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Exercise 3.9 | Exercise 3.9 Show that any continuous function \( f : X \rightarrow Y \), when \( Y \) is given the discrete topology and \( X \) is given the indiscrete topology, is a constant function. | No |
Exercise 1.4.2 | Exercise 1.4.2. Let \( w = \left( {{\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{m}}\right) \in {\Delta }_{m} \) . Prove that
\[
{\begin{Vmatrix}\mathop{\sum }\limits_{{i = 1}}^{m}{\omega }_{i}{x}_{i}\end{Vmatrix}}^{2} = \mathop{\sum }\limits_{{i = 1}}^{m}{\omega }_{i}{\begin{Vmatrix}{x}_{i}\end{Vmatrix}}^{2} - \frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{m}{\omega }_{i}{\omega }_{j}{\begin{Vmatrix}{x}_{i} - {x}_{j}\end{Vmatrix}}^{2}. \tag{1.44}
\] | No |
Exercise 6.2.18 | Exercise 6.2.18. Establish such a continuity result for (6.15). | No |
Exercise 1.10 | Exercise 1.10. Prove that the following conditions on a connected graph \( \Gamma \) are equivalent.
1. \( \Gamma \) is a tree.
2. Given any two vertices \( v \) and \( w \) in \( \Gamma \), there is a unique reduced edge path from \( v \) to \( w \) .
3. For every edge \( e \in E\left( \Gamma \right) \), removing \( e \) from \( \Gamma \) disconnects the graph. (Note: Removing \( e \) does not remove its associated vertices.)
4. If \( \Gamma \) is finite then \( \# V\left( \Gamma \right) = \# E\left( \Gamma \right) + 1 \) . | No |
Exercise 3 | Exercise 3 (Local minimizers) | No |
Exercise 5.1 | Exercise 5.1 Derive the expected value of the Kriging predictor defined in (5.13). | No |
Exercise 8.2.5 | [This exercise gives some more details to show the uniqueness of the extension constructed in the proof of Lemma 8.38.
(a) Let \( X \) and \( Y \) be Hausdorff topological spaces and \( D \) a dense subspace of \( X \) . Prove that if \( h,{h}^{\prime };X \rightarrow Y \) are continuous and \( {\left. h\right| }_{D} = {h}_{D}^{\prime } \), then \( h = {h}^{\prime } \) .
(b) Recall from Exercise 8.2 .2 that in particular \( {A}^{ * } \) is dense in the dual space \( X \) of \( {\operatorname{Reg}}_{{A}^{\prime }} \) Use this to conclude that, for any function \( f : A \rightarrow M \) with \( M \) a finite monoid, there can be at most one continuous homomorphism \( X \rightarrow M \) extending \( f \) .] | No |
Exercise 3.8 | Exercise 3.8 Derive the following argument:
\( \begin{array}{l} \text{ 1. }\forall x\left( {\operatorname{study}\left( x\right) \vee \neg \operatorname{pass}\left( x\right) }\right) \\ \forall x\neg \left( {\operatorname{pass}\left( x\right) \land \neg \operatorname{study}\left( x\right) }\right) \end{array} \) | No |
Exercise 2.9.5 | Let \( \mu \) and \( \sigma \) denote the mean and the standard deviation of the random variable \( X \) . Show that
\[
\mathbb{E}\left\lbrack {X}^{2}\right\rbrack = {\mu }^{2} + {\sigma }^{2}.
\] | No |
Exercise 9.4.7 | Find the derivative of \( \left( {{t}^{2} + {2t} - 8}\right) \exp \left( {5t}\right) \) . | No |
Exercise 8.5.30 | [Exercise 8.5.30. Verify that the basis in Example 8.15 is indeed an orthonormal basis of \( {\mathbb{R}}^{n \times n} \).] | No |
Exercise 6.7 | Exercise 6.7. For the EHIA data (Section 6.2.4.5), implement the join count test in R using a spatial weights matrix derived using (a) queen's move contiguity and (b) rook's move contiguity. Compare results. | No |
Exercise 2.7.9 | Exercise 2.7.9. Let \( C\ell \left( {0,5}\right) \cong \mathcal{A} \) where \( \mathcal{A} = \mathcal{M}\left( {4,\mathbb{C}}\right) \) for \( d = 2 \) . The \( C\ell \left( {0,5}\right) \) 1-vectors can be represented \( {}^{43} \) by the following matrices:
\[
{e}_{1} = \left( \begin{matrix} 0 & - 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix}\right) ,{e}_{2} = \left( \begin{matrix} 0 & - i & 0 & 0 \\ - i & 0 & 0 & 0 \\ 0 & 0 & 0 & - i \\ 0 & 0 & - i & 0 \end{matrix}\right) ,{e}_{3} = \left( \begin{matrix} - i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & 0 & 0 & - i \end{matrix}\right) ,
\]
\[
{e}_{4} = \left( \begin{matrix} 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \end{matrix}\right) ,{e}_{5} = \left( \begin{matrix} 0 & 0 & - i & 0 \\ 0 & 0 & 0 & i \\ - i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \end{matrix}\right) . \tag{2.7.104}
\]
Like for \( C\ell \left( {4,1}\right) \), we have five conjugacy classes of the roots \( {f}_{k} \) of \( - 1 \) in \( C\ell \left( {0,5}\right) \) for \( k \in \{ 0, \pm 1, \pm 2\} \) : four exceptional and one ordinary. Using the same notation as in Example 2.7.8, we find the following representatives of the conjugacy classes.
1. For \( k = 2 \), we have \( {\Delta }_{2}\left( t\right) = {\left( t - i\right) }^{4},{m}_{2}\left( t\right) = t - i \) and \( {\mathcal{F}}_{2} = \operatorname{diag}\left( {i, i, i, i}\right) \) which in the above representation (2.7.104) corresponds to the non-trivial central element \( {f}_{2} = \omega = {e}_{12345} \) . Then, \( \operatorname{Spec}\left( {f}_{2}\right) = 1 = \frac{k}{d};\operatorname{Scal}\left( {f}_{2}\right) = 0 \) ; the \( \mathbb{C} \) -dimension of the centralizer \( \operatorname{Cent}\left( {f}_{2}\right) \) is 16; and the \( \mathbb{R} \) -dimension of the conjugacy class of \( {f}_{2} \) is zero as it contains only \( {f}_{2} \) since \( {f}_{2} \in \mathrm{Z}\left( \mathcal{A}\right) \) . Thus, the \( \mathbb{R} \) -dimension of the class is again zero in agreement with (2.7.99).
2. For \( k = - 2 \), we have \( {\Delta }_{-2}\left( t\right) = {\left( t + i\right) }^{4},{m}_{-2}\left( t\right) = t + i \) and \( {\mathcal{F}}_{-2} = \) \( \operatorname{diag}\left( {-i, - i, - i, - i}\right) \) which corresponds to the central element \( {f}_{-2 = } - \omega = \)
13For the computations of this example in the Maple package CLIFFORD we have used the identification \( i = {e}_{3} \) . Yet the results obtained for the square roots of -1 are independent of this setting (we can alternatively use, e.g., \( i = {e}_{12345} \) or the imaginary unit \( i \in \mathbb{C} \) ), as can easily be checked for \( {f}_{1} \) of (2.7.105), \( {f}_{0} \) of (2.7.106) and \( {f}_{-1} \) of (2.7.107) by only assuming the standard Clifford product rules for \( {e}_{1} \) to \( {e}_{5} \) .
\( - {e}_{12345} \) . Again, \( \operatorname{Spec}\left( {f}_{-2}\right) = - 1 = \frac{k}{d};\operatorname{Scal}\left( {f}_{-2}\right) = 0 \) ; the \( \mathbb{C} \) -dimension of the centralizer \( \operatorname{Cent}\left( {f}_{-2}\right) \) is 16 and the conjugacy class of \( {f}_{-2} \) contains only \( {f}_{-2} \) since \( {f}_{-2} \in \mathrm{Z}\left( \mathcal{A}\right) \) . Thus, the \( \mathbb{R} \) -dimension of the class is again zero in agreement with (2.7.99).
3. For \( k \neq \pm 2 \), we consider three subcases when \( k = 1, k = 0 \) and \( k = - 1 \) . When \( k = 1 \), then \( {\Delta }_{1}\left( t\right) = {\left( t - i\right) }^{3}\left( {t + i}\right) \) and \( {m}_{1}\left( t\right) = \left( {t - i}\right) \left( {t + i}\right) \) . Then the root \( {\mathcal{F}}_{1} = \operatorname{diag}\left( {i, i, i, - i}\right) \) corresponds to
\[
{f}_{1} = \frac{1}{2}\left( {{e}_{3} + {e}_{12} + {e}_{45} + {e}_{12345}}\right) . \tag{2.7.105}
\]
Since \( \operatorname{Spec}\left( {f}_{1}\right) = \frac{1}{2} = \frac{k}{d},{f}_{1} \) is an exceptional root of -1 .
When \( k = 0 \), then \( {\Delta }_{0}\left( t\right) = {\left( t - i\right) }^{2}{\left( t + i}\right) }^{2} \) and \( {m}_{0}\left( t\right) = \left( {t - i}\right) \left( {t + i}\right) \) . Thus the root of -1 is this case is \( {\mathcal{F}}_{0} = \operatorname{diag}\left( {i, i, - i, - i}\right) \) which corresponds to just
\[
{f}_{0} = {e}_{45} \tag{2.7.106}
\]
Note that \( \operatorname{Spec}\left( {f}_{0}\right) = 0 \) thus \( {f}_{0} = {e}_{45} \) is an ordinary root of -1 .
When \( k = - 1 \), then \( {\Delta }_{-1}\left( t\right) = \left( {t - i}\right) {\left( t + i}\right) }^{3} \) and \( {m}_{-1}\left( t\right) = \left( {t - i}\right) \left( {t + i}\right) \) . Then,
the root of -1 in this case is \( {\mathcal{F}}_{-1} = \operatorname{diag}\left( {i, - i, - i, - i}\right) \) which corresponds to
\[
{f}_{-1} = \frac{1}{2}\left( {-{e}_{3} + {e}_{12} + {e}_{45} - {e}_{12345}}\right) . \tag{2.7.107}
\]
Since \( \operatorname{Scal}\left( {f}_{-1}\right) = - \frac{1}{2} = \frac{k}{d} \), we gather that \( {f}_{-1} \) is an exceptional root.
