the upper \( \alpha \) -quantile of \( N\left( {0,1}\right) \) . | Yes |
Theorem 6.3.2. Under the assumption of Lemma 6.3.1, and if \( \sqrt{n}\parallel \delta {\parallel }^{2} \rightarrow c \in \lbrack 0,\infty ) \) , then \[ \beta \left( \delta \right) = \Phi \left( {-{z}_{\alpha } + \sqrt{\frac{{ny}\left( {1 - y}\right) }{2}}\frac{m}{p}\parallel \delta {\parallel }^{2}}\right) . \] | Proof: As \( \frac{n - p + 1}{np}{T}_{{n}_{1},{n}_{2}}^{2} \sim {F}_{p, n - p + 1}\left( {m\parallel \delta {\parallel }^{2}}\right) \) under \( {H}_{1} : {\mu }_{1} - {\mu }_{2} = \tau = {\sum }^{-1/2}\delta \), from Lemma 6.3.1 and Corollary 6.3.1, \[ \beta \left( \delta \right) = P\left( {{F}_{p, n - p + 1}\left( {m... | Yes |
Theorem 7.1.1. (i). Ridge penalty \( \left( {\ell }_{2}\right. \) penalty \( )P\left( \mathbf{\beta }\right) = \parallel \mathbf{\beta }{\parallel }_{2}^{2} \), it leads to OLS shrinkage estimation: | \[ {\widehat{\beta }}_{i}^{\text{Ridge }} = {\widehat{\beta }}_{i}^{\text{OLS }}/\left( {1 + \lambda }\right) \mathrel{\text{:=}} {\operatorname{Sr}}_{\lambda }\left( {\widehat{\beta }}_{i}^{\text{OLS }}\right) \] where \( {\operatorname{Sr}}_{M}\left( t\right) = t/\left( {1 + M}\right), M > 0 \) is the shrinkage funct... | Yes |
Theorem 7.1.5. (i). Let \( \widehat{\beta } \) be the Elastic-net estimates of the Logistic regression defined in (7.1.8) and Poisson regression given in (7.1.9). Suppose that \( {\lambda }_{2} > 0 \) . Then for any \( k, l \in \{ 1,2,\ldots, p\} \), Using the gradient of the negative average log-likelihood 3675 functi... | \[ \left| {{\widehat{\beta }}_{k}^{\mathrm{{Lo}}} - {\widehat{\beta }}_{l}^{\mathrm{{Lo}}}}\right| \leq \frac{1}{{\lambda }_{2}}\sqrt{\left( {1 - {\rho }_{kl}}\right) \frac{1}{2n}\mathop{\sum }\limits_{{i = 1}}^{n}{\left| {Y}_{i} - \frac{{\mathbf{X}}_{i}{}^{T}{\widehat{\mathbf{\beta }}}^{\mathrm{{Lo}}}}{1 + {\mathbf{X}... | Yes |
Corollary 7.3.3. Let \( \delta \in \left( {0,1}\right) \) be a fixed number. Suppose that the assumption of Theorem 7.3.2 is satisfied, and the weakest signal and strongest signal meet the condition: \( {B}_{0} \mathrel{\text{:=}} \frac{{4\lambda }{d}^{ * }}{2sk} + \left( \frac{\lambda + {2s}}{\lambda }\right) {\vareps... | Proof: It is directly followed from the inequalities \[ P\left( {H \subset \widehat{H}}\right) \geq P\left( {j \in \widehat{H}\text{ for all }j \notin H}\right) \] \[ \geq P\left( {{\widehat{\beta }}_{j} \neq 0}\right. \text{and}\left. {{\beta }_{j}^{ * } = 0\text{, for all}j \notin H}\right) \] \[ \geq P\left( {\left|... | Yes |
By (7.4.5) and Slutsky’s theorem, please show the asymptotic pivot\n\n\[\n\sqrt{n}\left( {\widehat{\mathbf{b}} - {\mathbf{\beta }}^{ * }}\right) = \mathbf{W} + {o}_{p}\left( 1\right) ,\;\mathbf{W} \mid \mathbf{X} \sim {N}_{p}\left( {0,{\sigma }^{2}\widehat{\mathbf{\Theta }}\widehat{\mathbf{\sum }}{\widehat{\mathbf{\The... | Proof: To show \( \mathbf{\Delta } \) is negligible in probability as \( n, p \rightarrow \infty \), you have to use matrix norm inequality \( \parallel \mathbf{A}\mathbf{l}{\parallel }_{1} \leq \parallel \mathbf{A}{\parallel }_{\infty }\parallel \mathbf{l}{\parallel }_{1} \) for matrix \( \mathbf{A} \) and vector \( \... | No |
Rothman et al (2010) propose a sparse estimator for \( \mathbf{B} \) that accounts for correlated errors using penalized normal likelihood by regularizing \( \left( {\mathbf{B},{\mathbf{\sum }}^{-1}}\right) \) simultaneously which construct a sparse estimator of \( \mathbf{B} \) depending on \( \mathbf{\Omega } = : {\m... | \[ \left( {\widehat{\mathbf{B}},\widehat{\mathbf{\Omega }}}\right) = \mathop{\operatorname{argmin}}\limits_{{\mathbf{B},\Omega }}\left\{ {\operatorname{tr}\left\lbrack {{\mathrm{n}}^{-1}{\left( \mathbf{Y} - \mathbf{{XB}}\right) }^{\mathrm{T}}\Omega \left( {\mathbf{Y} - \mathbf{{XB}}}\right) }\right\rbrack - \log \left|... | Yes |
Theorem 8.1.1. Suppose IID random vectors \( {\left\{ {X}_{i}\right\} }_{i = 1}^{n} \) are either Gaussian or satisfies (8.1.7). Then, uniformly for \( \sum \in \mathfrak{U}\left( {\alpha, C}\right) \) , \[ {\begin{Vmatrix}{B}_{k}\left( {S}_{n}\right) - \sum \end{Vmatrix}}_{2}^{2} = {O}_{p}\left\{ {\left( \log \left( p... | The above result shows that the banding estimator is consistent to \( \sum \) under the spectral norm if \( \log \left( p\right) = o\left( n\right) \) or \( p = o\left( {e}^{cn}\right) \) for any positive constant \( c \) . | Yes |
Theorem 8.2.3. Let \( {\mathbf{\Omega }}_{0} \in {\mathcal{U}}_{o}\left( {\alpha, B}\right) \) and \( {\lambda }_{n} = {CB}\sqrt{\log p/n} \) with sufficiently large \( C \) . If (C1) or (C2) holds, then with probability greater than \( 1 - O\left( {{n}^{-\delta /8} + {p}^{-\tau /2}}\right) \), \[ {\begin{Vmatrix}\wide... | Theorem 8.2.3 shows that our estimator has the same rate as that in ? by banding the Cholesky factor of the precision matrix for the ordered variables. | No |
For \( M = {\mathbb{R}}^{m} \), let \( U = M \) and \( {\varphi }_{U} \) be the identity map. Then \( \left\{ \left( {U,{\varphi }_{U}}\right) \right\} \) is a coordinate covering of \( {\mathbb{R}}^{m} \). | This provides a smooth differentiable structure on \( {\mathbb{R}}^{m} \), called the standard differentiable structure of \( {\mathbb{R}}^{m} \). | Yes |
Consider the \( m \)-dimensional unit sphere\n\n\[ \n{S}^{m} = \left\{ {x \in {\mathbb{R}}^{m + 1}\left| {\;{\left( {x}^{1}\right) }^{2} + \cdots + {\left( {x}^{m + 1}\right) }^{2} = 1}\right. }\right\} .\n\]\n\nFor \( m = 1 \), take the following four coordinate charts:\n\n\[ \n\left\{ \begin{array}{l} {U}_{1}\left\{ ... | Obviously, \( \left\{ {{U}_{1},{U}_{2},{V}_{1},{V}_{2}}\right\} \) is an open covering of \( {S}^{1} \) . In the intersection \( {U}_{1} \cap {V}_{2} \), we have (see Figure 2)\n\n\[ \n\left\{ \begin{array}{l} {x}^{2} = \sqrt{1 - {\left( {x}^{1}\right) }^{2}} > 0 \\ {x}^{1} = - \sqrt{1 - {\left( {x}^{2}\right) }^{2}} <... | Yes |
The \( m \) -dimensional projective space \( {P}^{m} \) . Define a relation \( \sim \) in \( {\mathbb{R}}^{m + 1} - \{ 0\} \) as follows: for \( x, y \in {\mathbb{R}}^{m + 1} - \{ 0\}, x \sim y \) if and only if there exists a real number \( a \) such that \( x = {ay} \) . Obviously, \( \sim \) is an equivalence relati... | The numbers of the \( \left( {m + 1}\right) \) -tuple \( \left( {{x}^{1},\ldots ,{x}^{m + 1}}\right) \) are called the homogeneous coordinates of \( \left\lbrack x\right\rbrack \) . They are determined by \( \left\lbrack x\right\rbrack \) up to a a nonzero factor. \( {P}^{m} \) is thus the space of all straight lines i... | Yes |
There may exist distinct differentiable structures on a single topological manifold. J. Milnor gave a famous example (Milnor 1956), which shows that there exist nonisomorphic smooth structures on homeomorphic topological manifolds (see the discussion following the remark to definition 1.3 below). Hence a differentiable... | Choose two antipodal points \( A \) and \( B \) in \( {S}^{4} \) . Let\n\n\[ \n{U}_{1} = {S}^{4} - \{ A\} ,\;{U}_{2} = {S}^{4} - \{ B\} .\n\]\n\n(1.13)\n\nThen \( {U}_{1} \) and \( {U}_{2} \) form an open covering of \( {S}^{4} \) . We wish to paste the trivial sphere bundles \( {U}_{1} \times {S}^{3} \) and \( {U}_{2}... | Yes |
Theorem 2.1. Suppose \( \left\lbrack f\right\rbrack \in {\mathcal{F}}_{p} \) . For an admissible coordinate chart \( \left( {U,{\varphi }_{U}}\right) \) , let\n\n\[ F\left( {{x}^{1},\ldots ,{x}^{m}}\right) = f \circ {\varphi }_{U}{}^{-1}\left( {{x}^{1},\ldots ,{x}^{m}}\right) .\n\]\n\nThen \( \left\lbrack f\right\rbrac... | Proof. Suppose \( \gamma \in {\Gamma }_{p} \), with coordinate representation\n\n\[ {\left( {\varphi }_{U} \circ \gamma \left( t\right) \right) }^{i} = {x}^{i}\left( t\right) ,\; - \delta < t < \delta .\n\]\n\n\[ \ll \gamma ,\left\lbrack f\right\rbrack \gg = {\left. \frac{d}{dt}\left( f \circ \gamma \right) \right| }_{... | Yes |
Theorem 2.2. Suppose \( {f}^{1},\cdots ,{f}^{s} \in {C}_{p}^{\infty } \) and \( F\left( {{y}^{1},\cdots ,{y}^{s}}\right) \) is a smooth function in a neighborhood of \( \left( {{f}^{1}\left( p\right) ,\cdots ,{f}^{s}\left( p\right) }\right) \in {\mathbb{R}}^{s} \) . Then \( f = \) \( F\left( {{f}^{1},\cdots ,{f}^{s}}\r... | Proof. Suppose the domain of \( {f}^{k} \) containing \( p \) is \( {U}_{k} \) . Then \( f \) is defined in \( \mathop{\bigcap }\limits_{{k = 1}}^{s}{U}_{k} \), and for \( q \in \mathop{\bigcap }\limits_{{k = 1}}^{s}{U}_{k} \n\n\[ \nf\left( q\right) = F\left( {{f}^{1}\left( q\right) ,\cdots ,{f}^{s}\left( q\right) }\ri... | Yes |
Corollary 1. For any \( f, g \in {C}_{p}^{\infty }, a \in \mathbb{R} \), we have\n\n\[ d{\left( f + g\right) }_{p} = {\left( df\right) }_{p} + {\left( dg\right) }_{p} \]\n\n(2.11)\n\n\[ d{\left( af\right) }_{p} = a \cdot {\left( df\right) }_{p}, \]\n\n(2.12)\n\n\[ d{\left( fg\right) }_{p} = f\left( p\right) \cdot {\lef... | We see that (2.11) and (2.12) are the same as (2.9), and (2.13) follows directly from Theorem 2.2. | Yes |
Corollary 2. \( \dim {T}_{p}^{ * } = m \) . | Proof. Choose an admissible coordinate chart \( \left( {U,{\varphi }_{U}}\right) \), and define local coordinates \( {u}^{i} \) by\n\n\[ \n{u}^{i}\left( q\right) = {\left( {\varphi }_{U}\left( q\right) \right) }^{i} = {x}^{i} \circ {\varphi }_{U}\left( q\right) ,\;q \in U, \n\] \n\n(2.14)\n\nwhere \( {x}^{i} \) is a gi... | Yes |
Theorem 2.3. Suppose \( X \in {T}_{p}, f, g \in {C}_{p}^{\infty },\alpha ,\beta \in \mathbb{R} \) . Then\n\n1) \( X\left( {{\alpha f} + {\beta g}}\right) = \alpha \cdot {Xf} + \beta \cdot {Xg} \) ;\n\n2) \( X\left( {fg}\right) = f\left( p\right) \cdot {Xg} + g\left( p\right) \cdot {Xf} \) . | Proof. These follow from Corollary 1 of Theorem 2.2 directly. | No |