Again | No |
Exercise 3.8 | Consider estimator (3.34).
- Calculate its first two moments.
- Find sufficient conditions making this estimator consistent, asymptotically normal and satisfying (3.35). | No |
Exercise 1.17.1 | Exercise 1.17.1 Divide the representation of \( 2/n \) above by an appropriate power of 2 . Be careful when \( b \) is a power of 2 . | No |
Exercise 4 | Exercise 4. Suppose \( {s}_{1},{s}_{2}, c \), and \( x \) are as in Theorem 11. Show that \( x \) is semicircular by computing \( \varphi \left( {\operatorname{tr}\left( {x}^{n}\right) }\right) \) directly using the methods of Lemma 1.9. | No |
Exercise 1.6.2 | Exercise 1.6.2. (a) Use this to show that \( {F}_{n + 1}{F}_{n} = {F}_{n}^{2} + {F}_{n - 1}^{2} + \cdots + {F}_{0}^{2} \), where \( {F}_{n} \) is the \( n \) th Fibonacci number (see section 0.1 for the definition and a discussion of Fibonacci numbers and exercise \( {0.4.12}\left( b\right) \) for a generalization of this exercise).
(b) \( {}^{ \dagger } \) Find the correct generalization to more general second-order linear recurrence sequences. | No |
Exercise 2.4 | Exercise 2.4. In Example 2.2.8, \( X \) is a standard normal random variable and \( Z \) is an independent random variable satisfying
\[
\mathbb{P}\{ Z = 1\} = \mathbb{P}\{ Z = - 1\} = \frac{1}{2}.
\]
We defined \( Y = {XZ} \) and showed that \( Y \) is standard normal. We established that although \( X \) and \( Y \) are uncorrelated, they are not independent. In this exercise, we use moment-generating functions to show that \( Y \) is standard normal and \( X \) and \( Y \) are not independent.
(i) Establish the joint moment-generating function formula
\[
\mathbb{E}{e}^{{uX} + v{Y}^{\prime }} = {e}^{\frac{1}{2}\left( {{u}^{2} + {v}^{2}}\right) } \cdot \frac{{e}^{uv} + {e}^{-{uv}}}{2}.
\]
(ii) Use the formula above to show that \( \mathbb{E}{e}^{vY} = {e}^{\frac{1}{2}{v}^{2}} \) . This is the moment-generating function for a standard normal random variable, and thus \( Y \) must be a standard normal random variable.
(iii) Use the formula in (i) and Theorem 2.2.7(iv) to show that \( X \) and \( Y \) are not independent. | No |
Exercise 9.1 | Exercise 9.1
Show that no (a) \( \left( {{17},9,2}\right) \) ,(b) \( \left( {{21},6,1}\right) \) design exists. | No |
Exercise 3.2 | For any sequence of partitions
\[
0 = {t}_{0}^{m} < {t}_{1}^{m} < \ldots < {t}_{n}^{m} < \ldots ;\;{t}_{n}^{m} \uparrow \infty \text{ as }n \uparrow \infty \tag{3.1.5}
\]
of \( \lbrack 0,\infty ) \) such that for all \( T < \infty \) ,
\[
{\delta }_{m}\left( T\right) = \left( {\mathop{\sup }\limits_{\left\{ n : {t}_{n}^{m} \leq T\right\} }\left( {{t}_{n + 1}^{m} - {t}_{n}^{m}}\right) }\right) \rightarrow 0\text{ as }m \uparrow \infty \tag{3.1.6}
\]
let
\[
{Q}_{t}^{m} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( {W}_{{t}_{n + 1}^{m} \land t} - {W}_{{t}_{n}^{m} \land t}\right) }^{2}. \tag{3.1.7}
\]
Show that for each \( t,{Q}_{t}^{m} \) converges in probability to \( t \) . | Yes |
Exercise 1.17 | Exercise 1.17. Give a simpler proof of Thm. 1.32 in the case when \( X \) and \( Y \) are \( {C}^{2} \) , by providing the details of the following argument:
(1) Given \( X, Y \) of class \( {C}^{2} \) on \( \Omega \), we let, for \( s = \left( {{s}_{1},{s}_{2},{s}_{3},{s}_{4}}\right) \) near \( 0 \in {\mathbb{R}}^{4} \) , \( F\left( s\right) \mathrel{\text{:=}} {\Psi }_{{s}_{4}}^{-Y} \circ {\Psi }_{{s}_{3}}^{-X} \circ {\Psi }_{{s}_{2}}^{Y} \circ {\Psi }_{{s}_{1}}^{X}\left( x\right) \) . Show that \( F \) is \( {C}^{2} \) near 0 .
(2) Prove that the Maclaurin polynomial of degree 2 of \( F\left( s\right) \) is
\[
x + {s}_{1}X\left( x\right) + {s}_{2}Y\left( x\right) - {s}_{3}X\left( x\right) - {s}_{4}Y\left( x\right)
\]
\[
+ \frac{{s}_{1}^{2}}{2}{X}^{2}I\left( x\right) + \frac{{s}_{2}^{2}}{2}{Y}^{2}I\left( x\right) + \frac{{s}_{3}^{2}}{2}{X}^{2}I\left( x\right) + \frac{{s}_{4}^{2}}{2}{Y}^{2}I\left( x\right)
\]
\[
+ {s}_{1}{s}_{2}\left( {X \circ Y}\right) I\left( x\right) - {s}_{1}{s}_{3}{X}^{2}I\left( x\right) - {s}_{1}{s}_{4}\left( {X \circ Y}\right) I\left( x\right)
\]
\[
- {s}_{2}{s}_{3}\left( {Y \circ X}\right) I\left( x\right) - {s}_{2}{s}_{4}{Y}^{2}I\left( x\right) + {s}_{3}{s}_{4}\left( {X \circ Y}\right) I\left( x\right) .
\]
To this end, use Exr. 1.16, taking into account that when one computes the second-order derivatives of \( F\left( {{s}_{1},{s}_{2},{s}_{3},{s}_{4}}\right) \) at \( \left( {0,0,0,0}\right) \), then two out of the four variables can be taken to be null...
(3) Recognize that the Maclaurin expansion of degree 2 of \( t \mapsto F\left( {t, t, t, t}\right) \) is obtained by taking \( {s}_{1} = \cdots = {s}_{4} = t \) ; this will give (1.37). | No |
Exercise 9.8 | Exercise 9.8 (See [98, Example 3.1.7]) Let \( Z \subseteq {\mathbb{P}}^{N} \) be a fat point subscheme, \( I = I\left( Z\right) \) . If \( \alpha \left( {I}^{\left( m\right) }\right) < {r\alpha }\left( I\right) \), then \( {I}^{\left( m\right) } \nsubseteq {I}^{r} \) . | No |
Exercise 3.15.2 | Exercise 3.15.2. \( {}^{ \dagger } \) Give an example of an additive group \( G \) and a subgroup \( H \) for which \( G \) is not isomorphic with \( H \oplus G/H \) . | No |
Exercise 13.1 | Exercise 13.1. Re-run both of the MCMC examples in this chapter, but increase the number of iterations to 10,000 . Analyze your results from both cases. How does increasing the number of iterations affect the posterior parameter estimates and their confidence intervals? Does the log-likelihood value change? | No |
Exercise 4.3.13 | Exercise 4.3.13. \( n \) is multiplicatively perfect if it equals the product of its proper divisors.
(a) Show that \( n \) is multiplicatively perfect if and only if \( \tau \left( n\right) = 4 \) .
(b) Classify exactly which integers \( n \) satisfy this.
The integers \( m \) and \( n \) are amicable if the sum of the proper divisors of \( m \) equals \( n \) and the sum of the proper divisors of \( n \) equals \( m \) . For example,220 and 284 are amicable, as are 1184 and \( {12102} \) | No |
Exercise 2 | Exercise 2. In Section 8.4, we constructed the joint density of two random variables using the conditional and marginal density. For that problem, (i) what is the probability distribution of the time at which the call is placed? (ii) When is the expected time for the call to be placed? | No |
Exercise 1.5.12 | Exercise 1.5.12. For every \( n \in \mathbb{N} \), define \( {s}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}\frac{1}{k} \) . Prove that \( \left\{ {s}_{n}\right\} \) diverges. | No |
Exercise 4.11.3 | Exercise 4.11.3 (Conditional distribution of the future outputs conditioned on past state and past outputs). Consider a time-invariant forward Gaussian system representation with the equations
\[
x\left( {t + 1}\right) = {Ax}\left( t\right) + {Mv}\left( t\right), x\left( 0\right) = {x}_{0} \in G\left( {0,{Q}_{0}}\right) ,
\]
\[
y\left( t\right) = {Cx}\left( t\right) + {Nv}\left( t\right), v\left( t\right) \in G\left( {0, I}\right) .
\]
(a) Calculate the conditional characteristic function of the vector,
\[
\bar{y} = \left( \begin{array}{l} y\left( t\right) \\ y\left( {t + 1}\right) \\ \vdots \\ y\left( {t + {n}_{x} - 1}\right) \end{array}\right) ,\;\bar{y} : \Omega \rightarrow {\mathbb{R}}^{{n}_{x}{n}_{y}},
\]
\[
E\left\lbrack {\exp \left( {i{w}^{T}\bar{y}}\right) \mid {F}_{t}^{x} \vee {F}_{t - 1}^{y}}\right\rbrack ,\forall w \in {\mathbb{R}}^{{n}_{x}{n}_{y}},\forall t \in T.
\]
It is not necessary to calculate a formula for the conditional variance.
(b) Which condition implies that from the conditional characteristic function one can uniquely determine the value of the random variable \( x\left( t\right) \) ? | No |
Exercise 20.6.2 | Exercise 20.6.2 Let \( f\left( {u, v}\right) = 2{u}^{2} - 5{u}^{2}{v}^{5} \) and let \( u\left( {s, t}\right) = s{t}^{2} \) and \( v\left( {s, t}\right) = \) \( {s}^{2} + {t}^{4} \) . Find \( {f}_{s} \) and \( {f}_{t} \). | No |
Exercise 5.9.1 | Either use Kummer's Theorem (Theorem 3.7) or consider directly how often \( p \) divides the numerator and denominator of \( \left( \begin{matrix} {2n} \\ n \end{matrix}\right) \) . | Yes |
Exercise 8.5.1 | Exercise 8.5.1. Let \( T = \inf \left\{ {{B}_{t} \notin \left( {-a, a}\right) }\right\} \) . Show that
\[
E\exp \left( {-{\lambda T}}\right) = 1/\cosh \left( {a\sqrt{2\lambda }}\right) .