Theorem 3.1. Suppose \( W \) is an open subset of \( {\mathbb{R}}^{n} \) and \( f : W \rightarrow {\mathbb{R}}^{n} \) is a smooth map. If at a point \( {x}_{0} \in W \) the determinant of the Jacobian matrix is nonzero, i.e., \[ {\left. \det \left( \frac{\partial {f}^{i}}{\partial {x}^{j}}\right) \right| }_{{x}_{0}} \n... | By the discussion in the last part of the preceding section, the Jacobian matrix \( \left( {\partial {f}^{i}/\partial {x}^{j}}\right) \) of \( f \) is precisely the matrix representation of \( {f}_{ * } \) under the natural basis. Therefore \( {\left. \left( \partial {f}^{i}/\partial {x}^{j}\right) \right| }_{{x}_{0}} ... | Yes |
Theorem 3.2. Suppose \( M \) and \( N \) are both \( n \) -dimensional smooth manifolds, and \( f : M \rightarrow N \) is a smooth map. If at a point \( p \in M \), the tangent map \( {f}_{ * } \) : \( {T}_{p}\left( M\right) \rightarrow {T}_{f\left( p\right) }\left( N\right) \) is an isomorphism, then there exists a ne... | Proof. Since \( f : M \rightarrow N \) is a smooth map, we can choose local coordinates \( \left( {{U}_{0},\varphi }\right) \) at \( p \in M \) and \( \left( {{V}_{0},\psi }\right) \) at \( q = f\left( p\right) \in N \) such that \( f\left( {U}_{0}\right) \subset {V}_{0} \) and\n\n\[ \widetilde{f} = \psi \circ f \circ ... | Yes |
Theorem 3.3. Suppose \( M \) is an \( m \) -dimensional and \( N \) an \( n \) -dimensional manifold, \( m < n \) . If \( f : M \rightarrow N \) is a smooth map and the tangent map \( {f}_{ * } \) is nondegenerate at a point \( p \in M \), then there exist local coordinate systems \( \left( {U;{u}^{i}}\right) \) near \... | Proof. Suppose \( \left( {U;{u}^{i}}\right) \) and \( \left( {V;{v}^{\alpha }}\right) \) are local coordinate systems at points \( p \) and \( q \), respectively, and the representation of \( f \) under these systems is\n\n\[ {v}^{\alpha } = {f}^{\alpha }\left( {{u}^{1},\ldots ,{u}^{m}}\right) ,\;1 \leq \alpha \leq n. ... | Yes |
The torus \( {T}^{2} = {S}^{1} \times {S}^{1} \) (see Figure 8) can be viewed as a 2- dimensional manifold obtained by identifying opposite sides of the unit square in the plane \( {\mathbb{R}}^{2} \) . Any point on the torus can then be expressed as an ordered pair of real numbers \( \left( {x, y}\right) {\;\operatorn... | Now take any two real numbers \( a, b \) such that \( \frac{a}{b} \) is irrational. Consider the map \( \varphi : {\mathbb{R}}^{1} \rightarrow {T}^{2} \), where \[ \varphi \left( t\right) = \left( {{at}{\;\operatorname{mod}\;1},{bt}{\;\operatorname{mod}\;1}}\right) . \] Clearly, \( \left( {\varphi ,\mathbb{R}}\right) \... | Yes |
Theorem 3.5. Suppose \( \left( {\varphi, M}\right) \) is a submanifold of a smooth manifold \( N \) . If \( M \) is compact, then \( \varphi : M \rightarrow N \) is a regular imbedding. | Proof. Because \( \varphi \left( M\right) \), as a topological subspace of \( N \), is a Hausdorff space, and \( \varphi : M \rightarrow \varphi \left( M\right) \subset N \) is a \( 1 - 1 \) continuous map from the compact space \( M \) to a Hausdorff space, \( \varphi : M \rightarrow \varphi \left( M\right) \subset N ... | Yes |
Lemma 1. Suppose \( {D}_{1} \) and \( {D}_{2} \) are two concentric balls in \( {\mathbb{R}}^{m} \), with \( \overline{{D}_{1}} \subset {D}_{2} \) . Then there exists a smooth, real-valued function \( f \) defined on \( {\mathbb{R}}^{m} \) such that\n\n1) \( 0 \leq f \leq 1 \) ;\n\n2) \( f\left( x\right) = \left\{ \beg... | Proof. We may assume that \( {D}_{1} \) and \( {D}_{2} \) are centered at the origin with radii \( a \) and \( b,0 < a < b \), respectively. Let\n\n\[ g\left( t\right) = \left\{ \begin{array}{ll} \exp \left\{ \frac{1}{\left( {t - {a}^{2}}\right) \left( {t - {b}^{2}}\right) }\right\} , & t \in \left( {{a}^{2},{b}^{2}}\r... | Yes |
Lemma 2. Suppose \( U \) and \( V \) are two nonempty open sets in \( {\mathbb{R}}^{m} \) and \( \bar{V} \) is compact, with \( \bar{V} \subset U \) . Then there exists a smooth real-valued function \( f \) defined on \( {\mathbb{R}}^{m} \) such that\n\n1) \( 0 \leq f \leq 1 \) ;\n\n2) \( f\left( x\right) = \left\{ \be... | Proof. Since \( \bar{V} \) is compact and \( \bar{V} \subset U \), there exist finitely many sets of balls \( {\left\{ {D}_{i}^{\left( 1\right) },{D}_{i}^{\left( 2\right) }\right\} }_{1 \leq i \leq r} \) such that \( {D}_{i}^{\left( 1\right) },{D}_{i}^{\left( 2\right) } \) are pairwise concentric,\n\n\[{\bar{D}}_{i}^{\... | Yes |
Lemma 3. Suppose \( \left( {U,{\varphi }_{U}}\right) \) is a coordinate chart in a smooth manifold \( M \) , \( V \neq \varnothing \) is an open set in \( M \) with \( \bar{V} \) compact, and \( \bar{V} \subset U \) . Then there exists a smooth function \( h : M \rightarrow {\mathbb{R}}^{1} \) such that\n\n1) \( 0 \leq... | Proof. Since \( \bar{V} \) is compact and \( \bar{V} \subset U \), we may use the local compactness of \( M \) to construct an open set \( {U}_{1} \) such that\n\n\[ \bar{V} \subset {U}_{1} \subset \overline{{U}_{1}} \subset U. \]\n\nSuppose \( \dim M = m \) . Then \( {\varphi }_{U}\left( V\right) \) and \( {\varphi }_... | Yes |
A necessary and sufficient condition for a tangent vector field \( X \) on a smooth manifold \( M \) to be a smooth tangent vector field is that, for any point \( p \in M \), there exists a local coordinate system \( \left( {U;{u}^{i}}\right) \) such that the restriction of \( X \) on \( U \) can be expressed as\n\n\[ ... | Proof. Sufficiency is obvious, so we only prove necessity. Since \( X \) is a smooth tangent vector field on \( M,{\left. X\right| }_{U} \) is a smooth tangent vector field on the sub-manifold \( U \) . The tangent vector field \( {\left. X\right| }_{U} \) can be expressed as\n\n\[ \n{\left. X\right| }_{U} = \mathop{\s... | Yes |
Theorem 4.2. Suppose \( X, Y, Z \) are smooth tangent vector fields on \( M \), and \( f, g \in {C}^{\infty }\left( M\right) \) . Then\n\n1) \( \left\lbrack {X, Y}\right\rbrack = - \left\lbrack {Y, X}\right\rbrack \) ;\n\n2) \( \left\lbrack {X + Y, Z}\right\rbrack = \left\lbrack {X, Z}\right\rbrack + \left\lbrack {Y, Z... | Proof. Each of these properties can be verified directly by using the definition of the Poisson bracket product. For example, to prove 3), suppose \( h \in {C}^{\infty }\left( M\right) \) . Then\n\n\[ \left\lbrack {{fX},{gY}}\right\rbrack h = \left( {fX}\right) \left( {\left( {gY}\right) h}\right) - \left( {gY}\right) ... | Yes |
Theorem 1.1. Suppose \( h : V \times W \rightarrow V \otimes W \) is the bilinear map obtained from the tensor product \( \otimes \), i.e., for \( v \in V, w \in W \), \[ h\left( {v, w}\right) = v \otimes w \] Then for any bilinear map \( f : V \times W \rightarrow Z \), there exists a unique linear map \( g : V \otime... | Proof. Define a linear map \( g : V \otimes W \rightarrow Z \) such that its action on a basis is \[ g\left( {{a}_{i} \otimes {b}_{j}}\right) = f\left( {{a}_{i},{b}_{j}}\right) ,\;1 \leq i \leq n,\;1 \leq j \leq m. \] Suppose \[ v = \mathop{\sum }\limits_{{i = 1}}^{n}{v}^{i}{a}_{i} \in V,\;w = \mathop{\sum }\limits_{{j... | Yes |
Corollary 1. The vector spaces \( \mathcal{L}\left( {V, W;Z}\right) \) and \( \mathcal{L}\left( {V \otimes W;Z}\right) \) are isomorphic. | Proof. Define a map\n\n\[ \varphi : \mathcal{L}\left( {V \otimes W;Z}\right) \rightarrow \mathcal{L}\left( {V, W;Z}\right) \]\n\nsuch that\n\n\[ \varphi \left( g\right) = g \circ h,\;g \in \mathcal{L}\left( {V \otimes W;Z}\right) ,\]\n\nwhere \( h \) is defined as in (1.20). Theorem 1.1 implies that \( \varphi \) is a ... | Yes |
Theorem 1.2. The tensor product \( \otimes \) is associative, that is, for any \( \varphi \in \) \( \mathcal{L}\left( {{V}_{1},\ldots ,{V}_{s};\mathbb{F}}\right) ,\psi \in \mathcal{L}\left( {{W}_{1},\ldots ,{W}_{r};\mathbb{F}}\right) ,\xi \in \mathcal{L}\left( {{Z}_{1},\ldots ,{Z}_{t};\mathbb{F}}\right) \), the followi... | Proof. To simplify the proof, we only consider the case \( s = r = t = 1 \) . The general cases are similar. Let \( v \in {V}_{1}, w \in {W}_{1}, z \in {Z}_{1} \) . We have\n\n\[ \left( {\varphi \otimes \psi }\right) \otimes \xi \left( {v, w, z}\right) = \varphi \otimes \psi \left( {v, w}\right) \cdot \xi \left( z\righ... | No |
Theorem 1.3. Suppose \( h : {V}_{1} \times \cdots \times {V}_{s} \rightarrow {V}_{1} \otimes \cdots \otimes {V}_{s} \) is the s-linear map defined by the tensor product \( \otimes \), i.e., for any \( {v}_{i} \in {V}_{i},1 \leq i \leq s \) ,\n\n\[ h\left( {{v}_{1},\ldots ,{v}_{s}}\right) = {v}_{1} \otimes \cdots \otime... | The proof is similar to that of Theorem 1.1, and should be carried out by the reader. | No |
Theorem 2.1. Suppose \( x \in {T}^{r}\left( V\right) \) . A necessary and sufficient condition for \( x \) to be a symmetric tensor is that all its components are symmetric with respect to all indices. A necessary and sufficient condition for \( x \) to be an alternating tensor is that all its components are alternatin... | Proof. Suppose \( \left\{ {{e}_{1},\ldots ,{e}_{n}}\right\} \) is a basis of \( V \), and \( x \) is a symmetric tensor. Then for any \( \sigma \in \mathrm{S}\left( r\right) \) ,\n\n\[ \n{x}^{{i}_{1}\cdots {i}_{r}} = x\left( {{e}^{*{i}_{1}},\ldots ,{e}^{*{i}_{r}}}\right) \n\]\n\n\[ \n= {\sigma x}\left( {{e}^{*{i}_{1}},... | Yes |
\[ {P}^{r}\left( V\right) = {S}_{r}\left( {{T}^{r}\left( V\right) }\right) \] \[ {\Lambda }^{r}\left( V\right) = {A}_{r}\left( {{T}^{r}\left( V\right) }\right) \] | Proof. First we show that the image of a tensor \( x \) under the symmetrizing mapping is a symmetric tensor, and the image of a tensor \( x \) under the alternating mapping is an alternating tensor. Suppose \( x \in {T}^{r}\left( V\right) \) . Then for any \( \tau \in \mathcal{S}\left( r\right) \)\n\n\[ \tau \left( {{... | Yes |