\] | No |
Exercise 24.7 | Show that every angle bounded monotone operator is \( {3}^{ * } \) monotone. | No |
Exercise 6.1.10 | Exercise 6.1.10. Verify that \( {\Delta }_{T} \) is a chain map. | No |
Exercise 4.17 | Exercise 4.17. Show that the analogous statement can fail when \( m = 1 \) . (Consider the case \( \dim X = 1 \) .) | No |
Exercise 11.1 | Exercise 11.1. Use Figure 11.3 to give a purely geometric proof of (11.2). (Hint: compare the sum of the areas of the two isosceles right triangles to the area of the shaded rectangle.) | No |
Exercise 11.8.2 | Exercise 11.8.2. Show that the billiard ball is at \( \left( {x, y}\right) \) after time \( t \), where \( x \) and \( y \) are given as follows:
Let \( m = \left\lbrack {u + t}\right\rbrack \) . If \( m \) is even, let \( x = \{ u + t\} \) ; if \( m \) is odd, let \( x = 1 - \{ u + t\} \) .
Let \( n = \left\lbrack {v + {\alpha t}}\right\rbrack \) . If \( n \) is even, let \( y = \{ v + {\alpha t}\} \) ; if \( n \) is odd, let \( y = 1 - \{ v + {\alpha t}\} \) . | No |
Exercise 6.2 | Exercise 6.2. (Exercise 5.2 continued).
(1) Derive the stochastic equation satisfied by the instantaneous forward rate \( f\left( {t, T}\right) \) .
(2) Check that the HJM absence of arbitrage condition is satisfied in the equation of Question (1). | No |
Exercise 5.32 | Exercise 5.32 (Tanaka's formula and local time) Let \( B \) be an \( \left( {\mathcal{F}}_{t}\right) \) -Brownian motion started from 0 . For every \( \varepsilon > 0 \), we define a function \( {g}_{\varepsilon } : \mathbb{R} \rightarrow \mathbb{R} \) by setting \( {g}_{\varepsilon }\left( x\right) = \sqrt{\varepsilon + {x}^{2}} \) .
## 1. Show that
\[
{g}_{\varepsilon }\left( {B}_{t}\right) = {g}_{\varepsilon }\left( 0\right) + {M}_{t}^{\varepsilon } + {A}_{t}^{\varepsilon }
\]
where \( {M}^{\varepsilon } \) is a square integrable continuous martingale that will be identified in the form of a stochastic integral, and \( {A}^{\varepsilon } \) is an increasing process.
2. We set \( \operatorname{sgn}\left( x\right) = {\mathbf{1}}_{\{ x > 0\} } - {\mathbf{1}}_{\{ x < 0\} } \) for every \( x \in \mathbb{R} \) . Show that, for every \( t \geq 0 \) ,
\[
{M}_{t}^{\varepsilon }\underset{\varepsilon \rightarrow 0}{\overset{{L}^{2}}{ \rightarrow }}{\int }_{0}^{t}\operatorname{sgn}\left( {B}_{s}\right) \mathrm{d}{B}_{s}
\]
Infer that there exists an increasing process \( L \) such that, for every \( t \geq 0 \) ,
\[
\left| {B}_{t}\right| = {\int }_{0}^{t}\operatorname{sgn}\left( {B}_{s}\right) \mathrm{d}{B}_{s} + {L}_{t}.
\]
3. Observing that \( {A}_{t}^{\varepsilon } \rightarrow {L}_{t} \) when \( \varepsilon \rightarrow 0 \), show that, for every \( \delta > 0 \), for every choice of \( 0 < u < v \), the condition \( \left( {\left| {B}_{t}\right| \geq \delta \text{for every}t \in \left\lbrack {u, v}\right\rbrack }\right) \) a.s. implies that \( {L}_{v} = {L}_{u} \) . Infer that the function \( t \mapsto {L}_{t} \) is a.s. constant on every connected component of the open set \( \left\{ {t \geq 0 : {B}_{t} \neq 0}\right\} \) .
4. We set \( {\beta }_{t} = {\int }_{0}^{t}\operatorname{sgn}\left( {B}_{s}\right) \mathrm{d}{B}_{s} \) for every \( t \geq 0 \) . Show that \( {\left( {\beta }_{t}\right) }_{t \geq 0} \) is an \( \left( {\mathcal{F}}_{t}\right) \) - Brownian motion started from 0 .
5. Show that \( {L}_{t} = \mathop{\sup }\limits_{{s < t}}\left( {-{\beta }_{s}}\right) \), a.s. (In order to derive the bound \( {L}_{t} \leq \mathop{\sup }\limits_{{s < t}}\left( {-{\beta }_{s}}\right) \), one may consider the last zero of \( B \) before time \( t \), and use question 3.) Give the law of \( {L}_{t} \) .
6. For every \( \varepsilon > 0 \), we define two sequences of stopping times \( {\left( {S}_{n}^{\varepsilon }\right) }_{n \geq 1} \) and \( {\left( {T}_{n}^{\varepsilon }\right) }_{n \geq 1} \) , by setting
\[
{S}_{1}^{\varepsilon } = 0,\;{T}_{1}^{\varepsilon } = \inf \left\{ {t \geq 0 : \left| {B}_{t}\right| = \varepsilon }\right\}
\]
and then, by induction,
\[
{S}_{n + 1}^{\varepsilon } = \inf \left\{ {t \geq {T}_{n}^{\varepsilon } : {B}_{t} = 0}\right\} ,\;{T}_{n + 1}^{\varepsilon } = \inf \left\{ {t \geq {S}_{n + 1}^{\varepsilon } : \left| {B}_{t}\right| = \varepsilon }\right\} .
\]
For every \( t \geq 0 \), we set \( {N}_{t}^{\varepsilon } = \sup \left\{ {n \geq 1 : {T}_{n}^{\varepsilon } \leq t}\right\} \), where \( \sup \varnothing = 0 \) . Show that
\[
\varepsilon {N}_{t}^{\varepsilon }\xrightarrow[{\varepsilon \rightarrow 0}]{{L}^{2}}{L}_{t}
\]
(One may observe that
\[
{L}_{t} + {\int }_{0}^{t}\left( {\mathop{\sum }\limits_{{n = 1}}^{\infty }{\mathbf{1}}_{\left\lbrack {S}_{n}^{\varepsilon },{T}_{n}^{\varepsilon }\right\rbrack }\left( s\right) }\right) \operatorname{sgn}\left( {B}_{s}\right) \mathrm{d}{B}_{s} = \varepsilon {N}_{t}^{\varepsilon } + {r}_{t}^{\varepsilon }
\]
where the "remainder" \( {r}_{t}^{\varepsilon } \) satisfies \( \left| {r}_{t}^{\varepsilon }\right| \leq \varepsilon \) .)
7. Show that \( {N}_{t}^{1}/\sqrt{t} \) converges in law as \( t \rightarrow \infty \) to \( \left| U\right| \), where \( U \) is \( \mathcal{N}\left( {0,1}\right) \) - distributed.
(Many results of Exercise 5.32 are reproved and generalized in Chap. 8.) | No |
Exercise 6.11 | Exercise 6.11 (A network extension of Aloha) We now consider a network extension of Aloha, where all transmissions do not interfere with each other. Let \( {\mathcal{O}}_{i} \) be the number of neighbors of node \( i \) that are within receiving range of transmissions from node \( i \), and let \( {\mathcal{I}}_{i} \) be the set of nodes whose transmissions can be heard by node \( i \) . We assume that a node cannot transmit and receive simultaneously, and that a node can transmit to only one other node at a time. Let \( {p}_{ij} \) be the probability that node \( i \) transmits to node \( j \) . The probability of successful transmission from node \( i \) to node \( j \) is given by
\[
{x}_{ij} = {p}_{ij}\left( {1 - {P}_{j}}\right) \mathop{\prod }\limits_{{l \in {\mathcal{I}}_{j}\smallsetminus \{ i\} }}\left( {1 - {P}_{l}}\right) ,
\]
where \( {P}_{j} \), the probability that node \( j \) transmits, is given by
\[
{P}_{j} = \mathop{\sum }\limits_{{k \in {\mathcal{O}}_{j}}}{p}_{jk}
\]
The goal in this exercise is to select \( \left\{ {p}_{ij}\right\} \) to maximize
\[
\mathop{\sum }\limits_{{i, j : i \neq j}}\log {x}_{ij} \tag{6.23}
\]
i.e., to achieve the proportional fairness in this network Aloha setting. The constraints are
\[
\mathop{\sum }\limits_{{k \in {\mathcal{O}}_{j}}}{p}_{jk} \leq 1\;\forall j
\]
\[
{p}_{jk} \geq 0\;\forall k, j
\]
Show that the optimal \( \left\{ {p}_{ij}\right\} \) ’s are given by
\[
{p}_{ij}^{ * } = \frac{1}{\left| {\mathcal{I}}_{i}\right| + \mathop{\sum }\limits_{{k \in {\mathcal{O}}_{i}}}\left| {\mathcal{I}}_{k}\right| }.
\]
Hint: It is helpful to show first that the following elements of the vector \( x \) are functions of \( {p}_{ij} : {x}_{ij},{x}_{ki} \) for \( k \in {\mathcal{I}}_{i} \), and \( {x}_{kh} \) for \( k \) and \( h \) such that \( h \in {\mathcal{O}}_{i} \cup {\mathcal{O}}_{k} \) and \( k \neq i \) . | No |
Exercise 4.1 | Exercise 4.1. Let \( t \geq 2 \) and \( d \geq 1 \) . Show that there are polynomial time algorithms
- associating with \( \alpha ,\beta \in {\Gamma }_{t, d} \) formulas in \( {\Gamma }_{t, d} \) equivalent to \( \left( {\alpha \land \beta }\right) \) and \( \left( {\alpha \vee \beta }\right) \)
- associating with \( \alpha ,\beta \in {\Delta }_{t, d} \) formulas in \( {\Delta }_{t, d} \) equivalent to \( \left( {\alpha \land \beta }\right) \) and \( \left( {\alpha \vee \beta }\right) \) . | No |
Exercise 4.3.2 | Let \( Q \) be a probability on \( \left( {\overline{\mathcal{M}},\overline{\mathcal{H}}}\right) \) . Show that \( {\bar{\Lambda }}_{\infty }^{ \circ } = \infty Q \) -a.s. on \( \mathop{\bigcap }\limits_{n}\left( {{\tau }_{n} < \infty }\right) \) . (Hint: find \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{e}^{-{\bar{\Lambda }}_{{\tau }_{n}}^{ \circ }}{1}_{\left( {\tau }_{n} < \infty \right) } \) ). | No |
Exercise 6.1 | Exercise 6.1. Let \( \mathbf{X} = {\left\{ {X}_{i}\right\} }_{i = 1}^{n} \) be an i.i.d. sample from a model of Gaussian shift \( \mathcal{N}\left( {\theta ,{\sigma }^{2}}\right) \) (here \( \sigma \) is a known parameter and \( \theta \) is a parameter of interest).