Theorem 3.2. Suppose \( f : V \rightarrow W \) is a linear map. Then \( {f}^{ * } \) commutes with the exterior product, that is, for any \( \varphi \in {\Lambda }^{r}\left( {W}^{ * }\right) \) and \( \psi \in {\Lambda }^{s}\left( {W}^{ * }\right) \) , \[ {f}^{ * }\left( {\varphi \land \psi }\right) = {f}^{ * }\varphi ... | Proof. Choose any \( {v}_{1},\ldots ,{v}_{r + s} \in V \) . Then \[ {f}^{ * }\left( {\varphi \land \psi }\right) \left( {{v}_{1},\ldots ,{v}_{r + s}}\right) = \varphi \land \psi \left( {f\left( {v}_{1}\right) ,\ldots, f\left( {v}_{r + s}\right) }\right) \] \[ = \frac{1}{\left( {r + s}\right) !}\mathop{\sum }\limits_{{\... | Yes |
A necessary and sufficient condition for the vectors \( {v}_{1},\ldots ,{v}_{r} \in V \) to be linearly dependent is\n\n\[ \n{v}_{1} \land \cdots \land {v}_{r} = 0 \n\] | Proof. If \( {v}_{1},\ldots ,{v}_{r} \) are linearly dependent, then we may assume without loss of generality that \( {v}_{r} \) can be expressed as a linearly combination of \( {v}_{1},\ldots ,{v}_{r - 1} \) :\n\n\[ \n{v}_{r} = {a}_{1}{v}_{1} + \cdots + {a}_{r - 1}{v}_{r - 1}. \n\]\n\nThen\n\n\[ \n{v}_{1} \land \cdots... | Yes |
Theorem 3.5. Suppose \( {v}_{1},\ldots ,{v}_{r} \) are \( r \) linearly independent vectors in \( V \) , and \( w \in {\Lambda }^{p}\left( V\right) \) . A necessary and sufficient condition for \( w \) to be expressible in the form\n\n\[ w = {v}_{1} \land {\psi }_{1} + \cdots + {v}_{r} \land {\psi }_{r} \]\n\n(3.25)\n\... | Proof. When \( p + r > n,\left( {3.25}\right) \) and (3.26) are trivially true. In the following we assume that \( p + r \leq n \) .\n\nNecessity is obvious, so we need only show sufficiency. Extend \( {v}_{1},\ldots ,{v}_{r} \) to a basis \( \left\{ {{v}_{1},\ldots ,{v}_{r},{v}_{r + 1},\ldots ,{v}_{n}}\right\} \) of \... | Yes |
Theorem 3.6. Suppose \( {v}_{\alpha },{w}_{\alpha };{v}_{\alpha }^{\prime },{w}_{\alpha }^{\prime }\left( {1 \leq \alpha \leq k}\right) \) are two sets of vectors in \( V \) . If \( \left\{ {{v}_{\alpha },{w}_{\alpha },1 \leq \alpha \leq k}\right\} \) is linearly independent, and\n\n\[ \mathop{\sum }\limits_{{\alpha = ... | Proof. Wedge-multiply (3.29) by itself \( k \) times to get\n\n\[ k!\left( {{v}_{1} \land {w}_{1} \land \cdots \land {v}_{k} \land {w}_{k}}\right) = k!\left( {{v}_{1}^{\prime } \land {w}_{1}^{\prime } \land \cdots \land {v}_{k}^{\prime } \land {w}_{k}^{\prime }}\right) . \]\n\n(3.30)\n\nSince \( \left\{ {{v}_{\alpha },... | Yes |
Theorem 1.1. Suppose \( M \) is an \( m \) -dimensional smooth manifold, \( {\left\{ {U}_{\alpha }\right\} }_{\alpha \in \mathcal{A}} \) is an open covering of \( M \), and \( V \) is a \( q \) -dimensional vector space. If for any pair of indices, \( \alpha ,\beta \in \mathcal{A} \) where \( {U}_{\alpha } \cap {U}_{\b... | For a detailed proof of Theorem 1.1, see p.14 of Steenrod 1951. The idea of the proof is to paste the local products \( {U}_{\alpha } \times V \) along the corresponding fibers. To describe it briefly, let\n\n\[ \widetilde{E} = \mathop{\bigcup }\limits_{{\alpha \in \mathcal{A}}}\{ \alpha \} \times {U}_{\alpha } \times ... | Yes |
Example 1 (The dual bundle \( {E}^{ * } \) of a vector bundle \( E \) ). Suppose \( {V}^{ * } \) is the dual space of \( V,{E}^{ * } \) is the vector bundle on \( M \) with \( {V}^{ * } \) as its typical fiber, and the bundle projection is denoted by \( \widetilde{\pi } \) . The structure of the local products of the b... | If we choose dual bases of \( V \) and \( {V}^{ * } \), and then denote any element \( y \) in \( V \) by a coordinate row and any element \( \lambda \) in \( {V}^{ * } \) by a coordinate column, then the pairing between \( V \) and \( {V}^{ * } \) can be expressed as multiplication of matrices:\n\n\[ \n< y,\lambda > =... | Yes |
Example 3 (The tensor product \( E \otimes {E}^{\prime } \) of vector bundles \( E \) and \( {E}^{\prime } \) ). Suppose \( E \) and \( {E}^{\prime } \) are the same as in Example 2. Let \( {h}_{UW} \) be the tensor product of \( {g}_{UW} \) and \( {g}_{UW}^{\prime } \), that is, \( {h}_{UW} = {g}_{UW} \otimes {g}_{UW}... | \[ \left( {v \otimes {v}^{\prime }}\right) \cdot {h}_{UW} = \left( {v \cdot {g}_{UW}}\right) \otimes \left( {{v}^{\prime } \cdot {g}_{UW}^{\prime }}\right) ,\] (1.36) where \( v \in V,{v}^{\prime } \in {V}^{\prime } \) . Obviously, \( \left\{ {h}_{UW}\right\} \) also satisfies the compatibility conditions of transition... | Yes |
Theorem 2.2 (Poincaré’s Lemma). \( {d}^{2} = 0 \), i.e., for any exterior differential form \( \omega, d\left( {d\omega }\right) = 0 \) . | Proof. Since \( d \) is a linear operator, we need only prove the lemma when \( \omega \) is a monomial. By the local properties of \( d \), it is sufficient to assume that\n\n\[ \omega = {ad}{u}^{1} \land \cdots \land d{u}^{r} \]\n\nHence\n\n\[ {d\omega } = {da} \land d{u}^{1} \land \cdots \land d{u}^{r} \]\n\nDiffere... | Yes |