(i) Fix some level \( \alpha \in \left( {0,1}\right) \) and find a number \( {t}_{\alpha } \in \mathbb{R} \) such that the function
\[
\phi \left( X\right) \overset{\text{ def }}{ = }\mathbf{1}\left( {\bar{X} \geq {t}_{\alpha }}\right)
\]
is a test of level \( \alpha \) for checking the hypothesis \( {H}_{0} : \theta = {\theta }_{0} \) against the alternative \( {H}_{1} : \theta = {\theta }_{1} < {\theta }_{0}\left( {\theta }_{0}\right. \) and \( \left. {\theta }_{1}\right) \) are two fixed values \( ) \) .
(ii) Find the power function \( W\left( {\theta }_{1}\right) \) for this test.
(iii) Compare \( \alpha \) and \( W\left( {\theta }_{1}\right) \) . How can you interpret the results of this comparison?
(iv) Why a test in the form
\[
\phi \left( X\right) \overset{\text{ def }}{ = }\mathbf{1}\left( {\bar{X} \leq {s}_{\alpha }}\right)
\]
where \( {s}_{\alpha } \in \mathbb{R} \) is not appropriate for testing the hypothesis \( {H}_{0} \) against the alternative \( {\mathrm{H}}_{1} \) ?
(i) Observe that \( \sqrt{n}\left( {\bar{X} - {\theta }_{0}}\right) \) has a standard normal distribution \( \mathcal{N}\left( {0,1}\right) \) under \( {H}_{0} \) . Then for any \( t \)
\[
{\mathbb{P}}_{0}\left( {\bar{X} \geq t}\right) = {\mathbb{P}}_{0}\left\{ {\frac{\sqrt{n}\left( {\bar{X} - {\theta }_{0}}\right) }{\sigma } \geq \frac{\sqrt{n}\left( {t - {\theta }_{0}}\right) }{\sigma }}\right\} = 1 - \Phi \left\{ \frac{\sqrt{n}\left( {t - {\theta }_{0}}\right) }{\sigma }\right\} ,
\]
where \( {\mathbb{P}}_{0} \) denotes the probability measure of the normal distribution \( \mathcal{N}\left( {{\theta }_{0},{\sigma }^{2}}\right) \) .
Let us fix the parameter \( t \) such that \( {\mathbb{P}}_{0}\left( {\bar{X} \geq t}\right) = \alpha \) :
\[
\alpha = 1 - \Phi \left\{ \frac{\sqrt{n}\left( {t - \theta }\right) }{\sigma }\right\}
\]
\[
t = {t}_{\alpha } = {\theta }_{0} + \sigma {z}_{1 - \alpha }/\sqrt{n},
\]
where \( {z}_{1 - \alpha } \) is the \( \left( {1 - \alpha }\right) \) -quantile of the standard normal distribution.
So, a test of level \( \alpha \) is
\[
\phi \left( \mathbf{X}\right) \overset{\text{ def }}{ = }\mathbf{1}\left( {\bar{X} \geq {t}_{\alpha }}\right) = \mathbf{1}\left( {\bar{X} \geq {\theta }_{0} + \sigma {z}_{1 - \alpha }/\sqrt{n}}\right) .
\]
(ii) By the definition of the error of the second kind,
\[
W\left( {\theta }_{1}\right) = 1 - {\mathbb{P}}_{1}\{ \phi \left( \mathbf{X}\right) = 0\} = {\mathbb{P}}_{1}\{ \phi \left( \mathbf{X}\right) = 1\}
\]
\[
= {\mathbb{P}}_{1}\left( {\bar{X} \geq {\theta }_{0} + \sigma {z}_{1 - \alpha }/\sqrt{n}}\right)
\]
\[
= {\mathbb{P}}_{1}\left\{ {\frac{\sqrt{n}\left( {\bar{X} - {\theta }_{1}}\right) }{\sigma } \geq {z}_{1 - \alpha } - \sqrt{n}\left( {{\theta }_{0} - {\theta }_{1}}\right) \sigma }\right\}
\]
\[
= 1 - \Phi \left\{ {{z}_{1 - \alpha } - \frac{\sqrt{n}\left( {{\theta }_{0} - {\theta }_{1}}\right) }{\sigma }}\right\}
\]
(iii) One should compare two expressions:
\[
\alpha = 1 - \Phi \left( {z}_{1 - \alpha }\right) \;\text{ and }\;W\left( {\theta }_{1}\right) = 1 - \Phi \left\{ {{z}_{1 - \alpha } - \frac{\sqrt{n}\left( {{\theta }_{0} - {\theta }_{1}}\right) }{\sigma }}\right\} .
\]
By assumption, \( {\theta }_{0} > {\theta }_{1} \) . This yields
\[
{z}_{1 - \alpha } > {z}_{1 - \alpha } - \frac{\sqrt{n}\left( {{\theta }_{0} - {\theta }_{1}}\right) }{\sigma }.
\]
and therefore \( \alpha < W\left( {\theta }_{1}\right) \) because the function \( \Phi \left( \cdot \right) \) is monotone increasing.
This fact can be interpreted in the following way: the probability of rejecting the hypothesis when it is true is less than the probability of rejecting the hypothesis when it is false. In other words, "true rejection" has larger probability than "false rejection".
(iv) In the case of the test
\[
\phi \left( X\right) \overset{\text{ def }}{ = }\mathbf{1}\left( {\bar{X} \leq {s}_{\alpha }}\right) ,
\]
the error of the first level is larger than the power function at any point \( {\theta }_{1} < {\theta }_{0} \) . This means that "false rejection" has larger probability than "true rejection". | No |
Exercise 92.6 | [Show that one can take for \( p \) some power of \( {p}^{\prime } \) . (Hint: Prove and use the fact that the kernel of \( \operatorname{End}X \rightarrow \operatorname{End}{X}_{E} \) is annihilated by a positive integer.)] | No |
Exercise 4.3 | Exercise 4.3. Let \( F = \mathbb{Q}\left( i\right) \) and \( x = 2 - i \) . Compute \( \parallel x{\parallel }_{v} \) for all of the places \( v \in {V}_{F} \) and verify that the product formula holds for \( x \) . | Yes |
Exercise 8.2.7 | Exercise 8.2.7. Restrict the range of \( {\mathfrak{H}}_{2} \) to be the subspace \( \mathbb{C}\left( {\left\lbrack {0, T}\right\rbrack ;{\mathfrak{L}}^{2}\left( {\Omega ;{\mathbb{L}}^{2}\left( \mathcal{D}\right) }\right) }\right) \) of \( {\mathbb{L}}^{2}\left( {0, T;{\mathfrak{L}}^{2}\left( {\Omega ;{\mathbb{L}}^{2}\left( \mathcal{D}\right) }\right) }\right) \) . Do the conclusions of Claim 3 and Exer. 8.2.6 still hold? If so, how do the growth constants \( {M}_{{\mathfrak{H}}_{2}},\overline{{M}_{{\mathfrak{H}}_{2}}} \) change? | No |
Exercise 3 | Exercise 3 - Convergence problem. If \( P \) is the set of all primes and \( {\delta }_{k} \) is replaced by +1, then the series in formula (1) will not converge if \( \sigma < 1 \) . Here, \( z = \sigma + {it} \) . That is, \( \sigma \) is the real part of the complex number \( z \) . | No |
Exercise 8.5.5 | Exercise 8.5.5. Find a martingale of the form \( {B}_{t}^{6} - {c}_{1}t{B}_{t}^{4} + {c}_{2}{t}^{2}{B}_{t}^{2} - {c}_{3}{t}^{3} \) and use it to compute the third moment of \( T = \inf \left\{ {t : {B}_{t} \notin \left( {-a, a}\right) }\right\} \) . | Yes |
Exercise 2.3.6 | Exercise 2.3.6. Integrate Eq. (2.3.13) imposing the conditions \( y = 0 \) when \( t = 0 \) and \( y = \alpha \) when \( t = T/4 \) . Here \( T \) is the period of anharmonic oscillations of the pendulum and \( \alpha \) is its angular amplitude, i.e., the maximum angular displacement, \( \alpha = {\left. y\right| }_{\max } \) . Evaluate the period \( T \) and compare it with the period \( \tau \) of harmonic oscillations. | Yes |
Exercise 5.10.3 | Exercise 5.10.3(a). If \( r \leq s/2 \), then by Bertrand’s postulate there is a prime \( p \in (s/2, s\rbrack \subset \) \( r, s\rbrack \) . Otherwise \( k = s - r \leq r \) . In either case, by Bertrand’s postulate or the Sylvester-Schur Theorem, one term has a prime factor \( p > k \), and so this is the only term that can be divisible by \( p \) . | No |
Exercise 7.1.21 | Exercise 7.1.21. Prove that if \( X \) and \( Y \) are not empty, then \( X \times Y \) is regular if and only if \( X \) and \( Y \) are regular. | No |
Exercise 1.6 | [A consequence of Baker's Theorem 1.6 is the transcendence of numbers like
\[
{\int }_{0}^{1}\frac{dt}{1 + {t}^{3}} = \frac{1}{3}\left( {\log 2 + \frac{\pi }{\sqrt{3}}}\right)
\]
Let \( P \) and \( Q \) be two nonzero polynomials with algebraic coefficients and \( \deg P < \deg Q \) . Assume \( Q \) has no multiple zero. Let \( \gamma \) be a contour in the complex plane, which is either closed, or has endpoints which are algebraic or infinite, and such that the definite integral
\[
{\int }_{\gamma }\frac{P\left( z\right) }{Q\left( z\right) }{dz}
\]
exists and is not zero. Then this integral is a transcendental number.
Hint. See [V 1971].] | Yes |
Exercise 2.19 | Exercise 2.19 Let \( X \sim N\left( {0,1}\right) \) and set \( Y = {X}^{2} \) . Find the covariance matrix of \( \left( {X, Y}\right) \) . Are \( X \) and \( Y \) correlated? Are \( X \) and \( Y \) independent? | No |
Exercise 6.5 | Exercise 6.5. Let \( \Lambda \) be a hyperbolic set for a diffeomorphism \( f \) . Show that the homeomorphism \( f \mid \Lambda : \Lambda \rightarrow \Lambda \) is two-sided expansive (see Definition 5.5). | No |
Exercise 11.28 | \[
\left\{ \begin{array}{ll} \frac{{\partial }^{2}u}{\partial {t}^{2}} - \mu \operatorname{div}\left( {{\left| \frac{\partial u}{\partial t}\right| }^{p - 2}\nabla \frac{\partial u}{\partial t}}\right) - {\Delta u} + c\left( u\right) = g & \text{ in }Q, \\ u\left( {0, \cdot }\right) = {u}_{0},\;\frac{\partial u}{\partial t}\left( {0, \cdot }\right) = {v}_{0}, & \text{ in }\Omega , \\ {\left. u\right| }_{\sum } = 0 & \text{ on }\sum , \end{array}\right. \tag{11.119}
\]
with \( \mu > 0 \) . Apply the Galerkin method, denote the approximate solution by \( {u}_{k} \), and prove a-priori estimates for \( {u}_{k} \) in \( {L}^{\infty }\left( {I;{W}^{1,2}\left( \Omega \right) }\right) \cap {W}^{1, p}\left( {I,{W}^{1, p}\left( \Omega \right) }\right) \cap \) \( {W}^{2,{p}^{\prime }}\left( {I;{W}^{\max \left( {2, p}\right) }\left( \Omega \right) }\right) \) and specify qualifications on \( {u}_{0},{v}_{0}, g \), and \( c\left( \cdot \right) {.}^{31} \) Prove convergence by monotonicity and the Minty trick. \( {}^{32} \) Eventually, denoting \( {u}_{\mu } \) the solution to (11.119), prove that \( {u}_{\mu } \) approaches the solution to the Klein-Gordon equation (11.118) if \( c\left( r\right) = {\left| r\right| }^{q - 2}r \) when \( \mu \rightarrow 0.{}^{33} \) | No |
Exercise 1.5.2 | Exercise 1.5.2. Determine all representations of 199 by the form \( f = \left( {3,5,7}\right) \) . | Yes |
Exercise 11.7 | Exercise 11.7 Of special interest for applications is the case of infinitesimal generators depending quadratically on \( \mu \) (see the next section), which leads to the system of quadratic equations
\[
{\dot{x}}_{j} = \left( {{A}^{j}x, x}\right) ,\;j = 1,2,\ldots, N, \tag{11.61}
\]
where \( N \) is a natural number (or more generally \( N = \infty \) ), the unknown \( x = \) \( \left( {{x}_{1},{x}_{2},\ldots }\right) \) is an \( N \) -dimensional vector and the \( {A}^{j} \) are given square \( N \times N \) matrices. Suppose that \( \mathop{\sum }\limits_{{j = 1}}^{N}{A}^{j} = 0 \) . Show that the system (11.61) defines a positivity-preserving semigroup (i.e. if all the coordinates of the initial vector \( {x}^{0} \) are non-negative then the solution \( x\left( t\right) \) is globally uniquely defined and all coordinates of this solution are non-negative for all times), if and only if for each \( j \) the matrix \( {\widetilde{A}}^{j} \) obtained from \( {A}^{j} \) by deleting its \( j \) th column and \( j \) th row is such that \( \left( {{\widetilde{A}}^{j}v, v}\right) \geq 0 \) for any \( v \in {\mathbf{R}}^{N - 1} \) with non-negative coordinates. Hint: this condition expresses the fact that if \( {x}_{j} = 0 \) and the other \( {x}_{i}, i \neq j \) , are non-negative, then \( {\dot{x}}_{j} \geq 0 \) . | No |
Exercise 2.21 | Exercise 2.21. Show that in Exercise 2.19 if for some probability measure \( \lambda \) whose support \( S\left( \lambda \right) \) consists of \( 2 \times 2 \) stochastic matrices with rank one and \( \lambda * \mu = \lambda \) , then \( \lambda = \left( w\right) \mathop{\lim }\limits_{{n \rightarrow \infty }}{\mu }^{n} \) . | No |
Exercise 6 | Exercise 6 (Square-root iteration) | No |
Exercise 5.6 | Exercise 5.6. Let \( X = \mathbb{D} \) and let \( K \) be the Szegö kernel. Describe the spaces \( \mathcal{H}\left( {K \circ \varphi }\right) \) for \( \varphi \left( z\right) = {z}^{2} \) and for \( \varphi \left( z\right) = \frac{z - \alpha }{1 - \bar{\alpha }z},\alpha \in \mathbb{D} \) a simple Möbius map. | No |
Exercise 5.15 | Exercise 5.15 (Weak compactness in \( {L}^{p} \) spaces, \( p > 1 \) ) Let \( \mu \) be a Radon measure on \( {\mathbb{R}}^{n} \) . If \( {\left\{ {u}_{h}\right\} }_{h \in \mathbb{N}} \subset {L}^{p}\left( {{\mathbb{R}}^{n},\mu }\right) \left( {1 < p \leq \infty }\right) \) satisfies
\[
\mathop{\sup }\limits_{{h \in \mathbb{N}}}{\begin{Vmatrix}{u}_{h}\end{Vmatrix}}_{{L}^{p}\left( {{\mathbb{R}}^{n},\mu }\right) } < \infty ,
\]
then there exist a sequence \( h\left( k\right) \rightarrow \infty \) as \( k \rightarrow \infty \) and \( u \in {L}^{p}\left( {{\mathbb{R}}^{n},\mu }\right) \) such that
\[
\mathop{\lim }\limits_{{k \rightarrow \infty }}{\int }_{{\mathbb{R}}^{n}}\varphi {u}_{h\left( k\right) }\mathrm{d}\mu = {\int }_{{\mathbb{R}}^{n}}{\varphi u}\mathrm{\;d}\mu ,
\]
for every \( \varphi \in {L}^{{p}^{\prime }}\left( {{\mathbb{R}}^{n},\mu }\right) \left( {{p}^{\prime } = 1}\right. \) if \( p = \infty ,{p}^{\prime } = p/\left( {p - 1}\right) \) if \( \left. {p \in \left( {1,\infty }\right) }\right) \) . Hint: By Corollary 4.34, there exists a signed Radon measure \( v \) such that \( {\mu }_{h\left( k\right) } = {u}_{h\left( k\right) }\mu \overset{ * }{ \rightharpoonup }\nu \) . Show that \( v \ll \mu \) and that, in fact, \( v = {u\mu } \) for \( u \in {L}^{p}\left( {{\mathbb{R}}^{n},\mu }\right) \) . | No |
Exercise 13.3 | Plot the Dirac delta function on the interval \( \left\lbrack {-1,1}\right\rbrack \) (you won’t see the impulse at 0 ). Evaluate the function at 0 and 1 . Integrate the function over the interval \( \left\lbrack {-1,1}\right\rbrack \) . | Yes |
Exercise 9.1.17 | Exercise 9.1.17. Prove the following are equivalent for a metrizable space \( X \) .
1. \( X \) is compact.
2. Every metric on \( X \) is bounded.
3. Every continuous function on \( X \) is bounded. | No |
Exercise 4.1.3 | For this function
\[
f\left( x\right) = \left\{ \begin{array}{ll} - 2, & \text{ if }0 \leq x \leq 3 \\ 1 & \text{ if }3 < x \leq 5 \end{array}\right.
\]
- find the right and left hand limits as \( x \rightarrow 3 \) .
- Show that for \( \epsilon = 1 \), there is no radius \( \delta \) so that the definition of the limit is satisfied.
- explain why \( f \) is not continuous at 3 . | No |
Exercise 24 | Exercise 24. Using nonstandard methods, prove that a continuous function whose domain is a compact subset of \( {\mathbb{R}}^{n} \) must be uniformly continuous. | No |
Exercise 5.18 | Exercise 5.18. The kernel \( I + W{I}_{h} \) satisfies the reinforced complete maximum principle. The kernel \( W \) satisfies the positive maximum principle, that is,
\[
\mathop{\sup }\limits_{{x \in E}}{Wf}\left( x\right) = \mathop{\sup }\limits_{{x \in \{ f > 0\} }}{Wf}\left( x\right) .
\] | No |
Exercise 11.9.1 | Explain why if \( \alpha \) has a finite length continued fraction, then the last term is an integer \( \geq 2 \) . | No |
Exercise 4.49 | Exercise 4.49. Generated scales: create a scale with 20 generators in some \( {\mathbb{Z}}_{n} \) . | No |
Exercise 1.22 | Exercise 1.22. (1) Principle of uniqueness of charges. If \( \left( {f, g}\right) \) are in \( {\mathcal{E}}_{ + } \) and \( {Gf} = {Gg} < \infty \), then \( f = g \) .
(2) If \( f \) and \( g \) are two positive functions vanishing outside \( A \) and if \( {Gf} = {Gg} \) \( < \infty \) on \( A \), then \( f = g \) . | No |
Exercise 4.21 | Exercise 4.21 (Stop-loss start-gain paradox). Let \( S\left( t\right) \) be a geometric Brownian motion with mean rate of return zero. In other words,
\[
{dS}\left( t\right) = {\sigma S}\left( t\right) {dW}\left( t\right)
\]
where the volatility \( \sigma \) is constant. We assume the interest rate is \( r = 0 \) .
Suppose we want to hedge a short position in a European call with strike price \( K \) and expiration date \( T \) . We assume that the call is initially out of the money (i.e., \( S\left( 0\right) < K \) ). Starting with zero capital \( \left( {X\left( 0\right) = 0}\right) \), we could try the following portfolio strategy: own one share of the stock whenever its price strictly exceeds \( K \), and own zero shares whenever its price is \( K \) or less. In other words, we use the hedging portfolio process
\[
\Delta \left( t\right) = {\mathbb{I}}_{\left( K,\infty \right) }\left( {S\left( t\right) }\right) .