Theorem 2.3. Suppose \( \omega \) is a differential 1-form on a smooth manifold \( M \) ; \( X \) and \( Y \) are smooth tangent vector fields on \( M \) . Then\n\n\[ \langle X \land Y,{d\omega }\rangle = X\langle Y,\omega \rangle - Y\langle X,\omega \rangle - \langle \left\lbrack {X, Y}\right\rbrack ,\omega \rangle . ... | Proof. Since both sides in (2.19) are linear with respect to \( \omega \), we may assume that \( \omega \) is a monomial:\n\n\[ \omega = {gdf} \]\n\nwhere \( f, g \) are smooth functions on \( M \) . Therefore\n\n\[ {d\omega } = {dg} \land {df} \]\n\nBy (3.15) in Chapter 2, the left hand side of (2.19) is\n\n\[ \langle... | Yes |
Theorem 2.6. Suppose \( f : M \rightarrow N \) is a smooth map from a smooth manifold \( M \) to a smooth manifold \( N \) . Then the induced map \( {f}^{ * } : A\left( N\right) \rightarrow A\left( M\right) \) commutes with the exterior derivative \( d \), that is,\n\n\[ \n{f}^{ * } \circ d = d \circ {f}^{ * } : A\left... | Proof. Since both \( {f}^{ * } \) and \( d \) are linear, we need only consider the operation of both sides of (2.43) on a monomial \( \beta \) .\n\nFirst suppose \( \beta \) is a smooth function on \( N \), i.e., \( \beta \in {A}^{0}\left( N\right) \) . Choose any smooth tangent vector field \( X \) on \( M \) . Then ... | Yes |
Theorem 3.2 (Partition of Unity Theorem). Suppose \( \sum \) is an open covering of a smooth manifold \( M \) . Then there exists a family of smooth functions \( \left\{ {g}_{\alpha }\right\} \) on \( M \) satisfying the following conditions:\n\n1) \( 0 \leq {g}_{\alpha } \leq 1 \), and supp \( {g}_{\alpha } \) is comp... | Proof. Because \( M \) is a manifold, there is a topological basis \( {\sum }_{0} = \left\{ {U}_{\alpha }\right\} \) such that each element \( {U}_{\alpha } \) is a coordinate neighborhood, \( {\bar{U}}_{\alpha } \) is compact, and there also exists \( {W}_{i} \in \sum \) such that \( {\bar{U}}_{\alpha } \subset {W}_{i... | Yes |
Example 1. Suppose \( D = \left\lbrack {a, b}\right\rbrack \) is a closed interval in \( {\mathbb{R}}^{1} \) and \( f \) is a continuously differentiable function on \( D \) . Then the Fundamental Theorem of Calculus holds: | \[ {\int }_{D}{df} = f\left( b\right) - f\left( a\right) \] | Yes |
Suppose \( D \) is a bounded domain in \( {\mathbb{R}}^{2} \) whose orientation is consistent with that of \( {\mathbb{R}}^{2} \). Use \( \partial D \) to denote the oriented boundary of \( D \) with the orientation induced by \( D \), that is, the positive orientation of \( \partial D \) together with the normal vecto... | If we let \( \omega = {Pdx} + {Qdy} \), then\n\n\[ \n{d\omega } = \left( {\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\right) {dx} \land {dy}\n\]\n\nThus (4.2) can be written as\n\n\[ \n{\int }_{\partial D}\omega = {\int }_{D}{d\omega }\n\] | Yes |
Suppose \( D \) is a bounded domain in \( {\mathbb{R}}^{3} \) whose orientation is consistent with that of \( {\mathbb{R}}^{3} \) . The outward normal as the positive direction then induces an orientation on the boundary \( \partial D \) . Suppose \( P, Q \), and \( R \) are continuously differentiable functions on \( ... | \[ {\int }_{\partial D}P\;{dy}\;{dz} + Q\;{dz}\;{dx} + R\;{dx}\;{dy} = {\int }_{D}\left( {\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}}\right) \;{dx}\;{dy}\;{dz} \] (4.4) or, \[ {\int }_{\partial D}\varphi = {\int }_{D}{d\varphi } \] (4.5) where \( \varphi = {Pdy} \land ... | Yes |
Suppose \( \sum \) is an oriented surface in \( {\mathbb{R}}^{3} \) whose boundary \( \partial \sum \) is an oriented closed curve. The positive orientation of \( \partial \sum \) along with any positive normal vector to \( \sum \) satisfy the right hand rule (assuming \( {\mathbb{R}}^{3} \) to be oriented by a right-h... | \[ {\int }_{\partial \sum }{Pdx} + {Qdy} + {Rdz} = {\iint }_{\sum }\left\{ {\left( {\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}}\right) {dydz}}\right. \]\n\n\[ \left. {+\left( {\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}}\right) {dzdx} + \left( {\frac{\partial Q}{\partial x} - \frac{... | Yes |
Theorem 4.1. The boundary \( B \) of a region \( D \) with boundary is a regular imbedded closed submanifold. If \( M \) is orientable, then \( B \) is also orientable. | Proof. The boundary \( B \) of the region \( D \) is obviously a closed subset of \( M \) . Suppose \( \left( {U;{u}^{i}}\right) \) is an adapted coordinate neighborhood. Then\n\n\[ U \cap B = \left\{ {q \in U \mid {u}^{m}\left( q\right) = 0}\right\} . \]\n\n(4.9)\n\nBy Definition 3.2 in Chapter \( 1, B \) is a regular... | Yes |
Theorem 1.2. Suppose \( D \) is a connection on a vector bundle \( E \), and \( p \in M \) . Then there exists a local frame field \( S \) in a coordinate neighborhood of \( p \) such that the corresponding connection matrix \( \omega \) is zero at \( p \) . | Proof. Choose a coordinate neighborhood \( \left( {U;{u}^{i}}\right) \) of \( p \) such that \( {u}^{i}\left( p\right) = 0,1 \leq \) \( i \leq m \) . Suppose \( {S}^{\prime } \) is a local frame field on \( U \) with corresponding connection matrix \( {\omega }^{\prime } = \left( {\omega }^{\prime \beta }\right) \), wh... | Yes |