\]
The value of this hedge follows the stochastic differential equation
\[
{dX}\left( t\right) = \Delta \left( t\right) {dS}\left( t\right) + r\left( {X\left( t\right) - \Delta \left( t\right) X\left( t\right) }\right) {dt},
\]
and since \( r = 0 \) and \( X\left( 0\right) = 0 \), we have
\[
X\left( T\right) = \sigma {\int }_{0}^{T}{\mathbb{I}}_{\left( K,\infty \right) }\left( {S\left( t\right) }\right) S\left( t\right) {dW}\left( t\right) .
\]
\( \left( {4.10.46}\right) \)
Executing this hedge requires us to borrow from the money market to buy a share of stock whenever the stock price rises across level \( K \) and sell the share, repaying the money market debt, when it falls back across level \( K \) . (Recall that we have taken the interest rate to be zero. The situation we are describing can also occur with a nonzero interest rate, but it is more complicated to set up.) At expiration, if the stock price \( S\left( T\right) \) is below \( K \) , there would appear to have been an even number of crossings of the level \( K \), half in the up direction and half in the down direction, so that we would have bought and sold the stock repeatedly, each time at the same price \( K \) , and at the final time have no stock and zero debt to the money market. In other words, if \( S\left( T\right) < K \), then \( X\left( T\right) = 0 \) . On the other hand, if at the final time \( S\left( T\right) \) is above \( K \), we have bought the stock one more time than we sold it, so that we end with one share of stock and a debt of \( K \) to the money market. Hence, if \( S\left( T\right) > K \), we have \( X\left( T\right) = S\left( T\right) - K \) . If at the final time \( S\left( T\right) = K \), then we either own a share of stock valued at \( K \) and have a money market debt \( K \) or we have sold the stock and have zero money market debt. In either case, \( X\left( T\right) = 0 \) . According to this argument, regardless of the final stock price, we have \( X\left( T\right) = {\left( S\left( T\right) - K\right) }^{ + } \) . This kind of hedging is called a stop-loss start-gain strategy.
(i) Discuss the practical aspects of implementing the stop-loss start-gain strategy described above. Can it be done?
(ii) Apart from the practical aspects, does the mathematics of continuous-time stochastic calculus suggest that the stop-loss start-gain strategy can be implemented? In other words, with \( X\left( T\right) \) defined by (4.10.46), is it really true that \( X\left( T\right) = {\left( S\left( T\right) - K\right) }^{ + } \) ? | No |
Exercise 6.7.3 | Exercise 6.7.3. Let \( A \) and \( B \) be arbitrary subsets of a vector space \( V \) over \( \mathbb{F} \) . Define their Minkowski sum to be
\[
A + B = \{ \mathbf{x} + \mathbf{y} \mid \mathbf{x} \in A,\;\mathbf{y} \in B\} .
\]
Show that if \( A \) and \( B \) are cosets of a subspace \( W \) of \( V \), then so is \( A + B \) . | No |
Exercise 1.4 | Exercise 1.4. What (if anything) do the entries of \( A{A}^{T} \) tell us? | No |
Exercise 6.1.2 | Consider the following functions \( T : {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{2} \) . Explain why each of these functions \( T \) is not linear.
(a) \( T\left\lbrack \begin{array}{l} x \\ y \\ z \end{array}\right\rbrack = \left\lbrack \begin{matrix} x + {2y} + {3z} + 1 \\ {2y} - {3x} + z \end{matrix}\right\rbrack \)
(b) \( T\left\lbrack \begin{array}{l} x \\ y \\ z \end{array}\right\rbrack = \left\lbrack \begin{matrix} x + 2{y}^{2} + {3z} \\ {2y} + {3x} + z \end{matrix}\right\rbrack \)
(c) \( T\left\lbrack \begin{array}{l} x \\ y \\ z \end{array}\right\rbrack = \left\lbrack \begin{matrix} \sin x + {2y} + {3z} \\ {2y} + {3x} + z \end{matrix}\right\rbrack \)
(d) \( T\left\lbrack \begin{array}{l} x \\ y \\ z \end{array}\right\rbrack = \left\lbrack \begin{matrix} x + {2y} + {3z} \\ {2y} + {3x} - \ln z \end{matrix}\right\rbrack \) | No |
Exercise 2.1 | Exercise 2.1 The first law for a system in contact with a work source and a hot and a cold heat bath reads \( \Delta {U}_{S} = W + {Q}_{H} + {Q}_{C} \), where \( {Q}_{H\left( C\right) } \) is the heat flow from the hot (cold) bath. Assuming that the baths are described throughout the process by constant temperatures \( {T}_{H} > {T}_{C} \), the second law (2.6) generalizes to \( \sum = \Delta {S}_{S} - {Q}_{H}/{T}_{H} - {Q}_{C}/{T}_{C} \geq 0 \) . Now, assume that we want to use this set-up as a heat engine, i.e. we want to extract work from it: \( W < 0 \) . We further consider a cyclically working heat engine, which has eventually reached a steady state characterized by \( \Delta {U}_{S} = 0 \) and \( \Delta {S}_{S} = 0 \) per cycle. First, show in this case that \( W < 0 \) implies \( {Q}_{H} > 0 \) . Hence, the following number is positive and called the heat engine’s efficiency per cycle:
\[
\eta \equiv \frac{-W}{{Q}_{H}}. \tag{2.7}
\]
Next, use \( \Delta {U}_{S} = 0 \) and \( \Delta {S}_{S} = 0 \) and the first and second laws of thermodynamics to show that the following relations hold:
\[
\eta = 1 - \frac{{T}_{C}}{{T}_{H}} - \frac{{T}_{C}\sum }{{Q}_{H}} \leq 1 - \frac{{T}_{C}}{{T}_{H}} \equiv {\eta }_{C}. \tag{2.8}
\]
Here, \( {\eta }_{C} \) denotes the Carnot efficiency, which is the maximum efficiency of any engine working between two heat baths with fixed temperatures. Thus, any excess in the entropy production \( \sum \) diminishes the efficiency of the engine since \( {\eta }_{C} - \eta = {T}_{C}\sum /{Q}_{H} \geq 0 \) . | Yes |
Exercise 9.3.6 | Exercise 9.3.6 (Young’s Inequality). Fix \( 1 \leq p \leq \infty \), and assume that
\( f \in {L}^{p}\left( \mathbb{T}\right) \) and \( g \in {L}^{1}\left( \mathbb{T}\right) \) . Prove that
(a) \( f * g \) is defined a.e.,
(b) \( f * g \) is 1-periodic,
(c) \( f * g \) is measurable and \( f * g \in {L}^{p}\left( \mathbb{T}\right) \) ,
(d) \( \parallel f * g{\parallel }_{p} \leq \parallel f{\parallel }_{p}\parallel g{\parallel }_{1} \), and
(e) \( {\left( f * g\right) }^{ \land }\left( n\right) = \widehat{f}\left( n\right) \widehat{g}\left( n\right) \) for all \( n \in \mathbb{Z} \) . \( \;\diamond \) | No |
Exercise 2.2 | Exercise 2.2 Find all \( \gamma \in \Gamma = {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), which commute with
(a) \( S = \left( \begin{matrix} - 1 \\ 1 \end{matrix}\right) \) ,
(b) \( T = \left( \begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \) ,
(c) \( {ST} \) . | No |
Exercise 4.4.3 | Exercise 4.4.3. Prove that any hyperbolic automorphism of \( {\mathbb{T}}^{n} \) is mixing. | No |
Exercise 24 | [Exercise 24. Show, using the conditions (2.4),(2.5) that \( S \) is symplectic if and only if \( {S}^{T} \) is.] | No |
Exercise 16.4.1 | Exercise 16.4.1 Let \( {\mathbf{E}}_{\mathbf{1} \rightarrow \mathbf{2}} = {0.05} \) and \( {\mathbf{E}}_{\mathbf{{11}} \rightarrow \mathbf{2}} = {0.9} \) and initialize \( {Y}^{1} = {1.0} \) and \( {Y}^{11} = - 1 \) . Compute 5 iterations of \( {y}^{2}\left( {t + 1}\right) = {\mathbf{E}}_{\mathbf{1} \rightarrow \mathbf{2}}{Y}_{1}\left( t\right) + {\mathbf{E}}_{\mathbf{1}\mathbf{1} \rightarrow \mathbf{2}}{Y}_{11}\left( t\right) \) . | Yes |
Exercise 1.1 | Exercise 1.1 Calculate the moment generating functions (1.4)
\[
\Psi \left( u\right) = \mathbf{E}\exp \left\{ {u{\int }_{0}^{T}g\left( t\right) \mathrm{d}{X}_{t}}\right\}
\]
\[
\Psi \left( {u, v}\right) = \mathbf{E}\exp \left\{ {{\int }_{0}^{T}\left\lbrack {{ug}\left( t\right) + {vh}\left( t\right) }\right\rbrack \mathrm{d}{X}_{t}}\right\} ,
\]
\[
\Psi \left( {u, v, w}\right) = \mathbf{E}\exp \left\{ {{\int }_{0}^{T}\left\lbrack {{ug}\left( t\right) + {vh}\left( t\right) + {wf}\left( t\right) }\right\rbrack \mathrm{d}{X}_{t}}\right\} .
\]
Using these generating functions calculate the moments:
\[
\mathbf{E}{\left( {\int }_{0}^{T}g\left( t\right) \mathrm{d}{X}_{t}\right) }^{k} = {\left. \frac{{\partial }^{k}}{\partial {u}^{k}}\Psi \left( u\right) \right| }_{u = 0},\;k = 1,2,3,
\]
\( \mathbf{E}{\int }_{0}^{T}g\left( t\right) \mathrm{d}{X}_{t}{\int }_{0}^{T}h\left( t\right) \mathrm{d}{X}_{t} = {\left. \frac{{\partial }^{2}}{\partial u\partial v}\Psi \left( u, v\right) \right| }_{u = 0, v = 0}, \)
\( \mathbf{E}{\int }_{0}^{T}g\left( t\right) \mathrm{d}{X}_{t}\exp \left\{ {{\int }_{0}^{T}h\left( t\right) \mathrm{d}{X}_{t}}\right\} = {\left. \frac{\partial }{\partial u}\Psi \left( u, v\right) \right| }_{u = 0, v = 1}, \)
\( \mathbf{E}{\int }_{0}^{T}g\left( t\right) \mathrm{d}{X}_{t}{\int }_{0}^{T}f\left( t\right) \mathrm{d}{X}_{t}\exp \left\{ {{\int }_{0}^{T}h\left( t\right) \mathrm{d}{X}_{t}}\right\} = {\left. \frac{\partial }{\partial u}\Psi \left( u, v, w\right) \right| }_{u = 0, v = 1, w = 0}. \) | Yes |
Exercise 6.7 | Exercise 6.7 The sheaves \( {O}_{\mathbf{P}}{\left( n\right) }^{\text{alg }} \) and \( O\left( n\right) \) .
In order to describe the analytic line bundle \( {O}_{\mathbf{P}}\left( n\right) \) in terms of meromorphic functions we identify \( {O}_{\mathbf{P}}\left( n\right) \) with the line bundle \( \mathcal{L}\left( {n,\left\lbrack \infty \right\rbrack }\right) \) corresponding to the divisor \( n.\left\lbrack \infty \right\rbrack \) on \( \mathbf{P} \) . Let \( S = \left\{ {{p}_{1},\ldots ,{p}_{m}}\right\} \) be a finite set not containing \( \infty \) and let \( {f}_{S} = \) \( \mathop{\prod }\limits_{{i = 1}}^{m}\left( {z - {p}_{i}}\right) \) . Show that for \( U = {P}^{1} \smallsetminus S,{O}_{\mathbf{P}}{\left( n\right) }^{alg}\left( U\right) \) consists of all rational functions of the form \( g/{f}_{S}^{k} \), where \( k \geq 0 \) and \( \deg g \leq n + {km} \) . Describe \( {O}_{\mathbf{P}}{\left( n\right) }^{alg}\left( U\right) \), where \( U = {P}^{1} \smallsetminus S \) and \( S \) contains the point at infinity. We denote the sheaf \( {O}_{\mathbf{P}}{\left( n\right) }^{alg} \) by \( O\left( n\right) \) . | No |
Exercise 2.3.7 | Exercise 2.3.7 Spells of Employment and the Exponential Distribution An economic consultant for a fast-food chain has been given a random sample of the chain’s service workers. In her model, \( A \), the length of time \( y \) between the time the worker is hired and the time a worker quits has an exponential distribution with parameter \( {\theta }^{-1}, p\left( {y \mid \theta }\right) = \theta \exp \left( {-{\theta y}}\right) \) . For the purposes of this problem, assume that no one is ever laid off or fired. The only way of leaving employment is by quitting. You can also assume that no one ever has more than one spell of employment with the company-if they quit, they never come back.
In this problem we consider the case in which the consultant's data consist entirely of "complete spells;" that is, for each individual \( t \) in the sample the consultant observes the length of time, \( {y}_{t} \), between hiring and quitting.
(a) Express the joint density of the observables and find a sufficient statistic vector for \( \theta \) .
(b) Show that the conjugate prior distribution of \( \theta \) has the form \( {\underline{s}}^{2}\theta \sim {\chi }^{2}\left( \underline{v}\right) \) , and provide an "artificial data" interpretation of \( \left( {{\underline{s}}^{2},\underline{v}}\right) \) .
(c) Using the prior density in (b), express the kernel of the posterior density. Show that the posterior distribution of \( \theta \) has a gamma distribution of the form \( {\bar{s}}^{2}\theta \sim {\chi }^{2}\left( \bar{v}\right) \) . Express \( {\bar{s}}^{2} \) and \( \bar{v} \) in terms of the sufficient statistics from (a) and \( \left( {{\gamma }_{1},{\gamma }_{2}}\right) \) from (b). | No |
Exercise 6 | Find extremals corresponding to \( F\left( {y,{y}^{\prime }}\right) = {y}^{n}{\left\{ 1 + {\left( {y}^{\prime }\right) }^{2}\right\} }^{\frac{1}{2}} \) when \( n = \frac{1}{2} \) and \( n = - 1 \) . | No |
Exercise 4.2 | Exercise 4.2. From Proposition 4.1 and (1.9) the bond pricing PDE is given by
\[
\left\{ \begin{array}{l} \frac{\partial F}{\partial t}\left( {t, x}\right) = {xF}\left( {t, x}\right) - \left( {\alpha - {\beta x}}\right) \frac{\partial F}{\partial x}\left( {t, x}\right) - \frac{1}{2}{\sigma }^{2}{x}^{2}\frac{{\partial }^{2}F}{\partial {x}^{2}}\left( {t, x}\right) \\ F\left( {T, x}\right) = 1. \end{array}\right.
\]
When \( \alpha = 0 \) we search for a solution of the form
\[
F\left( {t, x}\right) = {e}^{A\left( {T - t}\right) - {xB}\left( {T - t}\right) },
\]
with \( A\left( 0\right) = B\left( 0\right) = 0 \), which implies
\[
\left\{ \begin{array}{l} {A}^{\prime }\left( s\right) = 0 \\ {B}^{\prime }\left( s\right) + {\beta B}\left( s\right) + \frac{1}{2}{\sigma }^{2}{B}^{2}\left( s\right) = 1, \end{array}\right.
\]
hence in particular \( A\left( s\right) = 0, s \in \mathbb{R} \), and \( B\left( s\right) \) solves a Riccatti equation, whose solution is easily checked to be
\[
B\left( s\right) = \frac{2\left( {{e}^{\gamma s} - 1}\right) }{{2\gamma } + \left( {\beta + \gamma }\right) \left( {{e}^{\gamma s} - 1}\right) },
\]
with \( \gamma = \sqrt{{\beta }^{2} + 2{\sigma }^{2}} \). | No |
Exercise 5.2 | \[
\left\lbrack \begin{array}{lll} 1 & 5 & 2 \\ 2 & 1 & 5 \\ 4 & 8 & 0 \end{array}\right\rbrack
\] | No |
Exercise 16 | Exercise 16. The coffee chain Starbucks created an app that supports mobile ordering at 7,400 of its stores in the United States, giving users the opportunity to order and pay for their drinks before they even arrive at their local Starbucks. Starbucks estimates the typical wait time given in the app will average around 3-5 minutes at most stores, with variability depending on the number of mobile orders in the queue and what a call order entails. After the transaction is completed in the app, users can skip the line and instead head to the pick-up counter where they can ask a barista for their order. Suppose that at one of the stores the waiting time in seconds has moment generating function given by
\[
{M}_{X}\left( t\right) = {\left( 1 - {200}t\right) }^{-1}
\]
If you enter your order immediately after another customer, what is the probability that your order will be ready in 300 seconds? (ii) If 300 seconds have passed and you arrive at the counter and your coffee is not ready, what is the probability that you will have to wait an additional 50 seconds? | Yes |
Exercise 7.4.6 | Exercise 7.4.6. Verify the universality of the graded symmetric algebra \( S\left( V\right) \) and the graded exterior algebra \( \bigwedge \left( V\right) \) as defined in this section in case \( \operatorname{char}\left( k\right) \neq 2 \) :
(a) Given a graded vector space \( V \) and a \( k \) -algebra \( B \), a graded symmetric map \( f : V \rightarrow B \) is a \( k \) -linear map such that \( f\left( v\right) f\left( w\right) = \) \( {\left( -1\right) }^{\left| v\right| \left| w\right| }f\left( w\right) f\left( v\right) \) for all homogeneous \( v, w \in V \) . Show that for all \( k \) -algebras \( B \) and graded symmetric maps \( f : V \rightarrow B \), there is a unique \( k \) -algebra homomorphism \( F : S\left( V\right) \rightarrow B \) such that \( {\left. F\right| }_{V} = f \)
(b) Given a graded vector space \( V \) and a \( k \) -algebra \( B \), a graded antisymmetric map \( f : V \rightarrow B \) is a \( k \) -linear map such that \( f\left( v\right) f\left( w\right) = \) \( - {\left( -1\right) }^{\left| v\right| \left| w\right| }f\left( w\right) f\left( v\right) \) for all homogeneous \( v, w \in V \) . Show that for all \( k \) -algebras \( B \) and graded anti-symmetric maps \( f : V \rightarrow B \) , there is a unique \( k \) -algebra homomorphism \( F : \bigwedge \left( V\right) \rightarrow B \) such that \( {\left. F\right| }_{V} = f \) . | No |
Exercise 12.6 | Exercise 12.6. Carry out the group level NCC evaluation as presented in Sections 12.3.5.2 and 12.3.5.3. Then repeat the same analysis with the step change impact function replaced by (a) the linear impact function and (b) the generalised impact function. Is the evaluation result robust against different forms for the impact function? Modify the WinBUGS code in order to assess how well the model with each impact function describes the post-policy data from the NCC group. | No |
Exercise 2.4.4 | Fix an arbitrary commutative ring with unit, \( k \), and a small category \( \mathcal{C} \) . Functors from \( \mathcal{C} \) to the category of \( k \) -modules are called \( \mathcal{C} \) -modules. Note that the set of natural transformations between two \( \mathcal{C} \) -modules carries the structure of a \( k \) -module.
For any object \( C \) of \( \mathcal{C} \), the functor \( {C}^{\prime } \mapsto k\left\{ {\mathcal{C}\left( {C,{C}^{\prime }}\right) }\right\} \) is a \( \mathcal{C} \) -module. Prove the following \( k \) -linear version of the Yoneda lemma by using the free-forgetful adjunction between the category of Sets and the category of \( k \) -modules: | No |
Exercise 4.4.8 | Exercise 4.4.8. Prove that the measure \( \mu \) on \( {\mathbb{R}}^{\mathbb{Z}} \) constructed above for a stationary sequence \( \left( {f}_{i}\right) \) is invariant under the shift \( \sigma \) . | No |
Exercise 12.10.27 | [Exercise 12.10.27. If \( \chi ,\phi \) are two Dirichlet characters such that \( \left( {{F}_{\chi },{F}_{\phi }}\right) = 1 \), prove that \( {F}_{\chi \phi } = {F}_{\chi }{F}_{\phi } \) .] | No |
Exercise 7.7 | Exercise 7.7 (Zero-strike Asian call). Consider a zero-strike Asian call whose payoff at time \( T \) is
\[
V\left( T\right) = \frac{1}{T}{\int }_{0}^{T}S\left( u\right) {du}
\]
(i) Suppose at time \( t \) we have \( S\left( t\right) = x \geq 0 \) and \( {\int }_{0}^{t}S\left( u\right) {du} = y \geq 0 \) . Use the
fact that \( {e}^{-{ru}}S\left( u\right) \) is a martingale under \( \widetilde{\mathbb{P}} \) to compute
\[
{e}^{-r\left( {T - t}\right) }\widetilde{\mathbb{E}}\left\lbrack {\left. {\frac{1}{T}{\int }_{0}^{T}S\left( u\right) {du}}\right| \;\mathcal{F}\left( t\right) }\right\rbrack .