Theorem 1.3. Suppose \( X, Y \) are two arbitrary smooth tangent vector fields on the manifold \( M \) . Then\n\n\[ R\left( {X, Y}\right) = {D}_{X}{D}_{Y} - {D}_{Y}{D}_{X} - {D}_{\left\lbrack X, Y\right\rbrack }.\] | Proof. Because the absolute differential quotient and the curvature operator are local operators, we need only consider the operations of both sides of (1.32) on a local section. Suppose \( s \in \Gamma \left( E\right) \) has the local expression\n\n\[ s = \mathop{\sum }\limits_{{\alpha = 1}}^{q}{\lambda }^{\alpha }{s}... | Yes |
Theorem 1.4. The curvature matrix \( \Omega \) satisfies the Bianchi identity | Proof. Apply exterior differentiation to both sides of \( \Omega = {d\omega } - \omega \land \omega \) :\n\n\[ {d\Omega } = - {d\omega } \land \omega + \omega \land {d\omega } \]\n\n\[ = \; - \left( {\Omega + \omega \land \omega }\right) \land \omega + \omega \land \left( {\Omega + \omega \land \omega }\right) \]\n\n\[... | Yes |
Theorem 2.1. Suppose \( D \) is a torsion-free affine connection on \( M \). Then for any point \( p \in M \) there exists a local coordinate system \( {u}^{i} \) such that the corresponding connection coefficients \( {\Gamma }_{ik}^{j} \) vanish at \( p \). | Proof. Suppose \( \left( {W;{w}^{i}}\right) \) is a local coordinate system at \( p \) with connection coefficients \( {\Gamma }^{\prime j}{}_{ik} \). Let\n\n\[ \n{u}^{i} = {w}^{i} + \frac{1}{2}{\Gamma }^{\prime }{}_{jk}^{i}\left( p\right) \left( {{w}^{j} - {w}^{j}\left( p\right) }\right) \left( {{w}^{k} - {w}^{k}\left... | Yes |
Theorem 2.2. Suppose \( D \) is a torsion-free affine connection on \( M \) . Then we have the Bianchi identity:\n\n\[ \n{R}_{{ikl}, h}^{j} + {R}_{{ilh}, k}^{j} + {R}_{{ihk}, l}^{j} = 0. \n\]\n\n(2.43) | Proof. From Theorem 1.4 we have\n\n\[ \nd{\Omega }_{i}^{j} = {\omega }_{i}^{k} \land {\Omega }_{k}^{j} - {\Omega }_{i}^{k} \land {\omega }_{k}^{j} \n\]\n\nthat is,\n\n\[ \n\frac{\partial {R}_{ikl}^{j}}{\partial {u}^{h}}d{u}^{h} \land d{u}^{k} \land d{u}^{l} = \left( {{\Gamma }_{ih}^{p}{R}_{pkl}^{j} - {\Gamma }_{ph}^{j}... | Yes |
Theorem 1.1. There exists a Riemannian metric on any m-dimensional smooth manifold \( M \) . | Proof. Choose a locally finite coordinate covering \( \left\{ \left( {{U}_{\alpha };{u}_{\alpha }^{i}}\right) \right\} \) of \( M \) . Suppose \( \left\{ {h}_{\alpha }\right\} \) is the corresponding partition of unity so that supp \( {h}_{\alpha } \subset {U}_{\alpha } \) . Let\n\n\[ d{s}_{\alpha }^{2} = \mathop{\sum ... | Yes |
Theorem 1.2 (Fundamental Theorem of Riemannian Geometry). Suppose \( M \) is an \( m \) -dimensional generalized Riemannian manifold. Then there exists a unique torsion-free and metric-compatible connection on \( M \) , called the Levi-Civita connection of \( M \), or the Riemannian connection of \( M \) . | Proof. Suppose \( D \) is a torsion-free and metric-compatible connection on \( M \) . Denote the connection matrix of \( D \) under the local coordinates \( {u}^{i} \) by \( \omega = \left( {\omega }_{i}^{j}\right) \) , where\n\n\[{\omega }_{i}^{j} = {\Gamma }_{ik}^{j}d{u}^{k}\]\n\n(1.26)\n\nThen we have\n\n\[d{g}_{ij... | Yes |
Theorem 1.4. The curvature tensor \( {R}_{ijkl} \) of a generalized Riemannian manifold satisfies the following properties:\n\n1) \( {R}_{ijkl} = - {R}_{jikl} = - {R}_{ijlk} \) ;\n\n2) \( {R}_{ijkl} + {R}_{iklj} + {R}_{iljk} = 0 \) ;\n\n3) \( {R}_{ijkl} = {R}_{klij} \) . | Proof. 1) This is a direct corollary of (1.46) and (1.52).\n\n2) From the torsion-free property of the Levi-Civita connection we have\n\n\[ d{u}^{i} \land {\omega }_{ij} = 0 \]\n\n(1.53)\n\nExteriorly differentiating and using (1.47) we then have\n\n\[ d{u}^{i} \land \left( {{\Omega }_{ij} - {\omega }_{i}^{l} \land {\o... | Yes |
Theorem 2.1. If \( M \) is a torsion-free affine connection space, then with respect to a normal coordinate system \( {\alpha }^{i} \) at the point \( x \), the connection coefficients \( {\Gamma }_{ik}^{j} \) are zero at \( x \) . | Proof. Since the geodesic curve \( {\alpha }^{i} = t{\alpha }_{0}^{i} \) satisfies (2.4) under the normal coordinate system \( {\alpha }^{i} \), we have, for any \( {\alpha }_{0}^{k} \), \[ {\Gamma }_{jk}^{i}\left( 0\right) {\alpha }_{0}^{j}{\alpha }_{0}^{k} = 0. \] (2.13) Since \( {\Gamma }_{jk}^{i} \) is symmetric in... | Yes |
Theorem 2.2. For any point \( {x}_{0} \) in an affine connection space \( M \), there exists a neighborhood \( W \) of \( {x}_{0} \) such that every point in \( W \) has a normal coordinate neighborhood that contains \( W \) . | Proof. Suppose \( \left( {U;{u}^{i}}\right) \) is a normal coordinate system at a point \( {x}_{0} \) . Let\n\n\[ U\left( {{x}_{0};\rho }\right) = \left\{ {x \in U \mid \mathop{\sum }\limits_{{i = 1}}^{m}{\left( {u}^{i}\left( x\right) \right) }^{2} < {\rho }^{2}}\right\} .\n\]\n\n(2.15)\n\nBy the above discussion on th... | Yes |