\]
Call your answer \( v\left( {t, x, y}\right) \) .
(ii) Verify that the function \( v\left( {t, x, y}\right) \) you obtained in (i) satisfies the Black-Scholes-Merton equation (7.5.8) and the boundary conditions (7.5.9) and (7.5.11) of Theorem 7.5.1. (We do not try to verify (7.5.10) because the computation of \( v\left( {t, x, y}\right) \) outlined here works only for \( y \geq 0 \) .)
(iii) Determine explicitly the process \( \Delta \left( t\right) = {v}_{x}\left( {t, S\left( t\right), Y\left( t\right) }\right) \), and observe that it is not random.
(iv) Use the Itô-Doeblin formula to show that if you begin with initial capital \( X\left( 0\right) = v\left( {0, S\left( 0\right) ,0}\right) \) and at each time you hold \( \Delta \left( t\right) \) shares of the underlying asset, investing or borrowing at the interest rate \( r \) in order to do this, then at time \( T \) the value of your portfolio will be
\[
X\left( T\right) = \frac{1}{T}{\int }_{0}^{T}S\left( u\right) {du}
\] | Yes |
Exercise 3.5 | Exercise 3.5. Specify the estimating equation for the case of logit regression.
The log likelihood with canonical parametrization equals
\[
\ell \left( {y, v}\right) = {yv} - \log \left( {1 + {e}^{v}}\right) .
\]
Therefore \( \widetilde{\theta } = {\operatorname{argmax}}_{\theta }L\left( \theta \right) = {\operatorname{argmax}}_{\theta }\mathop{\sum }\limits_{i}\left\{ {{Y}_{i}{\psi }_{i}^{\top }\theta - \log \left( {1 + {e}^{{\psi }_{i}^{\top }\theta }}\right) }\right\} \) .
Differentiating w.r.t. \( \theta \) yields:
\[
\frac{\partial }{\partial \theta }L\left( \theta \right) = \mathop{\sum }\limits_{i}\left( {{Y}_{i}{\psi }_{i} - \frac{1}{1 + {e}^{{\psi }_{i}^{\top }\theta }}{e}^{{\psi }_{i}^{\top }\theta }{\psi }_{i}}\right) .
\]
Therefore, the estimation equation is:
\[
\mathop{\sum }\limits_{i}\left( {{Y}_{i} - \frac{{e}^{{\psi }_{i}^{\top }\theta }}{1 + {e}^{{\psi }_{i}^{\top }\theta }}}\right) {\psi }_{i} = 0
\] | No |
Exercise 5.27 | Exercise 5.27 Derive eqns (5.143) and (5.144) by using eqn (3.10) and by choosing an interaction time \( {t}_{e} = \pi /\left( {{2g}\sqrt{{n}_{T}}}\right) \) for the emitter case and \( {t}_{a} = \pi /\left( {{2g}\sqrt{{n}_{T} + 1}}\right) \) for the absorber case. Again, because of various imperfections and constraints, the experimental probabilities do not exactly match eqns (5.143) and (5.144). | No |
Exercise 4.5 | Exercise 4.5. Let \( \alpha = \frac{1}{2}, f\left( t\right) = {\sin }_{1}\left( {t,{t}_{0}}\right), t \in \mathbb{T}, t \geq {t}_{0} \) . Find
\[
\mathcal{L}\left( {{D}_{\Delta ,{t}_{0}}^{\frac{1}{2}}{\sin }_{1}\left( {\cdot ,{t}_{0}}\right) }\right) \left( {z,{t}_{0}}\right)
\]
Answer. \( \frac{{z}^{\frac{1}{2}}}{{z}^{2} + 1} \) . | Yes |
Exercise 9.1 | Show that if \( {\Lambda }_{2}^{h} \) is given by (1.38) then
\[
{\Lambda }_{2}^{h}F\left( Y\right)
\]
\[
= \frac{1}{2}{\int }_{{X}^{2}}{\int }_{\mathcal{X}}\left\lbrack {{\left( \frac{{\delta F}\left( Y\right) }{{\delta Y}\left( .\right) }\right) }^{ \oplus }\left( \mathbf{y}\right) - {\left( \frac{{\delta F}\left( Y\right) }{{\delta Y}\left( .\right) }\right) }^{ \oplus }\left( \mathbf{z}\right) }\right\rbrack P\left( {\mathbf{z}, d\mathbf{y}}\right) {Y}^{\otimes 2}\left( {d\mathbf{z}}\right)
\]
\[
- \frac{h}{2}{\int }_{X}{\int }_{\mathcal{X}}\left\lbrack {{\left( \frac{{\delta F}\left( Y\right) }{{\delta Y}\left( .\right) }\right) }^{ \oplus }\left( \mathbf{y}\right) - 2\frac{{\delta F}\left( Y\right) }{{\delta Y}\left( z\right) }}\right\rbrack P\left( {z, z, d\mathbf{y}}\right) Y\left( {dz}\right)
\]
\[
+ {h}^{3}\mathop{\sum }\limits_{{\{ i, j\} \subset \{ 1,\ldots, n\} }}{\int }_{0}^{1}\left( {1 - s}\right) {ds}{\int }_{\mathcal{X}}P\left( {{x}_{i},{x}_{j}, d\mathbf{y}}\right)
\]
\[
\times \left( {\frac{{\delta }^{2}F}{{\delta Y}\left( .\right) {\delta Y}\left( .\right) }\left( {Y + \operatorname{sh}\left( {{\delta }_{\mathbf{y}} - {\delta }_{{x}_{i}} - {\delta }_{{x}_{j}}}\right) }\right) ,{\left( {\delta }_{\mathbf{y}} - {\delta }_{{x}_{i}} - {\delta }_{{x}_{j}}\right) }^{\otimes 2}}\right) \tag{9.4} | No |
Exercise 6.3.1 | Exercise 6.3.1. Prove that a \( {C}^{2} \) Anosov diffeomorphism preserving a smooth measure is weak mixing. | No |
Exercise 8.6.2 | Exercise 8.6.2. Suppose \( {S}_{n} \) is one-dimensional simple random walk and let
\[
{R}_{n} = 1 + \mathop{\max }\limits_{{m \leq n}}{S}_{m} - \mathop{\min }\limits_{{m \leq n}}{S}_{m}
\]
be the number of points visited by time \( n \) . Show that \( {R}_{n}/\sqrt{n} \Rightarrow a \) limit. | No |
Exercise 32 | Exercise 32. Let \( F, G : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{p} \) be two differentiable functions. Define
\[
h : {\mathbb{R}}^{m} \times {\mathbb{R}}^{m} \rightarrow \mathbb{R}\;\text{ by }\;h\left( {\mathbf{x},\mathbf{y}}\right) = F\left( \mathbf{x}\right) \bullet G\left( \mathbf{y}\right) .
\]
Show that \( h \) is differentiable and
\[
\nabla h\left( {\mathbf{x},\mathbf{y}}\right) = \left( {G\left( \mathbf{y}\right) \cdot {J}_{F}\left( \mathbf{x}\right), F\left( \mathbf{x}\right) \cdot {J}_{G}\left( \mathbf{y}\right) }\right) \;\text{ for all }\mathbf{x},\mathbf{y} \in {\mathbb{R}}^{m}.
\] | No |
Exercise 8.4 | Exercise 8.4. Let \( I \) be a proper ideal of rank \( r \) of \( K\left\lbrack \underline{X}\right\rbrack \) and let \( I = {\mathfrak{q}}_{1} \cap \cdots \cap {\mathfrak{q}}_{s} \) be an irredundant primary decomposition of \( I \) . (a) Show that, for each \( \underline{D} \in {\mathbb{N}}^{k} \), there is an injective \( K \) -linear map
\[
\left( {K{\left\lbrack \underline{X}\right\rbrack }_{ \leq \underline{D}} + I}\right) /I \rightarrow \mathop{\prod }\limits_{{i = 1}}^{s}\left( {\left( {K{\left\lbrack \underline{X}\right\rbrack }_{ \leq \underline{D}} + {\mathfrak{q}}_{i}}\right) /{\mathfrak{q}}_{i}}\right) .
\]
(b) Suppose that \( {q}_{i} \) has rank \( r \) for \( i = 1,\ldots, t \) and rank \( > r \) for \( i = t + 1,\ldots, s \) . Deduce that we have
\[
\mathcal{H}\left( {I;\underline{D}}\right) \leq \mathop{\sum }\limits_{{i = 1}}^{t}\mathcal{H}\left( {{\mathfrak{q}}_{i};\underline{D}}\right) \tag{D}
\]
for any \( \underline{D} \in {\mathbb{N}}^{k} \) . | No |
Exercise 11.3.1 | Exercise 11.3.1 (a) Substitute \( t = h \) in formula
\[
N\left( t\right) = N\left( 0\right) {e}^{-{\lambda t}} + \sigma {e}^{-{\lambda t}}{\int }_{0}^{t}{e}^{\lambda s}d{W}_{s}.
\]
Then take the expectation in
\[
N\left( 0\right) + \sigma {\int }_{0}^{h}{e}^{\lambda s}d{W}_{s} = \frac{1}{2}{e}^{\lambda h}N\left( 0\right)
\]
and obtain \( N\left( 0\right) = \frac{1}{2}\mathbb{E}\left\lbrack {e}^{\lambda h}\right\rbrack N\left( 0\right) \), which implies the desired result.
(b) Jensen’s inequality for the random variable \( h \) becomes \( \mathbb{E}\left\lbrack {e}^{\lambda h}\right\rbrack \geq {e}^{\mathbb{E}\left\lbrack {\lambda h}\right\rbrack } \) . This can be written as \( 2 \geq {e}^{\lambda \mathbb{E}\left\lbrack h\right\rbrack } \) . | No |