Theorem 2.4. For every point \( O \) in a Riemannian manifold \( M \), there exists a normal coordinate neighborhood \( W \) such that\n\n1) Every point in \( W \) has a normal coordinate neighborhood that contains \( W \) .\n\n2) The geodesic curve that connects \( O \) and \( p \in W \) is the unique shortest curve i... | Proof. Apply Theorem 2.2 to the Levi-Civita connection of \( M \), and 1) follows. Now assume that \( {u}^{i} \) is the normal coordinate system of the point \( O \) given by (2.28). A normal coordinate neighborhood \( W \) as required in 1) is\n\n\[ W = \\left\\{ {p \in M \mid \\mathop{\\sum }\\limits_{{i = 1}}^{m}{\\... | Yes |
Theorem 2.6. The function \( \rho : M \times M \rightarrow \mathbb{R} \) has the following properties:\n\n1) for any \( p, q \in M,\rho \left( {p, q}\right) \geq 0 \), and the equality holds only when \( p = q \) ;\n\n2) \( \rho \left( {p, q}\right) = \rho \left( {q, p}\right) \) ;\n\n3) for any three points \( p, q, r... | Proof. According to definition (2.47), the above properties are obvious. We need only show that \( \rho \left( {p, q}\right) > 0 \) whenever \( p \neq q \) .\n\nSuppose \( p, q \) are any two points in \( M, p \neq q \) . Since \( M \) is a Hausdorff space, there exists a neighborhood \( U \) of \( \mathrm{p} \) such t... | Yes |
Theorem 2.7. There exists a \( \eta \) -ball neighborhood \( W \) at any point \( p \) in a Riemannian manifold \( M \), where \( \eta \) is a sufficiently small positive number, such that any two points in \( W \) can be connected by a unique geodesic curve. | Proof. Suppose \( p \in M \) . By Theorem 2.4 there exists a ball-shaped normal coordinate neighborhood \( U \) of \( p \) with radius \( \epsilon \) such that for any point \( q \) in \( U \) there is a normal coordinate neighborhood \( {V}_{q} \) that contains \( U \) . We may assume that \( \epsilon \) also satisfie... | Yes |
Theorem 3.1. The curvature tensor of a Riemannian manifold \( M \) at a point \( p \) is uniquely determined by the sectional curvatures of all the 2-dimensional tangent subspaces at \( p \) . | Proof. Suppose there is a 4-linear function \( \widetilde{R}\left( {X, Y, Z, W}\right) \) satisfying all the properties 1)-3) of the curvature tensor \( R\left( {X, Y, Z, W}\right) \), and that for any two linearly independent tangent vectors \( X, Y \) at \( p \) ,\n\n\[ \n\frac{\bar{R}\left( {X, Y, X, Y}\right) }{G\l... | Yes |
GL \( \left( {n;\mathbb{R}}\right) \) is the set of nondegenerate \( n \times n \) real matrices with matrix multiplication for its group operation. Since \( \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \) is an open subset of \( {\mathbb{R}}^{{n}^{2}} \), it has the differentiable structure induced from \( {\mathbb{R}}^{... | The right hand side is a polynomial of the elements of the matrices A and B. Hence the map\n\n\[ \n\varphi \left( {A, B}\right) = A \cdot B \n\]\n\n(1.7)\n\nis smooth. Moreover, since the elements of \( {A}^{-1} \) are rational functions of the elements \( {A}_{i}^{j} \), the inverse map is also smooth. Hence \( \mathr... | Yes |
Suppose \( {z}_{\alpha } = {x}_{\alpha } + i{y}_{\alpha },\alpha = 1,2,{z}_{\alpha } \neq 0 \) . Then the product of the \( {z}_{\alpha } \) in terms of coordinates is | \[ \left( {{x}_{1},{y}_{1}}\right) \cdot \left( {{x}_{2},{y}_{2}}\right) = \left( {{x}_{1}{x}_{2} - {y}_{1}{y}_{2},{x}_{1}{y}_{2} + {x}_{2}{y}_{1}}\right) \] (1.8) and the inverse of \( z = x + {iy} \) is \[ {\left( x, y\right) }^{-1} = \left( {\frac{x}{{x}^{2} + {y}^{2}}, - \frac{y}{{x}^{2} + {y}^{2}}}\right) . \] (1.... | Yes |
Theorem 1.1. Suppose \( \sigma : G \rightarrow G \) is a smooth map. Then a necessary and sufficient condition for \( \sigma \) to be a right translation of the Lie group \( G \) is that it preserves the right fundamental differential form, i.e., \[ {\sigma }^{ * }{\omega }^{i} = {\omega }^{i},\;1 \leq i \leq r. \] | Proof. Suppose \( \sigma \) is the right translation \( {R}_{x}, x \in G \) . Then for any \( X \in {G}_{a} \) we have \[ {\left( {R}_{x}\right) }^{ * }\omega \left( X\right) = \omega \left( {{\left( {R}_{x}\right) }_{ * }X}\right) \] \[ = {\left( {R}_{{\left( ax\right) }^{-1}}\right) }_{ * } \circ {\left( {R}_{x}\righ... | Yes |
Theorem 1.2. The structure constants \( {c}_{jk}^{i} \) satisfy the Jacobi identity\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{r}\left( {{c}_{jk}^{i}{c}_{hl}^{j} + {c}_{jh}^{i}{c}_{lk}^{j} + {c}_{jl}^{i}{c}_{kh}^{j}}\right) = 0. \] | Proof. Exteriorly differentiating (1.16) we get\n\n\[ 0 = - \frac{1}{2}\mathop{\sum }\limits_{{j, k}}{c}_{jk}^{i}\left( {d{\omega }^{j} \land {\omega }^{k} - {\omega }^{j} \land d{\omega }^{k}}\right) \]\n\n\[ = \;\frac{1}{2}\mathop{\sum }\limits_{{j, k, h, l}}{c}_{jk}^{i}{c}_{hl}^{j}{\omega }^{k} \land {\omega }^{h} \... | Yes |
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